Content-Type: multipart/mixed; boundary="-------------0806040948613"

This is a multi-part message in MIME format.

---------------0806040948613
Content-Type: text/plain; name="08-104.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="08-104.comments"

PACS: 03.75.Hh, 67.25.de, 67.85.Bc.
---------------0806040948613
Content-Type: text/plain; name="08-104.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="08-104.keywords"

Bose-Einstein Condensation, Cycles, Infinite range Bose-Hubbard Model
---------------0806040948613
Content-Type: application/x-tex; name="Hard-Core_Paper-JSP-final.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="Hard-Core_Paper-JSP-final.tex"

\documentclass[a4paper,12pt]{article}
\usepackage{amssymb, amsmath}
\usepackage{epsfig}
\textheight=24.7cm \textwidth=16.5cm \topmargin=-15mm %\topmargin=-10mm
\oddsidemargin=0mm
\parindent=0mm
\parskip=4mm plus .5mm minus .5 mm
\pagestyle{myheadings} \markboth{}{} \thispagestyle{empty}
\makeatletter
\renewcommand{\section}{{\setcounter{equation}{0}}\@startsection%
{section}%
{1}%
{0mm}%
{-\baselineskip}%
{0.5\baselineskip}%
{\normalfont\large\bfseries}%
%{\normalfont\normalsize\bfseries}%
} \makeatother
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\renewcommand{\baselinestretch}{1}
%\renewcommand{\thefigure}{\arabic{figure}\hskip-0.5cm{caption}}
%\renewcommand{\caption}{caption}
%
%
\makeatletter
\renewcommand{\subsection}{\@startsection%
{subsection}%
{2}%
{0mm}%
{-\baselineskip}%
{0.5\baselineskip}%
{\normalfont\normalsize\bfseries}}%

%\renewcommand\appendix{\par
%  \setcounter{section}{0}%
%  \setcounter{subsection}{0}%
%  \gdef\thesection{\appendixname\,\@Alph\c@section :\!\!\!}%
%  \gdef\thesubsection{\@Alph\c@section .\@arabic\c@subsection }%
%}

\newtheorem{remark}{Remark}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newenvironment{proof}{{\bf Proof:}}{\hfill$\square$\vskip.5cm}
\newtheorem{defn}{Definition}[section]
\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\setlength{\unitlength}{1mm}
\setlength\jot{6pt}  %set inter-equation vertical spacing
\abovedisplayskip=12pt plus 3pt minus 9pt
\abovedisplayshortskip=2pt plus 3pt
\belowdisplayskip=12pt plus 3pt minus 9pt
\belowdisplayshortskip=7pt plus 3pt minus 4pt



%*******************************************************************
\usepackage{color}
\newcommand{\bl}{\textcolor{blue}}
\newcommand{\rd}{\textcolor{red}}
\newcommand{\gn}{\textcolor{green}}
\newcommand{\tit}{\textit}
\def\*{{\phantom *}}
%*******************************************************************

%Commonly used symbols in the document
\newcommand{\sLambda}{{\hbox {\tiny{$\Lambda$}}}}
\newcommand{\sV}{{\hbox {\tiny{$V$}}}}
\newcommand{\sbeta}{{\hbox {\tiny{$\beta$}}}}

\newcommand{\Hone}{{\mathcal{H}_\sV}}
\newcommand{\Projn}{P^n_{+}}
\newcommand{\trace}{\text{\rm{trace}}\,}
\newcommand{\sym}{\sigma_{+}\,}
\newcommand{\Prob}{\mathbb{P}}
\newcommand{\Ex}{\mathbb{E}}
\newcommand{\Id}{\mathbb{I}}
\newcommand{\starr}{^{\phantom *}}

\newcommand{\cHam}{H_\sV^{(n)}}   %canonical hamiltonian
\newcommand{\hcHam}{H^{\mathsf{hc}}_{n,\sV}}      %hard-core hamiltonian
\newcommand{\cEx}{\Ex^n_\sV}   %canonical expectation symbol
\newcommand{\cProb}{\Prob^n_\sV}   %canonical probability symbol
\newcommand{\cPart}{Z_\sbeta(n,V)}           %canonical partition function
\newcommand{\ccPart}{Z_\sbeta\left({(V\!\!\!-\!q)}/{V},\, n\!-\!q,V\!\!\!-\!q\right)}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\KK}{\mathbb{K}}
\newcommand{\NN}{\mathbb{N}}

\newcommand{\bigbar}{\bigg|}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\utilde}[1]{\underset{\widetilde{}}{#1}}
\newcommand{\undertilde}[1]{\underset{\widetilde{}}{\mathbf{#1}}}


\newcommand{\Phc}{\mathcal{P}_n^{\mathsf{hc}}}
\newcommand{\Hcansym}[1]{{\mathcal{H}_{\sV,+}^{(#1)}}}
\newcommand{\Hcan}[1]{{\mathcal{H}_{\sV}^{(#1)}}}
\newcommand{\Hhccan}[1]{{\mathcal{H}_{#1,\sV}^{\mathsf{hc}}}}
\newcommand{\Hhccansym}[1]{{\mathcal{H}_{#1,\sV,+}^{{\mathsf{hc}}}}}
\newcommand{\Hhccyclespace}{{\mathcal{H}_{q,n,\sV}^{{\mathsf{hc}}}}}

\newcommand{\Htilde}{\widetilde{H}}
\newcommand{\ii}{\mathbf{i}}
\newcommand{\jj}{\mathbf{j}}
\newcommand{\kk}{\mathbf{k}}
\newcommand{\boldl}{\mathbf{l}}
\newcommand{\e}{\mathrm{e}}

\newcommand{\thermlim}{ \lim_{\stackrel{n,\sV \to \infty}{n/\sV=\rho}}}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%TITLE PAGE
\phantom{.} 
\textbf{To appear in JSP}\hfill
\textbf{04-06-08}
\vskip1.5cm
\begin{center}
{\Large Long Cycles
\vskip 0.1cm
in the Infinite-Range-Hopping Bose-Hubbard Model
\vskip 0.1cm
with Hard Cores}
\vskip 0.5cm
{\bf G. Boland}
\footnote{email: Gerry.Boland@ucd.ie}
\vskip 0.1cm
and
\vskip 0.1cm
{\bf
J.V. Pul\'e}
\footnote{email:
Joe.Pule@ucd.ie}
\vskip 0.5cm
School of Mathematical Sciences
\linebreak
University College Dublin\\Belfield, Dublin 4, Ireland
\end{center}
\vskip 1cm
\begin{abstract}
\vskip -0.4truecm
%\mbox{}
\noindent
In this paper we study the relation between long cycles and Bose-Condensation in the Infinite range Bose-Hubbard Model with a hard core interaction.
We calculate the density of particles on long cycles in the thermodynamic limit
and find that the existence
of a non-zero long cycle density coincides with the occurrence of Bose-Einstein
condensation but this density is not equal to that of the Bose condensate.
\\
\\
{\bf  Keywords:} Bose-Einstein Condensation, Cycles, Infinite range Bose-Hubbard Model \\
{\bf  PACS:} 03.75.Hh,	%Static properties of condensates; thermodynamical, statistical, and structural properties
67.25.de, %Thermodynamic properties
67.85.Bc.	%Static properties of condensates

%OLD OPTIONS, circa 1992
%05.30.Jp ,   % Boson systems
%03.75.Fi,   % Phase coherent atomic ensemble (Bose condensation)
%67.40.-w.   % Boson degeneracy and superfluidity of 4He
\end{abstract}
\newpage\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
In 1953, Feynman analysed the partition function of an interacting Bose gas in terms
of the statistical distribution
of permutation cycles of particles and emphasized the roles of long cycles at the
transition point \cite{Fey 1953 and stat.mech.}. Then Penrose and Onsager,
pursuing Feynman's arguments,
observed that there should be Bose condensation when the fraction of the total particle
number belonging to
long cycles is strictly positive \cite{Pen.Onsa.}.
These ideas are now commonly accepted and also discussed in various contexts
in systems showing analogous phase
transitions such as percolation, gelation and polymerization
(see e.g. \cite{Chandler}, \cite{Sear-Cuesta}, \cite{Schakel}), though
it has been recently argued by Ueltschi \cite{Ueltschi1}
that in fact the hypothesis is not always valid. To our knowledge, there had not
appeared a precise mathematical and quantitative formulation of the relation
between Bose condensate and long cycles until the work of S\"ut\"o \cite{Suto} and
its validity
has been checked only in a few models: the free and mean field Bose gas in \cite{Suto},
(see also Ueltschi \cite{Ueltschi2})
and the perturbed mean field model of a Bose gas studied in \cite{DMP}.
In these models it is shown that the density
of particles in long cycles is equal to the Bose condensate density.
The purpose of this paper is test the validity of the hypothesis in yet another
model of a Bose gas, the Infinite range Bose-Hubbard Model with a hard core.
Here we calculate the density of particles on long cycles in the thermodynamic limit
and find that though the existence
of a non-zero long cycle density coincides with the occurence of Bose-Einstein
condensation, this density is not equal to the Bose condensate density.
\par
The main simplifying feature in this model is the following. In general the density
of particles on cycles
of length $q$ for $n$ particles can be expressed (apart from normalization)
as the trace (see for example Proposition \ref{stat})
of the exponential of the Hamiltonian for $n-q$ bosons and $q$ distinguishable
particles (no statistics). In terms of the random walk representation (cf \cite{Toth}), the particles in this model are allowed to hop from one site
to another with equal probability. We can prove (Proposition \ref{c}) that in the thermodynamic limit
we can neglect the hopping of the $q$ particles so that bosons have to avoid
each other and the fixed positions of the
distinguishable particles. This is equivalent to a reduction of the
lattice by $q$ sites. Moreover the $q$ particles are on a cycle of length $q$.
For $q > 1$, this means for example, that the position of the second particle at the beginning of its path is same as the position of the first particle at the end of its path. But since they do not hop this is impossible by the hard core condition
and therefore among the short cycles only the cycle of unit length contributes. Since
the sum of all the cycle densities gives the particle density, this means
that in the thermodynamic limit
the sum of the long cycle densities is the particle density less the one-cycle contribution.
The one-cycle density, apart from some scaling and the normalization, is then the partition
function for the boson system with one site removed from the lattice, which can be calculated.
\par
The model without a hard-core will be treated in another paper. There we can again neglect the hopping of the  $q$ distinguishable
particles. However in that case cycles of all lengths contribute to the long-cycle density. It is relatively easy to see that when there is no condensation
the long-cycle density vanishes but we do not yet know what happens when there is Bose-Einstein condensation.
\par
In Section 2 we first describe the model and recall its thermodynamic properties
as stated by Penrose \cite{Penrose} (see also T\'oth \cite{Toth} and Kirson\cite{Kirson}).
We then apply the general framework for cycle statistics described in \cite{DMP}, following
\cite{Martin}.
Using standard properties of the decomposition of permutations into cycles,
the canonical sum is
converted into a sum on cycle lengths.
This makes it possible to decompose the total density $\rho=\rho_{{\rm short}}+\rho_{{\rm long}}$
into the number density of particles belonging to cycles of finite length
($\rho_{{\rm short}}$)
and to infinitely long cycles ($\rho_{{\rm long}}$) in the thermodynamic limit.
It is conjectured
that when there is Bose condensation, $\rho_{{\rm long}}$ is different from
zero and identical to the condensate density. The main purpose of the paper is to check the
validity of this conjecture in our model. At the end of Section 2 we state in the main theorem
describing the relation between Bose-Einstein condensation and the density of long
cycles for our model.
\par
In Section 3 we prove the main theorem and in Section 4 we discuss briefly \textit{Off-diagonal
Long-Range Order}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION: Model & Results
\section{The Model and Results}
The Bose-Hubbard Hamiltonian is given by
\begin{equation}\label{B-H}
    H^{\mathrm{BH}}=J\!\!\!\!\!\!\sum_{x,y\in\Lambda_\sV\,:|x-y|=1}(a^*_x-a^*_y)(a\starr_x-a\starr_y)
    +\lambda\sum_{x\in \Lambda_{\sV}} n_x(n_x-1)
\end{equation}
where $\Lambda_\sV$ is a lattice of $V$ sites, $a^*_x$ and $a\starr_x$ are the Bose
creation and annihilation operators satisfying the usual commutation relations
$[a^*_x,a\starr_y]=\delta_{x,y}$ and $n_x=a^*_xa^{\phantom *}_x$.
The first term with $J>0$ is the kinetic energy operator and the
second term with $\lambda>0$ describes a repulsive interaction, as it discourages the presence of more
than one particle at each site. This model was originally introduced by
Fisher \textit{et al.} \cite{Fisher}.

The infinite-range hopping model is given by the Hamiltonian
\begin{equation}\label{I-R}
    H^{\mathrm{IR}}=\frac{1}{2V}\!\!\!\sum_{x,y\in\Lambda_\sV}(a^*_x-a^*_y)(a\starr_x-a\starr_y)
    +\lambda\sum_{x\in \Lambda_\sV} n_x(n_x-1).
\end{equation}
This is in fact a mean-field version of (\ref{B-H}) but in terms of the kinetic energy rather
than the interaction. In particular as in all mean-field models, the
lattice structure is irrelevant and there is no dependence
on dimensionality, so we can take $\Lambda_\sV=\{1,2,3, \ldots, V\}$.
The non-zero temperature properties of this model have
been studied by Bru and Dorlas \cite{BruDorlas} and by Adams and Dorlas \cite{AdamsDorlas}. We shall study
a special case of (\ref{I-R}), introduced by T\'oth \cite{Toth} where $\lambda=+\infty$,
that is complete single-site exclusion (hard-core). The properties of this model
in the canonical ensemble were first obtained by T\'oth using probabilistic methods. Later
Penrose \cite{Penrose} and Kirson \cite{Kirson} obtained equivalent results.
In the grand-canonical
ensemble the model is equivalent to the strong-coupling BCS model
(see for example Angelescu \cite{Angelescu}). Here we recall the thermodynamic properties of the
model in the canonical ensemble as given by Penrose.
\par
For $\rho\in(0, 1)$, let
\begin{equation*}\label{g}
    g(\rho)=
    \begin{cases}
    {\displaystyle  \frac{1}{1-2\rho}\ln\left( \frac{1-\rho}{\rho}\right )} \vspace{0.2cm}
    & \mathrm{if}\ \  \rho\neq 1/2,\\
    2 & \mathrm{if}\ \ \rho=1/2.
    \end{cases}
\end{equation*}
For each $\beta\geq 2$ the equation $\beta=g(\rho)$ has a unique solution in $(0,1/2]$ denoted by
$\rho_\beta$ (see Fig.\ref{fig1}). We define $\rho_\beta:=1/2$ for $\beta<2 $.
%%%%%%%%%%%%%%%%%%
%FIGURE
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=10cm]{penrose01.eps}
\end{center}
\caption{\it Definition of $\rho_\sbeta$}
\label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Theorem: Penrose 1
\begin{theorem} $\mathrm{(Penrose\ [10],\ Theorem\ 1)}$\\
The free energy per site at inverse temperature $\beta$ as a function of the particle density
$\rho\in[0, 1]$, $f_\beta(\rho)$, is given by
\begin{equation*}\label{f}
    f_\beta(\rho)=
    \begin{cases}
    {\displaystyle \rho+\frac{1}{\beta}\left(\rho\ln\rho+(1-\rho)\ln(1-\rho)\right )}
    & \mathrm{if}\ \  \rho\in[0,\rho_\beta]\cup[1-\rho_\beta,1],\\
    {\displaystyle \rho^2 +\rho_\beta(1-\rho_\beta)+\frac{1}{\beta}
    \left(\rho_\beta\ln\rho_\beta+(1-\rho_\beta)\ln(1-\rho_\beta)\right )}
    & \mathrm{if}\ \ \rho\in[\rho_\beta,1-\rho_\beta].
    \end{cases}
\end{equation*}
\end{theorem}
The density of particles in the ground state in the thermodynamic limit is given by
\begin{equation}\label{}
    \rho_\beta^c=\thermlim\frac{1}{V^2}\hskip -0.2cm\sum_{x,y\in\Lambda_\sV}\la a^*_x a\starr_y\ra
\end{equation}
where $\la\,\cdot\,\ra$ denotes the canonical expectation for $n$ particles.
Penrose showed that for certain values of $\rho$ and $\beta$, Bose-Einstein condensation occurs,
that is, $\rho_\beta^c>0$. The Bose-condensate density is given in the following theorem.
\begin{theorem} $\mathrm{(Penrose\ [10],\ Theorem\ 2)}$\\
The Bose-condensate density, $\rho_\beta^c$ at inverse temperature $\beta$ as a function
of the particle density
$\rho\in[0, 1]$, is given by
\begin{equation*}\label{rho-c}
	\rho_\beta^c=
	\begin{cases}
		0
		& \mathrm{if}\ \  \rho\in[0,\rho_\beta]\cup[1-\rho_\beta,1],\\
		{\displaystyle (\rho-\rho_\beta)(1-\rho-\rho_\beta)}
		& \mathrm{if}\ \ \rho\in[\rho_\beta,1-\rho_\beta].
    	\end{cases}
\end{equation*}
\end{theorem}
We note that both $f_\beta(\rho)-\rho$ and the condensate density $\rho_\beta^c$ are
symmetric about $\rho=1/2$.
This can easily seen by interchanging particles and holes. The Boson states being symmetric can be labelled
unambiguously by the sites they occupy but equivalently they can be labelled by the sites they do not occupy (holes).
\par
Before proceeding to the study of cycle statistics we need to define the $n$-particle
Hamiltonian more carefully. The single particle Hilbert space is $\mathcal{H}_\sV:=\CC^\sV$
and on it we define the operator
\begin{equation*}
\label{ham1}
	H_\sV=I-P_\sV
\end{equation*}
where $P_\sV$ is the orthogonal projection onto the unit vector
\begin{equation*}
	\mathbf{g}_\sV = \frac{1}{\sqrt{V}} (1,1,\dots,1)\in \mathcal{H}_\sV.
\end{equation*}
In terms of the usual basis vectors of $\mathcal{H}_\sV$, $\{\mathbf{e}_i\, |\, i=1\ldots V\}$, $P_\sV$ is given by
\begin{equation*}
	P_\sV  \mathbf{e}_i  = \frac{1}{V} \sum_{j=1}^V  \mathbf{e}_j.
\end{equation*}
Thus $H_\sV$ is the orthogonal projection onto the subspace orthogonal to $\mathbf{g}_\sV$.
For an operator $A$ on $\Hone$, we define $A^{(n)}$ on
$\Hcan{n} = \underbrace{\Hone \otimes \Hone \otimes \dots
\otimes \Hone}_{n \text{ times}}$ by \vspace{-0.2cm}
\begin{equation*}\label{A-n}
	A^{(n)}
	=A\otimes I\otimes \ldots \otimes I+I\otimes A\otimes \ldots \otimes I+\ldots
		+I\otimes I\otimes \ldots \otimes A.
\end{equation*}
With this notation we can define the non-interacting $n$-particle Hamiltonian $\cHam$ acting
on the unsymmetrised Hilbert space $\Hcan{n}$ as:
\begin{align*}
	\cHam
	&= I^{(n)} - P_\sV^{(n)}
\\
 	&= n - P_\sV \otimes I \otimes \dots \otimes I - I \otimes P_\sV \otimes I \otimes \dots \otimes I
                	- \dots - I \otimes I \otimes \dots \otimes P_\sV.
\end{align*}
%
For bosons we have to consider the symmetric subspace of $\Hcan{n}$.
The symmetrisation projection $\sigma_{+}^n$ on $\Hcan{n}$ is defined by
\begin{equation}\label{sym}
	\sigma_{+}^n = \frac{1}{n!} \sum_{\pi \in S_n} U_\pi
\end{equation}
where $U_\pi: \Hcan{n} \mapsto \Hcan{n}$ is the unitary representation of the permutation group
$S_n$ on $\Hcan{n}$ defined by
\begin{equation*}
	U_\pi(\phi_1 \otimes \phi_2 \otimes \cdots \otimes \phi_n)
	= \phi_{\pi(1)} \otimes \phi_{\pi(2)} \otimes \cdots
		\otimes \phi_{\pi(n)}, \;\; \phi_j \in \Hone, \; j=1, \dots, n; \;  \pi \in S_n.
\end{equation*}
Then the symmetric $n$-particle subspace is $\Hcansym{n} := \sigma_{+}^n \Hcan{n}$.
%
\par
When $\cHam$ is restricted to $\Hcansym{n}$, we obtain
\begin{equation*}
	\frac{1}{2V}\!\!\!\sum_{x,y\in\Lambda_\sV}(a^*_x-a^*_y)(a\starr_x-a\starr_y).
\end{equation*}
We introduce the hard-core interaction by applying a projection
to $\Hcan{n}$ to forbid more than one particle from occupying each site.
Let $\{\mathbf{e}_i\}_{i=1}^\sV$ be the usual orthonormal basis for $\Hone$.
We then define the hard core projection $\Phc$
on $\Hcan{n}$ by
\begin{equation}\label{P-hardcore}
	\Phc ( \mathbf{e}_{i_1} \otimes \mathbf{e}_{i_2} \otimes \dots \otimes \mathbf{e}_{i_n} ) =
	\begin{cases}
 		0   & \text{if } \, \mathbf{e}_{i_k} = \mathbf{e}_{i_{k'}} \; \text{for some }\, k \ne k', \\
		\mathbf{e}_{i_1} \otimes \mathbf{e}_{i_2} \otimes \dots \otimes \mathbf{e}_{i_n}   & \text{otherwise}.
	\end{cases}
\end{equation}
%
We shall call $\Hhccan{n} := \Phc \Hcan{n}$  the unsymmetrised hard-core $n$-particle space
and \linebreak $\Hhccansym{n} := \Phc \Hcansym{n} $ the symmetric hard-core $n$-particle space.
Note that as $[U_\pi,\Phc ]=0$ for all $\pi\in S_n$, $\Phc$ commutes with the symmetrisation
and so $\Hhccansym{n} = \sigma_{+}^n \Hhccan{n}$.

The hard-core $n$-particle Hamiltonian is then
\begin{equation}\label{H-hc}
    \hcHam:=\Phc \cHam \Phc
\end{equation}
acting on the hard-core $n$-particle space $\Hhccan{n}$. Therefore the Hamiltonian for the
infinite-range Bose-Hubbard model with hard-core is (\ref{H-hc}) acting on the \textbf{symmetric} hard-core
$n$-particle space $\Hhccansym{n}$.
\par
We shall now analyse the cycle statistics of this model.
\par
%
Using (\ref{sym}), the canonical partition function for the hard-core boson model may be written as
\begin{equation*}\label{}
    \cPart = \trace_{\Hhccansym{n}} \left[ \e^{-\beta \hcHam} \right]
   = \trace_{\Hhccan{n}} \left[\sigma_{+}^n \e^{-\beta \hcHam} \right]
   = \frac{1}{n!} \sum_{\pi \in S_n} \trace_{\Hhccan{n}} \left[ U_\pi \e^{-\beta \hcHam} \right].
\end{equation*}
Following \cite{DMP}, we define a probability measure on the permutation group $S_n$ by
\begin{equation}
\label{cProb_defn} %%%LABEL: cProb_defn
  \cProb(\pi) = \frac{1}{\cPart} \frac{1}{n!}
  \trace_{\Hhccan{n}} \left[ U_\pi \e^{-\beta \hcHam} \right].
\end{equation}
From the random walk formulation (see for example \cite{Toth}) one can see that the kernel of $\e^{-\beta \hcHam}$
is positive and therefore the righthand side of (\ref{cProb_defn}) is positive.
\par
Each permutation $\pi \in S_n$ can be decomposed uniquely into a number of cyclic permutations of lengths
$q_1, q_2, \dots, q_r$ with $r \le n$ and $q_1 + q_2 + \dots + q_r = n$.
For $q \in \{1,2,\ldots,n\}$, let $N_q(\pi)$ be the random variable
corresponding to the number of cycles of length $q$ in $\pi$. Then
the expectation of the number of $q$-cycles in the canonical ensemble is:
\begin{equation*}\label{}
    \cEx(N_q) = \sum_{r=1}^n r\cProb(N_q\!=\!r)
\end{equation*}
and the average density of particles in $q$-cycles for the system of $n$ bosons is
\begin{equation}\label{}
    c_\sV^n(q) = \frac{q \; \cEx(N_q) }{V}.
\end{equation}
This brings us then to the following definition.
\begin{definition}
The expected density of particles on cycles of \textbf{infinite} length is given by
\begin{equation}\label{}
    \rho^{\mathrm{long}}_\sbeta = \lim_{Q \to \infty} \;\thermlim \; \sum_{q=Q+1}^n c_\sV^n(q).
\end{equation}
\end{definition}
For the free Bose gas, the mean field and the perturbed mean field Bose gas, it has been
shown that $\rho^{\mathrm{long}}_\sbeta=\rho_\sbeta^c$, the condensate density. For our model,
the situation is different. Below we state the main result of this paper:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Theorem: main
\begin{theorem}\label{Th-rho-long}
The expected density of particles on cycles of infinite length,
$\rho_\sbeta^{\mathrm{long}}$, at inverse temperature $\beta$ as a function of the particle density
$\rho\in[0, 1]$, is given by
\begin{equation*}\label{rho-long}
   \rho_\sbeta^{\mathrm{long}}=
    \begin{cases}
    0
    & \mathrm{if}\ \  \rho\in[0,\rho_\sbeta]\cup[1-\rho_\sbeta,1],\\
    {\displaystyle \rho-\rho_\sbeta\mathrm{e}^{\beta(\rho-\rho_\sbeta)}}
    & \mathrm{if}\ \ \rho\in[\rho_\sbeta,1-\rho_\sbeta].
    \end{cases}
\end{equation*}
\end{theorem}
We note that (see Fig.\ref{fig2}):
\begin{itemize}
    \item $\rho_\sbeta^{\mathrm{long}}=0$ if and only if $\rho_\sbeta^c=0$.
    \item $\rho_\sbeta^{\mathrm{long}}$ is not symmetric with respect to $\rho=1/2$. As mentioned above the symmetry of the model about
    $\rho=1/2$ is due to the particle-hole symmetry. But the equivalent labelling of states by sets of occupied or unoccupied
    sites (particles and holes) cannot be used for distinguishable particles. We shall
    see (Proposition \ref{stat}) that the
    $q$-cycle occupation density $c_\sV^n(q)$ involves $q$ distinguishable particles and $n-q$ bosons and
    therefore the particle-hole symmetry is broken.
    \item When $\rho_\sbeta^c>0$, $\rho_\sbeta^{\mathrm{long}}$ starts below $\rho_\sbeta^c$ since its
    slope at $\rho_\beta$ is equal to $1-2\rho_\sbeta$ while $\rho_\sbeta^c$ has slope $1-\beta\rho_\sbeta$
    and $\beta>2$. Conversely, $\rho_\sbeta^{\mathrm{long}}$ finishes above
    $\rho_\sbeta^c$ since its slope at $1-\rho_\sbeta$ is less than that of $\rho_\sbeta^c$.
    \end{itemize}
%%%%%%%%%%%%%%%%%%
%FIGURE
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=12cm]{penrose02.eps}
\end{center}
\caption{\it $\rho_\sbeta^c$ and $\rho_\sbeta^{\mathrm{long}}$ for $\beta>2$}
\label{fig2}
\end{figure}
%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%                   Proof of the Main Result
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of the Main Result}
In this section we shall prove Theorem \ref{Th-rho-long}. First we note that if $n/V=\rho$, then
\begin{equation}\label{n}
    \sum_{q=1}^n c^n_\sV(q)=\rho,
\end{equation}
so that if we define
\begin{equation*}\label{}
    \rho^{\mathrm{short}}_\sbeta = \lim_{Q \to \infty} \;\thermlim \; \sum_{q=1}^Q c_\sV^n(q)
\end{equation*}
we have
\begin{equation*}\label{}
    \rho^{\mathrm{long}}_\sbeta=\rho -\rho^{\mathrm{short}}_\sbeta.
\end{equation*}

For $\rho^{\mathrm{short}}_\sbeta$ we can interchange the sum over $q$ and $\thermlim$,
\begin{equation*}\label{}
    \rho^{\mathrm{short}}_\sbeta = \lim_{Q \to \infty} \; \sum_{q=1}^Q \thermlim \;c_\sV^n(q),
\end{equation*}
making it much easier to calculate. In fact we shall prove that:
\begin{equation}\label{rho-short}
   \rho_\sbeta^{\mathrm{short}}=
    \begin{cases}
    \rho
    & \mathrm{if}\ \  \rho\in[0,\rho_\sbeta]\cup[1-\rho_\sbeta,1],\\
    {\displaystyle \rho_\sbeta\mathrm{e}^{\beta(\rho-\rho_\sbeta)}}
    & \mathrm{if}\ \ \rho\in[\rho_\sbeta,1-\rho_\sbeta].
    \end{cases}
\end{equation}

The proof is in four steps. The first step is to obtain a convenient expression for
$c^n_\sV(q)$, the mean density of particles belonging to a cycle of length $q$.
We shall denote the unitary representation of a $q$-cycle by $U_q: \Hcan{q} \to \Hcan{q}$, that is
\begin{equation*}
	U_q ( \phi_{i_1} \otimes \dots \otimes \phi_{i_n}) =
	\phi_{i_2} \otimes \dots \phi_{i_q} \otimes \phi_{i_1} .
\end{equation*}
When there is no ambiguity we shall use the same notation $U_q$ for $U_q \otimes I^{(n-q)}: \Hcan{n} \to \Hcan{n}$
where $I$ is the identity operator. Note that $[U_q, \Phc] = 0$ and $[U_q, \sigma^n_{+}] = 0$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Proposition
\begin{proposition}\label{stat}
\begin{equation*}
	c^n_\sV(q) = \frac{1}{Z_\sbeta(n,V)} \frac{1}{V} \, \trace_{ \Hhccyclespace  }
				\left[ U_q \e^{-\beta H^{\mathsf{hc}}_{n,\sV} } \right]
\end{equation*}
where $\Hhccyclespace := \Phc ( \Hcan{q} \otimes \Hcansym{n-q} )$.
\end{proposition}
Note that though we write this proposition for our special case, in fact $c^n_\sV(q)$ can be expressed in this form
for any Boson model with a symmetric Hamiltonian.

By using cycle statistics, we split our symmetric hard-core Hilbert space $\Hhccansym{n}$
into a tensor product of two spaces, an unsymmetrised $q$-particle space $\Hcan{q}$ and a
symmetric $n-q$ particle space $\Hcansym{n-q}$, with the hard-core projection applied.
Writing
\begin{equation*}
    A^{(q)}:= A^{(q)} \otimes I^{(n-q)} \ \ \ \ \mathrm{and}
    \ \ \ \ A^{(n-q)} := I^{(q)} \otimes A^{(n-q)}
\end{equation*}
for any operator $A$ on $\mathcal{H}_\sV$,
we can express our Hamiltonian (\ref{H-hc}) on $\Hhccyclespace$ as follows:
\begin{equation*}\label{}
H^{\mathsf{hc}}_{n,\sV} = \Phc \left( n - P_\sV^{(q)} - P_\sV^{(n-q)} \right) \Phc.
\end{equation*}
Let $\widetilde{P}^{(q)}_\sV = \Phc P^{(q)}_\sV \Phc$ and
define the following reduced Hamiltonian
\begin{equation}\label{redHam}
   \widetilde{H}^{\mathsf{hc}}_{q,n,\sV}
   = \Phc \left(n - P_\sV^{(n-q)} \right) \Phc,
\end{equation}
so that
\begin{equation*}\label{}
    H^{\mathsf{hc}}_{n,\sV}
    = \widetilde{H}^{\mathsf{hc}}_{q,n,\sV} - \widetilde{P}^{(q)}_\sV.
\end{equation*}
The next step is to estimate the effect of neglecting the
action of the $\widetilde{P}^{(q)}_\sV$ term (equivalent to the hopping of the $q$ particles)
in the unsymmetrised space.
Let
\begin{equation}\label{c_tilde}
   \widetilde{c}\,^n_\sV(q) =  \frac{1}{Z_\sbeta(n,V)} \frac{1}{V} \trace_{\Hhccyclespace}
        \left[ U_q \mathrm{e}^{-\beta \widetilde{H}^{\mathsf{hc}}_{q,n,\sV} } \right],
\end{equation}
and define
\begin{equation*}\label{}
    Z_\sbeta(\lambda,n,V) = \trace_{\Hhccansym{n}}
    \left[ \mathrm{e}^{-\beta H^{\mathsf{hc}}_{\lambda, n,\sV} } \right]
\end{equation*}
where
\begin{equation}\label{h-lambda}
H^{\mathsf{hc}}_{\lambda, n,\sV} = \Phc \left( n - \lambda P_\sV^{(n)} \right) \Phc .
\end{equation}
Then we have the following estimate.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Theorem part2
\begin{proposition}\label{c}
\begin{equation*}
    \left| c\,^n_\sV(q)-\widetilde{c}\,^n_\sV(q) \right|
\leq \frac{(1-\mathrm{e}^{-\beta q})}{V}
\frac{Z_\sbeta(\frac{V-q}{V},n-q,V-q)}{Z_\sbeta(n,V)}.
\end{equation*}
\end{proposition}
In the third step we obtain the limit of the ratio on the righthand side of the last inequality:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Theorem part3
\begin{proposition}\label{Z}
\begin{equation}
   \thermlim\frac{Z_\sbeta(\frac{V-q}{V},n-q,V-q)}{Z_\sbeta(n,V)}=
    \begin{cases}
    \rho^q \, \e^{\beta q}
    & \mathrm{if}\ \  \rho\in[0,\rho_\sbeta]\cup[1-\rho_\sbeta,1],\\
    {\displaystyle \rho^q_\sbeta \, \mathrm{e}^{\beta q(1+\rho-\rho_\sbeta)}}
    & \mathrm{if}\ \ \rho\in[\rho_\sbeta,1-\rho_\sbeta].
    \end{cases}
\end{equation}
\end{proposition}
The final step is a simple proposition where we check the following:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{c-q}
$\widetilde{c}\,^n_\sV(q)=0$ if $q>1$ and
\begin{equation}
   \thermlim\widetilde{c}\,^n_\sV(1)=
    \begin{cases}
    \rho
    & \mathrm{if}\ \  \rho\in[0,\rho_\sbeta]\cup[1-\rho_\sbeta,1],\\
    {\displaystyle \rho_\sbeta \, \mathrm{e}^{\beta(\rho-\rho_\sbeta)}}
    & \mathrm{if}\ \ \rho\in[\rho_\sbeta,1-\rho_\sbeta].
    \end{cases}
\end{equation}
\end{proposition}
Using these four results the main result, Theorem \ref{Th-rho-long} follows very easily.
From Propositions \ref{c} and \ref{Z}
we have
\begin{equation*}
    \thermlim c\,^n_\sV(q)=\thermlim\widetilde{c}\,^n_\sV(q).
\end{equation*}
Since by Proposition \ref{c-q}, $\widetilde{c}\,^n_\sV(q)$=0 if $q>1$, it follows that
\begin{equation*}\label{rho-short-2}
   \rho_\sbeta^{\mathrm{short}}=\thermlim c\,^n_\sV(1)=
    \begin{cases}
    \rho
    & \mathrm{if}\ \  \rho\in[0,\rho_\sbeta]\cup[1-\rho_\sbeta,1],\\
    {\displaystyle \rho_\sbeta\mathrm{e}^{\beta(\rho-\rho_\sbeta)}}
    & \mathrm{if}\ \ \rho\in[\rho_\sbeta,1-\rho_\sbeta]
    \end{cases}
\end{equation*}
which is the required result.
\par
In the next four subsections we prove the results stated above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Subsection : Proving Theorem part 1
%
\subsection{Proof of Proposition \ref{stat}}
We recall the
following facts on the permutation group.
\begin{itemize}
\item The decomposition into cycles leads to a partition of $S_n$ into equivalence classes
of permutations with the same cycle structure
$C_{\mathbf{q}}$, where $\mathbf{q} = [q_1, q_2, \dots ,q_r]$ is an
unordered $r$-tuple of natural numbers with $q_1 + q_2 + \dots + q_r =n$.
\item Two permutation $\pi'$ and $\pi''$ belong to the same class if and only if they are conjugate in
$S_n$, i.e. if there exists a $\pi \in S_n$ such that
\begin{equation} \label{conjclass}
    \pi'' = \pi^{-1} \pi' \pi.
\end{equation}
\item The number of permutations belonging to the class $C_{\mathbf{q}}$ is
\begin{equation} \label{countingperms}
    \frac{n!}{n_{\mathbf{q}}! (q_1 q_2\dots q_r)}
\end{equation}
with $n_\mathbf{q}! = n_1! n_2! \dots n_j! \dots$ and $n_j$ is the number of cycles of length $j$
in $\mathbf{q}$.
\end{itemize}

We observe that since our Hamiltonian is symmetric ($[\hcHam,U_\pi] = 0, \pi \in S_n$) and therefore for
$\pi', \pi'' \in C_{\mathbf{q}}$, one has by (\ref{conjclass})
\begin{equation}
  \begin{split}                                 \label{invar_under_cycles} %%%LABEL:inver_under_cycles
    \trace_{\Hhccan{n}} \left[ U_{\pi''} \e^{-\beta \hcHam} \right]
        &= \trace_{\Hhccan{n}} \left[U_\pi^{-1} U_{\pi'} U_\pi \e^{-\beta \hcHam} \right] \\
        &= \trace_{\Hhccan{n}} \left[ U_\pi^{-1} U_{\pi'} \e^{-\beta \hcHam} U_\pi \right] \\
        &= \trace_{\Hhccan{n}} \left[ U_{\pi'} \e^{-\beta \hcHam} \right].
   \end{split}
\end{equation}

%We shall take $q \in \pi$ to mean that the permutation $\pi$ contains a cycle of length $q$.

For $q \in \mathbb{N}$, let $N_q(\pi)$ be the number of cycles of length $q$ in $\pi$.

Let $r_j$ denote the number of cycles of length $j$. Then $\sum_{j} j r_j = n$ and the corresponding
number of permutations this cycle structure is $n!/ \prod_{j} j^{r_j} r_j!$ (from (\ref{countingperms})).
Denote $(r_j)$ the class of permutations with such a cycle structure.
Then
\begin{align*}
	\cProb(N_q\!=\!r)
&=	\frac{1}{\cPart} \frac{1}{n!}
	\mathop{\sum_{(r_j)} }_{ r_q = r} \sum_{\pi \in (r_j)}
	\trace_{\Hhccan{n}} \left[ U_\pi \e^{-\beta \hcHam} \right]
\\
&=	\frac{1}{\cPart} \frac{1}{n!}
	\mathop{\sum_{(r_j)}}_{ r_q = r}
	\frac{n!}{\prod_{j \ge 1} j^{r_j} r_j !}
	\trace_{\Hhccan{n}} \left[ U_{\tilde{\pi}} \e^{-\beta \hcHam} \right]
\end{align*}
where $\tilde{\pi}$ is any permutation with cycle distribution $(r_j)$. Suppose that $r \ge 1$
and consider a permutation where the first $q$ indices belong to the same cycle of length $q$.
Let $\pi'$ denote the permutation of the remaining $n-q$ indices. We have
\[
	U_\pi = U_q \otimes U_{\pi'}
\]
and $\pi'$ has cycle structure $(r_j - \delta_{jq})$. Then
\begin{align*}
	\cProb(N_q\!=\!r)
&=	\frac{1}{\cPart} \frac{1}{n!}
	\mathop{\sum_{(r_j), \sum j r_j = n-q}}_{ r_q = r-1}
	\frac{(n-q)!}{\prod_{j \ge 1} j^{r_j} r_j!} \frac{n!}{qr(n-q)!}
	\trace_{\Hhccan{n}} \left[ (U_q \otimes U_{\pi'}) \e^{-\beta \hcHam} \right]
\\
&=	\frac{1}{\cPart} \frac{1}{qr(n-q)!}
	\sum_{\pi' \in S_{n-q}}
	\trace_{\Hhccan{n}} \left[ (U_q \otimes U_{\pi'}) \e^{-\beta \hcHam} \right]
\end{align*}	

Then the canonical expectation of the number of $q$-cycles is found to be
\begin{align*}
    \cEx(N_q) &= \sum_{r=0}^{\infty} r\cProb(N_q = r)
\\
    &=	\frac{1}{\cPart} \frac{1}{q(n-q)!}
    		\sum_{\pi' \in S_{n-q}}
		\trace_{\Hhccan{n}} \left[ (U_q \otimes U_{\pi'}) \e^{-\beta \hcHam} \right] \hspace{1.4cm}
\end{align*}
\begin{align*}
\phantom{\cEx(N_q)}
    &=	\frac{1}{\cPart} \frac{1}{q(n-q)!} \sum_{\pi' \in S_{n-q}}
		\trace_{\Hcan{n}} \left[ \Phc (U_q \otimes U_{\pi'}) \e^{-\beta \hcHam} \Phc \right]
\\
    &=	\frac{1}{\cPart} \frac{1}{q}
		\trace_{\Hcan{q} \otimes \Hcansym{n-q}}
		\left[ \Phc (U_q \otimes I^{(n-q)}) \e^{-\beta \hcHam} \Phc \right]
\\
    &=	\frac{1}{\cPart} \frac{1}{q}
		\trace_{\Phc(\Hcan{q} \otimes \Hcansym{n-q})}
		\left[ (U_q \otimes I^{(n-q)}) \e^{-\beta \hcHam} \right].
\end{align*}

Since
\[
 	c_\sV^n(q) = \frac{q \; \cEx(N_q) }{V}
\]
we have proved Proposition \ref{stat}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Subsection : Proving Theorem part 2
%
\subsection{Proof of Proposition \ref{c}}\label{sub c}
To prove Proposition \ref{c} we have to obtain an upper bound  for
\begin{equation*}
    \left |\trace_{\Hhccyclespace}
        \left[ U_q \e^{-\beta H^{\mathsf{hc}}_{n,\sV} } \right]  \\
    - \trace_{\Hhccyclespace}
        \left[ U_q \e^{-\beta \widetilde{H}^{\mathsf{hc}}_{q,n,\sV} } \right]\right |
\end{equation*}
In order to do this we first shall introduce some notation and make some remarks before
proceeding.
\par
Let $\Lambda_{\sV +}^{(n-q)}$ be the family of sets of $n-q$ distinct points
of $\Lambda_\sV$. For $\mathbf{k}=\{k_1, k_2, \dots , k_{n-q}\}\in \Lambda_{\sV +}^{(n-q)}$
let
\begin{equation*}\label{}
  |\mathbf{k}\ra := \sigma_{+}^{n-q}(\mathbf{e}_{k_1} \otimes \mathbf{e}_{k_2}
  \otimes \dots \otimes \mathbf{e}_{k_{n-q}}).
  \end{equation*}
Then $\{|\mathbf{k}\ra\,|\,\mathbf{k}\in \Lambda_{\sV +}^{(n-q)}\}$
is an orthonormal basis for
$\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}
:=\mathcal{P}^{\mathrm{hc}}_{n-q} \mathcal{H}_{\sV +}^{(n-q)}$.
\par
Similarly let $\Lambda_{\sV}^{(q)}$ be the set of
ordered $q$-tuples of distinct indices of $\Lambda_\sV$ and
for $\mathbf{i}=(i_1, i_2, \dots , i_q)\in \Lambda_{\sV}^{(q)}$
let
\begin{equation*}\label{}
  |\mathbf{i}\ra := \mathbf{e}_{i_1} \otimes \mathbf{e}_{i_2}
  \otimes \dots \otimes \mathbf{e}_{i_q}.
  \end{equation*}
Then $\{|\mathbf{i}\ra\,|\,\mathbf{i}\in \Lambda_{\sV}^{(q)}\}$ is an orthonormal basis for
$\mathcal{H}_{q,\sV}^{\mathrm{hc}}:=\mathcal{P}^{\mathrm{hc}}_q \mathcal{H}_{\sV}^{(q)}$.
\par
If $\mathbf{k}\in \Lambda_{\sV +}^{(n-q)}$ and
$\mathbf{i}\in \Lambda_{\sV}^{(q)}$
we shall write $\mathbf{k}\sim\mathbf{i}$
if $\{k_1, k_2, \dots , k_{n-q}\}\cap \{i_1, i_2, \dots , i_q\}=\emptyset$
and we shall use the notation
\begin{equation*}\label{}
    |\mathbf{i};\mathbf{k}\ra:=|\mathbf{i}\ra\otimes|\mathbf{k}\ra.
\end{equation*}
Then a basis for $\Hhccyclespace$ may
be formed by taking the tensor product of the bases
of $\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}$ and
$\mathcal{H}_{q,\sV}^{\mathrm{hc}}$ where we disallow particles from appearing in both
spaces simultaneously.
Thus the set
$\{|\mathbf{i};\mathbf{k}\ra\,|\,\mathbf{k}\in \Lambda_{\sV +}^{(n-q)},\
\mathbf{i}\in \Lambda_{\sV}^{(q)},\ \mathbf{k}\sim\mathbf{i}\}$
is an orthonormal basis for $\Hhccyclespace$.
\par
We shall need also the following facts. For simplicity we shall write $\Htilde$ and
$\widetilde{P}$ for $\widetilde{H}^{\mathsf{hc}}_{q,n,\sV}$ and $\widetilde{P}^{(q)}_\sV$
respectively, as defined in equation (\ref{redHam}).
\par
Let $\mathcal{P}_\ii^{(n-q)}$ be the projection
of $\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}$ onto a space
with none of the $n-q$ particles at the points $i_1, i_2, \dots , i_q$
(so there are $V-q$ available sites for $n-q$ particles) and not more than
one particle at any site. Then

\begin{remark}\label{rem a}
For $\ii \sim \kk$, if $s>0$
\begin{equation}\label{Htilde}
	\e^{-\beta \Htilde s} | \ii; \kk \ra = |\ii; \e^{-\beta H^\ii s} \kk \ra \e^{-\beta q s}
\end{equation}
where $H^\ii = \mathcal{P}_\ii^{(n-q)} ((n-q) - P_\sV^{(n-q)} ) \mathcal{P}_\ii^{(n-q)}$.
\par
\end{remark}
This can be seen as follows:
For $\ii \sim \kk$,
\begin{align*}
    \widetilde{H} |\ii; \kk \ra
&=  \Phc ( n - P_\sV^{(n-q)} ) \Phc |\ii; \kk \ra
\\
&=  q\Phc |\ii; \kk \ra + \Phc | \ii; ( (n-q)- P_\sV^{(n-q)} ) \kk \ra
\\
&=  q |\ii; \kk \ra +  |\ii; \mathcal{P}_\ii^{(n-q)} ( (n-q) - P_\sV^{(n-q)} ) \kk \ra
\\
&=  q |\ii; \kk\ra +  |\ii; H^\ii\kk \ra.
\end{align*}
%
\begin{remark}\label{rem b}
For $\ii \sim \kk$,
\begin{equation}\label{}
	H^\ii|\kk \ra=(n-q)|\kk \ra-\frac{1}{V}
		 \sum_{l=1}^{n-q}\sum_{j\notin\, \ii\,\cup\,\kk\setminus\{k_l\}}
		 |(k_1,k_2, \ldots,\widehat{k}_l,j,\ldots,k_{n-q}) \ra
\end{equation}
where the hat symbol implies the term is removed from the sequence,
while from (\ref{h-lambda}), for $\kk\in \mathcal{H}^{(n-q)}_{\sV-q,+}$ we have
\begin{equation}
	H^{\mathsf{hc}}_{\lambda, n-q,\sV-q} | \kk \ra = (n-q)|\kk \ra-\frac{\lambda}{V-q}
    		\sum_{l=1}^{n-q}
		\sum^{V-q}_{\stackrel{j=1}{j\notin\, \kk\setminus\{k_l\}}}
		|(k_1,k_2, \ldots,\widehat{k}_l,j,\ldots,k_{n-q}) \ra.
\end{equation}
Thus $H^\ii$ is unitarily equivalent to $H^{\mathsf{hc}}_{(\sV-q)/\sV,\, n-q,\sV-q}$ and
\begin{equation}
	\trace_{\Hhccyclespace}  \left[
		\mathcal{P}_\ii^{(n-q)} \e^{-\beta \widetilde{H}^\ii}  \mathcal{P}_\ii^{(n-q)}
	\right]
    	=\ccPart.
\end{equation}
\end{remark}
%
\begin{remark}\label{rem c}
For $s, \alpha \in \RR$
    \[
        \left( \mathcal{P}_\ii^{(n-q)} \e^{-s \widetilde{H}^\ii}  \mathcal{P}_\ii^{(n-q)} \right)^\alpha
        = \mathcal{P}_\ii^{(n-q)} \e^{-s \alpha \widetilde{H}^\ii}  \mathcal{P}_\ii^{(n-q)}.
    \]
\end{remark}
We expand
\begin{equation*}
    \trace_{\Hhccyclespace} \left[U_q \e^{-\beta H^{\mathsf{hc}}_{n,\sV} } \right]
    = \trace_{\Hhccyclespace} \left[U_q \e^{-\beta (\Htilde - \widetilde{P}) } \right]
\end{equation*}
in a Dyson series in powers of
$\widetilde{P}$. If $m\geq 1$, the $m^\text{th}$ term is
\begin{align*}
    X_m := \beta^m \int_0^1\hskip -0.3cm ds_1 \int_0^{s_1} \hskip -0.4cmds_2 \dots
    & \int_0^{s_{m-1}}\hskip -0.8cm ds_m  \;\;
    \trace_{\mathcal{H}^{\mathsf{hc}}_{q,n,\sV}}
        \Bigg[ \e^{-\beta \Htilde (1-s_1)} \widetilde{P}
        \e^{-\beta \Htilde (s_1-s_2)} \widetilde{P}
        \cdots \\
        & \hskip 6cm\cdots \widetilde{P}
        \e^{-\beta \Htilde (s_{m-1} -s_m)} \widetilde{P}
        \e^{-\beta \Htilde s_m}  U_q \Bigg].
\end{align*}
Recall that $\widetilde{P}:= \Phc P_\sV^{(q)} \Phc$ where
\begin{equation}\label{}
P_\sV^{(q)} =
P_\sV \otimes I \otimes \cdots \otimes I + I \otimes P_\sV \otimes I \otimes \cdots \otimes I + \cdots +
I \otimes \cdots \otimes I \otimes P_\sV
\end{equation}
has $q$ terms, so in the above trace we have $m$ instances of this form.
Let $P^{(q)}_r = I \otimes \cdots \otimes
\underbrace{P_\sV}_{\text{$r^{th}$ place}} \otimes \dots \otimes I$, and let
$\widetilde{P}_r = \Phc P^{(q)}_r \Phc$.
\par
Then we can write
\begin{align*}
    X_m &= \beta^m \int_0^1 \hskip -0.3cm ds_1 \int_0^{s_1}\hskip -0.4cm ds_2 \dots
    \int_0^{s_{m-1}}\hskip -0.8cm ds_m \;\; \sum_{r_1 = 1}^q \cdots \sum_{r_m = 1}^q \\
    & \qquad \times \trace_{\mathcal{H}^{\mathsf{hc}}_{q,n,\sV}}
        \Bigg[ \e^{-\beta \Htilde (1-s_1)} \widetilde{P}_{r_1}
        \e^{-\beta \Htilde (s_1-s_2)} \widetilde{P}_{r_2}
        \cdots \widetilde{P}_{r_{m-1}}
        \e^{-\beta \Htilde (s_{m-1} -s_m)} \widetilde{P}_{r_m}
        \e^{-\beta \Htilde s_m}  U_q \Bigg].
\end{align*}
In terms of the basis of $\mathcal{H}^{\mathsf{hc}}_{q,n,\sV}$ we may write the expression for $X_m$ as
\begin{multline}                            \label{trace_term}   %LABEL: trace_term
    \hspace{1cm} X_m=\beta^m \int_0^1\hskip -0.3cm ds_1 \int_0^{s_1} \hskip -0.4cm ds_2
    \dots  \int_0^{s_{m-1}}\hskip -0.8cm ds_m \;\;
    \sum_{r_1=1}^q \dots \sum_{r_m=1}^q \;\;
    \sum_{\kk^0, \dots\ ,\kk^m} \;\; \sum_{\ii^0 \sim \kk^0} \cdots
    \sum_{\ii^m \sim \kk^m} \\
        \la \ii^0; \kk^0 | \e^{-\beta \Htilde (1-s_1)} \widetilde{P}_{r_1}
        | \ii^1; \kk^1 \ra
        \la \ii^1; \kk^1 | \e^{-\beta \Htilde (s_1-s_2)} \widetilde{P}_{r_2}
        | \ii^2; \kk^2 \ra \cdots\\
        \cdots
        \la \ii^{m-1}; \kk^{m-1} |\e^{-\beta \Htilde (s_{m-1} -s_m)} \tilde{P}_{r_m}
        | \ii^m; \kk^m \ra
        \la \ii^m; \kk^m |\e^{-\beta \Htilde s_m}  U_q  | \ii^0; \kk^0 \ra
\end{multline}
where it is understood that the $\ii$ summations are over $\Lambda_{\sV}^{(q)}$
and the $\kk$ summations are over $\Lambda_{\sV +}^{(n-q)}$.
%
Note that for $\ii \sim \kk$
\begin{equation}\label{P-r}
	\widetilde{P}_r | \ii ; \kk \ra = \frac{1}{V}\hskip -0.8cm \sum_{\stackrel{l=1...V}
		{l \notin \kk;\ l \neq i_1 \dots \widehat{i}_r \dots i_q}}\hskip -0.8cm
		|(i_1,\cdots , \widehat{i}_r ,l , \cdots , i_q ); \kk \ra
\end{equation}
where again the hat symbol implies that the term is removed from the sequence.
\par
Consider one of the inner products in the expression (\ref{trace_term}) for $X_m$, using
(\ref{Htilde}) and (\ref{P-r}) above. For
$\ii \sim \kk$ and $\jj \sim \kk'$:
\begin{align*}
	\la \ii ; \kk\, |\, \e^{-\beta s \widetilde{H}} \widetilde{P}_r | \jj ; \kk' \ra
&= 	\frac{\e^{-\beta q s}}{V}\hskip -0.7cm  \sum_{\stackrel{l=1...V}{l \notin \kk';\, l \neq j_1, \dots
	\widehat{j_r}, \dots j_q}} \hskip -0.7cm\la \ii\,|\,( j_1,
	\cdots ,
	\widehat{j_r}, l, \cdots ,j_q) \ra
	\la \kk\, |\, \e^{-\beta s H^\ii} | \kk' \ra
\\[0.3cm]
&= 	\frac{\e^{-\beta q s}}{V}
	\hskip -0.7cm\sum_{\stackrel{l=1...V}{l \notin \kk';\, l \neq j_1, \dots ,
	\widehat{j_r} , \dots ,  j_q}}\hskip -0.7cm \delta_{i_1 j_1}
	\dots \widehat{\delta_{i_r j_r}} \, \delta_{i_r l} \,
	\dots \delta_{i_q j_q}
	\la \kk | \e^{-\beta s H^\ii} | \kk' \ra.
\end{align*}
In summing over $l$ we replace $l$ by $i_r$ and the result is non-zero
only if $i_r \notin \kk'$ and $i_r \neq j_1, \dots , \widehat{j_r},  \dots , j_q$.
However this last condition is not necessary because if $i_r=j_s\ (s\neq r)$
then $j_s\neq i_s$ and we get zero. Also if for some $s\neq r$, $i_s \in \kk'$ then once again
$j_s\neq i_s$. We can therefore replace the condition $i_r \notin \kk'$ by $\ii \sim \kk'$.
Using $\mathcal{I}$ for the indicator function, we have
\begin{align*}
	\la \ii ; \kk\, |\, \e^{-\beta s \widetilde{H}} \widetilde{P}_r | \jj ; \kk' \ra
&= \frac{\e^{-\beta q s}}{V}  \delta_{i_1 j_1}
    	\dots \widehat{\delta_{i_r j_r}} \, \dots \delta_{i_q j_q}
	\la \kk | \e^{-\beta s H^\ii} | \kk' \ra
	\mathcal{I}_{(\ii \sim \kk')}
\\[0.2cm]
&= 	\frac{\e^{-\beta q s}}{V}  \delta_{i_1 j_1} \dots
	\widehat{\delta_{i_r j_r}} \, \dots \delta_{i_q j_q}
	\la \kk | \mathcal{P}_\ii^{(n-q)} \e^{-\beta s H^\ii}
	\mathcal{P}_\ii^{(n-q)} | \kk' \ra
\end{align*}
Now if we sum over $\jj \sim \kk'$, with $\ii \sim \kk$ and for a fixed $r$:
\begin{align*}
	\sum_{ \jj \sim \kk'} \la \ii ; \kk | \e^{-\beta \widetilde{H} s}
	\widetilde{P}_r | \jj ; \kk' \ra \la \jj ; \kk' |
&= 	\frac{\e^{-\beta s q}}{V}  \la \kk | \mathcal{P}_\ii^{(n-q)}
	\e^{-\beta H^\ii s} \mathcal{P}_\ii^{(n-q)} | \kk' \ra \\
&	\quad \times \hspace{-0.4cm}
	\sum_{\stackrel{j_r = 1 \dots V}{j_r \notin \kk' \cup \ii\setminus \{ i_r \}}}
	\la (i_1, \dots ,i_{r-1}, j_r, i_{r+1}, \dots,i_q); \kk' |  .
\end{align*}
%
It is convenient to define the operation $[r,x](\ii)$ which inserts the value of $x$
in the $r^\text{th}$ position of $\ii$ instead of $i_r$.
So for example taking the ordered triplet $\ii = (5,4,1)$,
then $[2,8](\ii) = (5,8,1)$. For brevity we shall denote the composition of these
operators as $[r_k, x_k; \, \dots \, ; r_2, x_2 ; r_1, x_1] := [r_k, x_k]  \circ \cdots \circ [r_2, x_2] \circ [r_1, x_1]$.

Thus the final term in the above expression may be rewritten as $\la [r, j_r](\ii) ; \kk' | $.
\par
Performing two summations for fixed $r_1$ and $r_2$ we get:
\begin{align*}
    \sum_{ \ii^1 \sim \kk^1} \sum_{ \ii^2 \sim \kk^2} &
    \la \ii^0 ; \kk^0 |
    \e^{-\beta s\widetilde{H} } \widetilde{P}_{r_1} | \ii^1 ; \kk^1 \ra
    \la \ii^1 ; \kk^1 | \e^{-\beta t\widetilde{H} } \widetilde{P}_{r_2}
    | \ii^2 ; \kk^2 \ra
    \la \ii^2 ; \kk^2 |  \\
&   =  \frac{\e^{-\beta q (s+t)}}{V^2}
    \sum_{i^1_{r_1} \notin \kk^1 \cup \ii^0 \setminus \{ i^0_{r_1} \}} \hspace{0.4cm}
    \sum_{i^2_{r_2} \notin \kk^2 \cup [r_1,i^1_{r_1}](\ii^0) \setminus \{ i^0_{r_2} \}}
    \la \kk^0 | \mathcal{P}_{\ii^0}\
    \e^{-\beta sH^{\ii^0} }\ \mathcal{P}_{\ii^0} | \kk^1 \ra \\
&  \qquad \times\la \kk^1 | \mathcal{P}_{[r_1,i^1_{r_1}](\ii^0)}\
\e^{-\beta t H^{[r_1,i^1_{r_1}](\ii^0)} }
   \  \mathcal{P}_{[r_1,i^1_{r_1}](\ii^0)} | \kk^2 \ra
    \la [r_1, i^1_{r_1}; r_2, i^2_{r_2}](\ii^0); \kk^2 |  \\
& \\
&   =  \frac{\e^{-\beta q (s+t)}}{V^2}
    \sum_{i^1_{r_1} \notin  \ii^0 \setminus \{ i^0_{r_1} \}} \hspace{0.4cm}
    \sum_{i^2_{r_2} \notin \kk^2 \cup [r_1,i^1_{r_1}](\ii^0) \setminus \{ i^0_{r_2} \}}
    \la \kk^0 |\ \mathcal{P}_{\ii^0}\ \e^{-\beta s H^{\ii^0} }\ \mathcal{P}_{\ii^0}\
    | \kk^1 \ra
\\
&   \qquad \times \la \kk^1 |\ \mathcal{P}_{[r_1,i^1_{r_1}](\ii^0)}\ \e^{-\beta t H^{[r_1,i^1_{r_1}](\ii^0)}}
   \ \mathcal{P}_{[r_1,i^1_{r_1}](\ii^0)}\ | \kk^2 \ra
    \la [r_2, i^2_{r_2} ; r_1, i^1_{r_1}] (\ii^0); \kk^2|
\\
\end{align*}
due to the fact that $\mathcal{P}_{[r_1, i^1_{r_1}](\ii^0)}|\kk^1\ra=0$ if $i^1_{r_1}\in \kk^1$.
We may apply this to all inner product terms of (\ref{trace_term}) except the final one.
Note we sum over the $V$ sites of the lattice, with certain points excluded in each case.

For the final inner product of (\ref{trace_term}) we obtain:
\begin{align*}
& \la [r_m, i^m_{r_m}; \,  \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) ; \kk^m |
        \e^{-\beta s_m \widetilde{H}  } U_q | \ii^0 ; \kk^0 \ra
\\[0.2cm]
&=  \e^{-\beta q s_m} \la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0); \kk^m |
        \e^{-\beta  s_m H^{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)} }
        U_q | \ii^0 ; \kk^0 \ra
\\[0.2cm]
&=  \e^{-\beta q s_m} \la \kk^m |
        \e^{-\beta  s_m H^{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)} }
        | \kk^0 \ra \;
        \la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra
\\[0.2cm]
&=  \e^{-\beta q s_m} \la \kk^m |
        \mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}
        \e^{-\beta s_m H^{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}  }
        \mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) }
        | \kk^0 \ra
\\
    & \hspace{4cm} \times  \la [r_m, i^m_{r_m};  \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra . \\
\end{align*}
Applying this to the whole tracial expression of (\ref{trace_term}) we obtain
\begin{align*}
   	X_m
=& \; \e^{-\beta q} \frac{\beta^m}{V^m} \sum_{\kk^0 \dots \kk^m} \sum_{\ii^0}
	\sum_{i^1_{r_1} \notin \ii^0 \setminus \{i^0_{r_1}\}} \hspace{0.3cm}
	\sum_{i^2_{r_2} \notin [r_1,i^1_{r_1}](\ii^0) \setminus \{i^1_{r_2}\}}
	\cdots \sum_{i^m_{r_m} \notin [r_{m-1}, i^{m-1}_{r_{m-1}}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)
	\setminus \{i^{m-1}_{r_m}\}}
\\
&	\la \kk^0 | \mathcal{P}_{\ii^0} \e^{-\beta(1-s_1) \Htilde^{\ii^0} } \mathcal{P}_{\ii^0}
	| \kk^1 \ra
\\
&	\la \kk^1 | \mathcal{P}_{[r_1,i^1_{r_1}](\ii^0)}
	\e^{-\beta (s_1-s_2)\Htilde^{[r_1,i^1_{r_1}](\ii^0)}}
	\mathcal{P}_{[r_1,i^1_{r_1}](\ii^0)} | \kk^2 \ra
\\
&	\la \kk^2 | \mathcal{P}_{[r_2, i^2_{r_2}; r_1, i^1_{r_1}](\ii^0)} \e^{-\beta (s_2-s_3)
	\Htilde^{[r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)} }
	\mathcal{P}_{[r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)} | \kk^3 \ra
\\
&	\cdots
\\
&	\la \kk^m | \mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}
	\e^{-\beta s_m \Htilde^{ [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)} }
	\mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}
	| \kk^0 \ra \\
&   \la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra
\\[0.4cm]
=& 	\;\e^{-\beta q} \frac{\beta^m}{V^m} \sum_{\ii^0}
	\sum_{i^1_{r_1} \notin \ii^0 \setminus \{i^0_{r_1}\}} \hspace{0.3cm}
	\sum_{i^2_{r_2} \notin [r_1,i^1_{r_1}](\ii^0) \setminus \{i^1_{r_2}\}}
	\cdots \sum_{i^m_{r_m} \notin [r_{m-1}, i^{m-1}_{r_{m-1}}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)
	\setminus \{i^{m-1}_{r_m}\}}
\\
&	\la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra
\\
&	\trace_{\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}} \Bigg[ \mathcal{P}_{\ii^0} \e^{-\beta (1-s_1)\Htilde^{\ii^0} }
	\mathcal{P}_{\ii^0} \mathcal{P}_{i^1_{r_1}(\ii^0)}
	\e^{-\beta (s_1-s_2)\Htilde^{i^1_{r_1}(\ii^0)} }
	\mathcal{P}_{i^1_{r_1}(\ii^0)} \cdots
\\
&	\qquad \cdots \mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}
	\e^{-\beta s_m\Htilde^{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)} }
	\mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}  \Bigg].
\end{align*}
From the H\"older inequality (see Manjegani \cite{Manjegani}), for finite dimensional non-negative matrices
$A_1, A_2, \dots , A_{m+1}$ we have the inequality
\[
    \left| \trace \big( A_1 A_2 \dots A_{m+1} \big) \right|
        \le \trace \big| A_1 A_2 \dots A_{m+1} \big|
        \le \prod_{k=1}^{m+1} \big( \trace A_k^{p_k} \big)^{\tfrac{1}{p_k}}
\]
where $\sum_{k=1}^{m+1} \tfrac{1}{p_k} = 1$, $p_i > 0$.

Set $p_1 = \frac{1}{1-s_1},\ p_2 = \frac{1}{s_1 - s_2},\ \dots ,\ p_m = \frac{1}{s_{m-1}-s_m},\ p_{m+1} = \frac{1}{s_m}$.
Taking the modulus of the above trace
\begin{align*}
&	\Bigg| \trace_{\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}}  \Bigg[ \mathcal{P}_{\ii^0}
	\e^{-\beta \Htilde^{\ii^0} (1-s_1)} \mathcal{P}_{\ii^0}
	\mathcal{P}_{[r_1, i^1_{r_1}](\ii^0)} \e^{-\beta \Htilde^{[r_1, i^1_{r_1}](\ii^0)} (s_1-s_2)}
	\mathcal{P}_{[r_1, i^1_{r_1}](\ii^0)} \; \cdots \\
&	\qquad \qquad \qquad \cdots \mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}
	\e^{-\beta \Htilde^{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)} (s_m)}
	\mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}
	\Bigg]    \Bigg|
\\
&	\le \quad \trace_{\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}} \bigg[ \mathcal{P}_{\ii^0}
	\e^{-\beta \Htilde^{\ii^0}} \mathcal{P}_{\ii^0} \bigg]^{1-s_1}
	\trace_{\Hhccan{n-q}} \bigg[ \mathcal{P}_{[r_1, i^1_{r_1}](\ii^0)}
	\e^{-\beta \Htilde^{[r_1, i^1_{r_1}](\ii^0)}} \mathcal{P}_{[r_1, i^1_{r_1}](\ii^0)} \bigg]^{s_1-s_2} \cdots \\
&	\quad \cdot\!\cdot\!\cdot\! \trace_{\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}}
	\bigg[ \mathcal{P}_{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}
	\e^{-\beta \Htilde^{[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)}}
	\mathcal{P}_{ [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)} \bigg]^{s_m} \!\!\!.
\end{align*}
Since the trace is independent of the $V-q$ sites
$\{\ii^0, [r_1,i^1_{r_1}](\ii^0),\,  \dots, \, [r_m, i^m_{r_m}; \, \dots\\ \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)\}$,
and therefore using Remark \ref{rem c}, the product of all the trace terms above is equal to
\[
	\trace_{\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}} \left[ \mathcal{P}_{\boldl}
		\e^{-\beta \Htilde^{\boldl}} \mathcal{P}_{\boldl} \right]
\]
with $\boldl = \{V-q+1, V-q+2, \dots ,V\}$ and from Remark \ref{rem b},
\begin{equation}
	\trace_{\mathcal{H}_{(n-q),\sV, +}^{\mathrm{hc}}} \left[ \mathcal{P}_{\boldl}
		\e^{-\beta \Htilde^{\boldl}} \mathcal{P}_{\boldl} \right]
	=\ccPart.
\end{equation}
\par
Consider the sum
\begin{multline}  					\label{sum_of_cycled_inner_products} %%LABEL
	\sum_{\ii^0} \; \sum_{i^1_{r_1} \notin \ii^0 \setminus \{i^0_{r_1}\}} \hspace{0.2cm}
	\sum_{i^2_{r_2} \notin [r_1, i^1_{r_1}](\ii^0) \setminus \{i^1_{r_2}\}}
	\cdots
\\
	\cdots \sum_{i^m_{r_m} \notin [r_{m-1}, i^{m-1}_{r_{m-1}}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)
	\setminus \{i^{m-1}_{r_m}\}}  \hspace{-0.5cm}
	\la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra.
\end{multline}

If $\{r_1, r_2, \dots , r_m\} \ne \{1,2,\dots, q\} $,  then  $| [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) \ra$
is of the form
\[
	| j_1, j_2, \dots ,j_{n_1},i^0_{n_1+1}, \dots ,i^0_{n_2}, j_{n_2+1},
	\dots, j_{n_3}, i^0_{n_3+1}, \dots  ,
	i^0_{n_4}, j_{n_4+1}, \dots \dots\ra
\]
where $\{n_1, n_2, \dots \}$ is a non-empty ordered set of distinct integers between 0 and $q$. This state is
clearly orthogonal to $U_q \ii^0$ for any $q$. Note that this situation does not arise if $q=1$.
Note also that  this is always the case if $ m < q$.
\par
We may bound the remaining sum corresponding to terms for which
$\{r_1, r_2, \dots , r_m\} = \{1,2,\dots, q\}$ by
\[
	\le \sum_{\ii^0} \;\;
	\underbrace{ \sum_{i^1_{r_1}=1}^\sV \;\;\;
	\sum_{i^2_{r_2}=1}^\sV \;\;\;
	\cdots  \;\;\;
	\sum_{i^m_{r_m} = 1}^\sV }_{\stackrel{\text{where $[r_m, i^m_{r_m}; \, \dots \,
		; r_2, i^2_{r_2} ; r_1, i^1_{r_1}]$}}{\text{\tiny{has distinct indices}}}} \;\;
	\la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra .
\]
Observe that in this case $| [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) \ra$ is independent of
$\ii^0$ so we may take it to be
\[
	|[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\mathbf{s}^0) \ra
\]
where $\mathbf{s}^0 = (1,2,3,\dots,q)$. Then we can interchange the $\ii^0$ summation with the others, and
for each choice of $i^1_{r_1}, i^2_{r_2}, \dots , i^m_{r_m}$ there exists only one possible $\ii^0 \in \Lambda_\sV^{(q)}$
such that
\[
	\la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra \ne 0
\]
So we may conclude that
\begin{multline}
	\sum_{\ii^0} \; \sum_{i^1_{r_1} \notin \ii^0 \setminus \{i^0_{r_1}\}} \hspace{0.2cm}
	\sum_{i^2_{r_2} \notin [r_1, i^1_{r_1}](\ii^0) \setminus \{i^1_{r_2}\}}
	\cdots
\\
	\cdots \sum_{i^m_{r_m} \notin [r_{m-1}, i^{m-1}_{r_{m-1}}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)
	\setminus \{i^{m-1}_{r_m}\}}  \hspace{-0.5cm}
	\la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra \le V^m.
\end{multline}


\;

Applying this, we see that the modulus of the integrated $m^\text{th}$ term of the
Dyson series may bounded above by
\begin{align*}
    |X_m| & \le \e^{-\beta q} \frac{\beta^m}{ m!} \frac{1}{V^m}\
    \ccPart\sum_{r_1 = 1}^q \cdots \sum_{r_m = 1}^q
\\
&   \qquad \times  \sum_{\ii^0} \;
	\sum_{i^1_{r_1} \notin \ii^0 \setminus \{i^0_{r_1}\}} \hspace{0.2cm}
	\sum_{i^2_{r_2} \notin [r_1, i^1_{r_1}](\ii^0) \setminus \{i^1_{r_2}\}}
	\cdots
\\
&	\qquad \cdots \sum_{i^m_{r_m} \notin [r_{m-1}, i^{m-1}_{r_{m-1}}; \, \dots
		 \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0)
	\setminus \{i^{m-1}_{r_m}\}}  \hspace{-0.5cm}
	\la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) | U_q \ii^0 \ra.
\\
&\le	\e^{-\beta q} \frac{\beta^m}{m!} \frac{1}{V^m}\
	\ccPart  \sum_{r_1 = 1}^q \cdots \sum_{r_m = 1}^q V^m
\\
&=	\e^{-\beta q} \frac{q^m \beta^m}{m!}\ccPart.
\end{align*}
Noting that the zeroth term of the Dyson series is
\[
	X_0 = \trace_{\mathcal{H}^{\mathsf{hc}}_{q,n,V}} \left[U_q \e^{-\beta \Htilde } \right],
\]
we may re-sum the series to obtain
\[
    \Bigg| \trace_{\mathcal{H}^{\mathsf{hc}}_{q,n,V}}
            \left[U_q \e^{-\beta H^{\mathsf{hc}}_{n,V} } \right]
        - \trace_{\mathcal{H}^{\mathsf{hc}}_{q,n,V}}
            \left[U_q \e^{-\beta \Htilde } \right]  \Bigg|
\le \e^{-\beta q}\ccPart\ \sum_{m=1}^\infty \frac{q^m \beta^m}{m!}.
\]
Thus
\begin{align*}
   \left|c_\sV^n(q)-\widetilde{c}_\sV^n(q)\right |
   & = \frac{1}{V} \left| \frac{
        \trace_{\mathcal{H}^{\mathsf{hc}}_{q,n,V}}
            \left[U_q \e^{-\beta H^{\mathsf{hc}}_{n,V} } \right]
        - \trace_{\mathcal{H}^{\mathsf{hc}}_{q,n,V}}
            \left[U_q \e^{-\beta \Htilde } \right]}{Z_\sbeta(n,V)} \right|
\\
    & \le \frac{\e^{-\beta q}}{V} \,
    \frac{ \ccPart}{Z_\sbeta(n,V)} \sum_{m=1}^\infty \frac{q^m \beta^m}{m!}
\\
    & = \frac{\e^{-\beta q}}{V} (\e^{\beta q} - 1)
    \frac{\ccPart}{Z_\sbeta(n,V)}.
    \end{align*}
\subsection{Proof of Proposition \ref{Z}}\label{sub Z}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                       proposition
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Recall that we have
\begin{equation}\label{Z1}
	Z_\sbeta(n-q,V-q)
	= \trace_{\mathcal{H}^{\mathrm{hc}}_{n-q,\sV-q,+}} [\e^{-\beta H_{n-q,\sV-q}^{\mathrm{hc}}}]
	= \e^{-\beta (n-q)}\ \trace_{\mathcal{H}^{\mathrm{hc}}_{n-q,\sV-q,+}}
	\left[ \e^{\beta \mathcal{P}^{\mathrm{hc}}_{n-q}
		P^{n-q}_{\sV-q}\mathcal{P}^{\mathrm{hc}}_{n-q}} \right]
\end{equation}
while
\begin{align}\label{Z2}
	\ccPart
	&= \trace_{\mathcal{H}^{\mathrm{hc}}_{n-q,\sV-q,+}}
  		[\e^{-\beta H_{(\sV-q)/V,n-q,\sV-q}^{\mathrm{hc}}}]\nonumber
\\
  	&= \e^{-\beta (n-q)}\ \trace_{\mathcal{H}^{\mathrm{hc}}_{n-q,\sV-q,+}}
  		\left[ \e^{\beta(\frac{\sV-q}{\sV}) \mathcal{P}^{\mathrm{hc}}_{n-q} P^{n-q}_{\sV-q}
  		\mathcal{P}^{\mathrm{hc}}_{n-q}} \right].
\end{align}
Comparison of (\ref{Z1}) and (\ref{Z2}) yields
\begin{equation*}\label{}
     \ccPart=\e^{-\beta \frac{q}{V}(n-q)}\ Z_{\beta(\frac{\sV-q}{\sV})}(n-q,V-q)
\end{equation*}
and thus we have to analyse the following ratio:
\begin{equation}                            \label{c_expression} %%LABEL
    \frac{ \e^{-\beta \frac{q}{V}(n-q)} Z_{\beta(\frac{\sV-q}{\sV})}(n-q, V-q)}{\cPart}.
\end{equation}
%
Penrose in \cite{Penrose} gave an explicit expression for $\cPart$:
\begin{equation*}\label{}
    \cPart=\hskip -0.3cm\sum_{r=0}^{\min(n, V-n)} z(r,n,V,\beta),
\end{equation*}
where
\begin{equation*}\label{}
    z(r,n,V,\beta):=\left( \frac{V-2r+1}{V-r+1} \right) \binom{V}{r}
    \exp\left\{ -\frac{\beta}{V} \left[ Vr - r^2 + r + n^2 - n \right] \right\}.
\end{equation*}
He also proved that if $h_\sV:[0,\min(\rho,1-\rho)]\to \RR$ converges uniformly in $[0,\min(\rho,1-\rho)]$
as $V\to\infty$ to a continuous  function $h:[0,\min(\rho,1-\rho)]\to \RR$, then
\begin{equation}\label{LD}
    \thermlim \frac{1}{\cPart} \sum_{r=0}^{\min(n,V-n)} h_\sV(\tfrac{r}{V})\
    z(r,n,V,\beta) =
    \begin{cases}
    h(\rho),
    & \mathrm{if}\ \  \rho\in[0,\rho_\beta],\\
    h(\rho_\beta),
    & \mathrm{if}\ \ \rho\in[\rho_\beta,1-\rho_\beta],\\
    h(1-\rho),
    & \mathrm{if}\ \  \rho\in[1-\rho_\beta,1].
    \end{cases}
\end{equation}
\par
We wish to express the ratio in (\ref{c_expression}) in the form of the lefthand side of
(\ref{LD}). We have
\begin{align*}
    Z_{\beta(\frac{\sV-q}{\sV})}(n-q, V-q)
    = \sum_{r=0}^{\min(n-q, V-n)} & \left( \frac{V-q-2r+1}{V-q-r+1} \right) \binom{V-q}{r} \\
    &   \times \exp\left\{ -\frac{\beta}{V} \left[ r(V-q) - r^2 + r + (n-q)^2 - (n-q) \right] \right\}
\end{align*}
For the case $\rho > \tfrac{1}{2}$, for large $V$, $n-q> V-n$ we must sum from
zero to $V-n$ and a straightforward calculation then gives
\begin{equation*}\label{}
    \e^{-\beta \tfrac{q}{V}(n-q)}Z_{\beta(\frac{\sV-q}{\sV})}(n-q, V-q)=\sum_{r=0}^{V-n} h_\sV(\tfrac{r}{V})\
    z(r,n,V,\beta)
\end{equation*}
where
\begin{multline} \label{h-rho>1/2}
    h_\sV(x) =  \left( \frac{1-2x-(q-1)/V}{1-2x+1/V} \right)
    \left( \frac{1-x+1/V}{1-x-(q-1)/V} \right) \\
        \times \prod_{s=0}^{q-1}\left(\frac{1-x-s/V}{1-s/V}\right)
        \exp\left\{ \beta q \left[ x + \rho - 1/V \right] \right\}.
\end{multline}
Therefore
\begin{equation} \label{h-lim>1/2}
    h(x)=\lim_{V \to\infty} h_\sV(x)=(1-x)^q \ \e^{q\beta (x+\rho)}.
\end{equation}
It is clear that the convergence is uniform since $h_\sV(x)$ is a product of terms each of which
converges uniformly on $[0,1-\rho]$ for $\rho>\tfrac{1}{2}$. Thus
\begin{equation*}
\thermlim \frac{ \e^{-\beta \frac{q}{V}(n-q)} Z_{\beta(\frac{\sV-q}{\sV})}(n-q, V-q)}{\cPart}
=
\begin{cases}
   (1-\rho_\sbeta)^q\ \e^{q\beta (\rho_\sbeta+\rho)}
    & \mathrm{if}\ \ \rho\in(1/2,1-\rho_\sbeta], \vspace{0.1cm}\\
    \rho^q \e^{\beta q}
    & \mathrm{if}\ \  \rho\in(1-\rho_\sbeta,1] .
    \end{cases}
\end{equation*}
Note that using the relation
\begin{equation*}
    \beta=\frac{1}{1-2\rho_\sbeta}\ln\left(\frac{1-\rho_\sbeta}{\rho_\sbeta}\right )
\end{equation*}
we get
\begin{equation*}
    (1-\rho_\sbeta)^q\ \e^{q\beta (\rho_\sbeta+\rho)}
    =\rho_\sbeta^q\ \e^{q\beta (1+\rho-\rho_\sbeta)}
\end{equation*}
and therefore we have proved Proposition \ref{Z} for $\rho>\tfrac{1}{2}$.
%%%%%%%%%%%%\rho <1/2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the case $\rho \le \tfrac{1}{2}$ we have that $n-q < V-n$, the sum
for $\e^{-\beta \frac{q}{V}(n-q)} Z_{\beta(\frac{\sV-q}{\sV})}(n-q, V-q)$ is up to $n-q$,
and therefore we need to shift the index by $q$ to get it into the required form.
After shifting we get
\begin{align*}
\e^{-\beta \tfrac{q}{V}(n-q)} Z_{\beta(\frac{\sV-q}{\sV})}(n-q, V-q)
 &= \sum_{r=q}^{n} z(r,n,V,\beta) \left( \frac{V+q-2r+1}{V-2r+1} \right)\\
    & \hspace{0cm} \times \frac{ r(r-1)(r-2) \cdots (r-q+1)}{V(V-1)(V-2)\cdots (V-q+1)}
    \exp\left\{ \frac{\beta q}{V} \left[ V + n - r \right] \right\}.
\end{align*}
Note that summand is zero if we put $r=0, \dots, q-1$. Thus we may sum
from zero to $n$ to get as before
\begin{equation*}
    \e^{-\beta \tfrac{q}{V}(n-q)}Z_{\beta(\frac{\sV-q}{\sV})}(n-q, V-q)=\sum_{r=0}^{n} h_\sV(\tfrac{r}{V})\
    z(r,n,V,\beta)
\end{equation*}
where this time
\begin{equation} \label{h-rho<=1/2}
    h_\sV(x) =  \left( \frac{1-2x +(q+1)/V}{1-2x+1/V} \right)
     \prod_{s=0}^{q-1}\left( \frac{x -s/V}{1-s/V} \right)
                        \exp\left\{ \beta q \left[ 1 + \rho - x  \right] \right\}
\end{equation}
so that
\begin{equation*}
   h(x)=\thermlim h_\sV(x) = x^q \exp\{\beta q(1 + \rho - x) \}.
\end{equation*}
Convergence is again uniform on $[0,\rho]$ for $\rho<\tfrac{1}{2}$
and therefore
\begin{equation*}\label{}
\thermlim \frac{ \e^{-\beta \frac{q}{V}(n-q)} Z_{\beta(\frac{\sV-q}{\sV})}(n-q, V-q)}{\cPart}
=
\begin{cases}
       \rho^q \e^{\beta q}
    & \mathrm{if}\ \  \rho\in[0,\rho_\sbeta),  \vspace{0.1cm}\\
    \rho_\sbeta^q\ \e^{q\beta (1+\rho-\rho_\sbeta)}
    & \mathrm{if}\ \ \rho\in[\rho_\sbeta,1/2),
    \end{cases}
\end{equation*}
proving Proposition \ref{Z} for $\rho < \tfrac{1}{2}$. The case $\rho = \tfrac{1}{2}$
is more delicate because the first term in (\ref{h-rho<=1/2}) does not converge uniformly.
We can write (taking $V=2n$)
\begin{equation*}\label{}
    h_{2n}(r/2n)=\widetilde{h}_{2n}(r/2n)+\frac{q}{2(n-r)+1}\widetilde{h}_{2n}(r/2n)
\end{equation*}
where
\begin{equation*} \label{h-rho=1/2}
    \widetilde{h}_{2n}(x) =
     \prod_{s=0}^{q-1}\left( \frac{x -s/{2n}}{1-s/{2n}} \right)
                        \exp\left\{ \beta q \left[ 3/2 - x  \right] \right\}.
\end{equation*}
Clearly $\widetilde{h}_{2n}(x)$ converges uniformly on $[0,1/2]$ and therefore
\begin{equation*}
    \lim_{n\to\infty}\frac{1}{Z_\sbeta(n,2n)}\sum_{r=0}^{n} \widetilde{h}_{2n}(\tfrac{r}{2n})\
    z(r,n,2n,\beta)=\rho^q_\sbeta \, \mathrm{e}^{\beta q(3/2-\rho_\sbeta)}.
\end{equation*}
We thus have to show that
\begin{equation*}
    \lim_{n\to\infty}\frac{1}{Z_\sbeta(n,2n)}
    \sum_{r=0}^{n} \frac{\widetilde{h}_{2n}(\tfrac{r}{2n})}{2(n-r)+1}\
    z(r,n,2n,\beta)=0.
\end{equation*}
Since $\widetilde{h}_{2n}(x)$ is bounded, by $C$ say,
\begin{equation*}
    \lim_{n\to\infty}\frac{1}{Z_\sbeta(n,2n)}
    \sum_{r<n-n^{1/4}} \frac{\widetilde{h}_{2n}(\tfrac{r}{2n})}{2(n-r)+1}\
    z(r,n,2n,\beta)\leq \lim_{n\to\infty}\frac{C}{2n^{1/4}}=0.
\end{equation*}
On the other hand one can prove that for $n-2n^{1/2}\leq r\leq n-n^{1/2}$ and $r'\geq n-n^{1/4}$
\begin{equation*}\label{}
    \ln z(r,n,2n,\beta)-\ln z(r',n,2n,\beta)>\frac{1}{8}\ln n
\end{equation*}
for $n$ large, so that $z(r',n,2n,\beta)/z(r,n,2n,\beta)<1$.
Therefore
\begin{eqnarray*}
  \lim_{n\to\infty}\frac{1}{Z_\sbeta(n,2n)}
    &&\hskip -1cm\sum_{r\geq n-n^{1/4}} \frac{\widetilde{h}_{2n}(\tfrac{r}{2n})}{2(n-r)+1}\
    z(r,n,2n,\beta)
    \leq
    \lim_{n\to\infty}
    C\frac{\displaystyle{\sum_{r\geq n-n^{1/4}}\hskip -0.4 cm z(r,n,2n,\beta)}}
    {\displaystyle{\sum_{n-2n^{1/2}\leq r\leq n-n^{1/2}}\hskip -1.1cm z(r,n,2n,\beta)}} \\
   &\leq &  \lim_{n\to\infty}\frac{C}{n^{1/4}}
    \frac{\displaystyle{\max_{r\geq n-n^{1/4}}\hskip -0.3 cm z(r,n,2n,\beta)}}
    {\displaystyle{\min_{n-2n^{1/2}\leq r\leq n-n^{1/2}}\hskip -1.0cm z(r,n,2n,\beta)}}
\leq\lim_{n\to\infty}\frac{C}{n^{1/4}}=0.
\end{eqnarray*}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SUBSECTION: lemma proof
\subsection{Proof of Proposition \ref{c-q}}\label{sub c-q}
Recall that
\begin{align*}
\widetilde{c}\,^n_\sV(q)
&   = \frac{1}{Z_\sbeta(n,V)} \frac{1}{V} \trace_{\Hhccyclespace}
        \left[ U_q \e^{-\beta \widetilde{H}^{\mathsf{hc}}_{q,n,\sV} } \right].
\end{align*}
Considering the trace over $\Hhccyclespace$, expanding it in
terms of its basis $\{|\ii; \kk\ra\}$ and using Remark \ref{rem a} above,
where $\ii \sim \kk$
\begin{align*}
 \trace_{\Hhccyclespace}
        \left[ U_q \e^{-\beta \widetilde{H}^{\mathsf{hc}}_{q,n,\sV} } \right]
&   = \sum_{\kk} \sum_{\ii \sim \kk}
\la \ii; \kk | U_q \e^{-\beta \Phc (n - P_\sV^{(n-q)} ) \Phc}
        | \ii; \kk \ra\\
&   = \e^{-\beta q} \sum_{\kk} \sum_{\ii \sim \kk} \la U_q \ii; \kk | \e^{-\beta H^{\ii}}
        | \ii; \kk \ra
= \e^{-\beta q} \sum_{\kk} \sum_{\ii \sim \kk} \la U_q \ii | \ii \ra
        \la \kk | \e^{-\beta H^{\ii}} | \kk \ra.
\end{align*}
For $q > 1$, an element of the basis of the unsymmetrised $q$-space $\Hcan{q}$ may
be written as an ordered $q$-tuple $\ii = (i_1, i_2, \dots , i_q)$ where the $i_l$'s are
all distinct.
Then we may write
\begin{align*}
\la U_q \ii | \ii \ra
&   = \la U_q (\mathbf{e}_{i_1} \otimes \mathbf{e}_{i_2} \otimes \dots \otimes \mathbf{e}_{i_q} ) \, |\,
        \mathbf{e}_{i_1} \otimes \mathbf{e}_{i_2} \otimes \dots \otimes \mathbf{e}_{i_q} \ra
\\
&   = \la \mathbf{e}_{i_2} \otimes \mathbf{e}_{i_3} \otimes \dots \otimes \mathbf{e}_{i_q} \otimes \mathbf{e}_{i_1}\, |\,
        \mathbf{e}_{i_1} \otimes \mathbf{e}_{i_2} \otimes \dots \otimes \mathbf{e}_{i_q}\ra=0.
\end{align*}
Hence $\widetilde{c}\,_\sV^n(q)$ is non-zero only if $q=1$.

For the second statement, note that we may re-express $\widetilde{c}\,_\sV^n(1)$ as follows:
\begin{align*}
    \widetilde{c}\,_\sV^n(1)
&   = \frac{1}{Z_\sbeta(n,V)} \frac{1}{V} \trace_{\Phc(\Hcan{1} \otimes \Hcansym{n-1})}
        \left[ \e^{-\beta \widetilde{H}^{\mathsf{hc}}_{1,n,\sV} } \right]
\\
&   = \frac{\e^{-\beta}}{Z_\sbeta(n,V)} \frac{1}{V} \sum_{i=1}^V \sum_{\kk\, /\hskip-0.17cm \ni\, i}
        \la \kk | \e^{-\beta H^i } | \kk \ra
\\
&   = \frac{\e^{-\beta}}{Z_\sbeta(n,V)} \frac{1}{V} \sum_{i=1}^V \trace_{\Hhccansym{n-1}}
        \left[ \mathcal{P}_i \e^{-\beta H^i} \mathcal{P}_i \right]
\\
&   = \e^{-\beta} \, \frac{ Z_\sbeta(\tfrac{V-1}{V}, n-1, V-1, )}{ Z_\sbeta(n,V) }
\end{align*}
and the result follows from Proposition \ref{Z}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ODLRO}
The one-body reduced density matrix for $x,x'\in \Lambda_\sV $ may be defined as
\begin{equation}\label{}
	D_{\sbeta,n,\sV}(x,x'):=\la a^*_x a\starr_{x'} \ra=\frac{1}{\cPart}
	\trace_{\Hhccansym{n}} \left[ K^{(n)}_{x,\,x'} \e^{-\beta \hcHam} \right].
\end{equation}
where for $\phi\in \mathcal{H}_\sV$, $K_{x,\,x'}\phi=\la \mathbf{e}_{x'}|\phi \ra \mathbf{e}_x$.

Penrose showed that for $x\neq x'$,
\begin{equation*}
	\thermlim D_{\sbeta,n,\sV}(x,x')=\rho_\sbeta^c ,
\end{equation*}
that is, whenever Bose-Einstein condensation occurs, there is
\textit{Off-diagonal long-range order} as defined by Yang \cite{Yang}.
It has been argued and proved in some cases (see for example \cite{Ueltschi2} and \cite{DMP})
that in the expansion of $D_{\sbeta,n,\sV}(x,x')$ in terms of permutation cycles, only infinite cycles contribute to
long-range order. Here we are able to show this explicitly.
\par
By the proposition in Appendix \ref{appendixA}, we have
\begin{equation*}
	D_{\sbeta,n,\sV}(x,x')= \sum_{q=1}^n C_\sV^n(q;K_{x,\,x'})
\end{equation*}
where
\begin{equation}
	C^n_\sV(q;K_{x,\,x'})
	= \frac{1}{Z_\sbeta(n,V)} \, \trace_{\Hhccyclespace}
	\left[ (K_{x,\,x'}\otimes I\otimes I\otimes \ldots\otimes I)
 	U_q \e^{-\beta \hcHam } \right].
\end{equation}
Note that this is equivalent to the expansion of $\sigma_\rho(x)$ in equations (2.14) and (2.16) in \cite{Ueltschi2}.

Applying the argument in Subsections \ref{sub c} and \ref{sub Z}, we can show that
\begin{equation} \label{proj_in_qspace_irrel}
	\thermlim C^n_\sV(q;K_{x,\,x'})=\thermlim \widetilde{C}^n_\sV(q;K_{x,\,x'})
\end{equation}
where we take
\begin{equation*}
	\widetilde{C}^n_\sV(q;K_{x,\,x'}) = \frac{1}{Z_\sbeta(n,V)} \, \trace_{ \Hhccyclespace  }
	\left[ (K_{x,\,x'}\otimes I\otimes I\otimes \ldots\otimes I)
 	U_q \e^{-\beta \widetilde{H}^{\mathsf{hc}}_{q,n,\sV} } \right].
\end{equation*}
The only difference is that instead of equation (\ref{sum_of_cycled_inner_products}), we obtain
\begin{multline} \label{summs}
	\qquad \sum_{\ii^0} \sum_{i^1_{r_1} \notin \ii^0 \setminus \{i^0_{r_1}\}} \hspace{0.1cm}
	\sum_{i^2_{r_2} \notin [r_1, i^1_{r_1}](\ii^0) \setminus \{i^1_{r_2}\}} \!\!
	\cdots
	 \!\! \sum_{i^m_{r_m} \notin [r_{m-1}, i^{m-1}_{r_{m-1}}; \, \dots \, ; r_1, i^1_{r_1}](\ii^0)
	\setminus \{i^{m-1}_{r_m}\}}
\\
	\la [r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\ii^0) |
		(K_{x,x'} \otimes I \otimes \cdots \otimes I)U_q \ii^0 \ra \qquad
\end{multline}
whose treatment is similar but slightly more complicated, as detailed below.

Let $q > 1$ and consider the case $\{r_1, r_2, \dots , r_m\} \neq \{1,2,\dots, q\} $.
When $1 \notin \{r_1, r_2, \dots , r_m\}$ we obtain inner products of the form:
\[
	\la i^0_1 | K_{x,x'} i^0_2 \ra \la j_2, j_3, \dots , j_q | i^0_3, i^0_4, \dots , i^0_q, i^0_1 \ra
\]
where $j_k \ne i^0_1$ for all $k$ by the hard-core condition, implying the second term is zero
as $j_q \ne i^0_1$. On the other hand, when $1 \in \{r_1, r_2, \dots , r_m\}$, then there exists at least one
$l \notin \{r_1, r_2, \dots , r_m\}$, yielding an inner product of the form
\[
	\la j_1 | K_{x,x'} i^0_2 \ra \la j_2, \dots , j_{l-1}, i_l, j_{l+1}, \dots, j_q | i^0_3, i^0_4, \dots , i^0_q, i^0_1 \ra
\]
which also results in the second term being zero as $\la i_l | i_{l+1} \ra = 0$. Note that the above cases do
not occur for $q=1$.

For the case $\{r_1, r_2, \dots , r_m\} = \{1,\dots, q\} $, as before, the remaining sum may be bounded by a similar
expression whose summations have slightly relaxed restrictions. Also the left hand side of the inner product is
independent of $\ii^0$, so again denoting $\mathbf{s}^0 = (1,2,3,\dots,q)$, we have
\begin{align*}
(\ref{summs})&
	\le
	\underbrace{ \sum_{i^1_{r_1}=1}^\sV \;\;
	\sum_{i^2_{r_2}=1}^\sV \;\;
	\cdots  \;\;
	\sum_{i^m_{r_m} = 1}^\sV }_{\stackrel{\text{where $[r_m, i^m_{r_m}; \, \dots
		\, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}](\mathbf{s}^0)$}}{\text{\tiny{has distinct indices}}}} \!\!
	\sum_{\ii^0} \;
	\la[r_m, i^m_{r_m}; \, \dots \, ; r_2, i^2_{r_2} ; r_1, i^1_{r_1}]
		(\mathbf{s}^0) | (K_{x,x'} i^0_2), i^0_3, \dots , i^0_q, i^0_1 \ra
\intertext{and as there is only one possible value for each $i^0_1, i^0_3, i^0_4, \dots , i^0_q$
giving a non-zero summand, we can bound above by}
&\le 	\sum_{i^1_{r_1}=1}^\sV \;
	\sum_{i^2_{r_2}=1}^\sV \;
	\cdots  \;
	\sum_{i^m_{r_m} = 1}^\sV
	\sum_{i^0_2=1}^\sV \la i^k_{r_k} | K_{x,x'} i^0_2 \ra \,
=	\, V^{m-1} \sum_{i^k_{r_k} = 1}^\sV \sum_{i^0_2=1}^\sV \la i^k_{r_k} | K_{x,x'} i^0_2 \ra \,
=	\,V^{m-1}
\end{align*}
where $k \in [1,m]$ is the smallest number such that $r_k = 1$, and for any $x, x' \in \Lambda_\sV$. Thus
the entire sum (\ref{summs}) is bounded above by $V^{m-1}$. Therefore one can conclude the argument
of Subsection \ref{sub Z}, proving (\ref{proj_in_qspace_irrel}).

Moreover, following the reasoning in Subsection \ref{sub c-q},
we can then check that for $q \ge 1$ and $x\neq x'$,
$\widetilde{C}^n_\sV(q;K_{x,\,x'})=0$, since for $q=1$, $\la \mathbf{e}_i | K_{x,x'} \mathbf{e}_i \ra = 0$, and for $q > 1$
\[
\la (K_{x,x'} \otimes I \otimes \dots \otimes I) U_q \ii | \ii \ra
    = \la (K_{x,x'} \mathbf{e}_{i_2}) \otimes \mathbf{e}_{i_3} \otimes \dots
    	\otimes \mathbf{e}_{i_q} \otimes \mathbf{e}_{i_1}\, |\,
	\mathbf{e}_{i_1} \otimes \mathbf{e}_{i_2} \otimes \dots \otimes \mathbf{e}_{i_q}\ra=0
\]
as the $i_l$'s are all distinct. So we have that
\begin{equation*}
	\thermlim C^n_\sV(q;K_{x,\,x'})=0
\end{equation*}
and that
\begin{equation*}
	\lim_{Q\to \infty} \thermlim \sum_{q=Q+1}^\infty C^n_\sV(q;K_{x,\,x'})=
	\thermlim D_{\sbeta,n,\sV}(x,x')=\rho_\sbeta^c.
\end{equation*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                       proposition
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\renewcommand{\theequation}{A.\arabic{equation}}
% redefine the command that creates the equation no.
\setcounter{equation}{0}  % reset counter
\section{Appendix: Expectation of Operator in Terms of Cycle Lengths}  \label{appendixA}
\begin{proposition}
	Given an operator $A$ on $\Hone$, the expectation of $A$ may be expressed in terms
	of cycle lengths
	\begin{equation}
		\langle A^{(n)} \rangle = \sum_{q=0}^n C^n_\sV(q;A)
	\end{equation}
	where
	\begin{equation}
		C^n_\sV(q;A)
		= \frac{1}{Z_\sbeta(n,V)} \, \trace_{\Hhccyclespace}
		\left[ (A\otimes I\otimes \ldots\otimes I)
	 	U_q \e^{-\beta \hcHam } \right].
	\end{equation}
	\end{proposition}
Note that $c^n_\sV(q)=C^n_\sV(q;I)/V$ where $c^n_\sV(q)$ is as defined in Proposition \ref{stat}.

\begin{proof}
\begin{equation*}
	\langle A^{(n)} \rangle
	= \frac{1}{\cPart}
				\trace_{\Hhccansym{n}} \left[ A^{(n)} \e^{-\beta \hcHam} \right] \\
	= \frac{1}{\cPart} \frac{1}{n!} \sum_{\pi \in S_n}
				\trace_{\Hhccan{n}} \left[ A^{(n)} U_\pi \e^{-\beta \hcHam} \right]
\end{equation*}
using the facts that $[U_\pi,\hcHam]=0$, $[U_\pi,\Phc]=0$ and by the cyclicity of the trace. Note we can simplify this
expression by the following method:
\begin{align*}
	\trace_{\Hhccan{n}} \left[ A^{(n)} U_\pi \e^{-\beta \hcHam} \right]
		&= 	\trace_{\Hhccan{n}} \left[ \sum_{i=1}^n
				(I \otimes \cdots \otimes \underbrace{A}_{\text{ $i$th position}} \otimes \cdots \otimes I)
				U_\pi \e^{-\beta \hcHam} \right] \\
		&= 	\trace_{\Hhccan{n}} \left[ \sum_{i=1}^n U_{(1i)} (A \otimes I \otimes \cdots \otimes I)
				U_{(1i)} U_\pi \e^{-\beta \hcHam}\right]
\intertext{\vspace{-0.4cm}where $U_{(1i)}$ represents the transposition $(1 \, i)$, so using cyclicity of the trace again}
		&= 	\trace_{\Hhccan{n}} \left[ \sum_{i=1}^n (A \otimes I \otimes \cdots \otimes I)
				U_{(1i)} U_\pi \e^{-\beta \hcHam} U_{(1i)} \right] \\
		&= 	\trace_{\Hhccan{n}} \left[ \sum_{i=1}^n (A \otimes I \otimes \cdots \otimes I)
				U_{(1i)} U_\pi U_{(1i)} \e^{-\beta \hcHam} \right] \\
		&= 	n \; \trace_{\Hhccan{n}} \left[ (A \otimes I \otimes \cdots \otimes I) U_{\pi'}  \e^{-\beta \hcHam} \right]
\end{align*}
where $\pi' = (1\,i) \, \pi \, (i\,1)$ using (\ref{invar_under_cycles}). Thus
\begin{equation}
	\langle A^{(n)} \rangle
	=  \frac{1}{\cPart} \frac{1}{(n-1)!} \sum_{\pi \in S_n}     \label{sum_perms}            %%%LABEL: sum_perms
		\trace_{\Hhccan{n}} \Big[ (A \otimes I \otimes \cdots \otimes I) U_\pi \e^{-\beta \hcHam} \Big] .
\end{equation}

Given distinct indices $i_2, \dots, i_q$, let
\[
	S_n^q(i_2, i_3, \dots i_q)
	= \Big\{ \pi \in  S_n : \pi(i_m) = i_{m+1},  1 \le m < q \text{ with }  i_1 = \pi(i_q) = 1 \Big\}.
\]
Then for any $\pi \in S_n^q(i_2, i_3, \dots i_q)$, there exists a
$\pi' \in S_{n-q}$ so that one can write
\[
	\trace_{\Hhccan{n}} \Big[ ( A \otimes I \otimes \cdots \otimes I ) U_\pi \e^{-\beta H} \Big]
	= \trace_{\Hhccan{n}} \Big[ ( A \otimes I \otimes \cdots \otimes I )(U_q \otimes U_{\pi'}) \e^{-\beta H} \Big].
\]
The set $S_n^q(i_2, i_3, \dots i_q)$ form a partition of the set of permutations where $1$ belongs
to a cycle of length $q$. There are $\tfrac{(n-1)!}{(n-q)!}$ such sets. Then
\begin{align*}
	\langle A^{(n)} \rangle
	&=  \frac{1}{\cPart} \frac{1}{(n-1)!} \sum_{\pi \in S_n}
			\trace_{\Hhccan{n}} \Big[ (A \otimes I \otimes \cdots \otimes I) U_\pi \e^{-\beta \hcHam} \Big]
\\
	&= \frac{1}{\cPart} \frac{1}{(n-1)!} \sum_{q=1}^n \frac{(n-1)!}{(n-q)!}
			\sum_{\pi' \in S_{n-q}}
			\trace_{\Hhccan{n}} \Big[ (A \otimes I \otimes \cdots \otimes I)
				(U_q \otimes U_{\pi'})  \e^{-\beta \hcHam} \Big]
\\
	&= \frac{1}{\cPart} \sum_{q=1}^n \frac{1}{(n-q)!}  \sum_{\pi' \in S_{n-q}}
			\trace_{\Hhccan{n}} \Big[ (A \otimes I \otimes \cdots \otimes I)
				(U_q \otimes U_{\pi'})  \e^{-\beta \hcHam} \Big]
\\
	&= \frac{1}{\cPart} \sum_{q=1}^n \trace_{\Phc (\Hcan{q} \otimes \Hcansym{n-q})}
			\Big[ (A \otimes I \otimes \cdots \otimes I) U_q  \e^{-\beta \hcHam} \Big]
\end{align*}
and recall that $\Hhccyclespace := \Phc ( \Hcan{q} \otimes \Hcansym{n-q} )$.
\end{proof}

\textbf{Acknowledgements:} The authors would like to thank T.C. Dorlas and S. Adams for some helpful discussions and 
they are grateful to the referees for their many useful suggestions. GB would like to thank the Irish Research Council for
Science, Engineering and Technology for their financial support.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%REFERENCES
%
%\newpage
\begin{thebibliography}{xx}%{xxxxxxxx}
%1
\bibitem{AdamsDorlas} Adams, S., Dorlas, T.C.: $C^\star$-algebraic approach to the Bose-Hubbard Model,
\textit{J. Math. Phys.}, \textbf{48}, 103304 (2007)
%2
\bibitem{Angelescu} Angelescu, N., Bundaru, M.: A Remark on the Condensation in the Hard-Core Lattice Bose Gas,
\textit{J. Stat. Phys.}, \textbf{69}, 897-903 (1992)
%3
\bibitem{BruDorlas} Bru, J.-B., Dorlas, T.C.: Exact Solution of the Infinite-Range-Hopping Bose-Hubbard Model,
\textit{J. Stat. Phys.}, \textbf{113}, 177-196 (2003)
%4
\bibitem{Chandler} Chandler, D., Wolynes, P. G.: Exploiting the isomorphism between quantum theory
and classical statistical mechanics of polyatomic fluids, {\it J. Chem. Phys.}, {\bf 74}, 4078 (1981)
%5
\bibitem{DMP}  Dorlas, T.C., Martin, Ph. A., Pul\'e, J.V.: Long Cycles in a Perturbed
Mean Field Model of a Boson Gas,
\textit{J. Stat. Phys.}, \textbf{121} 433-461 (2005)
%6
\bibitem{Feller} Feller, W.: \textit{An Introduction to Probability Theory and Its Applications},
Vol. 1, pp. 50-52 (Wiley, New York, 1957)
%7
\bibitem{Fey 1953 and stat.mech.} Feynman, R. P.: Atomic theory of the $\lambda$ transition in Helium,
{\it Phys. Rev.}, {\bf 91}, 1291, 1953,
and Feynman, R. P.: Statistical Mechanics, Chap. 11, {\it Benjamin} (1974)
%8
\bibitem{Fisher} Fisher, M. P. A., Weichman, P. B., Grinstein, G., Fisher, D. S.: Boson localization and the superfluid-insulator transition
\textit{Phys. Rev. B}, \textbf{40}, 546 (1989)
%9
\bibitem{Kirson} Kirson, M.W.: Bose-Einstein condensation in an exactly solvable model of strongly interacting
bosons, \textit{J. Phys. A: Math. Gen. A}, \textbf{33}, 731 (2000)
%10
\bibitem{Manjegani} Manjegani, S.M.: H\"older and Young Inequalities for the Trace of Operators,
\textit{Positivity} \textbf{11}, 239-250 (2007)
%11
\bibitem{Martin}  Martin, Ph. A.: Quantum Mayer graphs: application to Bose and Coulomb gases,
\textit{Acta Phys. Pol. B}, \textbf{34}, 3629 (2003)
%12
\bibitem{Pen.Onsa.}  Penrose, O., Onsager, L.: Bose-Einstein condensation and Liquid Helium,
{\it Phys. Rev.}, {\bf 104}, 576 (1956)
%13
\bibitem{Penrose} Penrose, O.: Bose-Einstein Condensation in an Exactly Soluble System of
Interacting Particles,
\textit{J. Stat. Phys.}, \textbf{63} 761-781 (1991)
%14
\bibitem{Schakel} Schakel, A. M. J.: Percolation, Bose-Einstein condensation and string proliferation,
{\it Phys. Rev. E\,}, {\bf 63}, 126115 (2001)
%15
\bibitem{Sear-Cuesta} Sear, R. P., Cuesta, J. A.: What do emulsification failure and Bose-Einstein
condensation have in common?, {\it  Europhys. Lett.}, {\bf 55}, 451 (2001)
%16
\bibitem{Suto} S\"ut\"o, A.: Percolation transition in the Bose gas,
{\it J. Phys. A: Math. Gen.},{\bf 26}, 4689 (1993)
and Percolation transition in the Bose gas: II
{\it J. Phys. A: Math. Gen.}, {\bf 35}, 6995 (2002)
%17
\bibitem{Toth} T\'oth, B.: Phase Transition in an Interacting Bose System. An Application of the Theory of Ventsel' and Freidlin
\textit{J. Stat. Phys.}, \textbf{61}, 749-64 (1990)
%18
\bibitem{Ueltschi2} Ueltschi, D.: Feynman Cycles in the Bose Gas,
\textit{J. Math. Phys.}, \textbf{47}, 123303 (2006)
%119
\bibitem{Ueltschi1} Ueltschi, D.: The Relation between Feynman Cycles and Off-Diagonal Long-Range Order,
\textit{Phys. Rev. Letters}, \textbf{97}, 170601 (2006)
%20
\bibitem{Yang} Yang, C.F.: Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He
and of Superconductors, \textit{Rev. Mod. Phys.}, \textbf{34}, 694 - 704 (1962)

\end{thebibliography}

\end{document}
---------------0806040948613
Content-Type: application/postscript; name="penrose01.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="penrose01.eps"

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Mayura Draw, Version 4.1
%%Title: Untitled
%%CreationDate: Wed Jan 02 21:24:38 2008
%%BoundingBox: 30 297 594 714
%%DocumentFonts: Symbol
%%+ Times-Italic
%%+ Times-Roman
%%EndComments
%%BeginProlog
%%BeginResource: procset MayuraDraw_ops
%%Version: 4.1
%%Copyright: (c) 1993-2001 Mayura Software
/PDXDict 100 dict def
PDXDict begin
% width height matrix proc key cache
% definepattern -\> font
/definepattern { %def
  7 dict begin
    /FontDict 9 dict def
    FontDict begin
      /cache exch def
      /key exch def
      /proc exch cvx def
      /mtx exch matrix invertmatrix def
      /height exch def
      /width exch def
      /ctm matrix currentmatrix def
      /ptm matrix identmatrix def
      /str
      (xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx)
      def
    end
    /FontBBox [ %def
      0 0 FontDict /width get
      FontDict /height get
    ] def
    /FontMatrix FontDict /mtx get def
    /Encoding StandardEncoding def
    /FontType 3 def
    /BuildChar { %def
      pop begin
      FontDict begin
        width 0 cache { %ifelse
          0 0 width height setcachedevice
        }{ %else
          setcharwidth
        } ifelse
        0 0 moveto width 0 lineto
        width height lineto 0 height lineto
        closepath clip newpath
        gsave proc grestore
      end end
    } def
    FontDict /key get currentdict definefont
  end
} bind def

% dict patternpath -
% dict matrix patternpath -
/patternpath { %def
  dup type /dicttype eq { %ifelse
    begin FontDict /ctm get setmatrix
  }{ %else
    exch begin FontDict /ctm get setmatrix
    concat
  } ifelse
  currentdict setfont
  FontDict begin
    FontMatrix concat
    width 0 dtransform
    round width div exch round width div exch
    0 height dtransform
    round height div exch
    round height div exch
    0 0 transform round exch round exch
    ptm astore setmatrix

    pathbbox
    height div ceiling height mul 4 1 roll
    width div ceiling width mul 4 1 roll
    height div floor height mul 4 1 roll
    width div floor width mul 4 1 roll

    2 index sub height div ceiling cvi exch
    3 index sub width div ceiling cvi exch
    4 2 roll moveto

    FontMatrix ptm invertmatrix pop
    { %repeat
      gsave
        ptm concat
        dup str length idiv { %repeat
          str show
        } repeat
        dup str length mod str exch
        0 exch getinterval show
      grestore
      0 height rmoveto
    } repeat
    pop
  end end
} bind def

% dict patternfill -
% dict matrix patternfill -
/patternfill { %def
  gsave
    eoclip patternpath
  grestore
  newpath
} bind def

/img { %def
  gsave
  /imgh exch def
  /imgw exch def
  concat
  imgw imgh 8
  [imgw 0 0 imgh neg 0 imgh]
  /colorstr 768 string def
  /colorimage where {
    pop
    { currentfile colorstr readhexstring pop }
    false 3 colorimage
  }{
    /graystr 256 string def
    {
      currentfile colorstr readhexstring pop
      length 3 idiv
      dup 1 sub 0 1 3 -1 roll
      {
        graystr exch
        colorstr 1 index 3 mul get 30 mul
        colorstr 2 index 3 mul 1 add get 59 mul
        colorstr 3 index 3 mul 2 add get 11 mul
        add add 100 idiv
        put
      } for
      graystr 0 3 -1 roll getinterval
    } image
  } ifelse
  grestore
} bind def

/arrowhead {
  gsave
    [] 0 setdash
    strokeC strokeM strokeY strokeK setcmykcolor
    2 copy moveto
    4 2 roll exch 4 -1 roll exch
    sub 3 1 roll sub
    exch atan rotate dup scale
    arrowtype
    dup 0 eq {
      -1 2 rlineto 7 -2 rlineto -7 -2 rlineto
      closepath fill
    } if
    dup 1 eq {
      0 3 rlineto 9 -3 rlineto -9 -3 rlineto
      closepath fill
    } if
    dup 2 eq {
      -6 -6 rmoveto 6 6 rlineto -6 6 rlineto
      -1.4142 -1.4142 rlineto 4.5858 -4.5858 rlineto
      -4.5858 -4.5858 rlineto closepath fill
    } if
    dup 3 eq {
      -6 0 rmoveto -1 2 rlineto 7 -2 rlineto -7 -2 rlineto
      closepath fill
    } if
    dup 4 eq {
      -9 0 rmoveto 0 3 rlineto 9 -3 rlineto -9 -3 rlineto
      closepath fill
    } if
    dup 5 eq {
      currentpoint newpath 3 0 360 arc
      closepath fill
    } if
    dup 6 eq {
      2.5 2.5 rmoveto 0 -5 rlineto -5 0 rlineto 0 5 rlineto
      closepath fill
    } if
    pop
  grestore
} bind def

/setcmykcolor where { %ifelse
  pop
}{ %else
  /setcmykcolor {
     /black exch def /yellow exch def
     /magenta exch def /cyan exch def
     cyan black add dup 1 gt { pop 1 } if 1 exch sub
     magenta black add dup 1 gt { pop 1 } if 1 exch sub
     yellow black add dup 1 gt { pop 1 } if 1 exch sub
     setrgbcolor
  } bind def
} ifelse

/RE { %def
  findfont begin
  currentdict dup length dict begin
    { %forall
      1 index /FID ne { def } { pop pop } ifelse
    } forall
    /FontName exch def dup length 0 ne { %if
      /Encoding Encoding 256 array copy def
      0 exch { %forall
        dup type /nametype eq { %ifelse
          Encoding 2 index 2 index put
          pop 1 add
        }{ %else
          exch pop
        } ifelse
      } forall
    } if pop
  currentdict dup end end
  /FontName get exch definefont pop
} bind def

/spacecount { %def
  0 exch
  ( ) { %loop
    search { %ifelse
      pop 3 -1 roll 1 add 3 1 roll
    }{ pop exit } ifelse
  } loop
} bind def

/WinAnsiEncoding [
  39/quotesingle 96/grave 130/quotesinglbase/florin/quotedblbase
  /ellipsis/dagger/daggerdbl/circumflex/perthousand
  /Scaron/guilsinglleft/OE 145/quoteleft/quoteright
  /quotedblleft/quotedblright/bullet/endash/emdash
  /tilde/trademark/scaron/guilsinglright/oe/dotlessi
  159/Ydieresis 164/currency 166/brokenbar 168/dieresis/copyright
  /ordfeminine 172/logicalnot 174/registered/macron/ring
  177/plusminus/twosuperior/threesuperior/acute/mu
  183/periodcentered/cedilla/onesuperior/ordmasculine
  188/onequarter/onehalf/threequarters 192/Agrave/Aacute
  /Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla
  /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute
  /Icircumflex/Idieresis/Eth/Ntilde/Ograve/Oacute
  /Ocircumflex/Otilde/Odieresis/multiply/Oslash
  /Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn
  /germandbls/agrave/aacute/acircumflex/atilde/adieresis
  /aring/ae/ccedilla/egrave/eacute/ecircumflex
  /edieresis/igrave/iacute/icircumflex/idieresis
  /eth/ntilde/ograve/oacute/ocircumflex/otilde
  /odieresis/divide/oslash/ugrave/uacute/ucircumflex
  /udieresis/yacute/thorn/ydieresis
] def

/SymbolEncoding [
  32/space/exclam/universal/numbersign/existential/percent
  /ampersand/suchthat/parenleft/parenright/asteriskmath/plus
  /comma/minus/period/slash/zero/one/two/three/four/five/six
  /seven/eight/nine/colon/semicolon/less/equal/greater/question
  /congruent/Alpha/Beta/Chi/Delta/Epsilon/Phi/Gamma/Eta/Iota
  /theta1/Kappa/Lambda/Mu/Nu/Omicron/Pi/Theta/Rho/Sigma/Tau
  /Upsilon/sigma1/Omega/Xi/Psi/Zeta/bracketleft/therefore
  /bracketright/perpendicular/underscore/radicalex/alpha
  /beta/chi/delta/epsilon/phi/gamma/eta/iota/phi1/kappa/lambda
  /mu/nu/omicron/pi/theta/rho/sigma/tau/upsilon/omega1/omega
  /xi/psi/zeta/braceleft/bar/braceright/similar
  161/Upsilon1/minute/lessequal/fraction/infinity/florin/club
  /diamond/heart/spade/arrowboth/arrowleft/arrowup/arrowright
  /arrowdown/degree/plusminus/second/greaterequal/multiply
  /proportional/partialdiff/bullet/divide/notequal/equivalence
  /approxequal/ellipsis/arrowvertex/arrowhorizex/carriagereturn
  /aleph/Ifraktur/Rfraktur/weierstrass/circlemultiply
  /circleplus/emptyset/intersection/union/propersuperset
  /reflexsuperset/notsubset/propersubset/reflexsubset/element
  /notelement/angle/gradient/registerserif/copyrightserif
  /trademarkserif/product/radical/dotmath/logicalnot/logicaland
  /logicalor/arrowdblboth/arrowdblleft/arrowdblup/arrowdblright
  /arrowdbldown/lozenge/angleleft/registersans/copyrightsans
  /trademarksans/summation/parenlefttp/parenleftex/parenleftbt
  /bracketlefttp/bracketleftex/bracketleftbt/bracelefttp
  /braceleftmid/braceleftbt/braceex
  241/angleright/integral/integraltp/integralex/integralbt
  /parenrighttp/parenrightex/parenrightbt/bracketrighttp
  /bracketrightex/bracketrightbt/bracerighttp/bracerightmid
  /bracerightbt
] def

/patarray [
/leftdiagonal /rightdiagonal /crossdiagonal /horizontal
/vertical /crosshatch /fishscale /wave /brick
] def
/arrowtype 0 def
/fillC 0 def /fillM 0 def /fillY 0 def /fillK 0 def
/strokeC 0 def /strokeM 0 def /strokeY 0 def /strokeK 1 def
/pattern -1 def
/mat matrix def
/mat2 matrix def
/nesting 0 def
/deferred /N def
/c /curveto load def
/c2 { pop pop c } bind def
/C /curveto load def
/C2 { pop pop C } bind def
/e { gsave concat 0 0 moveto } bind def
/F {
  nesting 0 eq { %ifelse
    pattern -1 eq { %ifelse
      fillC fillM fillY fillK setcmykcolor eofill
    }{ %else
      gsave fillC fillM fillY fillK setcmykcolor eofill grestore
      0 0 0 1 setcmykcolor
      patarray pattern get findfont patternfill
    } ifelse
  }{ %else
    /deferred /F def
  } ifelse
} bind def
/f { closepath F } bind def
/K { /strokeK exch def /strokeY exch def
     /strokeM exch def /strokeC exch def } bind def
/k { /fillK exch def /fillY exch def
     /fillM exch def /fillC exch def } bind def
/opc { pop } bind def
/Opc { pop } bind def
/L /lineto load def
/L2 { pop pop L } bind def
/m /moveto load def
/m2 { pop pop m } bind def
/n /newpath load def
/N {
  nesting 0 eq { %ifelse
    newpath
  }{ %else
    /deferred /N def
  } ifelse
} def
/S {
  nesting 0 eq { %ifelse
    strokeC strokeM strokeY strokeK setcmykcolor stroke
  }{ %else
    /deferred /S def
  } ifelse
} bind def
/s { closepath S } bind def
/Tx { fillC fillM fillY fillK setcmykcolor show
      0 leading neg translate 0 0 moveto } bind def
/T { grestore } bind def
/TX { pop } bind def
/Ts { pop } bind def
/tal { pop } bind def
/tld { pop } bind def
/tbx { pop exch pop sub /jwidth exch def } def
/tpt { %def
  fillC fillM fillY fillK setcmykcolor
  moveto show
} bind def
/tpj { %def
  fillC fillM fillY fillK setcmykcolor
  moveto
  dup stringwidth pop
  3 -1 roll
  exch sub
  1 index spacecount
  dup 0 eq { %ifelse
    pop pop show
  }{ %else
    div 0 8#040 4 -1 roll widthshow
  } ifelse
} bind def
/u {} def
/U {} def
/*u { /nesting nesting 1 add def } def
/*U {
  /nesting nesting 1 sub def
  nesting 0 eq {
    deferred cvx exec
  } if
} def
/w /setlinewidth load def
/d /setdash load def
/B {
  nesting 0 eq { %ifelse
    gsave F grestore S
  }{ %else
    /deferred /B def
  } ifelse
} bind def
/b { closepath B } bind def
/z { /align exch def pop /leading exch def exch findfont
     exch scalefont setfont } bind def
/tfn { exch findfont
     exch scalefont setfont } bind def
/Pat { /pattern exch def } bind def
/cm { 6 array astore concat } bind def
/q { mat2 currentmatrix pop } bind def
/Q { mat2 setmatrix } bind def
/Ah {
  pop /arrowtype exch def
  currentlinewidth 5 1 roll arrowhead
} bind def
/Arc {
  mat currentmatrix pop
    translate scale 0 0 1 5 -2 roll arc
  mat setmatrix
} bind def
/Arc2 { pop pop Arc } bind def
/Bx {
  mat currentmatrix pop
    concat /y1 exch def /x1 exch def /y2 exch def /x2 exch def
    x1 y1 moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
  mat setmatrix
} bind def
/Rr {
  mat currentmatrix pop
    concat /yrad exch def /xrad exch def
    2 copy gt { exch } if /x2 exch def /x1 exch def
    2 copy gt { exch } if /y2 exch def /y1 exch def
    x1 xrad add y2 moveto
    matrix currentmatrix x1 xrad add y2 yrad sub translate xrad yrad scale
    0 0 1 90 -180 arc setmatrix
    matrix currentmatrix x1 xrad add y1 yrad add translate xrad yrad scale
    0 0 1 180 270 arc setmatrix
    matrix currentmatrix x2 xrad sub y1 yrad add translate xrad yrad scale
    0 0 1 270 0 arc setmatrix
    matrix currentmatrix x2 xrad sub y2 yrad sub translate xrad yrad scale
    0 0 1 0 90 arc setmatrix
    closepath
  mat setmatrix
} bind def
/Ov {
  mat currentmatrix pop
    concat translate scale 1 0 moveto 0 0 1 0 360 arc closepath
  mat setmatrix
} bind def
end
%%EndResource
%%EndProlog
%%BeginSetup
%PDX g 3 3 1 0
PDXDict begin
%%EndSetup
%%Page: 1 1
%%BeginPageSetup
/_PDX_savepage save def

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  7.5 0 moveto 15 7.5 lineto
  0 7.5 moveto 7.5 15 lineto
  2 setlinewidth stroke
} bind
/rightdiagonal true definepattern pop

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  7.5 0 moveto 0 7.5 lineto
  15 7.5 moveto 7.5 15 lineto
  2 setlinewidth stroke
} bind
/leftdiagonal true definepattern pop

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  0 7.5 moveto 15 7.5 lineto
  2 setlinewidth stroke
} bind
/horizontal true definepattern pop

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  7.5 0 moveto 7.5 15 lineto
  2 setlinewidth stroke
} bind
/vertical true definepattern pop

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  0 7.5 moveto 15 7.5 lineto
  7.5 0 moveto 7.5 15 lineto
  2 setlinewidth stroke
} bind
/crosshatch true definepattern pop

30 30 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  0 7.5 moveto 30 7.5 lineto
  0 22.5 moveto 30 22.5 lineto
  7.5 0 moveto 7.5 7.5 lineto
  7.5 22.5 moveto 7.5 30 lineto
  22.5 7.5 moveto 22.5 22.5 lineto
  1 setlinewidth stroke
} bind
/brick true definepattern pop

30 30 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 2 scale
  2 setlinecap
  7.5 0 moveto 15 7.5 lineto
  0 7.5 moveto 7.5 15 lineto
  7.5 0 moveto 0 7.5 lineto
  15 7.5 moveto 7.5 15 lineto
  0.5 setlinewidth stroke
} bind
/crossdiagonal true definepattern pop

30 30 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 2 scale
  1 setlinecap
  0 7.5 moveto 0 15 7.5 270 360 arc
  7.5 15 moveto 15 15 7.5 180 270 arc
  0 7.5 moveto 7.5 7.5 7.5 180 360 arc
  0.5 setlinewidth stroke
} bind
/fishscale true definepattern pop

30 30 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  1 setlinecap 0.5 setlinewidth
  7.5 0 10.6 135 45 arcn
  22.5 15 10.6 225 315 arc
  stroke
  7.5 15 10.6 135 45 arcn
  22.5 30 10.6 225 315 arc
  stroke
} bind
/wave true definepattern pop

SymbolEncoding /_Symbol /Symbol RE
WinAnsiEncoding /_Times-Italic /Times-Italic RE
WinAnsiEncoding /_Times-Roman /Times-Roman RE

newpath 2 setlinecap 0 setlinejoin 2 setmiterlimit
[] 0 setdash
30 297 moveto 30 714 lineto 594 714 lineto 594 297 lineto closepath clip
newpath
%%EndPageSetup
u
0.5 w
q
1 0 0 1 29.49 0 cm
91.202 330.071 m
525.424 330.025 L
Q
S
q
1 0 0 1 0 0 cm
120.112 329.386 m
120.112 697.146 L
Q
S
u
q
1 0 0 1 -7.075 1.156 cm
329.995 405.19 m
324.973 405.142 322.805 405.142 321.322 405.19 c
313.132 405.454 290.749 407.196 280.27 409.816 c
271.019 412.128 259.065 416.643 251.939 420.801 c
243.704 425.608 225.079 438.809 217.826 445.085 c
209.397 452.379 192.964 471.682 186.025 482.09 c
178.967 492.677 164.632 514.468 158.85 526.61 c
153.982 536.833 143.239 564.771 140.348 576.335 c
137.757 586.699 133.988 608.135 132.832 620.855 c
132.663 622.711 132.224 625.262 132.254 631.263 c
Q
S
q
-1 0 0 1 652.6 0.8911 cm
329.995 405.19 m
324.973 405.142 322.805 405.142 321.322 405.19 c
313.132 405.454 290.749 407.196 280.27 409.816 c
271.019 412.128 259.065 416.643 251.939 420.801 c
243.704 425.608 225.079 438.809 217.826 445.085 c
209.397 452.379 192.964 471.682 186.025 482.09 c
178.967 492.677 164.632 514.468 158.85 526.61 c
153.982 536.833 143.239 564.771 140.348 576.335 c
137.757 586.699 133.988 608.135 132.832 620.855 c
132.663 622.711 132.224 625.262 132.254 631.263 c
Q
S
U
q
1 0 0 1 0 0 cm
322.478 330.025 m
322.478 330.604 L
Q
S
[3 3] 0 d
q
1 0 0 1 -0.2891 0 cm
323.057 330.025 m
323.057 406.347 L
Q
S
2 w
[] 0 d
q
1 0 0 1 88.46 -30.64 cm
464.713 360.669 m
455.652 360.669 471.652 360.669 L2
Q
S
q
1 0 0 1 88.46 -30.64 cm
464.713 360.669 471.652 360.669 4 2 Ah
Q
0.5 w
[3 3] 0 d
q
1 0 0 1 404.4 -0.2587 cm
120.112 329.386 m
120.112 697.146 L
Q
S
1 w
q
1 0 0 1 0 0 cm
120.112 464.942 m
524.267 464.942 L
Q
S
q
1 0 0 1 0 0 cm
192.964 464.942 m
192.964 329.645 L
Q
S
[1 0 0 1 -1.156 10.99] e
586.134 310.486 566.475 332.536 tbx
0 tal
19 tld
1 1 1 0 k
/_Symbol 18 tfn
(r) 566.475 314.446 tpt
T
-1.42109e-016 -1.42109e-016 -1.42109e-016 0 k
2 w
[] 0 d
q
0 1 -1 0 480.5 228.7 cm
464.713 360.669 m
455.652 360.669 471.652 360.669 L2
Q
S
q
0 1 -1 0 480.5 228.7 cm
464.713 360.669 471.652 360.669 4 2 Ah
Q
u
[1 0 0 1 -472.7 371.7] e
586.134 310.486 566.475 332.536 tbx
0 tal
19 tld
1 1 1 0 k
/_Symbol 18 tfn
(r) 566.475 314.446 tpt
T
[1 0 0 1 31.8 54.93] e
85.9983 628.378 47.8377 648.304 tbx
0 tal
19 tld
/_Times-Italic 18 tfn
(g) 47.8377 632.266 tpt
/_Times-Roman 18 tfn
(\(  \)) 56.8377 632.266 tpt
T
U
u
[1 0 0 1 -377.8 1.272] e
586.134 310.486 566.475 332.536 tbx
0 tal
19 tld
/_Symbol 18 tfn
(r) 566.475 314.446 tpt
T
[1 0 0 1 -368.6 -13.18] e
586.134 321.511 566.475 332.536 tbx
0 tal
10 tld
/_Symbol 9 tfn
(b) 566.475 323.491 tpt
T
U
[1 0 0 1 -460 145.4] e
586.134 310.486 566.475 332.536 tbx
0 tal
19 tld
/_Symbol 18 tfn
(b) 566.475 314.446 tpt
T
-1.42109e-016 -1.42109e-016 -1.42109e-016 0 k
1 w
[3 3] 0 d
q
1 0 0 1 0 0 cm
322.478 405.586 m
120.69 405.586 L
Q
S
[1 0 0 1 0 0] e
56.5106 547.203 39.743 558.228 tbx
0 tal
10 tld
1 1 1 0 k
/_Symbol 9 tfn
() 39.743 549.183 tpt
T
[1 0 0 1 494.9 -213.4] e
51.885 520.567 26.4446 542.617 tbx
0 tal
19 tld
/_Symbol 18 tfn
(1) 26.4446 524.527 tpt
T
[1 0 0 1 75.74 -73.43] e
53.0414 469.686 32.2265 491.736 tbx
0 tal
19 tld
/_Symbol 18 tfn
(2) 32.2265 473.646 tpt
T
[1 0 0 1 73.43 -252.1] e
58.2451 559.306 42.634 581.356 tbx
0 tal
19 tld
/_Symbol 18 tfn
(0) 42.634 563.266 tpt
T
[1 0 0 1 249.8 -206.4] e
84.7925 513.629 62.2925 535.679 tbx
0 tal
19 tld
/_Symbol 18 tfn
(0.5) 62.2925 517.589 tpt
T
U
%%PageTrailer
_PDX_savepage restore
%%Trailer
end
% showpage
%%EOF
---------------0806040948613
Content-Type: application/postscript; name="penrose02.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="penrose02.eps"

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Mayura Draw, Version 4.3
%%Title: penrose02.eps
%%CreationDate: Thu May 22 14:58:23 2008
%%BoundingBox: 20 364 592 716
%%DocumentFonts: Helvetica
%%+ Symbol
%%EndComments
%%BeginProlog
%%BeginResource: procset MayuraDraw_ops
%%Version: 4.3
%%Copyright: (c) 1993-2003 Mayura Software
/PDXDict 100 dict def
PDXDict begin
% width height matrix proc key cache
% definepattern -\> font
/definepattern { %def
  7 dict begin
    /FontDict 9 dict def
    FontDict begin
      /cache exch def
      /key exch def
      /proc exch cvx def
      /mtx exch matrix invertmatrix def
      /height exch def
      /width exch def
      /ctm matrix currentmatrix def
      /ptm matrix identmatrix def
      /str
      (xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx)
      def
    end
    /FontBBox [ %def
      0 0 FontDict /width get
      FontDict /height get
    ] def
    /FontMatrix FontDict /mtx get def
    /Encoding StandardEncoding def
    /FontType 3 def
    /BuildChar { %def
      pop begin
      FontDict begin
        width 0 cache { %ifelse
          0 0 width height setcachedevice
        }{ %else
          setcharwidth
        } ifelse
        0 0 moveto width 0 lineto
        width height lineto 0 height lineto
        closepath clip newpath
        gsave proc grestore
      end end
    } def
    FontDict /key get currentdict definefont
  end
} bind def

% dict patternpath -
% dict matrix patternpath -
/patternpath { %def
  dup type /dicttype eq { %ifelse
    begin FontDict /ctm get setmatrix
  }{ %else
    exch begin FontDict /ctm get setmatrix
    concat
  } ifelse
  currentdict setfont
  FontDict begin
    FontMatrix concat
    width 0 dtransform
    round width div exch round width div exch
    0 height dtransform
    round height div exch
    round height div exch
    0 0 transform round exch round exch
    ptm astore setmatrix

    pathbbox
    height div ceiling height mul 4 1 roll
    width div ceiling width mul 4 1 roll
    height div floor height mul 4 1 roll
    width div floor width mul 4 1 roll

    2 index sub height div ceiling cvi exch
    3 index sub width div ceiling cvi exch
    4 2 roll moveto

    FontMatrix ptm invertmatrix pop
    { %repeat
      gsave
        ptm concat
        dup str length idiv { %repeat
          str show
        } repeat
        dup str length mod str exch
        0 exch getinterval show
      grestore
      0 height rmoveto
    } repeat
    pop
  end end
} bind def

% dict patternfill -
% dict matrix patternfill -
/patternfill { %def
  gsave
    eoclip patternpath
  grestore
  newpath
} bind def

/img { %def
  gsave
  /imgh exch def
  /imgw exch def
  concat
  imgw imgh 8
  [imgw 0 0 imgh neg 0 imgh]
  /colorstr 768 string def
  /colorimage where {
    pop
    { currentfile colorstr readhexstring pop }
    false 3 colorimage
  }{
    /graystr 256 string def
    {
      currentfile colorstr readhexstring pop
      length 3 idiv
      dup 1 sub 0 1 3 -1 roll
      {
        graystr exch
        colorstr 1 index 3 mul get 30 mul
        colorstr 2 index 3 mul 1 add get 59 mul
        colorstr 3 index 3 mul 2 add get 11 mul
        add add 100 idiv
        put
      } for
      graystr 0 3 -1 roll getinterval
    } image
  } ifelse
  grestore
} bind def

/arrowhead {
  gsave
    [] 0 setdash
    strokeC strokeM strokeY strokeK setcmykcolor
    2 copy moveto
    4 2 roll exch 4 -1 roll exch
    sub 3 1 roll sub
    exch atan rotate dup scale
    arrowtype
    dup 0 eq {
      -1 2 rlineto 7 -2 rlineto -7 -2 rlineto
      closepath fill
    } if
    dup 1 eq {
      0 3 rlineto 9 -3 rlineto -9 -3 rlineto
      closepath fill
    } if
    dup 2 eq {
      -6 -6 rmoveto 6 6 rlineto -6 6 rlineto
      -1.4142 -1.4142 rlineto 4.5858 -4.5858 rlineto
      -4.5858 -4.5858 rlineto closepath fill
    } if
    dup 3 eq {
      -6 0 rmoveto -1 2 rlineto 7 -2 rlineto -7 -2 rlineto
      closepath fill
    } if
    dup 4 eq {
      -9 0 rmoveto 0 3 rlineto 9 -3 rlineto -9 -3 rlineto
      closepath fill
    } if
    dup 5 eq {
      currentpoint newpath 3 0 360 arc
      closepath fill
    } if
    dup 6 eq {
      2.5 2.5 rmoveto 0 -5 rlineto -5 0 rlineto 0 5 rlineto
      closepath fill
    } if
    pop
  grestore
} bind def

/setcmykcolor where { %ifelse
  pop
}{ %else
  /setcmykcolor {
     /black exch def /yellow exch def
     /magenta exch def /cyan exch def
     cyan black add dup 1 gt { pop 1 } if 1 exch sub
     magenta black add dup 1 gt { pop 1 } if 1 exch sub
     yellow black add dup 1 gt { pop 1 } if 1 exch sub
     setrgbcolor
  } bind def
} ifelse

/RE { %def
  findfont begin
  currentdict dup length dict begin
    { %forall
      1 index /FID ne { def } { pop pop } ifelse
    } forall
    /FontName exch def dup length 0 ne { %if
      /Encoding Encoding 256 array copy def
      0 exch { %forall
        dup type /nametype eq { %ifelse
          Encoding 2 index 2 index put
          pop 1 add
        }{ %else
          exch pop
        } ifelse
      } forall
    } if pop
  currentdict dup end end
  /FontName get exch definefont pop
} bind def

/spacecount { %def
  0 exch
  ( ) { %loop
    search { %ifelse
      pop 3 -1 roll 1 add 3 1 roll
    }{ pop exit } ifelse
  } loop
} bind def

/WinAnsiEncoding [
  39/quotesingle 96/grave 130/quotesinglbase/florin/quotedblbase
  /ellipsis/dagger/daggerdbl/circumflex/perthousand
  /Scaron/guilsinglleft/OE 145/quoteleft/quoteright
  /quotedblleft/quotedblright/bullet/endash/emdash
  /tilde/trademark/scaron/guilsinglright/oe/dotlessi
  159/Ydieresis 164/currency 166/brokenbar 168/dieresis/copyright
  /ordfeminine 172/logicalnot 174/registered/macron/ring
  177/plusminus/twosuperior/threesuperior/acute/mu
  183/periodcentered/cedilla/onesuperior/ordmasculine
  188/onequarter/onehalf/threequarters 192/Agrave/Aacute
  /Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla
  /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute
  /Icircumflex/Idieresis/Eth/Ntilde/Ograve/Oacute
  /Ocircumflex/Otilde/Odieresis/multiply/Oslash
  /Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn
  /germandbls/agrave/aacute/acircumflex/atilde/adieresis
  /aring/ae/ccedilla/egrave/eacute/ecircumflex
  /edieresis/igrave/iacute/icircumflex/idieresis
  /eth/ntilde/ograve/oacute/ocircumflex/otilde
  /odieresis/divide/oslash/ugrave/uacute/ucircumflex
  /udieresis/yacute/thorn/ydieresis
] def

/SymbolEncoding [
  32/space/exclam/universal/numbersign/existential/percent
  /ampersand/suchthat/parenleft/parenright/asteriskmath/plus
  /comma/minus/period/slash/zero/one/two/three/four/five/six
  /seven/eight/nine/colon/semicolon/less/equal/greater/question
  /congruent/Alpha/Beta/Chi/Delta/Epsilon/Phi/Gamma/Eta/Iota
  /theta1/Kappa/Lambda/Mu/Nu/Omicron/Pi/Theta/Rho/Sigma/Tau
  /Upsilon/sigma1/Omega/Xi/Psi/Zeta/bracketleft/therefore
  /bracketright/perpendicular/underscore/radicalex/alpha
  /beta/chi/delta/epsilon/phi/gamma/eta/iota/phi1/kappa/lambda
  /mu/nu/omicron/pi/theta/rho/sigma/tau/upsilon/omega1/omega
  /xi/psi/zeta/braceleft/bar/braceright/similar
  161/Upsilon1/minute/lessequal/fraction/infinity/florin/club
  /diamond/heart/spade/arrowboth/arrowleft/arrowup/arrowright
  /arrowdown/degree/plusminus/second/greaterequal/multiply
  /proportional/partialdiff/bullet/divide/notequal/equivalence
  /approxequal/ellipsis/arrowvertex/arrowhorizex/carriagereturn
  /aleph/Ifraktur/Rfraktur/weierstrass/circlemultiply
  /circleplus/emptyset/intersection/union/propersuperset
  /reflexsuperset/notsubset/propersubset/reflexsubset/element
  /notelement/angle/gradient/registerserif/copyrightserif
  /trademarkserif/product/radical/dotmath/logicalnot/logicaland
  /logicalor/arrowdblboth/arrowdblleft/arrowdblup/arrowdblright
  /arrowdbldown/lozenge/angleleft/registersans/copyrightsans
  /trademarksans/summation/parenlefttp/parenleftex/parenleftbt
  /bracketlefttp/bracketleftex/bracketleftbt/bracelefttp
  /braceleftmid/braceleftbt/braceex
  241/angleright/integral/integraltp/integralex/integralbt
  /parenrighttp/parenrightex/parenrightbt/bracketrighttp
  /bracketrightex/bracketrightbt/bracerighttp/bracerightmid
  /bracerightbt
] def

/patarray [
/leftdiagonal /rightdiagonal /crossdiagonal /horizontal
/vertical /crosshatch /fishscale /wave /brick
] def
/arrowtype 0 def
/fillC 0 def /fillM 0 def /fillY 0 def /fillK 0 def
/strokeC 0 def /strokeM 0 def /strokeY 0 def /strokeK 1 def
/pattern -1 def
/mat matrix def
/mat2 matrix def
/nesting 0 def
/deferred /N def
/c /curveto load def
/c2 { pop pop c } bind def
/C /curveto load def
/C2 { pop pop C } bind def
/e { gsave concat 0 0 moveto } bind def
/F {
  nesting 0 eq { %ifelse
    pattern -1 eq { %ifelse
      fillC fillM fillY fillK setcmykcolor eofill
    }{ %else
      gsave fillC fillM fillY fillK setcmykcolor eofill grestore
      0 0 0 1 setcmykcolor
      patarray pattern get findfont patternfill
    } ifelse
  }{ %else
    /deferred /F def
  } ifelse
} bind def
/f { closepath F } bind def
/K { /strokeK exch def /strokeY exch def
     /strokeM exch def /strokeC exch def } bind def
/k { /fillK exch def /fillY exch def
     /fillM exch def /fillC exch def } bind def
/opc { pop } bind def
/Opc { pop } bind def
/L /lineto load def
/L2 { pop pop L } bind def
/m /moveto load def
/m2 { pop pop m } bind def
/n /newpath load def
/N {
  nesting 0 eq { %ifelse
    newpath
  }{ %else
    /deferred /N def
  } ifelse
} def
/S {
  nesting 0 eq { %ifelse
    strokeC strokeM strokeY strokeK setcmykcolor stroke
  }{ %else
    /deferred /S def
  } ifelse
} bind def
/s { closepath S } bind def
/Tx { fillC fillM fillY fillK setcmykcolor show
      0 leading neg translate 0 0 moveto } bind def
/T { grestore } bind def
/TX { pop } bind def
/Ts { pop } bind def
/tal { pop } bind def
/tld { pop } bind def
/tbx { pop exch pop sub /jwidth exch def } def
/tpt { %def
  fillC fillM fillY fillK setcmykcolor
  moveto show
} bind def
/tpj { %def
  fillC fillM fillY fillK setcmykcolor
  moveto
  dup stringwidth pop
  3 -1 roll
  exch sub
  1 index spacecount
  dup 0 eq { %ifelse
    pop pop show
  }{ %else
    div 0 8#040 4 -1 roll widthshow
  } ifelse
} bind def
/u {} def
/U {} def
/*u { /nesting nesting 1 add def } def
/*U {
  /nesting nesting 1 sub def
  nesting 0 eq {
    deferred cvx exec
  } if
} def
/w /setlinewidth load def
/d /setdash load def
/B {
  nesting 0 eq { %ifelse
    gsave F grestore S
  }{ %else
    /deferred /B def
  } ifelse
} bind def
/b { closepath B } bind def
/z { /align exch def pop /leading exch def exch findfont
     exch scalefont setfont } bind def
/tfn { exch findfont
     exch scalefont setfont } bind def
/Pat { /pattern exch def } bind def
/cm { 6 array astore concat } bind def
/q { mat2 currentmatrix pop } bind def
/Q { mat2 setmatrix } bind def
/Ah {
  pop /arrowtype exch def
  currentlinewidth 5 1 roll arrowhead
} bind def
/Arc {
  mat currentmatrix pop
    translate scale 0 0 1 5 -2 roll arc
  mat setmatrix
} bind def
/Arc2 { pop pop Arc } bind def
/Bx {
  mat currentmatrix pop
    concat /y1 exch def /x1 exch def /y2 exch def /x2 exch def
    x1 y1 moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
  mat setmatrix
} bind def
/Rr {
  mat currentmatrix pop
    concat /yrad exch def /xrad exch def
    2 copy gt { exch } if /x2 exch def /x1 exch def
    2 copy gt { exch } if /y2 exch def /y1 exch def
    x1 xrad add y2 moveto
    matrix currentmatrix x1 xrad add y2 yrad sub translate xrad yrad scale
    0 0 1 90 -180 arc setmatrix
    matrix currentmatrix x1 xrad add y1 yrad add translate xrad yrad scale
    0 0 1 180 270 arc setmatrix
    matrix currentmatrix x2 xrad sub y1 yrad add translate xrad yrad scale
    0 0 1 270 0 arc setmatrix
    matrix currentmatrix x2 xrad sub y2 yrad sub translate xrad yrad scale
    0 0 1 0 90 arc setmatrix
    closepath
  mat setmatrix
} bind def
/Ov {
  mat currentmatrix pop
    concat translate scale 1 0 moveto 0 0 1 0 360 arc closepath
  mat setmatrix
} bind def
end
%%EndResource
%%EndProlog
%%BeginSetup
%PDX g 3 3 1 0
%%IncludeFont: Helvetica
%%IncludeFont: Symbol
PDXDict begin
%%EndSetup
%%Page: 1 1
%%BeginPageSetup
/_PDX_savepage save def

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  7.5 0 moveto 15 7.5 lineto
  0 7.5 moveto 7.5 15 lineto
  2 setlinewidth stroke
} bind
/rightdiagonal true definepattern pop

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  7.5 0 moveto 0 7.5 lineto
  15 7.5 moveto 7.5 15 lineto
  2 setlinewidth stroke
} bind
/leftdiagonal true definepattern pop

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  0 7.5 moveto 15 7.5 lineto
  2 setlinewidth stroke
} bind
/horizontal true definepattern pop

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  7.5 0 moveto 7.5 15 lineto
  2 setlinewidth stroke
} bind
/vertical true definepattern pop

15 15 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  0 7.5 moveto 15 7.5 lineto
  7.5 0 moveto 7.5 15 lineto
  2 setlinewidth stroke
} bind
/crosshatch true definepattern pop

30 30 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 setlinecap
  0 7.5 moveto 30 7.5 lineto
  0 22.5 moveto 30 22.5 lineto
  7.5 0 moveto 7.5 7.5 lineto
  7.5 22.5 moveto 7.5 30 lineto
  22.5 7.5 moveto 22.5 22.5 lineto
  1 setlinewidth stroke
} bind
/brick true definepattern pop

30 30 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 2 scale
  2 setlinecap
  7.5 0 moveto 15 7.5 lineto
  0 7.5 moveto 7.5 15 lineto
  7.5 0 moveto 0 7.5 lineto
  15 7.5 moveto 7.5 15 lineto
  0.5 setlinewidth stroke
} bind
/crossdiagonal true definepattern pop

30 30 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  2 2 scale
  1 setlinecap
  0 7.5 moveto 0 15 7.5 270 360 arc
  7.5 15 moveto 15 15 7.5 180 270 arc
  0 7.5 moveto 7.5 7.5 7.5 180 360 arc
  0.5 setlinewidth stroke
} bind
/fishscale true definepattern pop

30 30 [300 72 div 0 0 300 72 div 0 0]
{ %definepattern
  1 setlinecap 0.5 setlinewidth
  7.5 0 10.6 135 45 arcn
  22.5 15 10.6 225 315 arc
  stroke
  7.5 15 10.6 135 45 arcn
  22.5 30 10.6 225 315 arc
  stroke
} bind
/wave true definepattern pop

WinAnsiEncoding /_Helvetica /Helvetica RE
SymbolEncoding /_Symbol /Symbol RE

newpath 2 setlinecap 0 setlinejoin 2 setmiterlimit
[] 0 setdash
20 364 moveto 20 716 lineto 592 716 lineto 592 364 lineto closepath clip
newpath
%%EndPageSetup
1 1 1 0 k
1 w
q
1.6 0 0 1.6 94.35 395.8 cm
28.7695 21.5977 m
28.7695 186.398 28.7695 194.398 L2
Q
S
q
1.6 0 0 1.6 94.35 395.8 cm
28.7695 21.5977 28.7695 194.398 4 2 Ah
Q
0.129 w
q
1.6 0 0 1.6 103.6 394.8 cm
25.3125 24.8828 m
28.7695 24.8828 L
Q
S
1 w
q
1.6 0 0 1.6 99.84 394.8 cm
22.965 24.9766 m
275.491 24.5941 283.491 24.582 L2
Q
S
q
1.6 0 0 1.6 99.84 394.8 cm
22.965 24.9766 283.491 24.582 4 2 Ah
Q
0.129 w
q
1.6 0 0 1.6 103.6 394.8 cm
33.3047 23.5859 m
33.3047 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
42.8945 23.5859 m
42.8945 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
52.5273 23.5859 m
52.5273 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
62.1641 23.5859 m
62.1641 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
71.7969 22.2891 m
71.7969 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
81.3867 23.5859 m
81.3867 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
91.0195 23.5859 m
91.0195 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
100.652 23.5859 m
100.652 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
110.289 23.5859 m
110.289 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
119.922 22.2891 m
119.922 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
129.512 23.5859 m
129.512 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
139.145 23.5859 m
139.145 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
148.777 23.5859 m
148.777 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
158.414 23.5859 m
158.414 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
168.004 22.2891 m
168.004 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
177.637 23.5859 m
177.637 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
187.27 23.5859 m
187.27 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
196.902 23.5859 m
196.902 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
206.539 23.5859 m
206.539 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
216.129 22.2891 m
216.129 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
225.762 23.5859 m
225.762 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
235.395 23.5859 m
235.395 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
245.027 23.5859 m
245.027 24.8828 L
Q
S
q
1.6 0 0 1.6 103.6 394.8 cm
254.617 23.5859 m
254.617 24.8828 L
Q
S
[0.06912 0 0 0.06912 205.8 415.6] e
0 219.915 0 219.915 tbx
0 tal
12 tld
/_Helvetica 243 tfn
(0.2) 0 0 tpt
T
[0.06912 0 0 0.06912 282.8 415.6] e
0 219.915 0 219.915 tbx
0 tal
12 tld
/_Helvetica 243 tfn
(0.4) 0 0 tpt
T
[0.06912 0 0 0.06912 359.8 415.6] e
0 219.915 0 219.915 tbx
0 tal
12 tld
/_Helvetica 243 tfn
(0.6) 0 0 tpt
T
[0.06912 0 0 0.06912 436.7 415.6] e
0 219.915 0 219.915 tbx
0 tal
12 tld
/_Helvetica 243 tfn
(0.8) 0 0 tpt
T
u
-1.42109e-016 -1.42109e-016 -1.42109e-016 0 k
0.5 w
q
0.9 0 0 1.25 57.84 -107.7 cm
120.69 434.8 m
126.646 444.886 129.248 449.223 131.097 452.145 c
134.909 458.169 144.24 472.394 148.443 478.164 c
155.192 487.428 171.571 510.543 179.665 519.216 c
181.49 521.171 184.301 525.22 186.025 527.31 c
189.097 531.034 197.562 538.544 201.058 541.765 c
203.256 543.789 208.606 548.509 210.888 550.438 c
213.607 552.737 220.177 558.129 223.03 560.267 c
226.369 562.769 234.6 568.357 238.063 570.675 c
240.28 572.159 245.588 575.679 247.892 577.035 c
249.826 578.173 253.882 579.653 255.986 580.504 c
258.69 581.596 265.94 584.339 268.707 585.129 c
272.395 586.183 281.107 588.991 284.896 589.755 c
288.172 590.415 296.012 591.352 299.351 591.489 c
302.933 591.637 311.405 591.25 314.962 590.911 c
318.425 590.582 326.593 589.334 329.995 588.598 c
333.352 587.873 341.212 585.717 344.45 584.551 c
348.135 583.224 356.561 579.33 360.061 577.613 c
363.559 575.897 371.803 571.65 375.094 569.518 c
378.749 567.151 386.733 560.63 390.127 557.954 c
393.585 555.229 401.922 548.835 405.16 545.812 c
408.862 542.357 416.768 533.311 420.193 529.623 c
423.617 525.935 431.947 517.265 435.226 513.434 c
438.831 509.221 446.963 498.806 450.259 494.353 c
453.697 489.708 461.5 478.337 464.713 473.538 c
467.418 469.499 473.677 459.72 476.277 455.615 c
478.58 451.978 481.905 446.63 489.576 434.221 c
Q
S
q
0.9 0 0 1 57.84 0.5782 cm
120.69 434.8 m
133.375 455.25 138.868 464.067 142.661 470.069 c
145.921 475.228 153.776 487.511 157.116 492.619 c
158.935 495.402 163.427 501.953 165.21 504.761 c
168.762 510.351 177.121 524.113 180.822 529.623 c
184.032 534.404 191.94 545.182 195.276 549.86 c
198.799 554.8 206.834 566.86 210.309 571.831 c
213.556 576.475 221.385 587.517 224.764 592.068 c
228.237 596.745 236.769 607.732 240.375 612.304 c
243.625 616.424 251.389 626.258 254.83 630.228 c
258.254 634.178 266.852 643.206 270.441 646.996 c
273.836 650.581 281.854 659.225 285.474 662.607 c
288.627 665.553 296.502 672.132 299.929 674.749 c
303.248 677.283 311.435 682.924 314.962 685.156 c
318.309 687.276 326.359 692.169 329.995 693.829 c
333.328 695.351 341.409 698.281 345.028 699.033 c
348.458 699.746 356.541 700.454 360.061 700.189 c
363.663 699.919 371.726 697.872 375.094 696.72 c
378.708 695.484 386.898 691.939 390.127 689.782 c
394.117 687.117 401.98 679.365 405.16 675.905 c
407.209 673.676 411.495 667.939 413.254 665.498 c
414.923 663.182 418.688 657.514 420.193 655.09 c
422.281 651.727 426.991 643.543 428.866 640.057 c
430.123 637.72 432.942 632.051 434.069 629.65 c
435.6 626.391 439.026 618.509 440.429 615.195 c
441.928 611.658 445.435 603.181 446.79 599.584 c
448.068 596.189 450.848 587.99 451.993 584.551 c
453.086 581.27 455.526 573.377 456.619 570.096 c
457.764 566.657 460.772 558.539 461.822 555.063 c
462.887 551.54 464.913 543 465.87 539.452 c
466.76 536.15 469.071 528.313 469.917 524.997 c
470.787 521.59 472.659 513.405 473.386 509.964 c
474.108 506.551 475.493 498.335 476.277 494.932 c
477.075 491.47 479.539 483.364 480.325 479.899 c
481.126 476.364 482.479 467.831 483.216 464.287 c
483.931 460.844 486.021 452.712 486.685 449.254 c
487.176 446.698 487.754 442.94 488.997 434.221 c
Q
S
U
[1 0 0 1 0 0] e
57.0887 408.065 43.2122 679.496 tbx
0 tal
267 tld
1 1 1 0 k
/_Helvetica 243 tfn
() 43.2122 459.581 tpt
T
[1 0 0 1 0 0] e
52.4632 408.065 36.8521 679.496 tbx
0 tal
267 tld
/_Helvetica 243 tfn
() 36.8521 459.581 tpt
T
[1 0 0 1 0 0] e
37.4302 380.814 34.5393 652.245 tbx
0 tal
267 tld
/_Helvetica 243 tfn
() 34.5393 432.33 tpt
T
[1 0 0 1 0 0] e
40.3212 373.876 35.6957 645.307 tbx
0 tal
267 tld
/_Helvetica 243 tfn
() 35.6957 425.392 tpt
T
[1 0 0 1 0 0] e
44.9467 392.956 30.492 664.387 tbx
0 tal
267 tld
/_Helvetica 243 tfn
() 30.492 444.472 tpt
T
-1.42109e-016 -1.42109e-016 -1.42109e-016 0 k
q
1 0 0 1 0 0 cm
524.845 433.643 m
524.845 430.174 L
Q
S
q
1 0 0 1 19.52 -8.368 cm
148.43 438.24 147.704 438.928 m2
174.035 413.985 L
Q
S
q
1 0 0 1 19.52 -8.368 cm
174.035 413.985 147.704 438.928 2 1 Ah
Q
q
1 0 0 1 346.9 -4.031 cm
147.633 431.323 148.443 431.909 m2
117.662 409.648 L
Q
S
q
1 0 0 1 346.9 -4.031 cm
117.662 409.648 148.443 431.909 2 1 Ah
Q
u
u
u
[1.6 0 0 1.6 211.8 -128.6] e
32.8047 500.239 16.6154 519.839 tbx
0 tal
17 tld
1 1 1 0 k
/_Symbol 16 tfn
(r) 16.6154 503.759 tpt
T
[1.6 0 0 1.6 222.3 -150.5] e
32.8047 510.039 16.6154 519.839 tbx
0 tal
9 tld
/_Symbol 8 tfn
(b) 16.6154 511.799 tpt
T
U
[1 0 0 1 -125.8 304.4] e
409.359 381.5 376.402 394.904 tbx
0 tal
13 tld
/_Helvetica 12 tfn
(long) 376.402 384.044 tpt
T
U
-1.42109e-016 -1.42109e-016 -1.42109e-016 0 k
q
1 0 0 1 21.6 16.2 cm
242.262 665.802 m
279.643 635.784 280.423 635.158 L2
Q
S
q
1 0 0 1 21.6 16.2 cm
242.262 665.802 280.423 635.158 2 2 Ah
Q
U
u
u
[1 0 0 1 409.4 117.8] e
41.6297 516.797 18.5021 530.201 tbx
0 tal
13 tld
1 1 1 0 k
/_Helvetica 12 tfn
(c) 18.5021 519.341 tpt
T
u
[1.6 0 0 1.6 387.6 -179.4] e
32.8047 500.239 16.6154 519.839 tbx
0 tal
17 tld
/_Symbol 16 tfn
(r) 16.6154 503.759 tpt
T
[1.6 0 0 1.6 398.1 -201.2] e
32.8047 510.039 16.6154 519.839 tbx
0 tal
9 tld
/_Symbol 8 tfn
(b) 16.6154 511.799 tpt
T
U
U
-1.42109e-016 -1.42109e-016 -1.42109e-016 0 k
q
1 0 0 1 -17.28 42.12 cm
427.861 584.277 m
407.868 570.392 407.047 569.822 L2
Q
S
q
1 0 0 1 -17.28 42.12 cm
427.861 584.277 407.047 569.822 2 2 Ah
Q
U
u
[1 0 0 1 106.3 30.4] e
369.312 353.405 335.777 373.511 tbx
0 tal
19 tld
1 1 1 0 k
/_Helvetica 18 tfn
(1 -) 335.777 357.221 tpt
T
u
[1.6 0 0 1.6 441.6 -418.1] e
32.8047 500.239 16.6154 519.839 tbx
0 tal
17 tld
/_Symbol 16 tfn
(r) 16.6154 503.759 tpt
T
[1.6 0 0 1.6 452.1 -439.9] e
32.8047 510.039 16.6154 519.839 tbx
0 tal
9 tld
/_Symbol 8 tfn
(b) 16.6154 511.799 tpt
T
U
U
[0.06912 0 0 0.06912 514.8 416.3] e
0 219.915 0 219.915 tbx
0 tal
12 tld
/_Helvetica 243 tfn
(1.0) 0 0 tpt
T
[0.06912 0 0 0.06912 126.8 416.3] e
0 219.915 0 219.915 tbx
0 tal
12 tld
/_Helvetica 243 tfn
(0.0) 0 0 tpt
T
[1.6 0 0 1.6 530 -377.9] e
32.8047 500.239 16.6154 519.839 tbx
0 tal
17 tld
/_Symbol 16 tfn
(r) 16.6154 503.759 tpt
T
u
[1.6 0 0 1.6 170.5 -418.1] e
32.8047 500.239 16.6154 519.839 tbx
0 tal
17 tld
/_Symbol 16 tfn
(r) 16.6154 503.759 tpt
T
[1.6 0 0 1.6 181 -439.9] e
32.8047 510.039 16.6154 519.839 tbx
0 tal
9 tld
/_Symbol 8 tfn
(b) 16.6154 511.799 tpt
T
U
%%PageTrailer
_PDX_savepage restore
%%Trailer
end
% showpage
%%EOF
---------------0806040948613--
