%Paper: 
%From: tilo@nordita.dk (Tilo Wettig)
%Date: Thu, 5 Aug 93 15:17:58 MET DST

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\title{Factorial Moments in a Generalized Lattice Gas Model}

\author{T. Wettig and A. D. Jackson}

\address{NORDITA, Blegdamsvej 17, DK--2100 Copenhagen \O, Denmark and\\
Department of Physics, State University of New York, Stony Brook,
NY 11794-3800}

\date{\today}

\maketitle

\begin{abstract}
We construct a simple multicomponent lattice gas model in one
dimension in which each site can either be empty or occupied by at
most one particle of any one of $D$ species. Particles interact with a
nearest neighbor interaction which depends on the species involved.
This model is capable of reproducing the relations between factorial
moments observed in high--energy scattering experiments for moderate
values of $D$.  The factorial moments of the negative binomial
distribution can be obtained exactly in the limit as $D$ becomes
large, and two suitable prescriptions involving randomly drawn nearest
neighbor interactions are given.  These results indicate the need for
considerable care in any attempt to extract information regarding
possible critical phenomena from empirical factorial moments.
\end{abstract}

\pacs{PACS numbers: 13.85.Hd,13.65.+i,24.60.Lz,25.75.+r}


\section{Introduction}
\label{sect1}

Factorial moments provide a useful tool in the analysis of high energy
scattering data such as obtained in $e^+ e^-$ scattering \cite{ALEPH},
$p{\bar p}$ scattering (at energies up to 900 GeV) \cite{UA5} or the
scattering of protons or heavy ions ({\em e.g.,} $^{16}$O and
$^{32}$S) by heavy nuclei at a projectile energy of 200 GeV/A
\cite{KLM}.  One considers the full range of some variable (usually
the rapidity) for which one knows the average multiplicity, $\langle n
\rangle$, and its dispersion, $\langle \Delta n^2 \rangle = \langle
n^2 \rangle - \langle n \rangle^2$.  The data is then broken into $M$
equal bins, and one constructs the factorial moments as a suitably
normalized average of a combination of moments in each individual bin.
Specifically,
\begin{equation}
F_q (M) = \left[\frac{1}{M} \sum_{i=1}^M \langle n (n-1) (n-2) \ldots
(n-q+1) \rangle_i \right] \left/ \left[ \frac{1}{M} \sum_{i=1}^M
\langle n \rangle_i \right]^q \right. \ \ .
\label{1.1}
\end{equation}
Although interest in these forms has been motivated by theoretical
considerations \cite{Bialas1}, factorial moments are merely one way to
arrange the experimental data.  Evidently, $F_q (M)$ probes certain
combinations of the properties of the 1--, 2--, 3--, \ldots, $q$--body
correlations between particles in each bin.  If non--interacting
particles are distributed in the various bins in a purely statistical
fashion, all factorial moments are $1$ for all $M$.  This result is
completely inconsistent with the data.  Consider, for example, $p{\bar
p}$ scattering at 900 GeV.  Here, $F_2 (M)$ is seen to grow from 1
(for small $M$) to roughly 1.7 (for the largest values of $M$
considered).  Over the same range, $F_3 (M)$ ranges from 1 to 4.2,
$F_4 (M)$ ranges from 1 to 15 and $F_5 (M)$ ranges from 1 to 80.  This
growth in the factorial moments with $M$ is often called
`intermittency' in the literature and is frequently regarded as
indicating the presence of fluctuations of many different sizes.

It has been observed \cite{Giovannini} that the relations {\em
between\/} the various factorial moments for $p {\bar p}$ scattering
(and all of the other physical processes mentioned above) can be
reproduced with remarkable accuracy by the `negative binomial
distribution' (NB) for which
\begin{equation}
F_q^{NB} (M) = (1 + cM)(1 + 2cM) \ldots (1+[q-1]cM)
\label{1.2}
\end{equation}
with
\begin{equation}
c = \frac{\langle \Delta n^2 \rangle - \langle n \rangle}{\langle n
\rangle^2} \ \ .
\label{1.3}
\end{equation}
Plots of $F_q (M)$ versus $M$ evidently depend on the global averages
$\langle n \rangle$ and $\langle \Delta n^2 \rangle$.  Such plots will
be different for the various physical processes considered and are not
reproduced by the negative binomial distributions as given by
Eq.~(\ref{1.2}) \cite{footnote1}. However, since $F_2^{NB}$ is simply
$(1 + cM)$, it is tempting to consider the more `universal' plots of
$F_q$ versus $F_2$ \cite{Carruthers1}.  Such plots no longer depend on
the parameter $c$ and invite the comparison of data from very
different processes \cite{footnote2}. Such plots have been made over
the available range $1 < F_2 < 1.7$.  They do reveal universal
behaviour \cite{Ochs} and are in striking agreement with the curves
obtained from the negative binomial distribution.

The dramatic growth of the factorial moments with $M$ led Bialas and
Peschanski to note that `an observation of a variation in $\langle F_i
\rangle$ (our $F_q$) with $\delta y$ (our $M$) indicates the presence
of genuine fluctuations which must have some physical origin'
\cite{Bialas1}.  Many authors have taken up the challenge of
describing this physical origin \cite{Carruthers2}.  Some have
concentrated on the apparent presence of fluctuations of many
different sizes and considered models incorporating a variety of
critical phenomena \cite{Satz}.  Others have studied both schematic
and more realistic versions of cascade models
\cite{Bialas2,Giovannini}.

In two interesting papers, Chau and Huang have offered a different
kind of insight \cite{Chau}.  They imagine that the full range of
rapidity corresponds to the $N$ sites of a one--dimensional Ising (or
lattice gas) model \cite{footnote3}.  This model is exactly solvable.
They consider the $q$--body correlations and the factorial moments
which come from a lattice gas model (in the limit $N \rightarrow
\infty$).  The two parameters of the Hamiltonian are determined by
fixing the global values of $\langle n \rangle$ and $\langle \Delta
n^2 \rangle$.  The resulting factorial moments have the form
\begin{equation}
F_q^{LG} (M) = \sum_{k=0}^{q-1} \frac{q!(q-1)!}{(q-k)!k!(q-1-k)! 2^k}
(cM)^k
\label{1.4}
\end{equation}
where $c$ is given by Eq.~(\ref{1.3}).  This result is both rather
more and rather less than meets the eye.

Let us address the `less' first.  While the lattice gas factorial
moments are not identical to the corresponding moments of the
negative binomial distribution, both share the common
small $M$ expansion
\begin{equation}
F_q (M) = 1 + \frac{q(q-1)}{2} cM + {\cal O}(c^2 M^2 ) \ \ .
\label{1.5}
\end{equation}
We have already noted that the leading term is due to `one--body
effects'.  It should come as no surprise that the term of order $M$
is precisely due to two--body correlations.  Under the assumption that
genuine two--body correlations have a finite range, the $q$--dependence
of the term of order $M$ is uniquely determined.  The coefficient $c$,
which can be expressed as a suitable integral of the two--body
correlation function, is also fixed by the global dispersion, $\langle
\Delta n^2 \rangle$.  Thus, the observation of Chau
and Huang that the factorial moments `given by single negative
binomials are almost exactly the same as ours for $F_2 \rightarrow
1$' is a trivial consequence of the rules of the game ({\em i.e.,}
the fixing of $\langle n \rangle$ and $\langle \Delta n^2 \rangle$)
and the fact that the Ising model predicts many--body correlations of
a finite range.

Chau and Huang further note that the negative binomial distributions
are more successful than the lattice gas model for values of $F_2$
larger than 1.5.  This is the domain where terms of order $M^2$ and
higher --- which are not model independent --- play a role.  A few
comments on the existing data are in order before continuing.  Data
for 200 GeV/A $^{16}$O and $^{32}$S scattering on emulsions is limited
to the range $1 < F_2 < 1.35$.  Thus, this data never really probes
the interesting (model dependent) regions of $F_2$.  (Over this range,
the difference between $F_3^{LG}$ and $F_3^{NB}$ is less than 2.8\%
which is small compared with the experimental uncertainties.)  The
data for 200 GeV $p$ on emulsion and for 900 GeV $p{\bar p}$
scattering has a somewhat wider range covering $1 < F_2 \le 1.7$.  At
the upper end of this range, the difference between lattice gas and
negative binomial forms for $F_3$ grows to 6.2\% while the non--linear
terms in the negative binomial form account for 24\% of $F_3$.
Uncertainties in the data in this region (both systematic and
statistical) are approximately 7.2\%.  The situation is similar for the
existing data on $F_4$ and $F_5$.  It is possible to find convincing
empirical evidence for higher than linear terms in $F_q$.  It is more
difficult to distinguish empirically between negative binomial and
lattice gas models although the former does provide a better global
fit.

We believe that Chau and Huang have provided something very positive
in their work.  Their model admits an elementary generalization which
shows one possible route which leads smoothly from the lattice gas
model to the negative binomial distributions \cite{JWB}.  While the
nature of this route may be of limited empirical value given the
current status of the data, it strikes us as being of some theoretical
importance.  This generalization, which is the primary focus of this
paper, is simply stated: Consider a one dimensional lattice gas model
(again in the limit where the number of sites, $N$, is infinite) where
each site can either be empty or occupied by one particle which can be
of any one of $D$ species.  Each species has a chemical potential,
$\mu_d$, and each pair of species has a nearest neighbor interaction
of strength $\epsilon_{dd'}$.  Like the ordinary lattice gas (with $D
= 1$), this model allows for the analytic determination of the
\mbox{$q$--body} correlation functions and the factorial moments,
$F_q (M)$, using the obvious generalization of standard techniques.
As usual, this involves the construction and diagonalization of a
matrix, ${\cal M}$, related to the partition function for one pair of
adjacent sites.

The factorial moments for this generalized lattice gas model can then
be specified exactly in terms of certain (weighted) moments of an
elementary function of the eigenvalues of ${\cal M}$.  We shall show
that, for any value of $D$, these moments can be chosen to reproduce
all factorial moments of the ordinary lattice gas model.  Further, we
shall show that these moments can also be chosen to approach the
factorial moments of the negative binomial distribution.  In the limit
$D \rightarrow \infty$, this model permits the exact reproduction of
{\em all\/} factorial moments of the negative binomial distribution.

Given the specific form of ${\cal M}$, there is no guarantee that
these conditions on its eigenvalues can be met for any choice of the
parameters appearing in the related Hamiltonian.  We shall, thus, give
two methods for the selection of the various chemical potentials and
nearest neighbor interaction parameters in ${\cal M}$ which explicitly
reproduce the factorial moments of the negative binomial distribution.
This choice is essentially a `random dynamics' model in which the $D$
chemical potentials are set equal and in which the various nearest
neighbor interactions are drawn at random according to a certain
distribution.  The random nature of this model and the smallness of
the dispersion in its factorial moments for $D$ finite and small
provides some understanding for the success of cascade model
calculations of these processes and for the relative insensitivity of
such calculations to the details of input parameters.

The organization of this paper will be as follows.  In
Sec.~\ref{sect2} we summarize the results of Chau and Huang.  One
purpose of this summary is to establish the notation of the lattice
gas model (as opposed to the Ising model) which will later be
generalized.  In the process we shall establish a useful (and more
general) theorem regarding the factorial moments and shall explain
precisely which features of the Ising model are responsible for the
form of Eq.~(4).  Along the way, we shall point out why the general form of
Eq.~(5) is more a matter of definition than of physical content.

In Sec.~\ref{sect3} we shall establish our generalization to a
multicomponent lattice gas model containing $D$ species, outline the
techniques for its analytic solution, and present the general form for
the resulting factorial moments.

In Sec.~\ref{sect4} we shall consider the constraints which must be
imposed on this generalized model if it is (i) to reproduce the
results of Chau and Huang or (ii) to reproduce the factorial moments
of the negative binomial distribution.  These constraints will be
established analytically as a well--defined distribution of the
eigenvalues of a certain $(D+1)$--dimensional matrix.  We shall also
report the results of numerical studies which allow us to express
these constraints in terms of a well--defined but random distribution of
the $D(D+1)/2$ nearest--neighbor interaction strengths between the $D$
species.

A variety of conclusions will be drawn in Sec.~\ref{sect5}.

Two short appendices are provided to deal with more technical matters.



\section{The Lattice Gas Model}
\label{sect2}

Here, we follow the arguments of Chau and Huang with the notational
difference that we consider a one--dimensional lattice gas rather than
an Ising model.  We consider $N$ microscopic sites which can have an
occupation number of either 0 or 1.  Particles occupying adjacent
sites will experience a nearest neighbor interaction of strength
$\epsilon$.  The system is assumed to be cyclic in the sense that
particle $N$ interacts with particle $1$ as well as with particle $(N-1)$.
The Hamiltonian for the system is simply
\begin{equation}
H = \epsilon [n_1 n_2 + n_2 n_3 + \ldots + n_{N-1} n_N + n_N n_1 ] \ \ .
\label{2.1}
\end{equation}
The partition function for this system is \cite{footnote4}
\begin{equation}
\sum \exp{\left[-H - \mu \sum_{i=1}^N n_i \right] } \ \ .
\label{2.2}
\end{equation}
The external summation here covers the $2^N$ terms where each
$n_i$ has the value 0 or 1.  This problem is rendered trivial by
constructing a two--dimensional matrix, ${\cal M}$, such that
\begin{equation}
{\cal M}_{n_1 n_2} = \exp{\left[-\epsilon n_1 n_2 - \frac{1}{2} \mu n_1
- \frac{1}{2} \mu n_2 \right]} \ \ .
\label{2.3}
\end{equation}
It is convenient to let the matrix indices run over 0 and 1.

The partition function for the system is now simply the trace of
${\cal M}^N$.  This trace is most easily constructed by defining the
orthogonal matrix, $\theta$, which diagonalizes ${\cal M}$.
\begin{equation}
{{\cal M}_d} = \theta^T {\cal M} \theta \ \ .
\label{2.4}
\end{equation}
The diagonal form ${{\cal M}_d}$ has two eigenvalues, $\lambda_{\pm}$,
and we shall choose $\theta$ such that $({{{\cal M}_d}})_{00}$ is the
eigenvalue of larger magnitude, $\lambda_+$.  (Given the form of ${\cal
M}$, it is clear that $\lambda_+ > 0$.)  We then find that
\begin{equation}
{\rm Tr}[{\cal M}^N ] = {\rm Tr}[\theta {{\cal M}_d}^N \theta^T ] =
{\rm Tr}[{{\cal M}_d}^N ] = \lambda_+^N + \lambda_-^N \ \ .
\label{2.5}
\end{equation}
In the limit $N \rightarrow \infty$, we can neglect the term
$\lambda_-^N$ at the cost of introducing an error which is
exponentially small in $N$.

In order to calculate the various correlation functions, we introduce
the number operator matrix for site $i$, $n_i$.  Clearly,
$(n_i )_{n_1 n_2} = \delta_{1 n_1} \delta_{1 n_2}$, {\em i.e.,}
\begin{equation}
n_i = \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right) \ \ .
\label{2.6}
\end{equation}
The number operator at each site is an idempotent with $n_i^2 = n_i$.
Since all sites are equivalent (due to the cyclic form of
Eq.~(\ref{2.1})), the average number of particles per site is simply
\begin{equation}
\langle n_i \rangle = \left. {\rm Tr} \left[ n_i {\cal M}^N \right]
\right/ {\rm Tr} \left[ {\cal M}^N \right] \ \ .
\label{2.7}
\end{equation}
Given the fact that ${\rm Tr}[{\cal M}^N]=\lambda_+^N$ in the large
$N$ limit, we see that it is useful to replace ${{\cal M}_d}$ by
another diagonal matrix, ${{\bar {\cal M}_d}}$, in which each diagonal
element is simply divided by $\lambda_+$.  Thus, $({{\bar {\cal
M}_d}})_{00} = 1$ and $({{\bar {\cal M}_d}})_{11} = {{\bar
\lambda}_{-}}$ where $|{\bar \lambda}_{-}| < 1$.  With this notation,
\begin{equation}
\langle n_i \rangle = {\rm Tr} \left[ n_i \theta {{\bar {\cal M}_d}}^N
\theta^T \right] \ \ .
\label{2.8}
\end{equation}
Given the trivial structure of $n_i$,
\begin{equation}
\langle n_i \rangle = (\theta {{\bar {\cal M}_d}}^N \theta^T )_{11}
\rightarrow (\theta_{10})^2
\label{2.9}
\end{equation}
where the last form applies in the limit $N \rightarrow \infty$.
While $\theta_{10}$ and ${{\bar \lambda}_{-}}$ can be determined in
terms of $\epsilon$ and $\mu$ following an explicit diagonalization of
${\cal M}$, this is not necessary.  It is our intention to impose
global constraints on $\langle n \rangle = N \langle n_i \rangle$ and
$\langle \Delta n^2 \rangle$.  This can be done with equal ease
working with $\theta_{10}$ and ${{\bar \lambda}_{-}}$.

The two--body correlation function can now be written immediately as
\begin{equation}
\langle n_i n_{i+j} \rangle = (\theta {{\bar {\cal M}_d}}^j \theta^T )_{11}
(\theta {{\bar {\cal M}_d}}^{N-j} \theta^T )_{11} \ \ .
\label{2.10}
\end{equation}
Taking the large $N$ limit and making the usual assumption that $N-j$
is always ${\cal O}(N)$, we find
\begin{equation}
\langle n_i n_{i+j} \rangle = \langle n_i \rangle [ \langle n_i \rangle
+ (1 - \langle n_i \rangle ) {{\bar \lambda}_{-}}^j ] \ \ .
\label{2.11}
\end{equation}
Eq.~(\ref{2.11}) is valid for $j \ge 0$.  The structure of this
two--body correlation function is instructive and very general.  For
large separations (large $j$), $\langle n_i n_{i+j} \rangle$
approaches the uncorrelated, statistical value of $\langle n_i
\rangle^2$.  The approach to this asymptotic value is exponentially
fast.  For small $j$, there are significant short--range correlations.
For $j=0$, we find $\langle n_i n_{i+j} \rangle = \langle n_i \rangle$
independent of ${\bar \lambda}_{-}$ which is merely a
reflection of the fact that the number operator is idempotent.

It is elementary to determine the global dispersion, $\langle \Delta
n^2 \rangle$.  Since it is our intention to constrain $\langle \Delta
n^2 \rangle$ to be consistent with data, let us deal with this problem
immediately.  We first construct
\begin{equation}
\langle n^2 \rangle = \sum_{i_1 , i_2  = 1}^N \langle n_{i_1} n_{i_2}
\rangle = N \langle n_i \rangle + 2 \sum_{i_1 < i_2 } \langle n_{i_1}
n_{i_2} \rangle \ \ .
\label{2.12}
\end{equation}
It is, of course, possible to do the summations in Eq.~(\ref{2.12})
exactly given the form of Eq.~(\ref{2.11}).  For purposes of later
arguments, it is more useful to obtain $\langle n^2 \rangle$
approximately by making approximations which neglect terms of order
$1/N$.  Thus, we write
\begin{equation}
\langle n^2 \rangle = \langle n \rangle + 2 \langle n_i \rangle^2
\sum_{i_1 < i_2} 1 + 2\langle n_i \rangle (1 - \langle n_i \rangle)
\sum_{i_1 < i_2 } {{\bar \lambda}_{-}}^{i_2 - i_1} \ \ .
\label{2.13}
\end{equation}
The first sum in Eq.~(\ref{2.13}) will be approximated by $N^2 /2$.
The second sum is more interesting.  Since ${{\bar \lambda}_{-}}^j$
converges exponentially with $j$, we shall allow the $i_2$ sum to
extend over all values $1 \le (i_2 - i_1 ) \le \infty$.  The sum over
$i_1$ then merely introduces a factor of $N$.  Further, we neglect
$\langle n_i \rangle$ compared to $1$.  These approximations each
introduce errors of order $1/N$ which are acceptable as $N \rightarrow
\infty$.  We immediately obtain
\begin{equation}
\langle n^2 \rangle = \langle n \rangle + \langle n \rangle^2 +
2 \langle n \rangle \frac{{{\bar \lambda}_{-}}}{1 - {{\bar \lambda}_{-}}}
\ \ .
\label{2.14}
\end{equation}
We are now able to set ${{\bar \lambda}_{-}}$ in order to reproduce
any desired dispersion.

The calculation of factorial moments can be simplified significantly
when the number operator is idempotent.  This issue is addressed in
Appendix~\ref{app1} where appropriate operator (and ensemble average)
identities are established.  Using the definition of the factorial
moments, Eq.~(\ref{1.1}), using the simplifying result of
Eq.~(\ref{a.6}), and adopting the same large $N$ approximations, we
can obtain $F_2 (M)$.  (Now, the sums analogous to those in
Eq.~(\ref{2.12}) extend to an upper limit of $N_B$ where $N_B = N /
M$.  Since the number of bins, $M$, is finite, we are also concerned
with the limit $N_B \rightarrow \infty$.)  We find
\begin{equation}
F_2^{LG} (M) = 1 + c M
\label{2.15}
\end{equation}
where
\begin{equation}
c = \frac{2}{\langle n \rangle} \frac{{{\bar \lambda}_{-}}}{1 - {{\bar
\lambda}_{-}}} \ \ .
\label{2.16}
\end{equation}
Returning to Eqs.~(\ref{2.10}) and (\ref{2.11}), it is now elementary
to construct the $q$--body correlation functions as
\begin{equation}
\langle n_{i_1} n_{i_1 + i_2} \ldots n_{i_1 + \ldots + i_q} \rangle
= \langle n_i \rangle [\langle n_i \rangle + (1 - \langle n_i \rangle )
{{\bar \lambda}_{-}}^{i_2} ] [\langle n_i \rangle + (1-\langle n_i \rangle )
{{\bar \lambda}_{-}}^{i_3} ] \ldots [\langle n_i \rangle + (1 -
\langle n_i \rangle ) {{\bar \lambda}_{-}}^{i_q} ] \ \ .
\label{2.17}
\end{equation}
Eq.~(\ref{2.17}) applies for $i_1,i_2,\ldots,i_q \ge 0$.  This form,
which is not unique to the lattice gas model, determines the form of
the factorial moments and merits comment.  The $q$--body correlation
function is completely determined by the two--body correlation
function and $\langle n_i \rangle$.  More precisely, the $q$--body
correlation function is a product of the $(q -1 )$ two--body
correlators between adjacent particles.  Finally, the individual
two--body correlators are given as the sum of a statistical term and a
short--range piece.  These qualitative features are sufficient to
determine the associated factorial moments uniquely without any
detailed information about the precise nature of the short--range part
of the two--body correlator.  To emphasize this independence, we shall
make the replacement of $(1 - \langle n_i \rangle ) {{\bar
\lambda}_{-}}^j$ by $g(j)$ where the only restriction on $g(j)$ is
that it is of short range.

Now let us turn to an arbitrary factorial moment in the limit $N
\rightarrow \infty$.  We again use the operator identity derived in
Appendix~\ref{app1}, Eq.~(\ref{a.6}), and replace sums by integrals to
obtain
\begin{eqnarray}
F_q^{LG} (M) & = & q! \left( \frac{M}{\langle n \rangle} \right)^{q}
\int_0^{N_B} dx_1 \int_0^{N_B - x_1} dx_2 \ldots \int_0^{N_B - x_1 -
\ldots - x_{q-1}} dx_q \nonumber \\
& {} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \frac{\langle n \rangle}{N}
[\frac{\langle n \rangle}{N} + g (x_2 ) ] [\frac{\langle n \rangle}{N} +
g (x_3 ) ] \ldots [\frac{\langle n \rangle}{N} + g (x_q ) ] \ \ .
\label{2.18}
\end{eqnarray}
The structure of this result is clear.  We now imagine expanding the
$(q-1)$ square brackets.  Since the function $g(x_i )$ is short range,
we can extend the $x_i$ integration to infinity for any of those
$2^{q-1}$ terms which contains a factor of $g(x_i )$.  Let us
introduce the notation
\begin{equation}
G = \int_0^{\infty} \ dx \ g(x) \ \ .
\label{2.19}
\end{equation}
The term ${\cal O}(M^0 )$ can only come from picking the statistical
factor in each term.  There is $1$ way to do this.  We obtain no
factors of $G$.  The remaining integrals give a factor of $N_B^q /q!$.
Hence, the leading term in $F_q^{LG} (M)$ is always one, as expected.
The term ${\cal O}(M)$ comes from picking $(q-1)$ statistical factors
and $1$ factor of $g$.  There are $(q-1)$ ways to do this, and we
obtain a factor of $G$.  The remaining integrals give a factor of
$N_B^{q-1}/(q-1)!$.  Hence, the next term in $F_q^{LG} (M)$ is
precisely $(2GM / \langle n \rangle) q(q-1)/2$.  The general result is
now obvious:
\begin{equation}
F_q^{LG} (M) = \sum_{k=0}^{q-1} \left[ \frac{2 G M}{\langle n \rangle}
\right]^k
\frac{q! (q-1)!}{(q-k)!k!(q-1-k)! 2^k } \ \ .
\label{2.20}
\end{equation}
Comparing Eq.~(\ref{2.20}) with $q=2$ with the result of
Eq.~(\ref{2.15}), we see that we can make the identification
\begin{equation}
c = \frac{2G}{\langle n \rangle} \ \ .
\label{2.21}
\end{equation}

Eqs.~(\ref{2.20}) and (\ref{2.21}) are precisely the forms found by
Chau and Huang as quoted in Eq.~(\ref{1.4}).  The present derivation
makes it clear that these results {\em do\/} depend on the fact that
the $q$--body correlator is a product of the $(q-1)$ consecutive
two--body correlators and that they do {\em not\/} depend on the
specific form of the short--range two--body correlations.

The fact that
\begin{equation}
F_q (M) = 1 + \frac{q(q-1)}{2} cM + {\cal O}(c^2 M^2 )
\label{2.22}
\end{equation}
is even more general.  It depends only on the fact that the $q$--body
correlation function contains no long--range terms and thus includes
all one--dimensional models.  For any model we can consider
\begin{equation}
\langle n_{i_1} n_{i_1 + i_2} \ldots n_{i_1 + \ldots + i_q}  \rangle
\label{2.23}
\end{equation}
in the limit where one of the indices ({\em e.g.}, $i_k$) is small and
all others are large.  In the absence of long--range correlations,
this limit must reduce to
\begin{equation}
\langle n_i \rangle^{q-2} \langle n_i n_{i + i_k }  \rangle \ \ .
\label{2.24}
\end{equation}
(Translational invariance ensures that the final term here depends
only on $i_k$.)  Such terms are the only ones which contribute to the
linear piece of $F_q (M)$.  Their $q$ dependence is protected by
elementary combinatorics.  The coefficient $c$ is protected by the
fact that the two--body correlation has been adjusted to fit the
empirical global dispersion, $\langle \Delta n^2 \rangle$.  In a
similar fashion, the terms ${\cal O}(M^2 )$ receive contributions from
both two-- and three--body correlations.  If we had insisted on
establishing global constraints on $\langle n^3 \rangle$, these terms
would be similarly protected.

Thus, the fact that lattice gas factorial moments agree with those
coming from the negative binomial distribution in the limit $cM
\rightarrow 0$ provides {\em no\/} indication that the lattice gas
model is correct.  It is merely an indication of the fact that there
are no long--range correlations and that the `rules of the game' are
to consider only models with fixed global $\langle n \rangle$ and
$\langle \Delta n^2 \rangle$.


\section{A Generalized Lattice Gas Model}
\label{sect3}

As noted in Sec.~\ref{sect2}, the ordinary lattice gas model does
not reproduce the factorial moments of the negative binomial
distribution.  The fact that there is agreement in the small $cM$
limit is a consequence of general properties of one--dimensional
models and is not indicative of any particular merit of the lattice
gas model.  In this section we shall construct a simple generalization
of this lattice gas model which has the capacity to reproduce the
negative binomial factorial moments {\em exactly}.  This model is a
conceptually simple one--dimensional model which is also exactly
solvable (in a sense which is appropriate for constructing factorial
moments).

Consider a lattice gas in which site $i$ is either unoccupied or is
occupied by at most one particle which can be of any one of $D$
species.  Each species has its own chemical potential, $\mu_d$.
Particles again have a nearest neighbor interaction.  The strength of
the interaction between a particle of species $d$ and a particle of
species $d'$ is $\epsilon_{d d'}$.

This model can be solved using precisely the standard techniques of
Sec.~\ref{sect2}.  The matrix ${\cal M}$ now becomes a real,
symmetric matrix of dimension $(D+1)$.  The elements in this matrix
are
\begin{eqnarray}
{\cal M}_{00} & = & 1 \nonumber \\
{\cal M}_{0d} & = & \exp{[-\mu_d / 2]} \label{3.1} \\
{\cal M}_{dd'} & = & \exp{[-\epsilon_{dd'} -\mu_d / 2 - \mu_{d'} / 2]}
\nonumber
\end{eqnarray}
(We allow the matrix indices to run from $0$ to $D$. Again, the
inverse temperature, $\beta$, has been set equal to $1$.)  The number
operator at site $i$ is also a $(D+1)$--dimensional matrix having the
form $n_i = \openone - T$ with $T_{dd'} = \delta_{d0}
\delta_{d'0}$, {\em i.e.,}
\begin{equation}
n_i = \left( \begin{array}{cccccc}
                0 & & & & & \\
                & 1 & & & & \\
                & & 1 & & & \\
                & & & 1 & & \\
                & & & & \ddots & \\
                & & & & & 1
             \end{array} \right) \ \ .
\label{3.2}
\end{equation}
Both this number operator and $T$ are idempotent, so that the results
of Eqs.~(\ref{a.3}) and (\ref{a.6}) remain valid both at the operator
level and at the level of ensemble averages.

We again find the transformation $\theta$ which diagonalizes ${\cal
M}$ and places the eigenvalue of largest magnitude in $({\cal
M}_d)_{00}$ \cite{footnote5}. As before, we construct the
matrix ${\bar {\cal M}}_d$ by dividing each element of ${\cal M}_d$ by
the eigenvalue of largest magnitude.  (The form of ${\cal M}$ again
ensures that this largest eigenvalue will be positive.)  Thus,
$({\bar{\cal M}}_d)_{00}$ is again $1$. While we shall retain this
notation, it is not completely necessary in practice. The spirit of
the model is, ultimately, to take the limit as $N \rightarrow \infty$
for fixed $\langle n \rangle$.  In this limit, the average occupation
per site, $\langle n_i \rangle$, tends to zero.  This limit will be
realized by taking the limit as all of the $\mu_d \rightarrow
+\infty$.  In this limit, the coupling of the $D \times D$ submatrix
(with $d,d' \ne 0$) to the remaining elements of ${\cal M}$ becomes
arbitrarily weak.  In the limit, it is legitimate to treat this
coupling in lowest--order perturbation theory. We shall return to this
point in Sec.~\ref{sect4}.

With these preliminaries in hand, the construction of the various
correlation functions and factorial moments proceeds as before.  The
average number of particles per site is simply
\begin{equation}
\langle n_i  \rangle = \frac{\langle n  \rangle}{N} = Tr[(\openone - T)
\theta {{\bar {\cal M}_d}}^N \theta^T ]
\label{3.4}
\end{equation}
In the limit $N \rightarrow \infty$, ${{\bar {\cal M}_d}}^N$ may be
set equal to $T$ with exponentially small errors.  The two terms in
the factor $(\openone -T)$ must be treated separately
\cite{footnote6}. One immediately finds
\begin{equation}
\langle n_i  \rangle = 1 - (\theta_{00})^2
\label{3.5}
\end{equation}
As usual, $\langle n \rangle$ and hence $(\theta_{00})^2$ will be
fixed by experiment.

The two--body correlator is readily calculated as
\begin{equation}
\langle n_i n_{i+j}  \rangle = Tr[(\openone - T) \theta
{{\bar {\cal M}_d}}^j \theta^T (\openone - T) \theta T \theta^T ] \ \ .
\label{3.6}
\end{equation}
Expanding the factors of $(\openone -T)$, one immediately obtains
\begin{equation}
\langle n_i n_{i+j}  \rangle = \langle n_i  \rangle [ \langle n_i  \rangle
+ (1 - \langle n_i  \rangle ) \sum_{\ell=1}^D
(\theta_{0 \ell})^2 {{\bar \lambda}_{\ell}}^j ] \ \ .
\label{3.7}
\end{equation}
This form is virtually identical to that of Eq.~(\ref{2.11}).  The
first term is again the asymptotic statistical probability of finding one
particle (of any species) on each of the two sites.  Since $|{\bar
\lambda}_{\ell}| < 1$, the second term here again vanishes exponentially
for large $j$.  However, in this case, we are confronted with a sum of
exponentials rather than a single term.

Following the arguments from Eq.~(\ref{2.12}) to (\ref{2.16}) we find
that \cite{footnote7}
\begin{equation}
F_2^{GLG} (M) = 1 + cM
\label{3.8}
\end{equation}
with
\begin{equation}
c = \frac{\langle n^2 \rangle - \langle n \rangle^2 - \langle n \rangle}
{\langle n \rangle^2} = \frac{2}{\langle n \rangle} \sum_{\ell=1}^D
b_\ell^2 \frac{{\bar \lambda}_\ell}{1 - {\bar \lambda}_\ell} \ \ .
\label{3.9}
\end{equation}
Here, we have taken the notational liberty of exploiting the fact that
$\theta$ is an orthogonal matrix and made the substitution
\begin{equation}
b_\ell^2 = \frac{(\theta_{0\ell})^2}{1-\langle n_i \rangle} \ \ \ \
{\rm with} \ \ \ \ \sum_{\ell=1}^D b_\ell^2 = 1 \ \ .
\label{3.10}
\end{equation}
Again, the similarities between Eqs.~(\ref{3.9}) and (\ref{2.16}) are
striking.  The only difference is the presence of the sum over $\ell$.

Since certain differences arise at the level of the three--body
correlation function, this term is worth discussing specifically.
Obviously,
\begin{equation}
\langle n_i n_{i+j} n_{i+j+k}  \rangle = Tr[(\openone -T) \theta {{\bar
{\cal M}_d}}^j \theta^T (\openone -T) \theta {{\bar {\cal M}_d}}^k
\theta^T (\openone -T) \theta T \theta^T ]
\label{3.11}
\end{equation}
Expanding the factors of $(\openone -T)$ and constructing the traces
gives rise to the form
\begin{eqnarray}
\langle n_i n_{i+j} n_{i+j+k}  \rangle & = & \langle n_i  \rangle^3 +
\langle n_i  \rangle^2 (1 - \langle n_i  \rangle )
[\sum b_{\ell}^2 {{\bar \lambda}_{\ell}}^j +
\sum b_{\ell}^2 {{\bar \lambda}_{\ell}}^k -
\sum b_{\ell}^2 {{\bar \lambda}_{\ell}}^j \sum b_{\ell}^2
{{\bar \lambda}_{\ell}}^k ] \nonumber \\
& {} & + \langle n_i  \rangle (1 - \langle n_i  \rangle) \sum b_{\ell}^2
{{\bar \lambda}_{\ell}}^{j+k}
\label{3.12}
\end{eqnarray}
This result differs from the expression for the ordinary lattice gas.
The three--body correlator {\em cannot} be expressed as a product of
consecutive two--body correlators.  This is important since it will
give us precisely the freedom we need to proceed from the lattice gas
factorial moments to those of the negative binomial distribution.
Nevertheless, this correlator is still determined by the two--body
correlation function.  It merely requires the addition of $\langle n_i
n_{i+j+k} \rangle$.  This fact is also important, since we seek
factorial moments which depend on the global properties of the
distribution through $c$ only.  Note that in the special case of $D=1$
appropriate for the ordinary lattice gas, the sums collapse to a
single term with $b_1^2 = 1$.

It is now straightforward to perform the sums necessary to determine
$F_3^{GLG} (M)$ in the limit as $N \rightarrow \infty$.  Using
Eq.~(\ref{a.6}) we find
\begin{equation}
F_3^{GLG} (M) = 1 + 3cM + \frac{3}{2} d^2 M^2 \ \ .
\label{3.13}
\end{equation}
The presence of the term involving $\langle n_i n_{i+j+k} \rangle$ in
the three--body correlation function has forced the introduction of
the new term
\begin{equation}
d^2 = \frac{4}{\langle n \rangle^2} \sum_{\ell=1}^D b_{\ell}^2 \left(
\frac{{\bar \lambda}_{\ell}}{1 - {\bar \lambda}_{\ell}} \right)^2
\label{3.14}
\end{equation}
We note that this third factorial moment will equal that of the
original lattice gas model provided that $d^2 = c^2$.  There are (at
least) two situations where this will happen.  One is the case where
precisely one of the $b_{\ell}^2$ is equal to $1$ (with the remaining
$b_{\ell}^2$ equal to $0$).  The other is the case where all of the
${\bar \lambda}_{\ell}$ are equal.

We now turn to the general case of $F_q^{GLG} (M)$.  The strategy is
now perfectly straightforward and exceptionally tedious.  We construct
the $q$--body correlation function as the obvious generalization of
Eqs.~(\ref{3.6}) and (\ref{3.11}).  We expand the various factors of
$(\openone -T)$ and perform the requisite traces.  Using
Eq.~(\ref{a.6}), we then perform the requisite bin sums.  The final
result follows from some algebra and combinatorics.  We find
\begin{equation}
F_q^{GLG} (M) = \sum_{k=0}^{q-1} f_k^{(q)} M^k
\label{3.15}
\end{equation}
where
\begin{equation}
f_k^{(q)} = \frac{q!}{k! (q-k)!} \left[ \frac{d^k}{dz^k} [ 1 +
\sum_{\mu=1}^{\infty} z^{\mu} s_{\mu}] ^{(q-k)} \right]_{z=0} \ \ .
\label{3.16}
\end{equation}
Here, we have introduced the definition
\begin{equation}
s_{\mu} = \frac{1}{\langle n \rangle^{\mu}} \sum_{\ell=1}^D
{b_{\ell}}^2 \left( \frac{{{\bar \lambda}}_{\ell}}{1-{{\bar
\lambda}}_{\ell}} \right)^{\mu} \ \
\label{3.17}
\end{equation}
which implies $s_1 = c/2$.

It is also desirable to derive a generating function from which the
factorial moments can be obtained by simple differentiation.  To this
end, we first observe that $f_k^{(q)}$ can be re--written as
\begin{equation}
f_k^{(q)} = \frac{1}{(q-k)!} \left[ \frac{d^q}{dz^q} z^{q-k}
[ 1 + \sum_{\mu=1}^{\infty} z^{\mu} s_{\mu} ] ^{(q-k)} \right]_{z=0} \ \ .
\label{3.18}
\end{equation}
We now change the summation index in Eq.~(\ref{3.15}) from $k$ to
$(q-k)$ to obtain
\begin{equation}
F_q^{GLG}(M) = M^q \left[ \frac{d^q}{dz^q} \sum_{k=1}^{q}
\frac{1}{k!} \left( \frac{z}{M} [ 1+\sum_{\mu=1}^{\infty}
s_{\mu} z^{\mu}] \right) ^k \right]_{z=0} \ \ .
\label{3.19}
\end{equation}
The sum over $k$ can be extended to range from $0$ to $\infty$ since
the corresponding derivatives are zero. The factorial moments thus
become
\begin{equation}
F_q^{GLG}(M) = M^q \left[ \frac{d^q}{dz^q} \exp{\left( \frac{z}{M}
[1+\sum_{\mu=1}^{\infty} s_{\mu} z^{\mu}] \right)} \right]_{z=0} \ \ .
\label{3.20}
\end{equation}
Eqs.~(\ref{3.15})--(\ref{3.17}) and (\ref{3.20}) represent the final
results of our generalized lattice gas model.

The various terms, $f_k^{(q)}$, which characterize the factorial
moments of our generalized lattice gas model are completely determined
by the leading terms $f_{k'}^{(k'+1)}$ for $k' \le k$.  This follows
from combinatoric arguments and has virtually nothing to do with the
underlying microscopic details of the model.  It does involve two
basic ingredients of the model.  First, that each site has an
occupancy of either $0$ or $1$.  Second, that the non--statistical
pieces of the various correlations are of short range
\cite{footnote8}. It was not {\em a priori\/} obvious that any
one--dimensional model would yield the factorial moments of the
negative binomial distribution. It is now clear, however, that the
generalized lattice gas model is capable of reproducing the factorial
moments of {\em any\/} distribution (including the NB) through a
suitable choice of the $s_\mu$. It remains to be demonstrated that
there exists a choice of the underlying Hamiltonian which will provide
the desired $s_\mu$. Such demonstration is the topic of the next
section.


\section{Obtaining the Factorial Moments of the Negative Binomial
Distribution}
\label{sect4}

As we have repeatedly emphasized, we are certainly able to reproduce
the factorial moments of the ordinary lattice gas model. We will show
in Appendix~\ref{app2} that this happens in the special case where
\begin{equation}
s_{\mu} = \left( \frac{c}{2} \right)^{\mu} \ \ .
\label{3.21}
\end{equation}
As noted, this special case can be realized either when the sums over
$\ell$ contain only one term or when all the reduced eigenvalues are
equal.

We are also able to pick the various terms, $s_{\mu}$, in such a way
as to reduce Eqs.~(\ref{3.15})--(\ref{3.17}) to Eq.~(\ref{1.2})
exactly.  It is readily verified that the choice
\begin{equation}
s_{\mu} = \frac{c^{\mu}}{\mu+1}
\label{4.1}
\end{equation}
establishes the desired equality.  (The demonstration of this fact is
given in Appendix~\ref{app2}.)  With this choice, the factorial
moments of our generalized lattice gas model are rendered {\em
identical\/} to the factorial moments of the negative binomial
distribution.  This answer poses a question.  Is it possible to pick
the dimension $D$ (the number of species) and to find a nearest
neighbor Hamiltonian and its related matrix, ${\cal M}$, such that the
constraints of Eq.~(\ref{4.1}) are satisfied?  As we shall see, it is.

As noted in the previous section, we are ultimately interested in the
$N \rightarrow \infty$ limit.  We shall arbitrarily set all chemical
potentials equal to $\mu_{\rm o}$ and realize this limit by taking
$\mu_{\rm o} \rightarrow \infty$.  It is now sufficient to consider
the $D$--dimensional submatrix, $\tilde{\cal M}$, obtained by
neglecting the 0-th row and column of ${\cal M}$. Using first order
perturbation theory, the coupling of $\tilde{\cal M}$ to the remaining
elements of ${\cal M}$ can then be treated exactly.  This will result
in a slightly different expression for $s_\mu$.  We obtain
\begin{equation}
s_\mu = \frac{1}{\langle n \rangle ^\mu} \sum_{k=1}^D a_k^2 \left(
\frac{{\tilde{\lambda}}_k}{1-{\tilde{\lambda}}_k} \right) ^\mu
\label{4.7.1}
\end{equation}
where the ${\tilde \lambda}_k$ are the eigenvalues of ${\tilde {\cal
M}}$ \cite{footnote9} and the $a_k^2$ are normalized coefficients
which follow from the eigenvectors of ${\tilde {\cal M}}$ as
\begin{equation}
{a_k} = {\cal N} \frac{1}{1 - {\tilde \lambda}_k } \sum_{i=1}^D
{\tilde \theta}_{ik} \ \ .
\label{4.7.2}
\end{equation}
Here, $\tilde{\theta}$ is the matrix that diagonalizes $\tilde{\cal
M}$ and ${\cal N}$ is chosen such that $\sum a_k^2 = 1$.

We now re--write Eq.~(\ref{4.1}) as
\begin{equation}
\sum_{k=1}^D a_{k}^2 \left( \frac{{{\tilde \lambda}}_{k}}
{1-{{\tilde \lambda}}_{k}} \right)^{\mu}
= \frac{[c \langle n \rangle]^{\mu}}{\mu + 1}  \ \ .
\label{4.2}
\end{equation}
The purpose of this re--writing is to emphasize that, in order to
reproduce the negative binomial factorial moments, it is necessary to
satisfy an infinite number of constraints, Eq.~(\ref{4.2}).  There are
only a finite number of parameters in the Hamiltonian for
finite $D$.  Specifically, there are $D(D+1)/2$ interaction parameters
to be specified \cite{footnote10}. The satisfaction of an infinite number
of constraints will evidently require the limit $D \rightarrow
\infty$.  (We shall return to this point below.)

It is interesting to note that in the $D \rightarrow \infty$ limit, we
can replace the summation in Eq.~(\ref{4.2}) by an integral in order
to obtain
\begin{equation}
\int_{-1}^{+1} d\lambda \ {a(\lambda)}^2
\left( \frac{\lambda}{1-\lambda} \right)^{\mu}
= \frac{[c \langle n \rangle]^{\mu}}{\mu + 1}
\label{4.3}
\end{equation}
where the coefficients ${a(\lambda)}^2$ now play the role of a
(non--negative) continuous weighting function.  Working with a new
variable
\begin{equation}
\eta = \frac{\lambda}{1-\lambda} \ \ ,
\label{4.4}
\end{equation}
we thus seek another weighting function, $W(\eta)$, such that
\begin{equation}
\int_{-1/2}^{+\infty} d\eta \ W(\eta) \  \eta^{\mu} =
\frac{[c \langle n \rangle]^{\mu}}{\mu + 1} \ \ .
\label{4.5}
\end{equation}
As it happens, this set of constraints allows us to determine
$W(\eta)$ by inspection. We simply obtain a step function,
\begin{equation}
W(\eta ) = \theta (c \langle n \rangle - \eta ) \theta( \eta) \ \ .
\label{4.6}
\end{equation}
This choice (barring questions of uniform convergence) is unique.  It
is readily transformed into a statement about the weighting function
${a(\lambda)}^2$.  We find ${a(\lambda)}^2 = 0$ except
\begin{equation}
{a(\lambda)}^2  =  \frac{1}{c\langle n \rangle(1 - \lambda)^2} \ \ \ \
{\rm for} \ \ 0 \le \lambda \le \frac{c\langle n \rangle}
{1+c\langle n \rangle} \ \ .
\label{4.7}
\end{equation}
This statement of the constraints will be extremely useful in finding a
scheme for picking the parameters in the Hamiltonian in order to
reproduce the negative binomial moments.

It is clear that the form of $\tilde{{\cal M}}$ as
given by Eq.~(\ref{3.1}) (for $d,d' \ne 0$) is somewhat special.
It is necessary to demonstrate
that there exists at least one scheme for picking the $D(D+1)/2$
parameters of $\tilde{{\cal M}}$ in such a way that the various
constraints for obtaining the negative binomial distribution are
satisfied.  Here, we shall simply propose two different prescriptions
and demonstrate by numerical example that these prescriptions indeed
yield the desired result.  Of course, we shall make no claims
regarding the uniqueness of our prescriptions.  The first prescription
is as follows:
\begin{quote}
Draw $D$ random numbers, $x_d$, from the interval $[0,1]$. Set
${\tilde {\cal M}}_{dd}$ equal to $x_d L$ where $L>0$.  Set all
off--diagonal elements of ${\tilde {\cal M}}$ equal to zero. For each
draw, choose $L$ such that $s_1 = c/2$.  (This sets the dispersion to
its empirical value for each draw.)
\end{quote}
The physical content of this prescription is clear. While identical
particles experience a nearest neighbor interaction which ranges from
$\approx - \mu_{\rm o}$ to $+\infty$, inequivalent particles experience a
nearest neighbor interaction in the range $-\mu_{\rm o} \ll
\epsilon_{dd'} \leq +\infty$.  The success of this prescription is
guaranteed by a comparison of Eqs.~(\ref{4.7.2}) and (\ref{4.7}).
Choosing ${\tilde {\cal M}}$ to be diagonal guarantees that the various
sums in Eq.~(\ref{4.7.2}) will be independent of $k$.  The
$\lambda$--dependence of Eq.~(\ref{4.7}) will be obtained from
Eq.~({\ref{4.7.2}) provided that the eigenvalues are distributed
uniformly over a finite, positive range.  This is ensured by our
prescription.


For our numerical studies, we consider the values of $\langle n
\rangle = 20$ and $\langle\Delta n^2 \rangle = 110$ appropriate for
the description of $p{\bar p}$ scattering at 200 GeV.  (For this
reaction, empirical values for the factorial moments are readily
available.  Qualitatively similar results are obtained for other
high--energy scattering experiments.)  These data indicate that we
should set
\begin{equation}
c \langle n \rangle = \frac{\langle \Delta n^2  \rangle - \langle n \rangle}
{\langle n \rangle} = 4.5 \ \ .
\label{4.8}
\end{equation}
We now construct the ratios
\begin{equation}
r_q = \frac{(q+1) \sum_{k=1}^D a_k^2 \left( \frac{{{\tilde
\lambda}}_{k}} {1-{{\tilde \lambda}}_{k}}\right)^{q}}{\left[c \langle
n\rangle \right]^q} \ \ .
\label{4.9}
\end{equation}
Our only constraint, $s_1=c/2$, ensures $r_1=1$.  When we have
satisfied the constraints of Eq.~(\ref{4.2}), all of these ratios
should be equal to one \cite{footnote11}.  Since there are no
parameters to adjust, this is an extremely stringent test of our
prescription.  It succeeds.  In Table~\ref{table1} we report the
results of our numerical studies for $2 \le q \le 6$ and $D = 1$, $2$,
$4$, \ldots , $512$. The notation `$\langle\langle r_q \rangle
\rangle$' is a reminder that, since our theory is now randomly drawn,
we must perform an `ensemble average' over theories as well as the
usual thermodynamic ensemble average.  For each value of $D$, this
ensemble average was performed over $10^5$ theories.

Several comments are in order.  Our prescription meets the remaining
conditions of Eq.~(\ref{4.1}) with increasing accuracy as $D
\rightarrow \infty$.  For fixed $q$, the value of $\langle\langle r_q
\rangle\rangle$ approaches $1$ like $1/D$ as $D \rightarrow \infty$.
The dispersion also vanishes (like $1/\sqrt{D}$).  Thus, as $D$
becomes large, our simple prescription converges to the results of the
NB for any fixed $q$.  For fixed $D$, the error in $\langle\langle r_q
\rangle\rangle$ and its dispersion grow as $q \rightarrow \infty$.
Our point here is that there exists at least one
simple prescription for satisfying Eq.~(\ref{4.1}) exactly.

Given the limited scope and quality of existing data, we do not need
particularly large $D$ to fit the relations between measured factorial
moments.  The value of $D = 16$ results in sufficiently small errors
and dispersions that randomly drawn dynamics have a high probability
of reproducing the relations between empirical factorial moments for
$p{\bar p}$ scattering at 200 GeV within existing experimental
uncertainties.  This value of $D=16$ is also sufficient to provide a
quantitative description of relations between factorial moments for
900 GeV $p{\bar p}$ scattering and, indeed, of all other high--energy
scattering experiments for which factorial moments are known.  To
demonstrate this, Fig.~\ref{fig1} shows the factorial moments $F_3$ to
$F_5$ for $p{\bar p}$ scattering at 200, 546 and 900 GeV as a function
of $F_2$.  We have suppressed the model--independent constant and
linear pieces of these moments, {\em i.e.}, we plot
\begin{equation}
{\tilde F}_q(F_2)=F_q(F_2)-1-\frac{q(q-1)}{2}(F_2-1) \ \ .
\label{4.10}
\end{equation}
The theoretical `bands' shown on each figure correspond to the
theoretical dispersion obtained with $D=16$.  (More than 50\% of all
randomly drawn theories lie inside this band.)  These bands were
obtained with parameters chosen to reproduce the 200 GeV data.  The
use of parameters chosen to reproduce the higher energy data would
result in somewhat narrower bands that are slightly shifted downwards.

The error bars in Fig.~\ref{fig1} represent the published
uncertainties and have roughly equal statistical and systematic
components.  Some comment regarding the statistical error is in order.
Since $F_2 (M)$ and $F_q (M)$ are drawn from the {\em same\/} data
set, significant statistical correlations can exist.  Unfortunately,
it is impossible to make a quantitative statement about such
correlations without a detailed investigation of the data reduction
techniques actually used.  We have, however, performed simulations
based on a suitably modified form of the negative binomial
distribution. The results suggest that the uncertainties shown in
Fig.~\ref{fig1} actually represent a significant overestimate of the
real uncertainties in the determination of $F_q (F_2 )$.  This belief
is also supported by the observation that the data points appear to
have a far smaller spread than their errors would indicate.  A
reduction of these errors by a factor of approximately $3$ would make
it possible to provide a convincing discrimination between the
negative binomial distribution and the Ising model distribution. Our
simulations suggest that an error reduction of this magnitude is not
out of the question.

This first prescription has the virtues of simplicity and guaranteed
success.  It can, however, be criticized on the grounds that it has
singled out the species--diagonal interactions, $\epsilon_{dd}$, for
special treatment.  We thus consider a second and more democratic
prescription in which all pairs of species are treated equivalently:
\begin{quote}
Draw the various terms $\epsilon_{dd'}$ at random over the uniform
interval
\begin{displaymath}
\epsilon_{\rm o} - \mu_{\rm o} \le \epsilon_{dd'} \le \epsilon_{\rm o}
- \mu_{\rm o} + \Delta \epsilon \ \ .
\end{displaymath}
For each choice of ${\tilde {\cal M}}$, adjust $\epsilon_{\rm o}$ in order
to reproduce the desired value of $c$.  (This can always be done.)
Adjust $\Delta \epsilon$ to reproduce the value $r_2 = 1$.  In the
event that this cannot be done, discard the draw \cite{footnote12}.
\end{quote}
This more democratic selection of ${\tilde {\cal M}}$ means that we must
actually perform the requisite diagonalizations.  This means that one
must be content with studying a smaller number of samples.  More
seriously, it is no longer elementary to establish rules which guarantee
the desired equivalence between Eqs.~(\ref{4.7.2}) and (\ref{4.7}).
It is for this reason that we have imposed the somewhat artificial
constraint that $r_2=1$.

The results of numerical studies with this second prescription are
summarized in Table~\ref{table2} for $D = 32$, $64$, $128$ and $256$.
For each value of $D$ we considered $10^{3}$ matrices.  In this case,
we plot the average values (and the corresponding dispersions) for
$\epsilon_{\rm o}$ and $\Delta \epsilon$ as well as the various ratios
$r_q$ for $3 \le q \le 6$.  The large values of $\Delta \epsilon$
required indicate that we are dealing with sparse matrices.  This has
two apparently negative consequences.  First, there will be a great
many eigenvalues close to zero.  Second, there will be only a small
preference for positive eigenvalues \cite{footnote13}. These facts
would appear to make it difficult to satisfy Eq.~(\ref{4.7}). At the
very least, we anticipate much slower convergence (as a function of
$D$) when using this second prescription.  On the other hand, the
eigenvectors associated with negative eigenvalues necessarily involve
terms of mixed signs so that there will be a greater tendency for
cancellation in the sums over ${\tilde \theta}_{ik}$ in
Eq.~(\ref{4.7.2}) for negative eigenvalues.

Somewhat surprisingly, these effects conspire to a remarkable degree.
The numerical results of Table~\ref{table2} suggest that our second
prescription also leads to the factorial moments of the negative
binomial distribution in the large $D$ limit.  This claim of success
is necessarily somewhat tentative.  The slower convergence of this
prescription suggests the need to study values of $D$ larger than
those in Table~\ref{table1}.  The need to diagonalize many large
matrices restricts us to smaller values of $D$ and smaller statistical
samples.  We have performed other tests which also suggest that this
second prescription works.  For example, we have made histograms of
the $a^{2}_{k}$ of Eq.~(\ref{4.7.2}) as a function of the variable
$\eta$ defined above.  Except for the expected `Gibbs phenomenon', we
find remarkable agreement with the distribution of Eq.~(\ref{4.6}) as
$D$ becomes large.  We have also studied the cases $D = 512$ and
$1028$ with very poor statistics.  These results are also consistent
with our assertion that the factorial moments of the negative binomial
distribution will result in the large $D$ limit.

The purpose of this second prescription has been to demonstrate the
existence of at least one democratic way to reproduce the factorial
moments of the negative binomial distribution.  More efficient schemes
may well exist.


\section{Discussions and Conclusions}
\label{sect5}

We have demonstrated that a simple extension of the one--dimensional
lattice gas model to include nearest neighbor interactions between
pairs of $D$ species of particles can reproduce the factorial moments
of the negative binomial distribution exactly provided that a number
of constraints are met.  We have further demonstrated that these
constraints can be met by simple models of the microscopic dynamics in
which all particles have the same chemical potential and in which the
nearest neighbor interactions are drawn at random according to the
prescriptions of the previous section.

Since our model approaches the negative binomial distribution as $D
\rightarrow \infty$, it will reproduce the relations between empirical
factorial moments including those obtained in $p{\bar p}$ scattering,
$e^+ e^-$ scattering and relativistic heavy ion collisions.  The only
question is whether the number of species required to provide an
adequate fit is sufficiently small to be considered `physically
reasonable'.  Given the results of Fig.~\ref{fig1} for $16$ species,
we believe that the present model is physically reasonable.  These
results also offer some understanding for both the success of cascade
calculations and the anecdotal observation that the results of
calculations are often surprisingly insensitive to the details of the
model.  According to our picture, any randomly drawn theory would be
likely to do as well (at least at the level of the factorial moments).

It has been suggested that the slopes, $\alpha_q / \alpha_2$, in plots
of $\ln{F_q}$ versus $\ln{F_2}$ can be used to distinguish between
cascade systems and systems exhibiting some critical phenomenon such
as the transition to a quark-gluon plasma \cite{Ochs}. It has been
shown that systems at the critical temperature of a second--order
phase transition display monofractal structure with $\alpha_q /
\alpha_2 \sim (q-1)$ \cite{Satz}. In contrast, cascade models lead
to a multifractal structure with $\alpha_q / \alpha_2 \sim q(q-1)$ (in
Gaussian approximation) \cite{Bialas2}. Our model (along with the
Ising model) shows that the monofractal structure of a critical
phenomenon can also be obtained from a simple but heterogeneous
(equilibrium) system. It indicates that some caution is required
before claiming insight into the nature of microscopic mechanisms on
such a basis. Indeed, the present model may be a useful test case for
proposed schemes for the detection of critical phenomena from the
study of, {\em e.g.}, factorial moments.

Before closing, we would like to offer one speculative remark
regarding the existence (or possible non--existence) of signatures
which would honestly discriminate between `interesting' critical
systems and those `uninteresting' heterogeneous systems considered
here with which they can be confused.  We suspect that the study of
global properties --- such as the factorial moments --- can only
provide convincing tests of criticality for systems whose time
evolution can be studied.  The present results suggest that it may
always be possible to devise equilibrium, heterogeneous models which
can simulate the manifestations of critical phenomena to arbitrary
accuracy in systems where the only information available is that
provided at $t = \infty$.  Evidently, this class of systems includes
all scattering experiments in nuclear and high--energy physics.  The
only way to distinguish between these classes of models is to provide
{\em independent\/} arguments that the heterogeneous alternatives to
critical systems are `too artificial' or `too unphysical' to be
accepted.  Given the present results, we suspect that this case cannot
be made.  Thus, in physics as in the rest of life, much more
information is to be gained from the study of `living' systems than
from performing autopsies on `dead' systems.


\acknowledgments

We would like to acknowledge the hospitality of NORDITA during the
summer of 1993.  We would also like to thank Nandor Balazs, Kim
Sneppen, and Peter Orland for helpful discussions.  This work was
partially supported by the U.S.  Department of Energy under grant no.
\mbox{DE-FG02-88ER 40388}.


\appendix

\section{Simplifying Factorial Moments}
\label{app1}

Consider any model for which the number operator at any site $i$ is an
idempotent:
\begin{equation}
n_i^2 = n_i \ \ .
\label{a.1}
\end{equation}
Such models include either the ordinary lattice gas model of
Sec.~\ref{sect2} or the generalized lattice gas model of
Sec.~\ref{sect3}.  Define the number operator for a single bin,
\begin{equation}
n = \sum n_i \ \ ,
\label{a.2}
\end{equation}
where $i$ runs over every site in the bin.

Now assume that we know that the following operator identity holds for
some fixed order $q$:
\begin{equation}
n(n-1)(n-2) \ldots (n-q+1) = \sum \protect\raisebox{1.2ex}{\small
$\prime$} \, n_{i_1} n_{i_2} n_{i_3} \ldots n_{i_q} \ \ .
\label{a.3}
\end{equation}
The sum extends over all sites in a given bin.  The prime denotes that
no two of the site indices, $i_1$ to $i_q$, are equal.  Note that
Eq.~(\ref{a.3}) applies at the operator level before ensemble averages
are carried out.  Our aim is to show that, if Eq.~(\ref{a.3}) applies
for $q$, it necessarily applies for $q+1$ as well.  To see this,
multiply Eq.~(\ref{a.3}) by the number operator, $n$.  This leads to
\begin{equation}
n[n(n-1)(n-2) \ldots (n-q+1)] = \sum \protect\raisebox{1.2ex}{\small
$\prime$} \, n_{i_1} n_{i_2} n_{i_3} \ldots n_{i_q} n_{i_{q+1}} + q
\sum \protect\raisebox{1.2ex}{\small $\prime$} \, n_{i_1} n_{i_2}
n_{i_3} \ldots n_{i_q} \ \ .
\label{a.4}
\end{equation}
The second term here simply corresponds to the fact that the index
$i_{q+1}$ can be equal to any one of the $q$ indices $i_1$ to $i_q$.
Using Eq.~(\ref{a.3}), we can move the final term in Eq.~(\ref{a.4})
to the left of the equation to obtain
\begin{equation}
n(n-1)(n-2) \ldots (n-q+1)(n-q) = \sum \protect\raisebox{1.2ex}
{\small $\prime$} \, n_{i_1} n_{i_2} n_{i_3} \ldots n_{i_q}
n_{i_{q+1}} \ \ .
\label{a.5}
\end{equation}
This is the desired extension of the result.  Thus, Eq.~(\ref{a.3})
will apply for all $q$ if it applies for $q=1$.  It applies trivially
for $q=1$ since the primed sum is of no meaning in this case (there
being only one term in the product) and Eq.~(\ref{a.3}) collapses to
the definition, Eq.~(\ref{a.2}).

Given the form in which the various correlation functions can be
expressed in our models, one final rearrangement of Eq.~(\ref{a.3}) is
useful.  Specifically,
\begin{equation}
n(n-1)(n-2) \ldots (n-q+1) = q! \sum_{i_1 < i_2 < \ldots < i_q}
n_{i_1} n_{i_2} n_{i_3} \ldots n_{i_q} \ \ .
\label{a.6}
\end{equation}

The final result is that Eqs.~(\ref{a.3}) and (\ref{a.6}) apply for
all $q$ for all problems in which the number operator is idempotent.
Further, one can take ensemble averages of these operator identities,
and the ensemble averages can be taken under the summations.  These
results provide a considerable technical simplification in the
evaluation of factorial moments.



\section{Special Cases of the Generalized Lattice Gas Model}
\label{app2}

Here, we wish to reduce the factorial moments of our generalized
lattice gas model to (i) the factorial moments of the ordinary lattice
gas model and (ii) the factorial moments of the negative binomial
distribution by simplifying the general expressions,
Eqs.~(\ref{3.15})--(\ref{3.17}), with the use of special assumptions
regarding the choice of the terms $s_{\mu}$.

Consider, first, the ordinary lattice gas results.  The purpose here is
an exercise in method since we already know the result to be true.
Set the various $s_{\mu}$ as
\begin{equation}
s_{\mu} = \left( \frac{c}{2} \right) ^{\mu} \ \ .
\label{b.1}
\end{equation}
We can then perform the sum in the square brackets in Eq.~(\ref{3.16})
exactly to yield
\begin{equation}
1 + \sum_{\mu=1}^{\infty} z^{\mu} s_{\mu} = \frac{1}{1-zc/2} \ \ .
\label{b.2}
\end{equation}
The desired derivative is immediately obtained as
\begin{equation}
\left[ \frac{d^k}{dz^k} \left( \frac{1}{1-zc/2} \right)^{(q-k)}
\right]_{z=0} = (q-k)(q-k+1) \ldots (q -1) \left( \frac{c}{2} \right)
^k = \frac{(q-1)!}{(q-k-1)!} \left( \frac{c}{2} \right) ^k \ \ .
\label{b.3}
\end{equation}
This immediately gives
\begin{equation}
f_k^{(q)} = \frac{q!(q-1)!}{(q-k)!k!(q-1-k)! 2^k} c^k \ \ .
\label{b.4}
\end{equation}
Comparison with Eq.~(\ref{1.4}) reveals the expected agreement.

For the case of the negative binomials, it is more convenient to start
with the generalized generating function of Eq.~(\ref{3.20}). We now
wish to set
\begin{equation}
s_{\mu} = \frac{c^{\mu}}{\mu+1} \ \ .
\label{b.5}
\end{equation}
The sum over $\mu$ can again be performed exactly to yield
\begin{equation}
1+\sum_{\mu=1}^{\infty} z^{\mu} s_{\mu} = -\frac{\ln{[1-zc]}}{zc} \ \ .
\label{b.6}
\end{equation}
Substitution of this expression into Eq.~(\ref{3.20}) yields
\begin{eqnarray}
F_q(M) & = & M^q \left[ \frac{d^q}{dz^q} \exp{\left( -
\frac{\ln{[1-zc]}} {cM} \right)} \right]_{z=0} \nonumber \\
 & = & (cM)^q \left[ \frac{d^q}{dy^q} (1-y)^{-\frac{1}{cM}}
\right]_{y=0} \nonumber \\
 & = & (cM)^q \left( \frac{1}{cM} \right) \left( \frac{1}{cM} + 1 \right)
\ldots \left( \frac{1}{cM} + [q-1] \right) \nonumber \\
 & = & (1+cM)(1+2cM) \ldots (1+[q-1]cM)
\label{b.7}
\end{eqnarray}
which is seen to be identical with the NB result of Eq.~(\ref{1.2}).


\begin{references}

\bibitem{ALEPH} ALEPH Collaboration, Z. Phys. C {\bf 53}, 21 (1992).
\bibitem{UA5} UA5 Collaboration, Phys. Rep. {\bf 154}, 247 (1987); Z. Phys.
  C {\bf 43}, 357 (1989).
\bibitem{KLM} KLM Collaboration, R. Holynski {\em et al.\/}, Phys. Rev.
  C {\bf 40}, 2449 (1989).
\bibitem{Bialas1} A. Bialas and R. Peschanski, Nucl. Phys. {\bf B273}, 703
  (1986).
\bibitem{Giovannini} A. Giovannini and L. Van Hove, Z. Phys. C {\bf 30},
  391 (1986).
\bibitem{footnote1} A reasonably successful description can be
  obtained with the replacement of $M$ by $M^{\alpha}$ in
  Eq.~(\ref{1.2}) with $\alpha$ on the order of $1/5$.
\bibitem{Carruthers1} P. Carruthers, H. C. Eggers and I. Sarcevic,
  Phys. Lett. B {\bf 254}, 258 (1991).
\bibitem{footnote2} While such plots are again theoretically motivated,
  they do not introduce theoretical bias into what remains a purely
  phenomenological activity.
\bibitem{Ochs} W. Ochs, Z. Phys. C {\bf 50}, 339 (1991).
\bibitem{Carruthers2} For a review, see P. Carruthers and C. C. Shih, Int.
  J. Mod. Phys.  A {\bf 2}, 1447 (1987) and references contained therein.
\bibitem{Satz} H. Satz, Nucl. Phys. {\bf B326}, 613 (1989).
\bibitem{Bialas2} A. Bialas and R. Peschanski, Nucl. Phys. {\bf B308}, 857
  (1988).
\bibitem{Chau} L. L. Chau and D. W. Huang, Phys. Lett. B {\bf 283}, 1 (1992);
  Phys. Rev. Lett. {\bf 70}, 3380 (1993).
\bibitem{footnote3} This idea was first suggested by Dias de Deus and
  Seixas who studied the lattice gas model numerically. See J. Dias de Deus
  and J. C. Seixas, Phys. Lett. B {\bf 246}, 506 (1990); J. C. Seixas, in
  {\em Intermittency in high energy collisions}, edited by F. Cooper, R. C.
  Hwa and I. Sarcevic (World Scientific, Singapore, 1991).
\bibitem{JWB} For a brief overview, see A. D. Jackson, T. Wettig and N. L.
  Balazs, to be published.
\bibitem{footnote4} Here and in the following, we will set the inverse
  temperature, $\beta$, equal to $1$ without loss of generality.
\bibitem{footnote5} Here, the subscript $d$ means diagonal and is
  not to be confused with species $d$.
\bibitem{footnote6} This is the only real complication relative to the
  $D=1$ lattice gas model of the previous section.
\bibitem{footnote7} Again, we are taking the $N \rightarrow \infty$ limit
  for fixed $\langle n \rangle$ for which it is justified to approximate
  $1-\langle n_i \rangle$ by $1$.
\bibitem{footnote8} Given the absence of long--range order in
  one--dimensional systems, this requirement is readily satisfied.
\bibitem{footnote9} We consider only the physically interesting case where
  $|{{\tilde \lambda}_{k}}| < 1$ for all $k$.
\bibitem{footnote10} The chemical potentials have already been
  set equal to a common $\mu_{\rm o}$.
\bibitem{footnote11}  For the ordinary lattice gas, $D=1$, these ratios
  are simply $(q+1)/2^q$.
\bibitem{footnote12} This possibility becomes increasingly unlikely as
  $D$ becomes large.
\bibitem{footnote13} Since ${\tilde {\cal M}}$ contains only positive
  elements, we are guaranteed that some preference for positive
  eigenvalues will remain.  Further, if the eigenvalue of largest
  magnitude is non--degenerate, it will be positive.
\end{references}


\begin{figure}
\caption{Plots of ${\tilde F}_q$ as defined in Eq.~(\protect\ref{4.10})
versus $F_2$ for $3 \le q \le 5$.  Data are taken from
Ref.~\protect\cite{UA5}. The dashed line corresponds to the
predictions of the ordinary lattice gas (Ising) model. The bold face
line represents the negative binomial distribution. The shaded area is
the prediction of the generalized lattice gas model according to our
first prescription for $D=16$.  This `band' around the NB becomes
narrower as $D$ increases.  The error bars shown are representative
for the quoted uncertainties in the data (see the text for further
comments on the error bars).}
\label{fig1}
\end{figure}

\begin{table}
\caption{The ensemble average and dispersion of $r_q$ according to
our first prescription for $2 \le q \le 6$ and various values of the
number of particle species, $D$. The average was taken over $10^5$
randomly drawn theories with $c \langle n \rangle = 4.5$ for each
draw.  The case $D=1$ corresponds to the ordinary lattice gas (Ising)
model \protect\cite{Chau} and has no dispersion.}
\label{table1}
\begin{tabular}{rr@{${}\pm{}$}lr@{${}\pm{}$}lr@{${}\pm{}$}l
                 r@{${}\pm{}$}lr@{${}\pm{}$}l}
 \multicolumn{1}{c}{$D$} &
 \multicolumn{2}{c}{$\langle\langle r_2 \rangle\rangle$} &
 \multicolumn{2}{c}{$\langle\langle r_3 \rangle\rangle$} &
 \multicolumn{2}{c}{$\langle\langle r_4 \rangle\rangle$} &
 \multicolumn{2}{c}{$\langle\langle r_5 \rangle\rangle$} &
 \multicolumn{2}{c}{$\langle\langle r_6 \rangle\rangle$} \\ \tableline
   1 & \multicolumn{2}{c}{0.75} & \multicolumn{2}{c}{0.5} &
       \multicolumn{2}{c}{0.3125} & \multicolumn{2}{c}{0.1875} &
       \multicolumn{2}{c}{0.109375} \\
   2 & 0.812 & 0.023 & 0.598 & 0.035 & 0.417 & 0.037 & 0.280 & 0.033 &
       0.184 & 0.027 \\
   4 & 0.881 & 0.046 & 0.723 & 0.078 & 0.569 & 0.091 & 0.435 & 0.091 &
       0.326 & 0.083 \\
   8 & 0.941 & 0.067 & 0.851 & 0.132 & 0.751 & 0.179 & 0.652 & 0.207 &
       0.558 & 0.218 \\
  16 & 0.981 & 0.076 & 0.951 & 0.173 & 0.916 & 0.270 & 0.879 & 0.360 &
       0.842 & 0.441 \\
  32 & 0.998 & 0.069 & 0.999 & 0.170 & 1.005 & 0.295 & 1.017 & 0.442 &
       1.039 & 0.612 \\
  64 & 1.002 & 0.050 & 1.007 & 0.128 & 1.019 & 0.229 & 1.040 & 0.355 &
       1.071 & 0.515 \\
 128 & 1.001 & 0.035 & 1.005 & 0.088 & 1.013 & 0.154 & 1.026 & 0.233 &
       1.044 & 0.327 \\
 256 & 1.001 & 0.024 & 1.003 & 0.061 & 1.007 & 0.105 & 1.013 & 0.155 &
       1.023 & 0.210 \\
 512 & 1.000 & 0.017 & 1.002 & 0.042 & 1.004 & 0.072 & 1.007 & 0.106 &
       1.012 & 0.142 \\
\end{tabular}
\end{table}


\begin{table}
\caption{The ensemble average and dispersion of $r_q$ according to our
second prescription for $3 \le q \le 6$ and various values of the
number of particle species, $D$. The average was taken over $10^3$
randomly drawn theories with $c \langle n \rangle = 4.5$ for each
draw. Also shown are average value and dispersion of $\epsilon_{\rm o}$
and $\delta \epsilon$.}
\label{table2}
\begin{tabular}{rr@{${}\pm{}$}lr@{${}\pm{}$}lr@{${}\pm{}$}l
                 r@{${}\pm{}$}lr@{${}\pm{}$}lr@{${}\pm{}$}l}
 \multicolumn{1}{c}{$D$} & \multicolumn{2}{c}{$\epsilon_{\rm o}$} &
 \multicolumn{2}{c}{$\delta \epsilon$} &
 \multicolumn{2}{c}{$\langle\langle r_3 \rangle\rangle$} &
 \multicolumn{2}{c}{$\langle\langle r_4 \rangle\rangle$} &
 \multicolumn{2}{c}{$\langle\langle r_5 \rangle\rangle$} &
 \multicolumn{2}{c}{$\langle\langle r_6 \rangle\rangle$} \\ \tableline
  32 & 0.21 & 0.37 & 114 & 75 & 0.945 & 0.025 & 0.862 & 0.059
     & 0.769 & 0.094 & 0.674 & 0.124 \\
  64 & 0.47 & 0.23 & 149 & 63 & 0.959 & 0.022 & 0.895 & 0.054
     & 0.820 & 0.089 & 0.741 & 0.122 \\
 128 & 0.62 & 0.18 & 223 & 71 & 0.973 & 0.021 & 0.930 & 0.054
     & 0.876 & 0.093 & 0.818 & 0.133 \\
 256 & 0.73 & 0.15 & 363 & 85 & 0.988 & 0.020 & 0.967 & 0.054
     & 0.941 & 0.098 & 0.910 & 0.148 \\
\end{tabular}
\end{table}



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74 658 M 74 680 L
74 693 M 74 715 L
29 751 M 32 741 L
39 735 L
49 731 L
55 731 L
65 735 L
71 741 L
74 751 L
74 757 L
71 767 L
65 773 L
55 776 L
49 776 L
39 773 L
32 767 L
29 757 L
29 751 L
32 744 L
39 738 L
49 735 L
55 735 L
65 738 L
71 744 L
74 751 L
74 757 M 71 763 L
65 770 L
55 773 L
49 773 L
39 770 L
32 763 L
29 757 L
29 802 M 74 802 L
29 805 M 74 805 L
39 805 M 32 812 L
29 821 L
29 828 L
32 837 L
39 840 L
74 840 L
29 828 M 32 834 L
39 837 L
74 837 L
29 792 M 29 805 L
74 792 M 74 815 L
74 828 M 74 850 L
7 873 M 74 873 L
7 876 M 74 876 L
7 863 M 7 876 L
74 863 M 74 885 L
7 908 M 10 905 L
13 908 L
10 911 L
7 908 L
29 908 M 74 908 L
29 911 M 74 911 L
29 898 M 29 911 L
74 898 M 74 921 L
29 943 M 74 943 L
29 946 M 74 946 L
39 946 M 32 953 L
29 962 L
29 969 L
32 978 L
39 982 L
74 982 L
29 969 M 32 975 L
39 978 L
74 978 L
29 934 M 29 946 L
74 934 M 74 956 L
74 969 M 74 991 L
49 1011 M 49 1049 L
42 1049 L
36 1046 L
32 1043 L
29 1036 L
29 1027 L
32 1017 L
39 1011 L
49 1007 L
55 1007 L
65 1011 L
71 1017 L
74 1027 L
74 1033 L
71 1043 L
65 1049 L
49 1046 M 39 1046 L
32 1043 L
29 1027 M 32 1020 L
39 1014 L
49 1011 L
55 1011 L
65 1014 L
71 1020 L
74 1027 L
36 1075 M 39 1075 L
39 1072 L
36 1072 L
32 1075 L
29 1081 L
29 1094 L
32 1100 L
36 1104 L
42 1107 L
65 1107 L
71 1110 L
74 1113 L
36 1104 M 65 1104 L
71 1107 L
74 1113 L
74 1116 L
42 1104 M 45 1100 L
49 1081 L
52 1072 L
58 1068 L
65 1068 L
71 1072 L
74 1081 L
74 1091 L
71 1097 L
65 1104 L
49 1081 M 52 1075 L
58 1072 L
65 1072 L
71 1075 L
74 1081 L
29 1139 M 74 1139 L
29 1142 M 74 1142 L
49 1142 M 39 1145 L
32 1152 L
29 1158 L
29 1168 L
32 1171 L
36 1171 L
39 1168 L
36 1165 L
32 1168 L
29 1129 M 29 1142 L
74 1129 M 74 1152 L
7 1245 M 61 1245 L
71 1248 L
74 1254 L
74 1261 L
71 1267 L
65 1271 L
7 1248 M 61 1248 L
71 1251 L
74 1254 L
29 1235 M 29 1261 L
49 1290 M 49 1328 L
42 1328 L
36 1325 L
32 1322 L
29 1315 L
29 1306 L
32 1296 L
39 1290 L
49 1287 L
55 1287 L
65 1290 L
71 1296 L
74 1306 L
74 1312 L
71 1322 L
65 1328 L
49 1325 M 39 1325 L
32 1322 L
29 1306 M 32 1299 L
39 1293 L
49 1290 L
55 1290 L
65 1293 L
71 1299 L
74 1306 L
29 1354 M 74 1354 L
29 1357 M 74 1357 L
49 1357 M 39 1360 L
32 1367 L
29 1373 L
29 1383 L
32 1386 L
36 1386 L
39 1383 L
36 1380 L
32 1383 L
29 1344 M 29 1357 L
74 1344 M 74 1367 L
29 1408 M 74 1408 L
29 1412 M 74 1412 L
39 1412 M 32 1418 L
29 1428 L
29 1434 L
32 1444 L
39 1447 L
74 1447 L
29 1434 M 32 1441 L
39 1444 L
74 1444 L
39 1447 M 32 1453 L
CS M
29 1463 L
29 1469 L
32 1479 L
39 1482 L
74 1482 L
29 1469 M 32 1476 L
39 1479 L
74 1479 L
29 1399 M 29 1412 L
74 1399 M 74 1421 L
74 1434 M 74 1457 L
74 1469 M 74 1492 L
36 1540 M 29 1543 L
42 1543 L
36 1540 L
32 1537 L
29 1530 L
29 1518 L
32 1511 L
36 1508 L
42 1508 L
45 1511 L
49 1518 L
55 1534 L
58 1540 L
61 1543 L
39 1508 M 42 1511 L
45 1518 L
52 1534 L
55 1540 L
58 1543 L
68 1543 L
71 1540 L
74 1534 L
74 1521 L
71 1514 L
68 1511 L
61 1508 L
74 1508 L
68 1511 L
29 1633 M 32 1624 L
39 1617 L
49 1614 L
55 1614 L
65 1617 L
71 1624 L
74 1633 L
74 1640 L
71 1649 L
65 1656 L
55 1659 L
49 1659 L
39 1656 L
32 1649 L
29 1640 L
29 1633 L
32 1627 L
39 1620 L
49 1617 L
55 1617 L
65 1620 L
71 1627 L
74 1633 L
74 1640 M 71 1646 L
65 1652 L
55 1656 L
49 1656 L
39 1652 L
32 1646 L
29 1640 L
10 1701 M 13 1697 L
16 1701 L
13 1704 L
10 1704 L
7 1701 L
7 1694 L
10 1688 L
16 1684 L
74 1684 L
7 1694 M 10 1691 L
16 1688 L
74 1688 L
29 1675 M 29 1701 L
74 1675 M 74 1697 L
7 1778 M 74 1778 L
7 1781 M 74 1781 L
26 1800 M 52 1800 L
7 1768 M 7 1819 L
26 1819 L
7 1816 L
39 1781 M 39 1800 L
74 1768 M 74 1790 L
72 1833 M 74 1835 L
76 1833 L
74 1831 L
72 1831 L
68 1833 L
66 1835 L
64 1841 L
64 1849 L
66 1855 L
70 1856 L
76 1856 L
80 1855 L
82 1849 L
82 1843 L
64 1849 M 66 1853 L
70 1855 L
76 1855 L
80 1853 L
82 1849 L
84 1853 L
88 1856 L
91 1858 L
97 1858 L
101 1856 L
103 1855 L
105 1849 L
105 1841 L
103 1835 L
101 1833 L
97 1831 L
95 1831 L
93 1833 L
95 1835 L
97 1833 L
86 1855 M 91 1856 L
97 1856 L
101 1855 L
103 1853 L
105 1849 L
CS [] 0 setdash M
CS [] 0 setdash M
1206 86 M 1222 153 M 1222 86 L
1226 153 M 1226 86 L
1245 134 M 1245 108 L
1213 153 M 1264 153 L
1264 134 L
1261 153 L
1226 121 M 1245 121 L
1213 86 M 1235 86 L
1278 88 M 1280 86 L
1278 84 L
1276 86 L
1276 88 L
1278 92 L
1280 93 L
1286 95 L
1294 95 L
1299 93 L
1301 92 L
1303 88 L
1303 84 L
1301 80 L
1296 76 L
1286 72 L
1282 70 L
1278 66 L
1276 61 L
1276 55 L
1294 95 M 1297 93 L
1299 92 L
1301 88 L
1301 84 L
1299 80 L
1294 76 L
1286 72 L
1276 59 M 1278 61 L
1282 61 L
1292 57 L
1297 57 L
1301 59 L
1303 61 L
1282 61 M 1292 55 L
1299 55 L
1301 57 L
1303 61 L
1303 65 L
CS [] 0 setdash M
409 542 M 426 547 L
443 551 L
460 555 L
477 560 L
494 565 L
511 570 L
528 575 L
545 580 L
562 585 L
579 590 L
596 596 L
613 601 L
630 607 L
647 612 L
664 618 L
681 624 L
698 630 L
715 637 L
731 643 L
748 650 L
765 656 L
782 663 L
799 670 L
816 677 L
833 684 L
850 691 L
867 698 L
884 705 L
901 713 L
918 720 L
935 728 L
952 736 L
969 744 L
986 752 L
1003 760 L
1020 769 L
1037 777 L
1054 786 L
1071 794 L
1088 803 L
1105 812 L
1122 821 L
1139 830 L
1156 839 L
1173 848 L
1190 858 L
1207 868 L
1224 877 L
1241 887 L
1258 897 L
1275 907 L
1292 917 L
1309 927 L
1325 938 L
1342 948 L
1359 959 L
1376 970 L
1393 981 L
1410 992 L
1427 1003 L
1444 1014 L
1461 1025 L
1478 1037 L
1495 1048 L
1512 1060 L
1529 1072 L
1546 1083 L
1563 1095 L
1580 1108 L
1597 1120 L
1614 1132 L
1631 1145 L
1648 1157 L
1665 1170 L
1682 1183 L
1699 1196 L
1716 1209 L
1733 1222 L
1750 1235 L
1767 1249 L
1784 1262 L
1801 1276 L
1818 1289 L
1835 1303 L
1852 1317 L
1869 1331 L
1886 1346 L
1903 1360 L
1920 1374 L
1936 1389 L
1953 1403 L
1970 1418 L
1987 1433 L
2004 1448 L
2021 1463 L
2038 1479 L
2055 1494 L
2072 1509 L
2089 1525 L
2106 1541 L
409 555 M 426 560 L
443 565 L
460 570 L
477 576 L
494 581 L
511 587 L
528 593 L
545 599 L
562 605 L
579 611 L
596 617 L
613 624 L
630 630 L
647 637 L
664 644 L
681 651 L
698 658 L
715 665 L
731 673 L
748 680 L
765 688 L
782 696 L
799 704 L
816 712 L
CS M
833 720 L
850 728 L
867 737 L
884 746 L
901 754 L
918 763 L
935 772 L
952 781 L
969 791 L
986 800 L
1003 810 L
1020 819 L
1037 829 L
1054 839 L
1071 849 L
1088 860 L
1105 870 L
1122 880 L
1139 891 L
1156 902 L
1173 913 L
1190 924 L
1207 935 L
1224 946 L
1241 958 L
1258 969 L
1275 981 L
1292 993 L
1309 1005 L
1325 1017 L
1342 1029 L
1359 1042 L
1376 1054 L
1393 1067 L
1410 1080 L
1427 1093 L
1444 1106 L
1461 1119 L
1478 1132 L
1495 1146 L
1512 1159 L
1529 1173 L
1546 1187 L
1563 1201 L
1580 1215 L
1597 1230 L
1614 1244 L
1631 1259 L
1648 1273 L
1665 1288 L
1682 1303 L
1699 1318 L
1716 1333 L
1733 1349 L
1750 1364 L
1767 1380 L
1784 1396 L
1801 1412 L
1818 1428 L
1835 1444 L
1852 1460 L
1869 1477 L
1886 1493 L
1903 1510 L
1920 1527 L
1936 1544 L
1953 1561 L
1970 1578 L
1987 1596 L
2004 1613 L
2021 1631 L
2038 1649 L
2055 1667 L
2072 1685 L
2089 1703 L
2106 1721 L
409 546 M 426 546 L
499 583 M 556 583 L
601 619 M 667 619 L
692 656 M 765 656 L
775 692 M 854 692 L
851 729 M 937 729 L
922 765 M 1013 765 L
989 802 M 1086 802 L
1052 838 M 1154 838 L
1113 875 M 1219 875 L
1170 911 M 1282 911 L
1226 948 M 1341 948 L
1279 984 M 1399 984 L
1330 1021 M 1454 1021 L
1380 1057 M 1508 1057 L
1428 1094 M 1560 1094 L
1475 1130 M 1611 1130 L
1521 1166 M 1660 1166 L
1565 1203 M 1708 1203 L
1609 1239 M 1755 1239 L
1651 1276 M 1801 1276 L
1692 1312 M 1846 1312 L
1733 1349 M 1889 1349 L
1772 1385 M 1932 1385 L
1811 1422 M 1974 1422 L
1850 1458 M 2016 1458 L
1887 1495 M 2056 1495 L
1924 1531 M 2096 1531 L
1960 1568 M 2106 1568 L
1996 1604 M 2106 1604 L
2031 1641 M 2106 1641 L
2065 1677 M 2106 1677 L
2099 1714 M 2106 1714 L
2078 1515 M 2078 1691 L
2042 1482 M 2042 1652 L
2005 1449 M 2005 1614 L
1969 1417 M 1969 1577 L
1932 1385 M 1932 1540 L
1896 1354 M 1896 1503 L
1859 1324 M 1859 1468 L
1823 1294 M 1823 1433 L
1786 1264 M 1786 1398 L
1750 1235 M 1750 1364 L
1714 1207 M 1714 1331 L
1677 1179 M 1677 1299 L
1641 1152 M 1641 1267 L
1604 1125 M 1604 1236 L
1568 1099 M 1568 1205 L
1531 1073 M 1531 1175 L
1495 1048 M 1495 1145 L
1458 1023 M 1458 1117 L
1422 999 M 1422 1088 L
1385 975 M 1385 1061 L
1349 952 M 1349 1034 L
1312 930 M 1312 1008 L
1276 908 M 1276 982 L
1239 886 M 1239 957 L
1203 865 M 1203 933 L
1166 845 M 1166 909 L
1130 825 M 1130 886 L
1094 806 M 1094 863 L
1057 787 M 1057 841 L
1021 769 M 1021 820 L
984 751 M 984 799 L
948 734 M 948 779 L
911 717 M 911 760 L
875 701 M 875 741 L
838 686 M 838 723 L
802 671 M 802 705 L
765 656 M 765 688 L
729 642 M 729 672 L
692 629 M 692 656 L
656 616 M 656 641 L
619 603 M 619 626 L
583 591 M 583 613 L
547 580 M 547 599 L
510 569 M 510 587 L
474 559 M 474 575 L
437 549 M 437 563 L
CS M 10 setlinewidth
/P { moveto 0 10.05 rlineto stroke } def
409 550 M 426 555 L
443 560 L
460 565 L
477 570 L
494 575 L
511 581 L
528 586 L
545 592 L
562 597 L
579 603 L
596 609 L
613 615 L
630 622 L
647 628 L
664 634 L
681 641 L
698 648 L
715 655 L
731 662 L
748 669 L
765 676 L
782 683 L
799 691 L
816 699 L
833 706 L
850 714 L
867 722 L
884 730 L
901 739 L
918 747 L
935 756 L
952 764 L
969 773 L
986 782 L
1003 791 L
1020 800 L
1037 810 L
1054 819 L
1071 829 L
1088 838 L
1105 848 L
1122 858 L
1139 868 L
1156 878 L
1173 889 L
1190 899 L
1207 910 L
1224 920 L
1241 931 L
1258 942 L
1275 953 L
1292 965 L
1309 976 L
1325 987 L
1342 999 L
1359 1011 L
1376 1023 L
1393 1035 L
1410 1047 L
1427 1059 L
1444 1071 L
1461 1084 L
1478 1096 L
1495 1109 L
1512 1122 L
1529 1135 L
1546 1148 L
1563 1162 L
1580 1175 L
1597 1188 L
1614 1202 L
1631 1216 L
1648 1230 L
1665 1244 L
1682 1258 L
1699 1272 L
1716 1287 L
1733 1301 L
1750 1316 L
1767 1331 L
1784 1346 L
1801 1361 L
1818 1376 L
1835 1391 L
1852 1407 L
1869 1422 L
1886 1438 L
1903 1454 L
1920 1470 L
1936 1486 L
1953 1502 L
1970 1518 L
1987 1535 L
2004 1551 L
2021 1568 L
2038 1585 L
2055 1602 L
2072 1619 L
2089 1636 L
2106 1653 L
CS M 2 setlinewidth
/P { moveto 0 2.05 rlineto stroke } def
CS [32 24] 0 setdash M
409 529 M 426 533 L
443 536 L
460 540 L
477 544 L
494 548 L
511 552 L
528 556 L
545 560 L
562 564 L
579 569 L
596 573 L
613 578 L
630 583 L
647 587 L
664 592 L
681 597 L
698 602 L
715 607 L
731 613 L
748 618 L
765 623 L
782 629 L
799 635 L
816 640 L
833 646 L
850 652 L
867 658 L
884 664 L
901 671 L
918 677 L
935 683 L
952 690 L
969 696 L
986 703 L
1003 710 L
1020 717 L
1037 724 L
1054 731 L
1071 738 L
1088 745 L
1105 753 L
1122 760 L
1139 768 L
1156 775 L
1173 783 L
CS M
1190 791 L
1207 799 L
1224 807 L
1241 815 L
1258 823 L
1275 831 L
1292 840 L
1309 848 L
1325 857 L
1342 866 L
1359 875 L
1376 883 L
1393 892 L
1410 902 L
1427 911 L
1444 920 L
1461 929 L
1478 939 L
1495 948 L
1512 958 L
1529 968 L
1546 978 L
1563 988 L
1580 998 L
1597 1008 L
1614 1018 L
1631 1028 L
1648 1039 L
1665 1049 L
1682 1060 L
1699 1071 L
1716 1081 L
1733 1092 L
1750 1103 L
1767 1114 L
1784 1126 L
1801 1137 L
1818 1148 L
1835 1160 L
1852 1171 L
1869 1183 L
1886 1195 L
1903 1207 L
1920 1219 L
1936 1231 L
1953 1243 L
1970 1255 L
1987 1268 L
2004 1280 L
2021 1292 L
2038 1305 L
2055 1318 L
2072 1331 L
2089 1344 L
2106 1357 L
CS [] 0 setdash M
1366 751 M B
1390 737 M 1366 779 L
1341 737 L
1390 737 L
CF M
1335 677 M B
1359 663 M 1335 705 L
1311 663 L
1359 663 L
CF M
1335 783 M B
1359 768 M 1335 811 L
1311 769 L
1359 769 L
CF M
1242 878 M B
1266 864 M 1242 906 L
1218 864 L
1266 864 L
CF M
1119 804 M B
1143 790 M 1119 832 L
1095 790 L
1143 790 L
CF M
1026 740 M B
1050 726 M 1026 768 L
1002 726 L
1050 726 L
CF M
903 730 M B
927 716 M 903 758 L
878 716 L
927 716 L
CF M
810 688 M B
834 674 M 810 716 L
786 674 L
834 674 L
CF M
718 624 M B
742 610 M 718 652 L
693 610 L
742 610 L
CF M
625 614 M B
649 600 M 625 642 L
601 600 L
649 600 L
CF M
563 603 M B
588 589 M 563 631 L
539 589 L
588 589 L
CF M
532 571 M B
557 557 M 532 599 L
508 557 L
557 557 L
CF M
532 571 M 625 571 L
625 581 M 625 562 L
625 581 L
532 571 M 440 571 L
440 581 M 440 562 L
440 581 L
532 571 M 532 761 L
542 761 M 523 761 L
542 761 L
532 571 M 532 381 L
542 381 M 523 381 L
542 381 L
1699 1225 M B
1718 1205 M 1719 1244 L
1679 1244 L
1679 1205 L
1719 1205 L
CF M
1636 1209 M B
1656 1189 M 1656 1229 L
1617 1229 L
1617 1189 L
1656 1189 L
CF M
1514 1038 M B
1533 1019 M 1533 1058 L
1494 1058 L
1494 1019 L
1533 1019 L
CF M
1380 1011 M B
1400 991 M 1400 1031 L
1360 1031 L
1360 991 L
1400 991 L
CF M
1279 931 M B
1299 912 M 1299 951 L
1259 951 L
1259 912 L
1299 912 L
CF M
1172 869 M B
1191 849 M 1191 889 L
1152 889 L
1152 849 L
1191 849 L
CF M
1048 828 M B
1068 808 M 1068 847 L
1028 847 L
1028 808 L
1068 808 L
CF M
975 755 M B
994 735 M 994 775 L
955 775 L
955 735 L
994 735 L
CF M
895 722 M B
915 702 M 915 742 L
876 742 L
876 702 L
915 702 L
CF M
842 664 M B
862 644 M 862 684 L
822 684 L
822 644 L
862 644 L
CF M
755 670 M B
775 650 M 775 690 L
735 690 L
735 650 L
775 650 L
CF M
1380 1011 M 1457 1011 L
1457 1020 M 1457 1002 L
1457 1020 L
1380 1011 M 1303 1011 L
1303 1020 M 1303 1002 L
1303 1020 L
1380 1011 M 1380 1291 L
1389 1291 M 1371 1290 L
1389 1291 L
1380 1011 M 1380 731 L
1389 731 M 1371 731 L
1389 731 L
1952 1627 M B
1980 1627 M 1952 1655 L
1924 1627 L
1952 1599 L
1980 1627 L
CF M
1921 1659 M B
1949 1659 M 1921 1687 L
1893 1659 L
1921 1631 L
1949 1659 L
CF M
1828 1437 M B
1856 1437 M 1828 1465 L
1800 1437 L
1828 1409 L
1856 1437 L
CF M
1705 1353 M B
1733 1353 M 1705 1381 L
1677 1353 L
1705 1325 L
1733 1353 L
CF M
1520 1120 M B
1548 1120 M 1520 1148 L
1492 1120 L
1520 1092 L
1548 1120 L
CF M
1396 1004 M B
1424 1004 M 1396 1032 L
1368 1004 L
1396 976 L
1424 1004 L
CF M
1273 930 M B
1301 930 M 1273 958 L
1245 930 L
1273 902 L
1301 930 L
CF M
1180 878 M B
1209 878 M 1180 906 L
1152 878 L
1180 850 L
1209 878 L
CF M
1088 793 M B
1116 793 M 1088 821 L
1060 793 L
1088 765 L
1116 793 L
CF M
995 751 M B
1023 751 M 995 779 L
967 751 L
995 723 L
1023 751 L
CF M
903 719 M B
931 719 M 903 747 L
875 719 L
903 691 L
931 719 L
CF M
810 698 M B
838 698 M 810 726 L
782 698 L
810 670 L
838 698 L
CF M
1952 1627 M 2106 1627 L
2106 1636 M 2106 1618 L
2106 1636 L
1952 1627 M 1798 1627 L
1798 1636 M 1798 1618 L
1798 1636 L
1952 1627 M 1952 2102 L
1961 2102 M 1943 2102 L
1961 2102 L
1952 1627 M 1952 1152 L
1961 1152 M 1943 1152 L
1961 1152 L
563 1896 M B
588 1882 M 563 1924 L
539 1882 L
588 1882 L
CF M
563 1786 M B
583 1767 M 583 1806 L
543 1806 L
543 1767 L
583 1767 L
CF M
563 1677 M B
591 1677 M 563 1705 L
535 1677 L
563 1649 L
591 1677 L
CF M
627 1874 M CS [] 0 setdash M
640 1929 M 643 1925 L
640 1922 L
636 1925 L
636 1929 L
640 1935 L
643 1938 L
652 1941 L
665 1941 L
675 1938 L
678 1935 L
681 1929 L
681 1922 L
678 1916 L
668 1909 L
652 1903 L
646 1900 L
640 1893 L
636 1884 L
636 1874 L
665 1941 M 672 1938 L
675 1935 L
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showpage
end
%%EndDocument

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 1057 194 a
 14917438 14768253 1184071 9472573 39008583 47231303 startTexFig
 1057 194 a
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32 828 M 35 834 L
41 837 L
77 837 L
32 792 M 32 805 L
77 792 M 77 815 L
77 828 M 77 850 L
9 873 M 77 873 L
9 876 M 77 876 L
9 863 M 9 876 L
77 863 M 77 885 L
9 908 M 12 905 L
16 908 L
12 911 L
9 908 L
32 908 M 77 908 L
32 911 M 77 911 L
32 898 M 32 911 L
77 898 M 77 921 L
32 943 M 77 943 L
32 946 M 77 946 L
41 946 M 35 953 L
32 962 L
32 969 L
35 978 L
41 982 L
77 982 L
32 969 M 35 975 L
41 978 L
77 978 L
32 934 M 32 946 L
77 934 M 77 956 L
77 969 M 77 991 L
51 1011 M 51 1049 L
45 1049 L
38 1046 L
35 1043 L
32 1036 L
32 1027 L
35 1017 L
41 1011 L
51 1007 L
57 1007 L
67 1011 L
73 1017 L
77 1027 L
77 1033 L
73 1043 L
67 1049 L
51 1046 M 41 1046 L
35 1043 L
32 1027 M 35 1020 L
41 1014 L
51 1011 L
57 1011 L
67 1014 L
73 1020 L
77 1027 L
38 1075 M 41 1075 L
41 1072 L
38 1072 L
35 1075 L
32 1081 L
32 1094 L
35 1100 L
38 1104 L
45 1107 L
67 1107 L
73 1110 L
77 1113 L
38 1104 M 67 1104 L
73 1107 L
77 1113 L
77 1116 L
45 1104 M 48 1100 L
51 1081 L
54 1072 L
61 1068 L
67 1068 L
73 1072 L
77 1081 L
77 1091 L
73 1097 L
67 1104 L
51 1081 M 54 1075 L
61 1072 L
67 1072 L
73 1075 L
77 1081 L
32 1139 M 77 1139 L
32 1142 M 77 1142 L
51 1142 M 41 1145 L
35 1152 L
32 1158 L
32 1168 L
35 1171 L
38 1171 L
41 1168 L
38 1165 L
35 1168 L
32 1129 M 32 1142 L
77 1129 M 77 1152 L
9 1245 M 64 1245 L
73 1248 L
77 1254 L
77 1261 L
73 1267 L
67 1271 L
9 1248 M 64 1248 L
73 1251 L
77 1254 L
32 1235 M 32 1261 L
51 1290 M 51 1328 L
45 1328 L
38 1325 L
35 1322 L
32 1315 L
32 1306 L
35 1296 L
41 1290 L
51 1287 L
57 1287 L
67 1290 L
73 1296 L
77 1306 L
77 1312 L
73 1322 L
67 1328 L
51 1325 M 41 1325 L
35 1322 L
32 1306 M 35 1299 L
41 1293 L
51 1290 L
57 1290 L
67 1293 L
73 1299 L
77 1306 L
32 1354 M 77 1354 L
32 1357 M 77 1357 L
51 1357 M 41 1360 L
35 1367 L
32 1373 L
32 1383 L
35 1386 L
38 1386 L
41 1383 L
38 1380 L
35 1383 L
32 1344 M 32 1357 L
77 1344 M 77 1367 L
32 1408 M 77 1408 L
32 1412 M 77 1412 L
41 1412 M 35 1418 L
32 1428 L
32 1434 L
35 1444 L
41 1447 L
77 1447 L
32 1434 M 35 1441 L
41 1444 L
77 1444 L
41 1447 M 35 1453 L
32 1463 L
32 1469 L
35 1479 L
41 1482 L
77 1482 L
32 1469 M 35 1476 L
41 1479 L
77 1479 L
32 1399 M 32 1412 L
77 1399 M 77 1421 L
77 1434 M 77 1457 L
77 1469 M 77 1492 L
38 1540 M 32 1543 L
45 1543 L
38 1540 L
35 1537 L
32 1530 L
32 1518 L
35 1511 L
38 1508 L
45 1508 L
48 1511 L
51 1518 L
57 1534 L
61 1540 L
64 1543 L
41 1508 M 45 1511 L
48 1518 L
54 1534 L
57 1540 L
61 1543 L
70 1543 L
73 1540 L
77 1534 L
77 1521 L
73 1514 L
70 1511 L
64 1508 L
77 1508 L
70 1511 L
32 1633 M 35 1624 L
41 1617 L
51 1614 L
57 1614 L
67 1617 L
73 1624 L
77 1633 L
77 1640 L
73 1649 L
67 1656 L
CS M
57 1659 L
51 1659 L
41 1656 L
35 1649 L
32 1640 L
32 1633 L
35 1627 L
41 1620 L
51 1617 L
57 1617 L
67 1620 L
73 1627 L
77 1633 L
77 1640 M 73 1646 L
67 1652 L
57 1656 L
51 1656 L
41 1652 L
35 1646 L
32 1640 L
12 1701 M 16 1697 L
19 1701 L
16 1704 L
12 1704 L
9 1701 L
9 1694 L
12 1688 L
19 1684 L
77 1684 L
9 1694 M 12 1691 L
19 1688 L
77 1688 L
32 1675 M 32 1701 L
77 1675 M 77 1697 L
9 1778 M 77 1778 L
9 1781 M 77 1781 L
29 1800 M 54 1800 L
9 1768 M 9 1819 L
29 1819 L
9 1816 L
41 1781 M 41 1800 L
77 1768 M 77 1790 L
71 1849 M 107 1849 L
67 1851 M 107 1851 L
67 1851 M 96 1829 L
96 1860 L
107 1843 M 107 1856 L
CS [] 0 setdash M
CS [] 0 setdash M
1206 86 M 1222 153 M 1222 86 L
1226 153 M 1226 86 L
1245 134 M 1245 108 L
1213 153 M 1264 153 L
1264 134 L
1261 153 L
1226 121 M 1245 121 L
1213 86 M 1235 86 L
1278 88 M 1280 86 L
1278 84 L
1276 86 L
1276 88 L
1278 92 L
1280 93 L
1286 95 L
1294 95 L
1299 93 L
1301 92 L
1303 88 L
1303 84 L
1301 80 L
1296 76 L
1286 72 L
1282 70 L
1278 66 L
1276 61 L
1276 55 L
1294 95 M 1297 93 L
1299 92 L
1301 88 L
1301 84 L
1299 80 L
1294 76 L
1286 72 L
1276 59 M 1278 61 L
1282 61 L
1292 57 L
1297 57 L
1301 59 L
1303 61 L
1282 61 M 1292 55 L
1299 55 L
1301 57 L
1303 61 L
1303 65 L
CS [] 0 setdash M
409 448 M 426 452 L
443 455 L
460 459 L
477 463 L
494 467 L
511 471 L
528 475 L
545 479 L
562 483 L
579 488 L
596 492 L
613 497 L
630 502 L
647 507 L
664 512 L
681 517 L
698 522 L
715 528 L
731 533 L
748 539 L
765 544 L
782 550 L
799 556 L
816 562 L
833 568 L
850 575 L
867 581 L
884 588 L
901 594 L
918 601 L
935 608 L
952 615 L
969 622 L
986 630 L
1003 637 L
1020 645 L
1037 652 L
1054 660 L
1071 668 L
1088 676 L
1105 684 L
1122 693 L
1139 701 L
1156 710 L
1173 719 L
1190 727 L
1207 736 L
1224 746 L
1241 755 L
1258 764 L
1275 774 L
1292 783 L
1309 793 L
1325 803 L
1342 813 L
1359 824 L
1376 834 L
1393 845 L
1410 855 L
1427 866 L
1444 877 L
1461 888 L
1478 899 L
1495 911 L
1512 922 L
1529 934 L
1546 946 L
1563 958 L
1580 970 L
1597 982 L
1614 994 L
1631 1007 L
1648 1020 L
1665 1033 L
1682 1046 L
1699 1059 L
1716 1072 L
1733 1086 L
1750 1099 L
1767 1113 L
1784 1127 L
1801 1141 L
1818 1155 L
1835 1170 L
1852 1184 L
1869 1199 L
1886 1214 L
1903 1229 L
1920 1244 L
1936 1259 L
1953 1275 L
1970 1291 L
1987 1306 L
2004 1322 L
2021 1339 L
2038 1355 L
2055 1372 L
2072 1388 L
2089 1405 L
2106 1422 L
409 457 M 426 461 L
443 465 L
460 469 L
477 474 L
494 478 L
511 483 L
528 488 L
545 493 L
562 498 L
579 503 L
596 508 L
613 514 L
630 519 L
647 525 L
664 531 L
681 537 L
698 543 L
715 550 L
731 556 L
748 563 L
765 569 L
782 576 L
799 583 L
816 590 L
833 598 L
850 605 L
867 613 L
884 620 L
901 628 L
918 636 L
935 645 L
952 653 L
969 661 L
986 670 L
1003 679 L
1020 688 L
1037 697 L
1054 706 L
1071 716 L
1088 726 L
1105 735 L
1122 745 L
1139 755 L
1156 766 L
1173 776 L
1190 787 L
1207 797 L
1224 808 L
1241 819 L
1258 831 L
1275 842 L
1292 854 L
1309 866 L
1325 878 L
1342 890 L
1359 902 L
1376 914 L
1393 927 L
1410 940 L
1427 953 L
1444 966 L
1461 980 L
1478 993 L
1495 1007 L
1512 1021 L
1529 1035 L
1546 1049 L
1563 1064 L
1580 1078 L
1597 1093 L
1614 1108 L
1631 1124 L
1648 1139 L
1665 1155 L
1682 1170 L
1699 1186 L
1716 1203 L
1733 1219 L
1750 1236 L
1767 1252 L
1784 1269 L
1801 1286 L
1818 1304 L
1835 1321 L
1852 1339 L
1869 1357 L
1886 1375 L
1903 1394 L
1920 1412 L
1936 1431 L
1953 1450 L
1970 1469 L
1987 1489 L
2004 1508 L
2021 1528 L
2038 1548 L
2055 1569 L
2072 1589 L
2089 1610 L
2106 1631 L
476 474 M 523 474 L
601 510 M 658 510 L
706 547 M 772 547 L
799 583 M 872 583 L
882 619 M 962 619 L
958 656 M 1045 656 L
1028 692 M 1121 692 L
1094 729 M 1192 729 L
CS M
1155 765 M 1259 765 L
1213 802 M 1323 802 L
1269 838 M 1383 838 L
1321 875 M 1441 875 L
1372 911 M 1496 911 L
1420 948 M 1549 948 L
1467 984 M 1600 984 L
1512 1021 M 1649 1021 L
1555 1057 M 1697 1057 L
1597 1094 M 1743 1094 L
1638 1130 M 1787 1130 L
1678 1166 M 1831 1166 L
1716 1203 M 1873 1203 L
1754 1239 M 1914 1239 L
1790 1276 M 1955 1276 L
1826 1312 M 1994 1312 L
1861 1349 M 2032 1349 L
1895 1385 M 2069 1385 L
1928 1422 M 2106 1422 L
1961 1458 M 2106 1458 L
1993 1495 M 2106 1495 L
2024 1531 M 2106 1531 L
2055 1568 M 2106 1568 L
2085 1604 M 2106 1604 L
2078 1394 M 2078 1596 L
2042 1358 M 2042 1552 L
2005 1323 M 2005 1509 L
1969 1289 M 1969 1467 L
1932 1256 M 1932 1426 L
1896 1223 M 1896 1386 L
1859 1191 M 1859 1347 L
1823 1160 M 1823 1309 L
1786 1129 M 1786 1272 L
1750 1099 M 1750 1236 L
1714 1070 M 1714 1200 L
1677 1042 M 1677 1166 L
1641 1014 M 1641 1132 L
1604 987 M 1604 1099 L
1568 961 M 1568 1068 L
1531 935 M 1531 1037 L
1495 910 M 1495 1006 L
1458 886 M 1458 977 L
1422 862 M 1422 949 L
1385 840 M 1385 921 L
1349 817 M 1349 894 L
1312 796 M 1312 868 L
1276 775 M 1276 843 L
1239 754 M 1239 819 L
1203 734 M 1203 795 L
1166 715 M 1166 772 L
1130 697 M 1130 750 L
1094 679 M 1094 729 L
1057 662 M 1057 708 L
1021 645 M 1021 688 L
984 629 M 984 669 L
948 613 M 948 651 L
911 598 M 911 633 L
875 584 M 875 616 L
838 570 M 838 600 L
802 557 M 802 584 L
765 544 M 765 569 L
729 532 M 729 555 L
692 521 M 692 541 L
656 510 M 656 528 L
619 499 M 619 516 L
583 489 M 583 504 L
547 479 M 547 493 L
510 471 M 510 483 L
474 462 M 474 473 L
437 454 M 437 464 L
CS M 10 setlinewidth
/P { moveto 0 10.05 rlineto stroke } def
409 454 M 426 458 L
443 461 L
460 466 L
477 470 L
494 474 L
511 478 L
528 483 L
545 488 L
562 492 L
579 497 L
596 502 L
613 508 L
630 513 L
647 518 L
664 524 L
681 530 L
698 535 L
715 541 L
731 547 L
748 554 L
765 560 L
782 567 L
799 573 L
816 580 L
833 587 L
850 594 L
867 601 L
884 608 L
901 616 L
918 623 L
935 631 L
952 639 L
969 647 L
986 655 L
1003 664 L
1020 672 L
1037 681 L
1054 689 L
1071 698 L
1088 707 L
1105 717 L
1122 726 L
1139 735 L
1156 745 L
1173 755 L
1190 765 L
1207 775 L
1224 785 L
1241 796 L
1258 806 L
1275 817 L
1292 828 L
1309 839 L
1325 850 L
1342 862 L
1359 873 L
1376 885 L
1393 897 L
1410 909 L
1427 921 L
1444 933 L
1461 946 L
1478 959 L
1495 972 L
1512 985 L
1529 998 L
1546 1011 L
1563 1025 L
1580 1039 L
1597 1052 L
1614 1067 L
1631 1081 L
1648 1095 L
1665 1110 L
1682 1125 L
1699 1140 L
1716 1155 L
1733 1170 L
1750 1186 L
1767 1201 L
1784 1217 L
1801 1233 L
1818 1249 L
1835 1266 L
1852 1282 L
1869 1299 L
1886 1316 L
1903 1333 L
1920 1351 L
1936 1368 L
1953 1386 L
1970 1404 L
1987 1422 L
2004 1440 L
2021 1459 L
2038 1478 L
2055 1496 L
2072 1516 L
2089 1535 L
2106 1554 L
CS M 2 setlinewidth
/P { moveto 0 2.05 rlineto stroke } def
CS [32 24] 0 setdash M
409 440 M 426 443 L
443 446 L
460 449 L
477 452 L
494 456 L
511 459 L
528 462 L
545 466 L
562 470 L
579 473 L
596 477 L
613 481 L
630 485 L
647 489 L
664 494 L
681 498 L
698 502 L
715 507 L
731 511 L
748 516 L
765 521 L
782 526 L
799 531 L
816 536 L
833 541 L
850 546 L
867 552 L
884 557 L
901 563 L
918 568 L
935 574 L
952 580 L
969 586 L
986 592 L
1003 598 L
1020 604 L
1037 611 L
1054 617 L
1071 624 L
1088 630 L
1105 637 L
1122 644 L
1139 651 L
1156 658 L
1173 665 L
1190 673 L
1207 680 L
1224 688 L
1241 695 L
1258 703 L
1275 711 L
1292 719 L
1309 727 L
1325 735 L
1342 743 L
1359 752 L
1376 760 L
1393 769 L
1410 777 L
1427 786 L
1444 795 L
1461 804 L
1478 813 L
1495 823 L
1512 832 L
1529 841 L
1546 851 L
1563 861 L
1580 871 L
1597 881 L
1614 891 L
1631 901 L
1648 911 L
1665 922 L
1682 932 L
1699 943 L
1716 954 L
1733 964 L
1750 975 L
1767 987 L
1784 998 L
1801 1009 L
1818 1021 L
1835 1032 L
1852 1044 L
1869 1056 L
1886 1068 L
1903 1080 L
1920 1092 L
1936 1104 L
1953 1117 L
1970 1129 L
1987 1142 L
2004 1155 L
2021 1168 L
2038 1181 L
2055 1194 L
2072 1208 L
2089 1221 L
2106 1235 L
CS [] 0 setdash M
1366 648 M B
1390 634 M 1366 676 L
1341 634 L
1390 634 L
CF M
1335 656 M B
1359 642 M 1335 684 L
1311 642 L
1359 642 L
CF M
1335 669 M B
1359 655 M 1335 697 L
1311 655 L
1359 655 L
CF M
1242 693 M B
1266 679 M 1242 721 L
1218 679 L
1266 679 L
CF M
1119 659 M B
1143 644 M 1119 687 L
1095 645 L
1143 645 L
CF M
1026 616 M B
1050 602 M 1026 644 L
1002 602 L
1050 602 L
CF M
903 594 M B
927 580 M 903 622 L
878 580 L
927 580 L
CF M
810 565 M B
834 551 M 810 593 L
786 551 L
834 551 L
CF M
718 509 M B
742 495 M 718 537 L
693 495 L
742 495 L
CF M
625 506 M B
649 492 M 625 534 L
601 492 L
649 492 L
CF M
563 482 M B
588 468 M 563 510 L
539 468 L
588 468 L
CF M
532 463 M B
557 449 M 532 491 L
508 449 L
557 449 L
CF M
532 463 M 625 463 L
625 473 M 625 454 L
625 473 L
532 463 M 440 463 L
440 473 M 440 454 L
440 473 L
532 463 M 532 528 L
542 528 M 523 527 L
542 528 L
532 463 M 532 399 L
542 399 M 523 399 L
542 399 L
1699 1292 M B
1718 1272 M 1719 1312 L
1679 1312 L
1679 1272 L
1719 1272 L
CF M
1636 1080 M B
1656 1060 M 1656 1100 L
1617 1100 L
1617 1060 L
1656 1060 L
CF M
1514 968 M B
1533 948 M 1533 988 L
1494 988 L
1494 948 L
1533 948 L
CF M
1380 891 M B
1400 871 M 1400 911 L
1360 911 L
1360 871 L
1400 871 L
CF M
1279 802 M B
1299 782 M 1299 822 L
1259 822 L
1259 782 L
1299 782 L
CF M
1172 743 M B
1191 723 M 1191 763 L
1152 763 L
1152 723 L
1191 723 L
CF M
1048 671 M B
1068 652 M 1068 691 L
1028 691 L
1028 652 L
1068 652 L
CF M
975 629 M B
994 609 M 994 649 L
955 649 L
955 609 L
994 609 L
CF M
895 595 M B
915 576 M 915 615 L
876 615 L
876 576 L
915 576 L
CF M
842 565 M B
862 545 M 862 584 L
822 584 L
822 545 L
862 545 L
CF M
755 553 M B
775 533 M 775 573 L
735 573 L
735 533 L
775 533 L
CF M
1380 891 M 1457 891 L
1457 900 M 1457 882 L
1457 900 L
1380 891 M 1303 891 L
1303 900 M 1303 882 L
1303 900 L
1380 891 M 1380 991 L
1389 991 M 1371 991 L
1389 991 L
1380 891 M 1380 791 L
1389 791 M 1371 791 L
1389 791 L
1952 1699 M B
1980 1699 M 1952 1727 L
1924 1699 L
1952 1671 L
1980 1699 L
CF M
1921 1573 M B
1949 1573 M 1921 1601 L
1893 1573 L
1921 1545 L
1949 1573 L
CF M
1828 1330 M B
1856 1330 M 1828 1358 L
1800 1330 L
1828 1302 L
1856 1330 L
CF M
1705 1215 M B
1733 1215 M 1705 1243 L
1677 1215 L
1705 1187 L
1733 1215 L
CF M
1520 955 M B
1548 955 M 1520 983 L
1492 955 L
1520 927 L
1548 955 L
CF M
1396 854 M B
1424 854 M 1396 882 L
1368 854 L
1396 826 L
1424 854 L
CF M
1273 779 M B
1301 779 M 1273 807 L
1245 779 L
1273 751 L
1301 779 L
CF M
1180 723 M B
1209 723 M 1180 751 L
1152 723 L
1180 695 L
1209 723 L
CF M
1088 667 M B
1116 667 M 1088 695 L
1060 667 L
1088 639 L
1116 667 L
CF M
995 624 M B
1023 624 M 995 652 L
967 624 L
995 596 L
1023 624 L
CF M
903 594 M B
931 594 M 903 622 L
875 594 L
903 566 L
931 594 L
CF M
810 565 M B
838 565 M 810 593 L
782 565 L
810 537 L
838 565 L
CF M
1952 1699 M 2106 1699 L
2106 1708 M 2106 1690 L
2106 1708 L
1952 1699 M 1798 1699 L
1798 1708 M 1798 1690 L
1798 1708 L
1952 1699 M 1952 2140 L
1961 2140 M 1943 2140 L
1961 2140 L
1952 1699 M 1952 1258 L
1961 1258 M 1943 1258 L
1961 1258 L
563 1896 M B
588 1882 M 563 1924 L
539 1882 L
588 1882 L
CF M
563 1786 M B
583 1767 M 583 1806 L
543 1806 L
543 1767 L
583 1767 L
CF M
563 1677 M B
591 1677 M 563 1705 L
535 1677 L
563 1649 L
591 1677 L
CF M
627 1874 M CS [] 0 setdash M
640 1929 M 643 1925 L
640 1922 L
636 1925 L
636 1929 L
640 1935 L
643 1938 L
652 1941 L
665 1941 L
675 1938 L
678 1935 L
681 1929 L
681 1922 L
678 1916 L
668 1909 L
652 1903 L
646 1900 L
640 1893 L
636 1884 L
636 1874 L
665 1941 M 672 1938 L
675 1935 L
678 1929 L
678 1922 L
675 1916 L
665 1909 L
652 1903 L
636 1880 M 640 1884 L
646 1884 L
662 1877 L
672 1877 L
678 1880 L
681 1884 L
646 1884 M 662 1874 L
675 1874 L
678 1877 L
681 1884 L
681 1890 L
720 1941 M 710 1938 L
704 1929 L
701 1912 L
701 1903 L
704 1887 L
710 1877 L
720 1874 L
726 1874 L
736 1877 L
742 1887 L
745 1903 L
745 1912 L
742 1929 L
736 1938 L
726 1941 L
720 1941 L
713 1938 L
710 1935 L
707 1929 L
704 1912 L
704 1903 L
707 1887 L
710 1880 L
713 1877 L
720 1874 L
726 1874 M 733 1877 L
736 1880 L
739 1887 L
742 1903 L
742 1912 L
739 1929 L
736 1935 L
733 1938 L
726 1941 L
784 1941 M 774 1938 L
768 1929 L
765 1912 L
765 1903 L
768 1887 L
774 1877 L
784 1874 L
790 1874 L
800 1877 L
806 1887 L
810 1903 L
810 1912 L
806 1929 L
800 1938 L
790 1941 L
784 1941 L
778 1938 L
774 1935 L
771 1929 L
768 1912 L
768 1903 L
771 1887 L
774 1880 L
778 1877 L
784 1874 L
790 1874 M 797 1877 L
800 1880 L
803 1887 L
806 1903 L
806 1912 L
803 1929 L
800 1935 L
797 1938 L
790 1941 L
925 1932 M 928 1922 L
928 1941 L
925 1932 L
919 1938 L
909 1941 L
903 1941 L
893 1938 L
887 1932 L
883 1925 L
880 1916 L
880 1900 L
883 1890 L
887 1884 L
893 1877 L
903 1874 L
909 1874 L
919 1877 L
925 1884 L
903 1941 M 896 1938 L
890 1932 L
887 1925 L
883 1916 L
883 1900 L
887 1890 L
890 1884 L
896 1877 L
903 1874 L
925 1900 M 925 1874 L
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showpage
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%%EndDocument

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 -32 1311 a
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223 2159 L
221 2171 L
218 2175 L
216 2178 L
211 2180 L
2261 255 M 2261 2261 L
2261 255 M 2235 255 L
2261 360 M 2235 360 L
2261 466 M 2209 466 L
2261 571 M 2235 571 L
2261 677 M 2235 677 L
2261 783 M 2235 783 L
2261 888 M 2209 888 L
2261 994 M 2235 994 L
2261 1099 M 2235 1099 L
2261 1205 M 2235 1205 L
2261 1310 M 2209 1310 L
2261 1416 M 2235 1416 L
2261 1522 M 2235 1522 L
2261 1627 M 2235 1627 L
2261 1733 M 2209 1733 L
2261 1838 M 2235 1838 L
2261 1944 M 2235 1944 L
2261 2049 M 2235 2049 L
2261 2155 M 2209 2155 L
2261 2261 M 2235 2261 L
CS [] 0 setdash M
CS [] 0 setdash M
77 651 M 32 667 M 77 667 L
32 670 M 77 670 L
41 670 M 35 677 L
32 686 L
32 693 L
35 702 L
41 706 L
77 706 L
32 693 M 35 699 L
41 702 L
77 702 L
32 658 M 32 670 L
77 658 M 77 680 L
77 693 M 77 715 L
32 751 M 35 741 L
41 735 L
51 731 L
57 731 L
67 735 L
73 741 L
77 751 L
77 757 L
73 767 L
67 773 L
57 776 L
51 776 L
41 773 L
35 767 L
32 757 L
32 751 L
35 744 L
41 738 L
51 735 L
57 735 L
67 738 L
73 744 L
77 751 L
77 757 M 73 763 L
67 770 L
57 773 L
51 773 L
41 770 L
35 763 L
32 757 L
32 802 M 77 802 L
32 805 M 77 805 L
41 805 M 35 812 L
32 821 L
32 828 L
35 837 L
41 840 L
77 840 L
CS M
32 828 M 35 834 L
41 837 L
77 837 L
32 792 M 32 805 L
77 792 M 77 815 L
77 828 M 77 850 L
9 873 M 77 873 L
9 876 M 77 876 L
9 863 M 9 876 L
77 863 M 77 885 L
9 908 M 12 905 L
16 908 L
12 911 L
9 908 L
32 908 M 77 908 L
32 911 M 77 911 L
32 898 M 32 911 L
77 898 M 77 921 L
32 943 M 77 943 L
32 946 M 77 946 L
41 946 M 35 953 L
32 962 L
32 969 L
35 978 L
41 982 L
77 982 L
32 969 M 35 975 L
41 978 L
77 978 L
32 934 M 32 946 L
77 934 M 77 956 L
77 969 M 77 991 L
51 1011 M 51 1049 L
45 1049 L
38 1046 L
35 1043 L
32 1036 L
32 1027 L
35 1017 L
41 1011 L
51 1007 L
57 1007 L
67 1011 L
73 1017 L
77 1027 L
77 1033 L
73 1043 L
67 1049 L
51 1046 M 41 1046 L
35 1043 L
32 1027 M 35 1020 L
41 1014 L
51 1011 L
57 1011 L
67 1014 L
73 1020 L
77 1027 L
38 1075 M 41 1075 L
41 1072 L
38 1072 L
35 1075 L
32 1081 L
32 1094 L
35 1100 L
38 1104 L
45 1107 L
67 1107 L
73 1110 L
77 1113 L
38 1104 M 67 1104 L
73 1107 L
77 1113 L
77 1116 L
45 1104 M 48 1100 L
51 1081 L
54 1072 L
61 1068 L
67 1068 L
73 1072 L
77 1081 L
77 1091 L
73 1097 L
67 1104 L
51 1081 M 54 1075 L
61 1072 L
67 1072 L
73 1075 L
77 1081 L
32 1139 M 77 1139 L
32 1142 M 77 1142 L
51 1142 M 41 1145 L
35 1152 L
32 1158 L
32 1168 L
35 1171 L
38 1171 L
41 1168 L
38 1165 L
35 1168 L
32 1129 M 32 1142 L
77 1129 M 77 1152 L
9 1245 M 64 1245 L
73 1248 L
77 1254 L
77 1261 L
73 1267 L
67 1271 L
9 1248 M 64 1248 L
73 1251 L
77 1254 L
32 1235 M 32 1261 L
51 1290 M 51 1328 L
45 1328 L
38 1325 L
35 1322 L
32 1315 L
32 1306 L
35 1296 L
41 1290 L
51 1287 L
57 1287 L
67 1290 L
73 1296 L
77 1306 L
77 1312 L
73 1322 L
67 1328 L
51 1325 M 41 1325 L
35 1322 L
32 1306 M 35 1299 L
41 1293 L
51 1290 L
57 1290 L
67 1293 L
73 1299 L
77 1306 L
32 1354 M 77 1354 L
32 1357 M 77 1357 L
51 1357 M 41 1360 L
35 1367 L
32 1373 L
32 1383 L
35 1386 L
38 1386 L
41 1383 L
38 1380 L
35 1383 L
32 1344 M 32 1357 L
77 1344 M 77 1367 L
32 1408 M 77 1408 L
32 1412 M 77 1412 L
41 1412 M 35 1418 L
32 1428 L
32 1434 L
35 1444 L
41 1447 L
77 1447 L
32 1434 M 35 1441 L
41 1444 L
77 1444 L
41 1447 M 35 1453 L
32 1463 L
32 1469 L
35 1479 L
41 1482 L
77 1482 L
32 1469 M 35 1476 L
41 1479 L
77 1479 L
32 1399 M 32 1412 L
77 1399 M 77 1421 L
77 1434 M 77 1457 L
77 1469 M 77 1492 L
38 1540 M 32 1543 L
45 1543 L
38 1540 L
35 1537 L
32 1530 L
32 1518 L
35 1511 L
38 1508 L
45 1508 L
48 1511 L
51 1518 L
57 1534 L
61 1540 L
64 1543 L
41 1508 M 45 1511 L
48 1518 L
54 1534 L
57 1540 L
61 1543 L
70 1543 L
73 1540 L
77 1534 L
77 1521 L
73 1514 L
70 1511 L
64 1508 L
77 1508 L
70 1511 L
32 1633 M 35 1624 L
41 1617 L
51 1614 L
57 1614 L
67 1617 L
73 1624 L
77 1633 L
77 1640 L
73 1649 L
67 1656 L
57 1659 L
51 1659 L
41 1656 L
35 1649 L
32 1640 L
32 1633 L
35 1627 L
41 1620 L
51 1617 L
57 1617 L
67 1620 L
73 1627 L
77 1633 L
77 1640 M 73 1646 L
67 1652 L
57 1656 L
51 1656 L
41 1652 L
35 1646 L
32 1640 L
12 1701 M 16 1697 L
19 1701 L
16 1704 L
12 1704 L
9 1701 L
9 1694 L
12 1688 L
19 1684 L
77 1684 L
9 1694 M 12 1691 L
19 1688 L
77 1688 L
32 1675 M 32 1701 L
77 1675 M 77 1697 L
9 1778 M 77 1778 L
9 1781 M 77 1781 L
29 1800 M 54 1800 L
9 1768 M 9 1819 L
29 1819 L
9 1816 L
41 1781 M 41 1800 L
77 1768 M 77 1790 L
67 1835 M 86 1831 L
82 1835 L
80 1841 L
80 1847 L
82 1853 L
86 1856 L
92 1858 L
96 1858 L
102 1856 L
106 1853 L
107 1847 L
107 1841 L
106 1835 L
104 1833 L
100 1831 L
98 1831 L
96 1833 L
98 1835 L
100 1833 L
80 1847 M 82 1851 L
86 1855 L
92 1856 L
96 1856 L
102 1855 L
106 1851 L
107 1847 L
67 1835 M 67 1855 L
69 1835 M 69 1845 L
67 1855 L
CS [] 0 setdash M
CS [] 0 setdash M
1206 86 M 1222 153 M 1222 86 L
1226 153 M 1226 86 L
1245 134 M 1245 108 L
1213 153 M 1264 153 L
1264 134 L
1261 153 L
1226 121 M 1245 121 L
1213 86 M 1235 86 L
1278 88 M 1280 86 L
1278 84 L
1276 86 L
1276 88 L
1278 92 L
1280 93 L
1286 95 L
1294 95 L
1299 93 L
1301 92 L
CS M
1303 88 L
1303 84 L
1301 80 L
1296 76 L
1286 72 L
1282 70 L
1278 66 L
1276 61 L
1276 55 L
1294 95 M 1297 93 L
1299 92 L
1301 88 L
1301 84 L
1299 80 L
1294 76 L
1286 72 L
1276 59 M 1278 61 L
1282 61 L
1292 57 L
1297 57 L
1301 59 L
1303 61 L
1282 61 M 1292 55 L
1299 55 L
1301 57 L
1303 61 L
1303 65 L
CS [] 0 setdash M
409 501 M 426 504 L
443 506 L
460 508 L
477 511 L
494 513 L
511 516 L
528 519 L
545 521 L
562 524 L
579 527 L
596 530 L
613 533 L
630 537 L
647 540 L
664 543 L
681 547 L
698 551 L
715 554 L
731 558 L
748 562 L
765 566 L
782 570 L
799 575 L
816 579 L
833 583 L
850 588 L
867 593 L
884 597 L
901 602 L
918 607 L
935 612 L
952 618 L
969 623 L
986 629 L
1003 634 L
1020 640 L
1037 646 L
1054 652 L
1071 658 L
1088 664 L
1105 671 L
1122 677 L
1139 684 L
1156 690 L
1173 697 L
1190 704 L
1207 712 L
1224 719 L
1241 726 L
1258 734 L
1275 742 L
1292 750 L
1309 758 L
1325 766 L
1342 774 L
1359 783 L
1376 791 L
1393 800 L
1410 809 L
1427 818 L
1444 828 L
1461 837 L
1478 847 L
1495 856 L
1512 866 L
1529 876 L
1546 887 L
1563 897 L
1580 908 L
1597 919 L
1614 930 L
1631 941 L
1648 952 L
1665 964 L
1682 975 L
1699 987 L
1716 999 L
1733 1011 L
1750 1024 L
1767 1036 L
1784 1049 L
1801 1062 L
1818 1075 L
1835 1089 L
1852 1102 L
1869 1116 L
1886 1130 L
1903 1144 L
1920 1159 L
1936 1174 L
1953 1188 L
1970 1203 L
1987 1219 L
2004 1234 L
2021 1250 L
2038 1266 L
2055 1282 L
2072 1298 L
2089 1315 L
2106 1332 L
409 507 M 426 509 L
443 512 L
460 515 L
477 518 L
494 521 L
511 524 L
528 527 L
545 530 L
562 534 L
579 537 L
596 541 L
613 545 L
630 549 L
647 553 L
664 557 L
681 561 L
698 565 L
715 570 L
731 574 L
748 579 L
765 584 L
782 589 L
799 594 L
816 600 L
833 605 L
850 611 L
867 616 L
884 622 L
901 628 L
918 634 L
935 641 L
952 647 L
969 654 L
986 661 L
1003 668 L
1020 675 L
1037 682 L
1054 690 L
1071 697 L
1088 705 L
1105 713 L
1122 721 L
1139 729 L
1156 738 L
1173 747 L
1190 755 L
1207 764 L
1224 774 L
1241 783 L
1258 793 L
1275 803 L
1292 813 L
1309 823 L
1325 833 L
1342 844 L
1359 855 L
1376 866 L
1393 877 L
1410 888 L
1427 900 L
1444 912 L
1461 924 L
1478 936 L
1495 949 L
1512 962 L
1529 975 L
1546 988 L
1563 1002 L
1580 1015 L
1597 1029 L
1614 1044 L
1631 1058 L
1648 1073 L
1665 1088 L
1682 1103 L
1699 1118 L
1716 1134 L
1733 1150 L
1750 1166 L
1767 1183 L
1784 1200 L
1801 1217 L
1818 1234 L
1835 1252 L
1852 1270 L
1869 1288 L
1886 1306 L
1903 1325 L
1920 1344 L
1936 1363 L
1953 1383 L
1970 1403 L
1987 1423 L
2004 1444 L
2021 1465 L
2038 1486 L
2055 1507 L
2072 1529 L
2089 1551 L
2106 1574 L
431 510 M 472 510 L
621 546 M 678 546 L
761 583 M 832 583 L
876 619 M 958 619 L
974 656 M 1065 656 L
1060 692 M 1161 692 L
1138 729 M 1246 729 L
1208 765 M 1324 765 L
1273 802 M 1396 802 L
1334 838 M 1463 838 L
1390 875 M 1526 875 L
1443 911 M 1586 911 L
1493 948 M 1641 948 L
1541 984 M 1695 984 L
1587 1021 M 1745 1021 L
1630 1057 M 1794 1057 L
1672 1093 M 1841 1093 L
1711 1130 M 1885 1130 L
1750 1166 M 1928 1166 L
1787 1203 M 1970 1203 L
1823 1239 M 2010 1239 L
1857 1276 M 2049 1276 L
1891 1312 M 2087 1312 L
1924 1349 M 2106 1349 L
1955 1385 M 2106 1385 L
1986 1422 M 2106 1422 L
2016 1458 M 2106 1458 L
2045 1495 M 2106 1495 L
2074 1531 M 2106 1531 L
2101 1568 M 2106 1568 L
2078 1304 M 2078 1537 L
2042 1269 M 2042 1490 L
2005 1235 M 2005 1445 L
1969 1202 M 1969 1401 L
1932 1170 M 1932 1359 L
1896 1139 M 1896 1318 L
1859 1109 M 1859 1278 L
1823 1080 M 1823 1240 L
1786 1051 M 1786 1202 L
1750 1024 M 1750 1166 L
1714 997 M 1714 1132 L
1677 972 M 1677 1098 L
1641 947 M 1641 1066 L
1604 923 M 1604 1035 L
1568 900 M 1568 1005 L
1531 878 M 1531 976 L
1495 856 M 1495 949 L
1458 835 M 1458 922 L
1422 815 M 1422 896 L
1385 796 M 1385 872 L
1349 777 M 1349 848 L
1312 760 M 1312 825 L
1276 742 M 1276 803 L
1239 726 M 1239 782 L
1203 710 M 1203 762 L
1166 695 M 1166 743 L
1130 680 M 1130 725 L
1094 666 M 1094 708 L
1057 653 M 1057 691 L
1021 640 M 1021 675 L
984 628 M 984 660 L
948 616 M 948 646 L
911 605 M 911 632 L
875 595 M 875 619 L
838 585 M 838 607 L
802 575 M 802 595 L
765 566 M 765 584 L
729 558 M 729 574 L
692 550 M 692 564 L
656 542 M 656 555 L
619 535 M 619 546 L
583 528 M 583 538 L
547 522 M 547 531 L
CS M
510 516 M 510 524 L
474 510 M 474 517 L
437 505 M 437 511 L
CS M 10 setlinewidth
/P { moveto 0 10.05 rlineto stroke } def
409 505 M 426 507 L
443 510 L
460 512 L
477 515 L
494 518 L
511 521 L
528 524 L
545 527 L
562 530 L
579 534 L
596 537 L
613 541 L
630 544 L
647 548 L
664 552 L
681 556 L
698 560 L
715 564 L
731 569 L
748 573 L
765 578 L
782 582 L
799 587 L
816 592 L
833 597 L
850 602 L
867 608 L
884 613 L
901 619 L
918 625 L
935 631 L
952 637 L
969 643 L
986 649 L
1003 656 L
1020 662 L
1037 669 L
1054 676 L
1071 683 L
1088 690 L
1105 698 L
1122 705 L
1139 713 L
1156 721 L
1173 729 L
1190 737 L
1207 745 L
1224 754 L
1241 763 L
1258 772 L
1275 781 L
1292 790 L
1309 799 L
1325 809 L
1342 819 L
1359 829 L
1376 839 L
1393 849 L
1410 860 L
1427 871 L
1444 882 L
1461 893 L
1478 904 L
1495 916 L
1512 927 L
1529 939 L
1546 952 L
1563 964 L
1580 977 L
1597 989 L
1614 1003 L
1631 1016 L
1648 1029 L
1665 1043 L
1682 1057 L
1699 1071 L
1716 1086 L
1733 1100 L
1750 1115 L
1767 1130 L
1784 1146 L
1801 1161 L
1818 1177 L
1835 1193 L
1852 1210 L
1869 1226 L
1886 1243 L
1903 1260 L
1920 1278 L
1936 1295 L
1953 1313 L
1970 1332 L
1987 1350 L
2004 1369 L
2021 1388 L
2038 1407 L
2055 1427 L
2072 1447 L
2089 1467 L
2106 1487 L
CS M 2 setlinewidth
/P { moveto 0 2.05 rlineto stroke } def
CS [32 24] 0 setdash M
409 497 M 426 498 L
443 500 L
460 502 L
477 504 L
494 507 L
511 509 L
528 511 L
545 513 L
562 516 L
579 518 L
596 521 L
613 523 L
630 526 L
647 529 L
664 532 L
681 535 L
698 538 L
715 541 L
731 544 L
748 547 L
765 550 L
782 554 L
799 557 L
816 561 L
833 564 L
850 568 L
867 572 L
884 576 L
901 580 L
918 584 L
935 588 L
952 592 L
969 597 L
986 601 L
1003 606 L
1020 610 L
1037 615 L
1054 620 L
1071 624 L
1088 629 L
1105 635 L
1122 640 L
1139 645 L
1156 650 L
1173 656 L
1190 661 L
1207 667 L
1224 673 L
1241 679 L
1258 685 L
1275 691 L
1292 697 L
1309 703 L
1325 710 L
1342 716 L
1359 723 L
1376 730 L
1393 737 L
1410 744 L
1427 751 L
1444 758 L
1461 765 L
1478 773 L
1495 780 L
1512 788 L
1529 796 L
1546 803 L
1563 812 L
1580 820 L
1597 828 L
1614 836 L
1631 845 L
1648 854 L
1665 862 L
1682 871 L
1699 880 L
1716 890 L
1733 899 L
1750 908 L
1767 918 L
1784 928 L
1801 937 L
1818 948 L
1835 958 L
1852 968 L
1869 978 L
1886 989 L
1903 1000 L
1920 1010 L
1936 1021 L
1953 1033 L
1970 1044 L
1987 1055 L
2004 1067 L
2021 1079 L
2038 1090 L
2055 1102 L
2072 1115 L
2089 1127 L
2106 1140 L
CS [] 0 setdash M
1366 759 M B
1390 745 M 1366 787 L
1341 745 L
1390 745 L
CF M
1335 656 M B
1359 642 M 1335 684 L
1311 642 L
1359 642 L
CF M
1335 656 M B
1359 642 M 1335 684 L
1311 642 L
1359 642 L
CF M
1242 662 M B
1266 648 M 1242 690 L
1218 648 L
1266 648 L
CF M
1119 650 M B
1143 636 M 1119 678 L
1095 636 L
1143 636 L
CF M
1026 614 M B
1050 600 M 1026 642 L
1002 600 L
1050 600 L
CF M
903 601 M B
927 587 M 903 629 L
878 587 L
927 587 L
CF M
810 571 M B
834 557 M 810 599 L
786 557 L
834 557 L
CF M
718 542 M B
742 528 M 718 570 L
693 528 L
742 528 L
CF M
625 533 M B
649 519 M 625 561 L
601 519 L
649 519 L
CF M
563 521 M B
588 507 M 563 549 L
539 507 L
588 507 L
CF M
532 512 M B
557 498 M 532 540 L
508 498 L
557 498 L
CF M
532 512 M 625 512 L
625 522 M 625 503 L
625 522 L
532 512 M 440 512 L
440 522 M 440 503 L
440 522 L
532 512 M 532 538 L
542 538 M 523 538 L
542 538 L
532 512 M 532 487 L
542 487 M 523 487 L
542 487 L
1699 1342 M B
1718 1323 M 1719 1362 L
1679 1362 L
1679 1323 L
1719 1323 L
CF M
1636 1101 M B
1656 1081 M 1656 1120 L
1617 1120 L
1617 1081 L
1656 1081 L
CF M
1514 912 M B
1533 892 M 1533 932 L
1494 932 L
1494 892 L
1533 892 L
CF M
1380 840 M B
1400 820 M 1400 860 L
1360 860 L
1360 820 L
1400 820 L
CF M
1279 767 M B
1299 747 M 1299 787 L
1259 787 L
1259 747 L
1299 747 L
CF M
1172 704 M B
1191 684 M 1191 724 L
1152 724 L
1152 684 L
1191 684 L
CF M
1048 652 M B
1068 632 M 1068 672 L
1028 672 L
1028 632 L
1068 632 L
CF M
975 623 M B
994 603 M 994 643 L
955 643 L
955 603 L
994 603 L
CF M
895 597 M B
915 578 M 915 617 L
876 617 L
876 578 L
915 578 L
CF M
842 575 M B
862 555 M 862 595 L
822 595 L
822 555 L
862 555 L
CF M
755 562 M B
775 542 M 775 582 L
735 582 L
735 542 L
775 542 L
CF M
1380 840 M 1457 840 L
1457 849 M 1457 830 L
1457 849 L
1380 840 M 1303 840 L
1303 849 M 1303 830 L
1303 849 L
1380 840 M 1380 930 L
1389 930 M 1371 929 L
1389 930 L
1380 840 M 1380 750 L
1389 750 M 1371 750 L
1389 750 L
1952 1986 M B
1980 1986 M 1952 2014 L
1924 1986 L
1952 1958 L
1980 1986 L
CF M
1921 1355 M B
1949 1355 M 1921 1383 L
1893 1355 L
1921 1327 L
1949 1355 L
CF M
1828 1213 M B
1856 1213 M 1828 1241 L
1800 1213 L
1828 1185 L
1856 1213 L
CF M
1705 1158 M B
1733 1158 M 1705 1186 L
1677 1158 L
1705 1130 L
1733 1158 L
CF M
1520 875 M B
1548 875 M 1520 903 L
1492 875 L
1520 847 L
1548 875 L
CF M
1396 799 M B
1424 799 M 1396 827 L
1368 799 L
1396 771 L
1424 799 L
CF M
1273 723 M B
1301 723 M 1273 751 L
1245 723 L
1273 695 L
1301 723 L
CF M
1180 688 M B
1209 688 M 1180 716 L
1152 688 L
1180 660 L
1209 688 L
CF M
1088 652 M B
1116 652 M 1088 680 L
1060 652 L
1088 624 L
1116 652 L
CF M
995 614 M B
1023 614 M 995 642 L
967 614 L
995 586 L
1023 614 L
CF M
903 593 M B
931 593 M 903 621 L
875 593 L
903 565 L
931 593 L
CF M
810 571 M B
838 571 M 810 599 L
782 571 L
810 543 L
838 571 L
CF M
1921 1355 M 2075 1355 L
2075 1364 M 2075 1345 L
2075 1364 L
1921 1355 M 1767 1355 L
1767 1364 M 1767 1345 L
1767 1364 L
1921 1355 M 1921 1788 L
1930 1788 M 1912 1787 L
1930 1788 L
1921 1355 M 1921 922 L
1930 922 M 1912 922 L
1930 922 L
563 1896 M B
588 1882 M 563 1924 L
539 1882 L
588 1882 L
CF M
563 1786 M B
583 1767 M 583 1806 L
543 1806 L
543 1767 L
583 1767 L
CF M
563 1677 M B
591 1677 M 563 1705 L
535 1677 L
563 1649 L
591 1677 L
CF M
627 1874 M CS [] 0 setdash M
640 1929 M 643 1925 L
640 1922 L
636 1925 L
636 1929 L
640 1935 L
643 1938 L
652 1941 L
665 1941 L
675 1938 L
678 1935 L
681 1929 L
681 1922 L
678 1916 L
668 1909 L
652 1903 L
646 1900 L
640 1893 L
636 1884 L
636 1874 L
665 1941 M 672 1938 L
675 1935 L
678 1929 L
678 1922 L
675 1916 L
665 1909 L
652 1903 L
636 1880 M 640 1884 L
646 1884 L
662 1877 L
672 1877 L
678 1880 L
681 1884 L
646 1884 M 662 1874 L
675 1874 L
678 1877 L
681 1884 L
681 1890 L
720 1941 M 710 1938 L
704 1929 L
701 1912 L
701 1903 L
704 1887 L
710 1877 L
720 1874 L
726 1874 L
736 1877 L
742 1887 L
745 1903 L
745 1912 L
742 1929 L
736 1938 L
726 1941 L
720 1941 L
713 1938 L
710 1935 L
707 1929 L
704 1912 L
704 1903 L
707 1887 L
710 1880 L
713 1877 L
720 1874 L
726 1874 M 733 1877 L
736 1880 L
739 1887 L
742 1903 L
742 1912 L
739 1929 L
736 1935 L
733 1938 L
726 1941 L
784 1941 M 774 1938 L
768 1929 L
765 1912 L
765 1903 L
768 1887 L
774 1877 L
784 1874 L
790 1874 L
800 1877 L
806 1887 L
810 1903 L
810 1912 L
806 1929 L
800 1938 L
790 1941 L
784 1941 L
778 1938 L
774 1935 L
771 1929 L
768 1912 L
768 1903 L
771 1887 L
774 1880 L
778 1877 L
784 1874 L
790 1874 M 797 1877 L
800 1880 L
803 1887 L
806 1903 L
806 1912 L
803 1929 L
800 1935 L
797 1938 L
790 1941 L
925 1932 M 928 1922 L
928 1941 L
925 1932 L
919 1938 L
909 1941 L
903 1941 L
893 1938 L
887 1932 L
883 1925 L
880 1916 L
880 1900 L
883 1890 L
887 1884 L
893 1877 L
903 1874 L
909 1874 L
919 1877 L
925 1884 L
903 1941 M 896 1938 L
890 1932 L
887 1925 L
883 1916 L
883 1900 L
887 1890 L
890 1884 L
896 1877 L
903 1874 L
925 1900 M 925 1874 L
928 1900 M 928 1874 L
916 1900 M 938 1900 L
957 1900 M 996 1900 L
996 1906 L
993 1912 L
989 1916 L
983 1919 L
973 1919 L
964 1916 L
957 1909 L
954 1900 L
954 1893 L
957 1884 L
964 1877 L
973 1874 L
980 1874 L
989 1877 L
996 1884 L
993 1900 M 993 1909 L
989 1916 L
973 1919 M 967 1916 L
961 1909 L
957 1900 L
957 1893 L
961 1884 L
967 1877 L
973 1874 L
1015 1941 M 1038 1874 L
1018 1941 M 1038 1884 L
1060 1941 M 1038 1874 L
1009 1941 M 1028 1941 L
1047 1941 M 1066 1941 L
1137 1919 M 1137 1852 L
1140 1919 M 1140 1852 L
1140 1909 M 1147 1916 L
1153 1919 L
1159 1919 L
1169 1916 L
1176 1909 L
1179 1900 L
1179 1893 L
1176 1884 L
1169 1877 L
1159 1874 L
1153 1874 L
1147 1877 L
1140 1884 L
1159 1919 M 1166 1916 L
1172 1909 L
1176 1900 L
1176 1893 L
1172 1884 L
1166 1877 L
1159 1874 L
1127 1919 M 1140 1919 L
1127 1852 M 1150 1852 L
CS [] 0 setdash M
1188 1933 M 1256 1933 L
CS [] 0 setdash M
1204 1919 M 1204 1852 L
1208 1919 M 1208 1852 L
1208 1909 M 1214 1916 L
1220 1919 L
1227 1919 L
1237 1916 L
1243 1909 L
1246 1900 L
1246 1893 L
1243 1884 L
1237 1877 L
1227 1874 L
1220 1874 L
1214 1877 L
1208 1884 L
1227 1919 M 1233 1916 L
1240 1909 L
1243 1900 L
1243 1893 L
1240 1884 L
1233 1877 L
1227 1874 L
1195 1919 M 1208 1919 L
1195 1852 M 1217 1852 L
CS [] 0 setdash M
627 1757 M CS [] 0 setdash M
643 1825 M 636 1793 L
643 1799 L
652 1802 L
662 1802 L
672 1799 L
678 1793 L
681 1783 L
681 1777 L
678 1767 L
672 1760 L
662 1757 L
652 1757 L
643 1760 L
640 1764 L
636 1770 L
636 1773 L
640 1777 L
643 1773 L
640 1770 L
662 1802 M 668 1799 L
675 1793 L
678 1783 L
678 1777 L
675 1767 L
668 1760 L
662 1757 L
643 1825 M 675 1825 L
643 1821 M 659 1821 L
675 1825 L
729 1818 M 729 1757 L
733 1825 M 733 1757 L
733 1825 M 697 1777 L
749 1777 L
720 1757 M 742 1757 L
803 1815 M 800 1812 L
803 1809 L
806 1812 L
806 1815 L
803 1821 L
797 1825 L
787 1825 L
778 1821 L
771 1815 L
768 1809 L
765 1796 L
765 1777 L
768 1767 L
774 1760 L
784 1757 L
790 1757 L
800 1760 L
806 1767 L
810 1777 L
810 1780 L
806 1789 L
800 1796 L
790 1799 L
787 1799 L
778 1796 L
771 1789 L
768 1780 L
787 1825 M 781 1821 L
774 1815 L
771 1809 L
768 1796 L
768 1777 L
771 1767 L
778 1760 L
784 1757 L
790 1757 M 797 1760 L
803 1767 L
806 1777 L
806 1780 L
803 1789 L
797 1796 L
790 1799 L
925 1815 M 928 1805 L
928 1825 L
925 1815 L
919 1821 L
909 1825 L
903 1825 L
893 1821 L
887 1815 L
883 1809 L
880 1799 L
CS M
880 1783 L
883 1773 L
887 1767 L
893 1760 L
903 1757 L
909 1757 L
919 1760 L
925 1767 L
903 1825 M 896 1821 L
890 1815 L
887 1809 L
883 1799 L
883 1783 L
887 1773 L
890 1767 L
896 1760 L
903 1757 L
925 1783 M 925 1757 L
928 1783 M 928 1757 L
916 1783 M 938 1783 L
957 1783 M 996 1783 L
996 1789 L
993 1796 L
989 1799 L
983 1802 L
973 1802 L
964 1799 L
957 1793 L
954 1783 L
954 1777 L
957 1767 L
964 1760 L
973 1757 L
980 1757 L
989 1760 L
996 1767 L
993 1783 M 993 1793 L
989 1799 L
973 1802 M 967 1799 L
961 1793 L
957 1783 L
957 1777 L
961 1767 L
967 1760 L
973 1757 L
1015 1825 M 1038 1757 L
1018 1825 M 1038 1767 L
1060 1825 M 1038 1757 L
1009 1825 M 1028 1825 L
1047 1825 M 1066 1825 L
1137 1802 M 1137 1735 L
1140 1802 M 1140 1735 L
1140 1793 M 1147 1799 L
1153 1802 L
1159 1802 L
1169 1799 L
1176 1793 L
1179 1783 L
1179 1777 L
1176 1767 L
1169 1760 L
1159 1757 L
1153 1757 L
1147 1760 L
1140 1767 L
1159 1802 M 1166 1799 L
1172 1793 L
1176 1783 L
1176 1777 L
1172 1767 L
1166 1760 L
1159 1757 L
1127 1802 M 1140 1802 L
1127 1735 M 1150 1735 L
CS [] 0 setdash M
1188 1817 M 1256 1817 L
CS [] 0 setdash M
1204 1802 M 1204 1735 L
1208 1802 M 1208 1735 L
1208 1793 M 1214 1799 L
1220 1802 L
1227 1802 L
1237 1799 L
1243 1793 L
1246 1783 L
1246 1777 L
1243 1767 L
1237 1760 L
1227 1757 L
1220 1757 L
1214 1760 L
1208 1767 L
1227 1802 M 1233 1799 L
1240 1793 L
1243 1783 L
1243 1777 L
1240 1767 L
1233 1760 L
1227 1757 L
1195 1802 M 1208 1802 L
1195 1735 M 1217 1735 L
CS [] 0 setdash M
627 1641 M CS [] 0 setdash M
678 1686 M 675 1676 L
668 1669 L
659 1666 L
656 1666 L
646 1669 L
640 1676 L
636 1686 L
636 1689 L
640 1698 L
646 1705 L
656 1708 L
662 1708 L
672 1705 L
678 1698 L
681 1689 L
681 1669 L
678 1657 L
675 1650 L
668 1644 L
659 1641 L
649 1641 L
643 1644 L
640 1650 L
640 1653 L
643 1657 L
646 1653 L
643 1650 L
656 1666 M 649 1669 L
643 1676 L
640 1686 L
640 1689 L
643 1698 L
649 1705 L
656 1708 L
662 1708 M 668 1705 L
675 1698 L
678 1689 L
678 1669 L
675 1657 L
672 1650 L
665 1644 L
659 1641 L
720 1708 M 710 1705 L
704 1695 L
701 1679 L
701 1669 L
704 1653 L
710 1644 L
720 1641 L
726 1641 L
736 1644 L
742 1653 L
745 1669 L
745 1679 L
742 1695 L
736 1705 L
726 1708 L
720 1708 L
713 1705 L
710 1702 L
707 1695 L
704 1679 L
704 1669 L
707 1653 L
710 1647 L
713 1644 L
720 1641 L
726 1641 M 733 1644 L
736 1647 L
739 1653 L
742 1669 L
742 1679 L
739 1695 L
736 1702 L
733 1705 L
726 1708 L
784 1708 M 774 1705 L
768 1695 L
765 1679 L
765 1669 L
768 1653 L
774 1644 L
784 1641 L
790 1641 L
800 1644 L
806 1653 L
810 1669 L
810 1679 L
806 1695 L
800 1705 L
790 1708 L
784 1708 L
778 1705 L
774 1702 L
771 1695 L
768 1679 L
768 1669 L
771 1653 L
774 1647 L
778 1644 L
784 1641 L
790 1641 M 797 1644 L
800 1647 L
803 1653 L
806 1669 L
806 1679 L
803 1695 L
800 1702 L
797 1705 L
790 1708 L
925 1698 M 928 1689 L
928 1708 L
925 1698 L
919 1705 L
909 1708 L
903 1708 L
893 1705 L
887 1698 L
883 1692 L
880 1682 L
880 1666 L
883 1657 L
887 1650 L
893 1644 L
903 1641 L
909 1641 L
919 1644 L
925 1650 L
903 1708 M 896 1705 L
890 1698 L
887 1692 L
883 1682 L
883 1666 L
887 1657 L
890 1650 L
896 1644 L
903 1641 L
925 1666 M 925 1641 L
928 1666 M 928 1641 L
916 1666 M 938 1666 L
957 1666 M 996 1666 L
996 1673 L
993 1679 L
989 1682 L
983 1686 L
973 1686 L
964 1682 L
957 1676 L
954 1666 L
954 1660 L
957 1650 L
964 1644 L
973 1641 L
980 1641 L
989 1644 L
996 1650 L
993 1666 M 993 1676 L
989 1682 L
973 1686 M 967 1682 L
961 1676 L
957 1666 L
957 1660 L
961 1650 L
967 1644 L
973 1641 L
1015 1708 M 1038 1641 L
1018 1708 M 1038 1650 L
1060 1708 M 1038 1641 L
1009 1708 M 1028 1708 L
1047 1708 M 1066 1708 L
1137 1686 M 1137 1618 L
1140 1686 M 1140 1618 L
1140 1676 M 1147 1682 L
1153 1686 L
1159 1686 L
1169 1682 L
1176 1676 L
1179 1666 L
1179 1660 L
1176 1650 L
1169 1644 L
1159 1641 L
1153 1641 L
1147 1644 L
1140 1650 L
1159 1686 M 1166 1682 L
1172 1676 L
1176 1666 L
1176 1660 L
1172 1650 L
1166 1644 L
1159 1641 L
1127 1686 M 1140 1686 L
1127 1618 M 1150 1618 L
CS [] 0 setdash M
1188 1700 M 1256 1700 L
CS [] 0 setdash M
1204 1686 M 1204 1618 L
1208 1686 M 1208 1618 L
1208 1676 M 1214 1682 L
1220 1686 L
1227 1686 L
1237 1682 L
CS M
1243 1676 L
1246 1666 L
1246 1660 L
1243 1650 L
1237 1644 L
1227 1641 L
1220 1641 L
1214 1644 L
1208 1650 L
1227 1686 M 1233 1682 L
1240 1676 L
1243 1666 L
1243 1660 L
1240 1650 L
1233 1644 L
1227 1641 L
1195 1686 M 1208 1686 L
1195 1618 M 1217 1618 L
CS [] 0 setdash M
1823 547 M CS [] 0 setdash M
1876 667 M 1866 657 L
1857 643 L
1847 624 L
1842 599 L
1842 580 L
1847 556 L
1857 537 L
1866 522 L
1876 513 L
1866 657 M 1857 638 L
1852 624 L
1847 599 L
1847 580 L
1852 556 L
1857 542 L
1866 522 L
1883 667 M 1873 657 L
1864 643 L
1854 624 L
1849 599 L
1849 580 L
1854 556 L
1864 537 L
1873 522 L
1883 513 L
1873 657 M 1864 638 L
1859 624 L
1854 599 L
1854 580 L
1859 556 L
1864 542 L
1873 522 L
1975 599 M 1970 595 L
1975 590 L
1979 595 L
1979 599 L
1970 609 L
1960 614 L
1946 614 L
1931 609 L
1922 599 L
1917 585 L
1917 575 L
1922 561 L
1931 551 L
1946 547 L
1955 547 L
1970 551 L
1979 561 L
1946 614 M 1936 609 L
1926 599 L
1922 585 L
1922 575 L
1926 561 L
1936 551 L
1946 547 L
1982 599 M 1977 595 L
1982 590 L
1987 595 L
1987 599 L
1977 609 L
1967 614 L
1953 614 L
1938 609 L
1929 599 L
1924 585 L
1924 575 L
1929 561 L
1938 551 L
1953 547 L
1963 547 L
1977 551 L
1987 561 L
1953 614 M 1943 609 L
1934 599 L
1929 585 L
1929 575 L
1934 561 L
1943 551 L
1953 547 L
2020 667 M 2030 657 L
2040 643 L
2049 624 L
2054 599 L
2054 580 L
2049 556 L
2040 537 L
2030 522 L
2020 513 L
2030 657 M 2040 638 L
2044 624 L
2049 599 L
2049 580 L
2044 556 L
2040 542 L
2030 522 L
2028 667 M 2037 657 L
2047 643 L
2056 624 L
2061 599 L
2061 580 L
2056 556 L
2047 537 L
2037 522 L
2028 513 L
2037 657 M 2047 638 L
2052 624 L
2056 599 L
2056 580 L
2052 556 L
2047 542 L
2037 522 L
CS [] 0 setdash M
stroke
grestore
showpage
end
%%EndDocument

 endTexFig
 73 2525 a Fe(FIG.)13 b(1.)k(Plots)d(of)386 2514 y(~)377 2525
y Fd(F)404 2531 y Fc(q)435 2525 y Fe(as)g(de\014ned)h(in)e(Eq.)g(\(61\))h(v)o
(ersus)h Fd(F)992 2531 y Fb(2)1024 2525 y Fe(for)e(3)e Fa(\024)h
Fd(q)g Fa(\024)g Fe(5.)18 b(Data)13 b(are)h(tak)o(en)g(from)e(Ref.)g([2].)17
b(The)73 2572 y(dashed)12 b(line)f(corresp)q(onds)j(to)e(the)g(predictions)g
(of)f(the)i(ordinary)e(lattice)h(gas)f(\(Ising\))h(mo)q(del.)k(The)c(b)q(old)
f(face)h(line)73 2620 y(represen)o(ts)19 b(the)f(negativ)o(e)e(binomial)e
(distribution.)26 b(The)17 b(shaded)h(area)f(is)g(the)g(prediction)g(of)f
(the)i(generalized)73 2668 y(lattice)13 b(gas)h(mo)q(del)e(according)i(to)f
(our)h(\014rst)h(prescription)f(for)g Fd(D)f Fe(=)e(16.)18
b(This)13 b(`band')g(around)h(the)g(NB)g(b)q(ecomes)73 2716
y(narro)o(w)o(er)g(as)h Fd(D)g Fe(increases.)20 b(The)15 b(error)g(bars)f
(sho)o(wn)h(are)f(represen)o(tativ)o(e)i(for)e(the)h(quoted)f(uncertain)o
(ties)h(in)f(the)73 2763 y(data)f(\(see)j(the)e(text)h(for)e(further)i
(commen)o(ts)d(on)i(the)g(error)h(bars\).)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF

