%Paper: hep-th/9303075
%From: Robert Lacaze <lacaze@amoco.saclay.cea.fr>
%Date: 12 Mar 93 09:52:10+0100


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%TITLE PAGE
\begingroup\titlefont\obeylines
\hfil The free energy of the Potts model :\hfil
\hfil from the continuous \hfil
\hfil to the first-order transition region.\hfil
\endgroup\bigskip
%

\medskip
\centerline{ T. Bhattacharya\footnote{*}%
{Present address: MS B285, Group T-8, Los Alamos National Laboratory,
NM 87544, U.S.A.}, R. Lacaze\footnote{**}%
{Chercheur au CNRS} and A. Morel}
\centerline{Service de Physique Th\'eorique de Saclay\footnote{***}%
{Laboratoire de la Direction des Sciences de la Mati\`ere du CEA }}
\centerline{91191 Gif-sur-Yvette Cedex, France}

%
\bigskip\bigskip\bigskip\centerline{{\sectnfont ABSTRACT}}\medskip
We present a large $q$ expansion of the 2d $q$-states Potts model free
energies up
to order 9 in $1/\sqrt{q}$. Its analysis leads us to an ansatz which,
in the first-order region, incorporates properties inferred from the known
critical regime at $q=4$, and predicts, for $q>4$, the $n^{\rm th}$ energy
 cumulant scales as the power $(3 n /2-2)$ of the correlation length.
The parameter-free energy distributions reproduce accurately, without
reference to any interface effect,  the numerical data obtained
in a simulation for $q=10$ with lattices of linear dimensions up to $L=50$.
The pure phase specific heats are predicted to be much larger, at $q\leq10$,
than the values extracted from current finite size scaling analysis of extrema.
Implications for safe numerical determinations of interface tensions are
discussed.

%
\bigskip\bigskip\bigskip\bigskip\banner
\bigskip\noindent Submitted for publication to {\sl Europhysics Letters}
\vfil\eject

%
Much effort has been recently devoted to the 2-d Potts model, both with
numerical and analytical techniques. In the former case, the goal was
either to test numerical
algorithms and criteria for distinguishing first-order from continuous
transitions, or to learn how to extract previously unknown quantities such as
the interface tension.
Although much progress was accomplished, there remains some unsatisfactory
issues such as, for example, slight inconsistencies in finite size scaling
analysis of the energy cumulants close to the transition temperature
$\beta_{\rm t}^{-1}$, and discrepancies between exact results and numerical
simulations for the interface tension.
This question is important since only numerical simulations can determine this
quantity in other cases of physical interest such as the 3-d $q=3$ Potts model
or QCD at the deconfinement transition.

Analytical works have shown \ref{boko} that close to $\beta_{\rm t}$, the
partition function $Z$ of the Potts model, in a box of volume $V=L^2$ with
periodic boundary conditions (used all through in this work),
is equal to the sum of the `partition functions' $Z_i$ of the $q+1$
pure phases, and a term that falls of exponentially faster in the linear
size of the system. We shall presently ignore
this latter term which contains, amongst others, the interface tension effects,
and concentrate on the $i^{\rm th}$ phase free energy
$F_i= \ln Z_i/V$, which is $V$ independent and differentiable  many times with
 respect to the inverse temperature $\beta$ at $\beta_{\rm t}$.

In this letter, we construct explicit formulae for the ordered and disordered
free energies $F_{\rm o}$ and $F_{\rm d}$ of the 2-d $q$-states Potts model
at $q>4$. At large $q$ we perform their expansion in power of $1/\sqrt{q}$.
At low $(q-4)$ we conjecture their behaviour from their known critical
behaviour at $q=4$. We show that these two descriptions match in a large
intermediate $q$ region. Then, using the above rigorous result, we add up the
$Z_i$'s so obtained and make absolute predictions on energy probability
densities in very good agreement with numerical data.
We show that the difficulties encountered in the finite size
scaling analysis of numerical data are due to very large high order cumulants,
which has consequences for the extraction of interface tensions.

Many properties of the model are known exactly \ref{wu}.
In particular, it exhibits a temperature driven phase
transition which occurs at $\beta_{\rm t}=\ln(\sqrt q+1)$.
The transition is second-order for $q \leq 4$, and its critical properties
are described, e.g., by the $\alpha$ and $\nu$ indices, which at $q=4$
take the common value 2/3. Accordingly, the correlation length and the
specific heat there diverge as
$$\xi_{q=4}\sim \mid \beta -\beta_{\rm t} \mid ^{-2/3}\qquad,\qquad
C_{q=4}\sim \mid \beta -\beta_{\rm t} \mid ^{-2/3}. \EQNO{xi4}$$
Hence in the vicinity of $\beta =\beta_{\rm t}$, the ratio ${C / \xi}$ remains
finite at $q=4_-$.

The first-order transition region is $q>4$. There the energies
$E_{\rm o}$ and $E_{\rm d}$ of the ordered and disordered phases
respectively are exactly known at $\beta_{\rm t}$ \ref{baxt}. Recent
works \refand{xi}{bogs} on the largest correlation
length at $\beta_{\rm t}$ have shown a common behaviour as $q\rightarrow 4_+$
$$\xi={1\over 8\sqrt{2}} \ x \ (1+{\cal O}(x^{-2}))\qquad {\rm with} \qquad
x=\exp({\pi^2\over 2 \ln{1\over 2}(\sqrt{q}+\sqrt{q-4})}). \EQNO{xi}$$
These formulae show not only that $\xi$ rapidly diverges as
$q\rightarrow 4_+$, but also that the leading behaviour of \eq{xi} {\it is
accurate over a very wide range of $q$ values}. For example the correction
term in \eq{xi} is still of the order of 1\% for $q$ as large as 75.
The pure phase specific heats $C_{\rm o}$ and $C_{\rm d}$ are unknown,
but their known difference vanishes when $q\rightarrow 4_+$ as
$x^{-{1\over2}}$. These properties are of course in accordance with
the point ( $q=4, \beta=\beta_{\rm t}$ )
being a second-order transition point, and lead one to speculate that
$C_{\rm d}\sim C_{\rm o}$ diverges as $q\rightarrow 4_+$, possibly
in such a way that the ratio $C / \xi$ is finite on both sides of $q=4$.

First we shall take advantage of the fact that, for the correlation length
and latent heat, the "small $q-4$ " region extends in practice up to large
$q$ values, and start from the opposite end. We compute the free energy of
 the model in the framework of a large $q$ expansion, extrapolate down in
$q$ as far as we can, and analyze the resulting energy cumulants as functions
of $x$ in an intermediate $q$-value region. Nice regularities emerge, among
which a smooth behaviour of $C / x$ is ascertained.

The large $q$ expansion of the ordered free energy $F_{\rm o}$ (that for
$F_{\rm d}$ in the disordered phase follows from duality), was obtained
through the Fortuin-Kasteleyn \ref{kast} representation of the Potts model
partition function
$$Z=\sum_{X} ({\rm e}^{\beta}-1)^l q^n \EQNO{FK}$$
where $X$ is any configuration of bonds on a cubic
lattice, $l$ its number of bonds and $n$ its number of connected components,
or clusters of sites ( two sites bound to each other belong to the same
cluster, an isolated site is a cluster). The completely ordered
configuration corresponds to $n=1$ and $l=2 V$.
So the partition function can be reorganized as an
expansion in $q^{-{1\over2}}$ about this configuration:
$$Z_{\rm o} = q ({\rm e}^{\beta}-1)^{2V} \sum_{l\geq 0,n\geq 0} N_{l,n}(V)
    ({{\rm e}^{\beta}-1\over\sqrt{q}})^{-l}q^{n-{l\over2}}, \EQNO{Zo}$$
where $N_{l,n}(V)$ is the number of configurations in a volume $V$ with $l$
removed bonds and $n+1$ clusters.
The enumeration of all the $N_{l,n}(V)$  such that $(l-2n)\leq M$
yields an expansion of $Z_{\rm o}$ to order $M$.
Details will be given elsewhere \ref{futur}.
To any given finite order $M$, a large enough volume $V$ can be chosen to
eliminate all  boundary terms, so that all
configurations retained correspond to {\it disordered islands} in a bulk
{\it ordered phase}. We check that the sum in \eq{Zo} exponentiates in $V$
up to terms of order $M+1$, defining a series for $F_{\rm o}$
truncated beyond order $M$. We have computed up to order $M=9$,
including terms up to $N_{49,20}(V)$.
This series, whose first terms can be compared to existing low
temperature series \ref{enting}, provides us with similar series for the
$k^{\rm th}$ derivative with respect to $\beta$, $F_{\rm o}^{(k)}$.
At $\beta=\beta_{\rm t}$  the $k=0$ (free energy) and $k=1$ (internal energy)
series match the exact results \ref{baxt} up to $M=9$.
The $k=2 \ {\rm and} \ 3$ cases give
$$F_{\rm o}^{(2)}={16\over q}+{34\over q^{3/2}}+{114\over q^2}
+{254\over q^{5/2}} +{882\over q^3}+{1944\over q^{7/2}}
+{6128\over q^4}+{13550\over q^{9/2}} , \EQNO{Foq}$$
$$F_{\rm o}^{(3)}=-{64\over q}-{430\over q^{3/2}}-{2654\over q^2}
-{12186\over q^{5/2}}-{57018\over q^3}-{224732\over q^{7/2}}
-{888024\over q^4}-{3164682\over q^{9/2}} .  \EQNO{Fdq}$$
Let us make a few comments on these expansions.
\item{(i)} $F_{\rm o}^{(2)}$ gives the specific heat
 $C_{\rm o}=\beta_{\rm t}^2 F_{\rm o}^{(2)}$. The $F_{\rm o}^{(3)}$ series
gives -- \avg{(E-E_{\rm o})^3}, the first odd moment of the
energy distribution in this phase.
\item{(ii)} All terms in each series have the same sign, $F^{(2)}$
being of course positive while $F^{(3)}<0$ means that $E>E_{\rm o}$ is
 favoured with respect to $E<E_{\rm o}$.
\item{(iii)} The coefficients are fastly increasing with the order,
the more so for larger values of $k$. This confirms the expectation that
huge energy fluctuations are associated with the large correlation
length, of order $x$ in \eq{xi} when $q$ decreases towards $q=4$.
This is expected to have a direct impact on the numerical analysis of these
models, large values of the high cumulants invalidating the commonly used two
gaussian formula \ref{challa}.
\bigskip
To proceed with a quantitative analysis of $F^{(2)}$ and $F^{(3)}$ as
functions of $q$, we conjecture for these quantities an essential
singularity at $q=4$ as $\xi$ has and construct Pad\'e approximants for the
series of $\ln F^{(k)}$ instead of $F^{(k)}$, i.e. take as estimates
of $F^{(k)}$
$$F_{\rm est}^{(k)}=\exp \bigl[ \hbox{ Pad\'e}(\ln F^{(k)}) \bigr ]
   \EQNO{FPad}$$
The result of this construction for $F^{(2)}$ and $F^{(3)}/F^{(2)}$
as functions of $x$ for $q=30,20,15,10,8,7 \ {\rm and} \ 6$ is summarized in
\fig{1} as a log-log plot.
The error bars are rough estimates of the uncertainties resulting from the
higher order terms and have been obtained by varying the degrees of the
numerator and denominator of the Pad\'e approximant. The lines represent
 our prejudices $F^{(2)}\simeq {\rm Cst} \ x$ and
$F^{(3)}/F^{(2)}\simeq {\rm Cst} \ x^{3/2}$ (see below), where the constants
are fixed by the $q=10$ values.
It is clear that the general trend of both quantities is well reproduced
over an astonishingly large range by such simple forms.

As a by-product of this study we obtain analytical estimates for the ordered
phase specific heat, which are compared to existing numerical data in Table 1.
\vskip 13.5 truecm
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\centerline{\bf Table 1}
\noindent{\it The ordered phase specific heat, our prediction compared
to numerical estimates}
\bigskip
At $q=20$ our prediction is in good agreement with the numerical estimate.
In contrast it strongly disagrees at $q\leq 10$ with the value of $C_{\rm o}$
obtained from a finite size analysis of the maximum of the specific heat
measured in the coexistence regime. However it agrees at $q=10$ with
the estimate obtained in \ref{bilcom} at $\beta=\beta_{\rm t}$, in accordance
with the rigorous statements of \ref{boko}.
\bigskip
The large $q$ expansion analysis supports the idea that not only does
the correlation length in a pure phase diverge at $\beta=\beta_{\rm t}$,
$q\rightarrow 4_+$, but also that the associated fluctuations imply
divergences of the energy cumulants. For example we obtain
$F_{\rm o}^{(3)} \sim -1800$ at $q=10$ (see \fig{1}).
Moreover the internal energy fluctuations behave
in a way consistent with $C / \xi$ being finite at $q=4_+$
as it is known to be at  $q=4_-$.
We then propose the following {\it ansatz}:
\item  {}There exists a region of $q>4$ where the free energies around
$\beta=\beta_{\rm t}$ reflect accurately the scaling properties associated with
the second-order point lying at $q=4 , \beta = \ln 3$ and characterized
 by the corresponding critical indices $\alpha$ and $\nu$.

\noindent Specifically, according to the known value 2/3 of $\alpha$,
we parametrize $F_{\rm o}^{(2)}(\beta)$  at
$q=4$ and for $\beta\rightarrow (\beta_{\rm t})_+$ as
$$F_{\rm o}^{(2)}(\beta)=A \ (\beta-\beta_{\rm t})^{-2/3} . \EQNO{F2S}$$
The higher derivatives are trivially deduced and in their
expressions we replace $(\beta-\beta_{\rm t})$ by ( Cst ~ $\xi^{-3/2}$ )
from \eq{xi4} and, boldly continuing above $q=4$ at $\beta=\beta_{\rm t}$,
reexpress $F^{(p+2)}_{\rm o}$ as a function of $x$ via \eq{xi} to get
$$F^{(p+2)}\equiv F_{\rm o}^{(p+2)}(\beta_{\rm t})=A \ (-)^p
 {\Gamma(2/3+p)\over\Gamma(2/3)} (Bx)^{1+3p/2} , \EQNO{Fx}$$
where the constant $B$ takes into account proportionality constants.
%between $(\beta-\beta_{\rm t}) \ {\rm and} \ \xi, \xi \ {\rm and} \ x$.
Thus we get the following representation for the ordered phase free energy
$$F_{\rm o}(\beta)=F(\beta_{\rm t})-E_{\rm o} (\beta-\beta_t) +
 \sum_{n=2}^{\infty} (\beta-\beta_{\rm t})^n {F^{(n)}\over n!} \EQNO{FoB}$$
where we introduced the known linear term ($F^{(1)}=-E_{\rm o}$), and took
$F^{(n)}$ as given by \eq{Fx} for $n>1$.
Note that all the odd cumulants are negative, as we found to be the case
for $F^{(3)}$ from the large $q$ expansion.
A similar expression holds for $F_{\rm d}$, starting from \eq{F2S} with
$(\beta-\beta_{\rm t}) \rightarrow (\beta_t-\beta)$, and replacing
 $E_{\rm o}$ by $E_{\rm d}$ in \eq{FoB}.

It is easy to sum the series \eq{FoB}. For later convenience, we introduce
scaled temperature ($v$), energy ($\epsilon$) and length (${\cal S}$) variables
$$v=(\beta-\beta_{\rm t}) (B x)^{3/2}\qquad ,\qquad
\epsilon={E (Bx)^{1/2}\over 3A} \qquad ,\qquad
{\cal S}^2={(Bx)^2\over 3A} \EQNO{scaled}$$
and end up with the following compact result
$$F_{\rm o}(\beta)=F(\beta_{\rm t})+{1\over {\cal S}^2} \bigl [
 -{3\over4}-(\epsilon_{\rm o}+1) v +{3\over4}(1+v)^{4/3} \bigr] \EQNO{final}$$
This equation is the central result of this letter. We claim that, although we
neglected all regular and less singular contributions to \eq{F2S}, \eq{final}
summarizes accurately, over a wide range of $q>4$ values, all the properties of
the model.

Let us justify this statement by comparing the predictions of \eq{final}
for the energy distribution to data \ref{bilcom} taken at $q=10$ with
various $L$ values. This distribution is obtained by inverse Laplace
 transform of the partition function
$$ P_V(E)=N_1\int_{\beta_0-i\infty}^{\beta_0+i\infty}{\rm d}\beta \Bigl [
 q\exp[V F_{\rm o}(\beta)]+\exp[V F_{\rm d}(\beta)]
\Bigr ] \exp[VE(\beta-\beta_{\rm t})] \EQNO{PV}$$
where $F_{\rm d}$ follows from \eq{final} by $v\rightarrow-v$ and
$\epsilon_{\rm o}\rightarrow\epsilon_{\rm d}$ (this is consistent with
duality up to terms of order $x^{-3/2}$ as compared to 1) and
with $N_1$ a factor ensuring
\vfill\eject
{}~~
\vskip 15.5 truecm
\special{psfile=fig2.ps voffset=-260 hoffset=-10 hscale=80 vscale=80}

\noindent the probability normalization.
Trading $(\beta-\beta_{\rm t})$ for $v$ and $E$ for $\epsilon$
of \eq{scaled}, we get
$$P_V^{\rm o}(E)=N_2\int_{v_{\rm o}-i\infty}^{v_{\rm o}+i\infty}
{\rm d}v\exp\bigl[({L\over{\cal S}})^2 [(\epsilon-\epsilon_{\rm o}-1)v
+{3\over4}(1+v)^{4/3})]\bigr] \EQNO{PVo}$$
Details on the computation of this integral will be given in \ref{futur}
and here we limit ourselves to short remarks.
\item{i)} As a function of $\epsilon-\epsilon_{\rm o}$, $P_V^{\rm o}$
depends on $q$ and $L$
only through the scaled volume $({L / {\cal S}})^2$.
\item{ii)} Any $v_0>-1$ is suitable and the integral converges exponentially.
\vfill\eject
{}~~
\vskip 15.5 truecm
\special{psfile=fig3.ps voffset=-260 hoffset=-10 hscale=80 vscale=80}

\item{iii)} At large $(L/{\cal S})$, a saddle point method can be valuable.
However $(L/{\cal S})$ is not large in practice and there exists an energy
value slightly above $E_{\rm o}$  where the saddle point value reaches $v=-1$
(metastability point).
\item{iv)} Actual computation requires numerical values of $A$ and $B$. At
$q=10$ we get $A=.193$ and $B=.386$ from the large $q$ expansion results
(at other $q$ values, slight changes have to be made according to \fig{1} ).
\item{v)} At $q=10$ the length scale is ${\cal S}\sim 60$,
nearly 6 times the correlation length, so that for current values of $L$
the ratio $(L/{\cal S})$ is hardly of order 1 !

For the above reasons, we compute the integral \eq{PVo} numerically.
The results are shown together with the data of \ref{bilcom} on \fig{2}
for $L=16,20,24$, and on \fig{3} for $L=36,44,50$.
Remembering that the continuous curves are absolute predictions
without any free parameter, it is quite striking to see how such a simple
ansatz as \eq{PVo} yields good results. Because $(L/{\cal S})$ is not large,
they are very different from what an asymptotic expansion would give.
For example $E_{\rm peak}-E_{\rm o}$ behaves effectively as $\sim 1/L$
over a wide range of $L$ values, whereas the asymptotic expectation (saddle
point method in \eq{PV}) is

$L^2(E_{\rm o}-E_{\rm peak})=-F_{\rm o}^{(3)} / ( 2F_{\rm o}^{(2)})\sim
 15,100,350,1000 \quad{\rm at}\quad q=20,10,8,7$.

Marked discrepancies only appear at the external edges of the ordered (left)
and disordered (right) peaks. In particular the theoretical curve levels out
unduly at the bottom right of \fig{2}. This is a spurious effect :
the tail of the ordered peak contributes more than the disordered phase.
However it appears at a negligibly small level at larger $L$'s,
being asymptotically of order $\exp[-c \ L^2]$.
As $L$ is increased (\fig{3}), the agreement between predictions and data
becomes better and better at nearly all values of $E$, but around the dip
between the peaks. There indeed mixed phase contributions with percolating
interfaces should finally win over the pure phase contributions.
We consider the small departure of the theoretical
curve below the data points at $L=50$ as an evidence for the emergence of
mixed phase contributions. Since the dip region is often
used for the determination of the interface tension $\sigma$ because
 strip configurations eventually yield an
energy independent plateau in $P_V(E)$ with \ref{binder}
$$2 \ \sigma \simeq -{1\over L} \ln {P_V(E_{{\rm dip}}) \over
 P_V(E_{{\rm peak}})} , \EQNO{2sig}$$
the value and $L$ dependence of the right hand side of \eq{2sig} are
interesting issues. Although in our construction this quantity diverges as $L$
asymptotically, we unexpectedly find it roughly constant for $L\leq 50$,
at a value around .11, not very far from the exact value
$1/\xi_{\rm d}=0.95$ \ref{bogs}.
This casts some doubt on attempts to determine $\sigma$ from \eq{2sig}
\refand{janke}{rummu} unless the plateau in $E$ is seen
\refand{plateau}.

Summarizing, we have shown, for the 2-d Potts model at $q>4$, up to large $q$
values, that the bulk properties are accurately described close to
$\beta_{\rm t}$ by free energies $F_{\rm o}(\beta)$
and $F_{\rm d}(\beta)$ inferred by a simple ansatz from the known properties
of the second-order point at $q=4$.
Definite expressions for $F_{\rm o}(\beta)$ and $F_{\rm d}(\beta)$
were obtained by matching this ansatz to a calculation of their large $q$
expansion, achieved at order $(1/\sqrt{q})^9$.
Our main result is expressed in \eq{final} and illustrated by the adequacy
of the corresponding energy distribution to explain most of the features
observed in numerical simulations at finite volume.

In the first-order region, each phase knows little about
the existence of the other ones, and rather feels the effectively close
critical point $q=4,\beta=\beta_{\rm t}$, from which it inherits nice
scaling properties. This is so as long as the linear size of the box
is smaller than the length scale ${\cal S}$, proportional to the correlation
length $\xi$, but many times larger ( ${\cal S}= 5 \ {\rm to} \ 7 \xi$ in
the range $q=7 \ {\rm to} \ 20$).
Only at $V\geq{\cal S}^2$ asymptotics takes place, so that interface tension
effects can show up and be measured.

Similar ideas could be applied to other situations. One such situation is
the 3-d Potts model at $q\geq3$ where the first-order regime could
also be influenced by the second-order point $q_{3 \rm d}$ situated
between $q=2$ (Ising) and $q=3$. Then one
might get some information on this critical point from numerical studies
at $q\geq3$. Another interesting case is QCD at finite temperature;
although the transition is first-order, universality might be nevertheless
invoked to relate its behaviour to that of the 3-d 3 states Potts model,
both models sharing the same (universal) behaviour at
a "close" point of parameter space.

We are aware of the fact that our ansatz describes the pure phase free energies
as analytic functions of $\beta$ at $\beta_{\rm t}$, which they probably are
not, in the same way as in field driven first-order transitions the zero-field
point is an essential singularity of the free energy \ref{isak}.
This question deserves further discussion, but we believe that our ansatz,
although not `analytically correct', takes into account the most significant
 features of the model.

It is a pleasure to thank R. Balian and A. Billoire for illuminating
 discussions.
We also acknowledge useful conversations with B. Grossmann and T. Neuhaus.
\vfill\eject
%
\bigskip\centerline{\sectnfont References}\bigskip
\item {\reftag{boko})} C.~Borgs and R.~Koteck\'y, J. Stat. Phys. 61 (1990) 79;
C.~Borgs, R.~Koteck\'y and S.~Miracle-Sol\'e, J. Stat. Phys. 62 (1991) 529.
\item{\reftag{wu})} F.~Y.~Wu, Rev. Mod. Phys. 54 (1982) 235.
\item{\reftag{baxt})} R.~J.~Baxter, J. Phys. C6 (1973) L445.
\item{\reftag{xi})} A.~Kl\"umper, A.~Schadschneider and J.~Zittartz,
Z Phys. B76 (1989) 247.
\item \    E.~Buffenoir and S.~Wallon, to appear in Journal of Phys. A.
\item{\reftag{bogs})} C.~Borgs and W.~Janke,
{\it An explicit formula for the interface tension of the 2D Potts Model}
Preprint FUB-HEP 13/92, HRLZ 54/92.
\item{\reftag{kast})} P.~W.~Kasteleyn and C.~M.~Fortuin, J. Phys. Soc. Japan
{\bf 26} (Suppl.), 11 (1969).
\item{\reftag{futur})} T.~ Bhattacharya, R.~Lacaze and A.~Morel,
 in preparation.
\item{\reftag{enting})} I.~G.~Enting, J. Phys. A10 (1977) 325.
\item {\reftag{challa})} M.~S.~S.~Challa, D.~P.~Landau and K.~Binder,
      Phys. Rev. B34 (1986) 1841.
\item {\reftag{bil20})} A.~Billoire, T.~Neuhaus and B.~Berg,
Nucl. Phys. B(in press).
\item {\reftag{bilcom})} A.~Billoire, R.~Lacaze and A.~Morel,
Nucl. Phys. B 370 (1992) 773.
\item{\reftag{koster})} J.~Lee and J.~M.~Kosterlitz,
 Phys. Rev. B43 (1991) 3265.
\item {\reftag{janke})}  W.~Janke, B.~Berg and M.~Katoot,
 Nucl. Phys. B382 (1992) 649.
\item {\reftag{rummu})}  K.~Rummukainen,
{\it Multicanonical Cluster Algorithm and the 2-D 7-State Potts Model},
Preprint CERN-TH.6654/92.
\item {\reftag{binder})} K.~Binder, Phys. Rev A25 (1982) 1699;
Z. Phys. B43 (1981) 119.
\item {\reftag{isak})} M.~E.~Fisher, Physics (N.Y.) 3 (1967) 255;
S.~N.~Isakov, Comm. Math. Phys. 95 (1984) 427.
\item {\reftag{plateau})} B.~Berg, U.~Hansmann and T.~Neuhaus,
to appear in Proceedings
"Computer Simulations Studies in Condensed Matter Physics", Athens 1992;
\item {} B.~Grossmann and M.~L.~Laursen, {\it The Confined-Deconfined Interface
Tension in Quenched QCD using the Histogram Method}, Preprint HLRZ-93-7;
\item {} A.~Billoire, T.~Neuhaus and B.~Berg, {\it A Determination of Interface
Free Energies}, in preparation.
\vfil\end
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844 972 lv
839 972 mv
839 977 lv
842 980 lv
847 980 lv
849 977 lv
849 972 lv
847 970 lv
842 970 lv
839 972 lv
827 1001 mv
828 1001 lv
827 998 mv
828 998 lv
828 1001 mv
828 998 lv
823 997 mv
823 1002 lv
825 1004 lv
830 1004 lv
833 1002 lv
833 997 lv
830 994 lv
825 994 lv
823 997 lv
811 1027 mv
812 1027 lv
811 1024 mv
812 1024 lv
811 1027 mv
811 1024 lv
806 1023 mv
806 1028 lv
809 1030 lv
814 1030 lv
816 1028 lv
816 1023 lv
814 1020 lv
809 1020 lv
806 1023 lv
794 1052 mv
795 1052 lv
794 1049 mv
795 1049 lv
795 1052 mv
795 1049 lv
790 1048 mv
790 1053 lv
792 1055 lv
797 1055 lv
800 1053 lv
800 1048 lv
797 1045 lv
792 1045 lv
790 1048 lv
778 1079 mv
779 1079 lv
778 1076 mv
779 1076 lv
779 1079 mv
779 1076 lv
774 1075 mv
774 1080 lv
776 1083 lv
781 1083 lv
784 1080 lv
784 1075 lv
781 1072 lv
776 1072 lv
774 1075 lv
762 1103 mv
763 1103 lv
762 1101 mv
763 1101 lv
762 1103 mv
762 1101 lv
757 1099 mv
757 1104 lv
760 1107 lv
765 1107 lv
767 1104 lv
767 1099 lv
765 1097 lv
760 1097 lv
757 1099 lv
746 1126 mv
746 1126 lv
746 1125 mv
746 1125 lv
746 1126 mv
746 1125 lv
741 1123 mv
741 1128 lv
743 1131 lv
748 1131 lv
751 1128 lv
751 1123 lv
748 1120 lv
743 1120 lv
741 1123 lv
729 1148 mv
729 1148 lv
729 1147 mv
729 1147 lv
729 1148 mv
729 1147 lv
724 1145 mv
724 1150 lv
727 1153 lv
732 1153 lv
734 1150 lv
734 1145 lv
732 1142 lv
727 1142 lv
724 1145 lv
712 1172 mv
713 1172 lv
712 1169 mv
713 1169 lv
713 1172 mv
713 1169 lv
708 1168 mv
708 1173 lv
710 1176 lv
715 1176 lv
718 1173 lv
718 1168 lv
715 1165 lv
710 1165 lv
708 1168 lv
696 1192 mv
697 1192 lv
696 1190 mv
697 1190 lv
697 1192 mv
697 1190 lv
692 1189 mv
692 1194 lv
694 1196 lv
699 1196 lv
702 1194 lv
702 1189 lv
699 1186 lv
694 1186 lv
692 1189 lv
680 1213 mv
681 1213 lv
680 1211 mv
681 1211 lv
680 1213 mv
680 1211 lv
675 1209 mv
675 1215 lv
678 1217 lv
683 1217 lv
685 1215 lv
685 1209 lv
683 1207 lv
678 1207 lv
675 1209 lv
663 1226 mv
664 1226 lv
663 1224 mv
664 1224 lv
664 1226 mv
664 1224 lv
659 1223 mv
659 1228 lv
661 1230 lv
666 1230 lv
669 1228 lv
669 1223 lv
666 1220 lv
661 1220 lv
659 1223 lv
647 1236 mv
647 1236 lv
647 1234 mv
647 1234 lv
647 1236 mv
647 1234 lv
642 1232 mv
642 1238 lv
645 1240 lv
650 1240 lv
652 1238 lv
652 1232 lv
650 1230 lv
645 1230 lv
642 1232 lv
630 1243 mv
631 1243 lv
630 1241 mv
631 1241 lv
631 1243 mv
631 1241 lv
626 1240 mv
626 1245 lv
628 1248 lv
633 1248 lv
636 1245 lv
636 1240 lv
633 1237 lv
628 1237 lv
626 1240 lv
614 1243 mv
615 1243 lv
614 1241 mv
615 1241 lv
614 1243 mv
614 1241 lv
609 1240 mv
609 1245 lv
612 1248 lv
617 1248 lv
619 1245 lv
619 1240 lv
617 1237 lv
612 1237 lv
609 1240 lv
598 1239 mv
598 1239 lv
598 1237 mv
598 1237 lv
598 1239 mv
598 1237 lv
593 1235 mv
593 1241 lv
596 1243 lv
601 1243 lv
603 1241 lv
603 1235 lv
601 1233 lv
596 1233 lv
593 1235 lv
581 1224 mv
582 1224 lv
581 1221 mv
582 1221 lv
582 1224 mv
582 1221 lv
577 1220 mv
577 1225 lv
579 1228 lv
584 1228 lv
587 1225 lv
587 1220 lv
584 1217 lv
579 1217 lv
577 1220 lv
565 1199 mv
566 1199 lv
565 1197 mv
566 1197 lv
565 1199 mv
565 1197 lv
560 1195 mv
560 1201 lv
563 1203 lv
568 1203 lv
570 1201 lv
570 1195 lv
568 1193 lv
563 1193 lv
560 1195 lv
548 1165 mv
549 1165 lv
548 1161 mv
549 1161 lv
549 1165 mv
549 1161 lv
544 1160 mv
544 1165 lv
546 1168 lv
551 1168 lv
554 1165 lv
554 1160 lv
551 1158 lv
546 1158 lv
544 1160 lv
532 1112 mv
533 1112 lv
532 1109 mv
533 1109 lv
532 1112 mv
532 1109 lv
527 1108 mv
527 1113 lv
530 1116 lv
535 1116 lv
537 1113 lv
537 1108 lv
535 1106 lv
530 1106 lv
527 1108 lv
515 1043 mv
516 1043 lv
515 1040 mv
516 1040 lv
516 1043 mv
516 1040 lv
511 1039 mv
511 1044 lv
513 1047 lv
518 1047 lv
521 1044 lv
521 1039 lv
518 1037 lv
513 1037 lv
511 1039 lv
498 960 mv
501 960 lv
498 953 mv
501 953 lv
500 960 mv
500 953 lv
494 954 mv
494 959 lv
497 962 lv
502 962 lv
505 959 lv
505 954 lv
502 952 lv
497 952 lv
494 954 lv
481 847 mv
485 847 lv
481 838 mv
485 838 lv
483 847 mv
483 838 lv
478 840 mv
478 845 lv
481 848 lv
486 848 lv
488 845 lv
488 840 lv
486 837 lv
481 837 lv
478 840 lv
463 711 mv
470 711 lv
463 692 mv
470 692 lv
467 711 mv
467 692 lv
461 699 mv
461 704 lv
464 707 lv
469 707 lv
472 704 lv
472 699 lv
469 697 lv
464 697 lv
461 699 lv
444 532 mv
456 532 lv
444 500 mv
456 500 lv
450 532 mv
450 500 lv
445 513 mv
445 518 lv
448 521 lv
453 521 lv
455 518 lv
455 513 lv
453 511 lv
448 511 lv
445 513 lv
1510 283 mv
1516 283 lv
1510 268 mv
1516 268 lv
1513 283 mv
1513 268 lv
1513 271 mv
1518 276 lv
1513 281 lv
1508 276 lv
1513 271 lv
1495 397 mv
1498 397 lv
1495 388 mv
1498 388 lv
1497 397 mv
1497 388 lv
1497 387 mv
1502 392 lv
1497 397 lv
1492 392 lv
1497 387 lv
1480 493 mv
1482 493 lv
1480 486 mv
1482 486 lv
1481 493 mv
1481 486 lv
1481 484 mv
1486 489 lv
1481 494 lv
1476 489 lv
1481 484 lv
1464 579 mv
1465 579 lv
1464 575 mv
1465 575 lv
1465 579 mv
1465 575 lv
1465 572 mv
1470 577 lv
1465 582 lv
1460 577 lv
1465 572 lv
1448 649 mv
1450 649 lv
1448 645 mv
1450 645 lv
1449 649 mv
1449 645 lv
1449 642 mv
1454 647 lv
1449 652 lv
1444 647 lv
1449 642 lv
1432 715 mv
1434 715 lv
1432 711 mv
1434 711 lv
1433 715 mv
1433 711 lv
1433 707 mv
1438 712 lv
1433 718 lv
1428 712 lv
1433 707 lv
1416 766 mv
1417 766 lv
1416 764 mv
1417 764 lv
1417 766 mv
1417 764 lv
1417 760 mv
1422 765 lv
1417 770 lv
1412 765 lv
1417 760 lv
1400 809 mv
1401 809 lv
1400 808 mv
1401 808 lv
1401 809 mv
1401 808 lv
1401 803 mv
1406 808 lv
1401 813 lv
1395 808 lv
1401 803 lv
1384 845 mv
1385 845 lv
1384 843 mv
1385 843 lv
1384 845 mv
1384 843 lv
1384 839 mv
1390 844 lv
1384 849 lv
1379 844 lv
1384 839 lv
1368 872 mv
1369 872 lv
1368 870 mv
1369 870 lv
1369 872 mv
1369 870 lv
1369 866 mv
1374 871 lv
1369 876 lv
1364 871 lv
1369 866 lv
1353 892 mv
1353 892 lv
1353 891 mv
1353 891 lv
1353 892 mv
1353 891 lv
1353 886 mv
1358 891 lv
1353 897 lv
1347 891 lv
1353 886 lv
1336 909 mv
1337 909 lv
1336 907 mv
1337 907 lv
1337 909 mv
1337 907 lv
1337 903 mv
1342 908 lv
1337 913 lv
1331 908 lv
1337 903 lv
1320 918 mv
1321 918 lv
1320 916 mv
1321 916 lv
1320 918 mv
1320 916 lv
1320 912 mv
1326 917 lv
1320 923 lv
1315 917 lv
1320 912 lv
1304 923 mv
1305 923 lv
1304 921 mv
1305 921 lv
1305 923 mv
1305 921 lv
1305 917 mv
1310 922 lv
1305 927 lv
1300 922 lv
1305 917 lv
1288 921 mv
1289 921 lv
1288 919 mv
1289 919 lv
1289 921 mv
1289 919 lv
1289 915 mv
1294 920 lv
1289 925 lv
1283 920 lv
1289 915 lv
1272 919 mv
1273 919 lv
1272 917 mv
1273 917 lv
1272 919 mv
1272 917 lv
1272 913 mv
1278 918 lv
1272 923 lv
1267 918 lv
1272 913 lv
1256 912 mv
1256 912 lv
1256 911 mv
1256 911 lv
1256 912 mv
1256 911 lv
1256 906 mv
1261 911 lv
1256 916 lv
1251 911 lv
1256 906 lv
1240 904 mv
1241 904 lv
1240 901 mv
1241 901 lv
1240 904 mv
1240 901 lv
1240 897 mv
1245 902 lv
1240 908 lv
1235 902 lv
1240 897 lv
1224 894 mv
1225 894 lv
1224 892 mv
1225 892 lv
1224 894 mv
1224 892 lv
1224 888 mv
1230 893 lv
1224 898 lv
1219 893 lv
1224 888 lv
1208 881 mv
1209 881 lv
1208 878 mv
1209 878 lv
1208 881 mv
1208 878 lv
1208 874 mv
1213 879 lv
1208 885 lv
1203 879 lv
1208 874 lv
1192 868 mv
1193 868 lv
1192 865 mv
1193 865 lv
1192 868 mv
1192 865 lv
1192 861 mv
1197 867 lv
1192 872 lv
1187 867 lv
1192 861 lv
1176 856 mv
1176 856 lv
1176 853 mv
1176 853 lv
1176 856 mv
1176 853 lv
1176 849 mv
1181 854 lv
1176 860 lv
1171 854 lv
1176 849 lv
1160 845 mv
1161 845 lv
1160 841 mv
1161 841 lv
1160 845 mv
1160 841 lv
1160 838 mv
1165 843 lv
1160 848 lv
1155 843 lv
1160 838 lv
1143 834 mv
1145 834 lv
1143 830 mv
1145 830 lv
1144 834 mv
1144 830 lv
1144 827 mv
1149 832 lv
1144 837 lv
1139 832 lv
1144 827 lv
1127 825 mv
1129 825 lv
1127 822 mv
1129 822 lv
1128 825 mv
1128 822 lv
1128 818 mv
1133 823 lv
1128 828 lv
1123 823 lv
1128 818 lv
1111 814 mv
1113 814 lv
1111 809 mv
1113 809 lv
1112 814 mv
1112 809 lv
1112 807 mv
1117 812 lv
1112 817 lv
1107 812 lv
1112 807 lv
1095 815 mv
1097 815 lv
1095 809 mv
1097 809 lv
1096 815 mv
1096 809 lv
1096 807 mv
1101 812 lv
1096 817 lv
1091 812 lv
1096 807 lv
1079 812 mv
1081 812 lv
1079 807 mv
1081 807 lv
1080 812 mv
1080 807 lv
1080 804 mv
1085 809 lv
1080 815 lv
1075 809 lv
1080 804 lv
1064 810 mv
1064 810 lv
1064 807 mv
1064 807 lv
1064 810 mv
1064 807 lv
1064 803 mv
1069 808 lv
1064 813 lv
1059 808 lv
1064 803 lv
1048 816 mv
1048 816 lv
1048 813 mv
1048 813 lv
1048 816 mv
1048 813 lv
1048 809 mv
1053 815 lv
1048 820 lv
1043 815 lv
1048 809 lv
1031 825 mv
1033 825 lv
1031 820 mv
1033 820 lv
1032 825 mv
1032 820 lv
1032 818 mv
1037 823 lv
1032 828 lv
1027 823 lv
1032 818 lv
1015 831 mv
1017 831 lv
1015 826 mv
1017 826 lv
1016 831 mv
1016 826 lv
1016 823 mv
1021 828 lv
1016 834 lv
1011 828 lv
1016 823 lv
999 836 mv
1001 836 lv
999 832 mv
1001 832 lv
1000 836 mv
1000 832 lv
1000 829 mv
1005 834 lv
1000 839 lv
995 834 lv
1000 829 lv
983 848 mv
985 848 lv
983 844 mv
985 844 lv
984 848 mv
984 844 lv
984 840 mv
989 845 lv
984 850 lv
979 845 lv
984 840 lv
967 862 mv
968 862 lv
967 857 mv
968 857 lv
968 862 mv
968 857 lv
968 854 mv
973 860 lv
968 865 lv
963 860 lv
968 854 lv
950 872 mv
953 872 lv
950 866 mv
953 866 lv
952 872 mv
952 866 lv
952 864 mv
957 869 lv
952 874 lv
946 869 lv
952 864 lv
935 890 mv
937 890 lv
935 885 mv
937 885 lv
936 890 mv
936 885 lv
936 882 mv
941 887 lv
936 892 lv
931 887 lv
936 882 lv
919 908 mv
921 908 lv
919 902 mv
921 902 lv
920 908 mv
920 902 lv
920 900 mv
925 905 lv
920 910 lv
915 905 lv
920 900 lv
903 924 mv
904 924 lv
903 920 mv
904 920 lv
904 924 mv
904 920 lv
904 917 mv
909 922 lv
904 927 lv
898 922 lv
904 917 lv
887 942 mv
888 942 lv
887 938 mv
888 938 lv
887 942 mv
887 938 lv
887 934 mv
893 939 lv
887 945 lv
882 939 lv
887 934 lv
871 958 mv
872 958 lv
871 955 mv
872 955 lv
872 958 mv
872 955 lv
872 952 mv
877 957 lv
872 962 lv
867 957 lv
872 952 lv
855 977 mv
856 977 lv
855 974 mv
856 974 lv
856 977 mv
856 974 lv
856 970 mv
861 975 lv
856 980 lv
850 975 lv
856 970 lv
839 1000 mv
840 1000 lv
839 997 mv
840 997 lv
839 1000 mv
839 997 lv
839 993 mv
845 998 lv
839 1003 lv
834 998 lv
839 993 lv
823 1019 mv
824 1019 lv
823 1014 mv
824 1014 lv
823 1019 mv
823 1014 lv
823 1011 mv
828 1016 lv
823 1022 lv
818 1016 lv
823 1011 lv
807 1040 mv
808 1040 lv
807 1037 mv
808 1037 lv
807 1040 mv
807 1037 lv
807 1033 mv
812 1038 lv
807 1043 lv
802 1038 lv
807 1033 lv
791 1061 mv
792 1061 lv
791 1057 mv
792 1057 lv
792 1061 mv
792 1057 lv
792 1054 mv
797 1059 lv
792 1064 lv
786 1059 lv
792 1054 lv
775 1082 mv
776 1082 lv
775 1079 mv
776 1079 lv
775 1082 mv
775 1079 lv
775 1076 mv
781 1081 lv
775 1086 lv
770 1081 lv
775 1076 lv
759 1103 mv
760 1103 lv
759 1100 mv
760 1100 lv
759 1103 mv
759 1100 lv
759 1096 mv
764 1101 lv
759 1106 lv
754 1101 lv
759 1096 lv
742 1121 mv
744 1121 lv
742 1117 mv
744 1117 lv
743 1121 mv
743 1117 lv
743 1113 mv
748 1119 lv
743 1124 lv
738 1119 lv
743 1113 lv
727 1139 mv
728 1139 lv
727 1136 mv
728 1136 lv
727 1139 mv
727 1136 lv
727 1132 mv
733 1138 lv
727 1143 lv
722 1138 lv
727 1132 lv
711 1157 mv
712 1157 lv
711 1155 mv
712 1155 lv
711 1157 mv
711 1155 lv
711 1151 mv
716 1156 lv
711 1161 lv
706 1156 lv
711 1151 lv
695 1175 mv
696 1175 lv
695 1172 mv
696 1172 lv
695 1175 mv
695 1172 lv
695 1168 mv
700 1174 lv
695 1179 lv
690 1174 lv
695 1168 lv
679 1189 mv
679 1189 lv
679 1186 mv
679 1186 lv
679 1189 mv
679 1186 lv
679 1183 mv
684 1188 lv
679 1193 lv
674 1188 lv
679 1183 lv
663 1204 mv
663 1204 lv
663 1202 mv
663 1202 lv
663 1204 mv
663 1202 lv
663 1198 mv
668 1203 lv
663 1208 lv
658 1203 lv
663 1198 lv
647 1214 mv
648 1214 lv
647 1211 mv
648 1211 lv
647 1214 mv
647 1211 lv
647 1208 mv
652 1213 lv
647 1218 lv
642 1213 lv
647 1208 lv
631 1219 mv
631 1219 lv
631 1216 mv
631 1216 lv
631 1219 mv
631 1216 lv
631 1213 mv
636 1218 lv
631 1223 lv
626 1218 lv
631 1213 lv
615 1225 mv
615 1225 lv
615 1223 mv
615 1223 lv
615 1225 mv
615 1223 lv
615 1219 mv
620 1224 lv
615 1229 lv
610 1224 lv
615 1219 lv
598 1226 mv
599 1226 lv
598 1224 mv
599 1224 lv
599 1226 mv
599 1224 lv
599 1220 mv
604 1225 lv
599 1230 lv
594 1225 lv
599 1220 lv
583 1221 mv
583 1221 lv
583 1219 mv
583 1219 lv
583 1221 mv
583 1219 lv
583 1215 mv
588 1220 lv
583 1225 lv
578 1220 lv
583 1215 lv
567 1210 mv
567 1210 lv
567 1207 mv
567 1207 lv
567 1210 mv
567 1207 lv
567 1203 mv
572 1208 lv
567 1213 lv
562 1208 lv
567 1203 lv
550 1191 mv
551 1191 lv
550 1189 mv
551 1189 lv
551 1191 mv
551 1189 lv
551 1185 mv
556 1190 lv
551 1195 lv
546 1190 lv
551 1185 lv
534 1164 mv
535 1164 lv
534 1160 mv
535 1160 lv
535 1164 mv
535 1160 lv
535 1157 mv
540 1162 lv
535 1167 lv
530 1162 lv
535 1157 lv
518 1130 mv
519 1130 lv
518 1125 mv
519 1125 lv
519 1130 mv
519 1125 lv
519 1122 mv
524 1127 lv
519 1132 lv
514 1127 lv
519 1122 lv
502 1080 mv
504 1080 lv
502 1076 mv
504 1076 lv
503 1080 mv
503 1076 lv
503 1073 mv
508 1078 lv
503 1083 lv
498 1078 lv
503 1073 lv
485 1024 mv
488 1024 lv
485 1016 mv
488 1016 lv
487 1024 mv
487 1016 lv
487 1015 mv
492 1020 lv
487 1025 lv
482 1020 lv
487 1015 lv
469 955 mv
472 955 lv
469 948 mv
472 948 lv
471 955 mv
471 948 lv
471 946 mv
476 952 lv
471 957 lv
466 952 lv
471 946 lv
451 834 mv
458 834 lv
451 818 mv
458 818 lv
455 834 mv
455 818 lv
455 821 mv
460 826 lv
455 831 lv
449 826 lv
455 821 lv
436 762 mv
442 762 lv
436 746 mv
442 746 lv
439 762 mv
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937 627 lv
934 624 lv
929 624 lv
927 627 lv
913 661 mv
918 661 lv
913 647 mv
918 647 lv
916 661 mv
916 647 lv
911 651 mv
911 656 lv
913 659 lv
918 659 lv
921 656 lv
921 651 lv
918 649 lv
913 649 lv
911 651 lv
898 694 mv
901 694 lv
898 685 mv
901 685 lv
900 694 mv
900 685 lv
894 687 mv
894 692 lv
897 694 lv
902 694 lv
905 692 lv
905 687 lv
902 684 lv
897 684 lv
894 687 lv
881 727 mv
886 727 lv
881 713 mv
886 713 lv
883 727 mv
883 713 lv
878 718 mv
878 723 lv
881 725 lv
886 725 lv
889 723 lv
889 718 lv
886 715 lv
881 715 lv
878 718 lv
865 767 mv
869 767 lv
865 758 mv
869 758 lv
867 767 mv
867 758 lv
862 760 mv
862 765 lv
865 767 lv
870 767 lv
872 765 lv
872 760 lv
870 757 lv
865 757 lv
862 760 lv
849 809 mv
853 809 lv
849 800 mv
853 800 lv
851 809 mv
851 800 lv
846 802 mv
846 807 lv
849 809 lv
854 809 lv
856 807 lv
856 802 lv
854 799 lv
849 799 lv
846 802 lv
833 843 mv
837 843 lv
833 831 mv
837 831 lv
835 843 mv
835 831 lv
830 835 mv
830 840 lv
833 842 lv
838 842 lv
840 840 lv
840 835 lv
838 832 lv
833 832 lv
830 835 lv
818 893 mv
820 893 lv
818 886 mv
820 886 lv
819 893 mv
819 886 lv
814 887 mv
814 892 lv
816 895 lv
822 895 lv
824 892 lv
824 887 lv
822 885 lv
816 885 lv
814 887 lv
801 939 mv
804 939 lv
801 931 mv
804 931 lv
803 939 mv
803 931 lv
798 933 mv
798 938 lv
800 941 lv
805 941 lv
808 938 lv
808 933 lv
805 930 lv
800 930 lv
798 933 lv
786 994 mv
788 994 lv
786 987 mv
788 987 lv
787 994 mv
787 987 lv
782 988 mv
782 993 lv
784 996 lv
789 996 lv
792 993 lv
792 988 lv
789 986 lv
784 986 lv
782 988 lv
769 1043 mv
772 1043 lv
769 1035 mv
772 1035 lv
771 1043 mv
771 1035 lv
765 1037 mv
765 1042 lv
768 1044 lv
773 1044 lv
776 1042 lv
776 1037 lv
773 1034 lv
768 1034 lv
765 1037 lv
753 1093 mv
756 1093 lv
753 1086 mv
756 1086 lv
755 1093 mv
755 1086 lv
749 1087 mv
749 1092 lv
752 1094 lv
757 1094 lv
760 1092 lv
760 1087 lv
757 1084 lv
752 1084 lv
749 1087 lv
737 1145 mv
739 1145 lv
737 1138 mv
739 1138 lv
738 1145 mv
738 1138 lv
733 1139 mv
733 1144 lv
736 1147 lv
741 1147 lv
744 1144 lv
744 1139 lv
741 1137 lv
736 1137 lv
733 1139 lv
721 1193 mv
723 1193 lv
721 1187 mv
723 1187 lv
722 1193 mv
722 1187 lv
717 1187 mv
717 1192 lv
720 1195 lv
725 1195 lv
727 1192 lv
727 1187 lv
725 1185 lv
720 1185 lv
717 1187 lv
705 1239 mv
707 1239 lv
705 1233 mv
707 1233 lv
706 1239 mv
706 1233 lv
701 1233 mv
701 1238 lv
704 1241 lv
709 1241 lv
711 1238 lv
711 1233 lv
709 1231 lv
704 1231 lv
701 1233 lv
689 1277 mv
691 1277 lv
689 1270 mv
691 1270 lv
690 1277 mv
690 1270 lv
685 1271 mv
685 1276 lv
687 1279 lv
693 1279 lv
695 1276 lv
695 1271 lv
693 1268 lv
687 1268 lv
685 1271 lv
673 1305 mv
675 1305 lv
673 1299 mv
675 1299 lv
674 1305 mv
674 1299 lv
669 1300 mv
669 1305 lv
671 1307 lv
676 1307 lv
679 1305 lv
679 1300 lv
676 1297 lv
671 1297 lv
669 1300 lv
657 1321 mv
659 1321 lv
657 1315 mv
659 1315 lv
658 1321 mv
658 1315 lv
653 1315 mv
653 1320 lv
655 1323 lv
660 1323 lv
663 1320 lv
663 1315 lv
660 1313 lv
655 1313 lv
653 1315 lv
641 1321 mv
643 1321 lv
641 1314 mv
643 1314 lv
642 1321 mv
642 1314 lv
637 1315 mv
637 1320 lv
639 1323 lv
644 1323 lv
647 1320 lv
647 1315 lv
644 1312 lv
639 1312 lv
637 1315 lv
624 1297 mv
627 1297 lv
624 1290 mv
627 1290 lv
626 1297 mv
626 1290 lv
620 1291 mv
620 1296 lv
623 1298 lv
628 1298 lv
631 1296 lv
631 1291 lv
628 1288 lv
623 1288 lv
620 1291 lv
608 1247 mv
611 1247 lv
608 1241 mv
611 1241 lv
609 1247 mv
609 1241 lv
604 1241 mv
604 1246 lv
607 1249 lv
612 1249 lv
615 1246 lv
615 1241 lv
612 1239 lv
607 1239 lv
604 1241 lv
592 1166 mv
594 1166 lv
592 1159 mv
594 1159 lv
593 1166 mv
593 1159 lv
588 1160 mv
588 1165 lv
591 1168 lv
596 1168 lv
598 1165 lv
598 1160 lv
596 1157 lv
591 1157 lv
588 1160 lv
576 1044 mv
579 1044 lv
576 1037 mv
579 1037 lv
577 1044 mv
577 1037 lv
572 1038 mv
572 1043 lv
575 1046 lv
580 1046 lv
582 1043 lv
582 1038 lv
580 1035 lv
575 1035 lv
572 1038 lv
559 868 mv
563 868 lv
559 857 mv
563 857 lv
561 868 mv
561 857 lv
556 860 mv
556 865 lv
559 868 lv
564 868 lv
566 865 lv
566 860 lv
564 858 lv
559 858 lv
556 860 lv
540 656 mv
550 656 lv
540 628 mv
550 628 lv
545 656 mv
545 628 lv
540 640 mv
540 645 lv
542 647 lv
548 647 lv
550 645 lv
550 640 lv
548 637 lv
542 637 lv
540 640 lv
1387 303 mv
1401 303 lv
1387 265 mv
1401 265 lv
1394 303 mv
1394 265 lv
1394 279 mv
1399 284 lv
1394 289 lv
1389 284 lv
1394 279 lv
1375 501 mv
1381 501 lv
1375 485 mv
1381 485 lv
1378 501 mv
1378 485 lv
1378 488 mv
1383 493 lv
1378 498 lv
1373 493 lv
1378 488 lv
1360 661 mv
1364 661 lv
1360 650 mv
1364 650 lv
1362 661 mv
1362 650 lv
1362 651 mv
1367 656 lv
1362 661 lv
1357 656 lv
1362 651 lv
1345 767 mv
1348 767 lv
1345 757 mv
1348 757 lv
1346 767 mv
1346 757 lv
1346 757 mv
1352 762 lv
1346 767 lv
1341 762 lv
1346 757 lv
1330 857 mv
1332 857 lv
1330 850 mv
1332 850 lv
1331 857 mv
1331 850 lv
1331 849 mv
1336 854 lv
1331 859 lv
1326 854 lv
1331 849 lv
1314 921 mv
1316 921 lv
1314 914 mv
1316 914 lv
1315 921 mv
1315 914 lv
1315 912 mv
1320 917 lv
1315 922 lv
1310 917 lv
1315 912 lv
1297 967 mv
1300 967 lv
1297 960 mv
1300 960 lv
1299 967 mv
1299 960 lv
1299 959 mv
1304 964 lv
1299 969 lv
1294 964 lv
1299 959 lv
1282 991 mv
1284 991 lv
1282 985 mv
1284 985 lv
1283 991 mv
1283 985 lv
1283 983 mv
1288 988 lv
1283 993 lv
1278 988 lv
1283 983 lv
1266 1003 mv
1268 1003 lv
1266 997 mv
1268 997 lv
1267 1003 mv
1267 997 lv
1267 995 mv
1272 1000 lv
1267 1005 lv
1262 1000 lv
1267 995 lv
1250 1001 mv
1253 1001 lv
1250 995 mv
1253 995 lv
1252 1001 mv
1252 995 lv
1252 993 mv
1257 998 lv
1252 1003 lv
1246 998 lv
1252 993 lv
1234 985 mv
1237 985 lv
1234 978 mv
1237 978 lv
1235 985 mv
1235 978 lv
1235 976 mv
1241 981 lv
1235 986 lv
1230 981 lv
1235 976 lv
1219 964 mv
1221 964 lv
1219 958 mv
1221 958 lv
1220 964 mv
1220 958 lv
1220 956 mv
1225 961 lv
1220 966 lv
1215 961 lv
1220 956 lv
1203 938 mv
1205 938 lv
1203 931 mv
1205 931 lv
1204 938 mv
1204 931 lv
1204 929 mv
1209 934 lv
1204 939 lv
1199 934 lv
1204 929 lv
1187 900 mv
1189 900 lv
1187 893 mv
1189 893 lv
1188 900 mv
1188 893 lv
1188 891 mv
1193 896 lv
1188 901 lv
1183 896 lv
1188 891 lv
1170 861 mv
1174 861 lv
1170 851 mv
1174 851 lv
1172 861 mv
1172 851 lv
1172 851 mv
1177 856 lv
1172 861 lv
1167 856 lv
1172 851 lv
1154 826 mv
1158 826 lv
1154 815 mv
1158 815 lv
1156 826 mv
1156 815 lv
1156 815 mv
1161 820 lv
1156 825 lv
1151 820 lv
1156 815 lv
1139 794 mv
1142 794 lv
1139 784 mv
1142 784 lv
1141 794 mv
1141 784 lv
1141 784 mv
1146 789 lv
1141 794 lv
1135 789 lv
1141 784 lv
1123 749 mv
1126 749 lv
1123 740 mv
1126 740 lv
1125 749 mv
1125 740 lv
1125 739 mv
1130 744 lv
1125 749 lv
1120 744 lv
1125 739 lv
1108 725 mv
1111 725 lv
1108 717 mv
1111 717 lv
1109 725 mv
1109 717 lv
1109 716 mv
1114 721 lv
1109 726 lv
1104 721 lv
1109 716 lv
1090 689 mv
1096 689 lv
1090 676 mv
1096 676 lv
1093 689 mv
1093 676 lv
1093 677 mv
1098 682 lv
1093 687 lv
1088 682 lv
1093 677 lv
1075 671 mv
1080 671 lv
1075 658 mv
1080 658 lv
1077 671 mv
1077 658 lv
1077 659 mv
1082 664 lv
1077 670 lv
1072 664 lv
1077 659 lv
1059 650 mv
1064 650 lv
1059 637 mv
1064 637 lv
1061 650 mv
1061 637 lv
1061 638 mv
1067 643 lv
1061 648 lv
1056 643 lv
1061 638 lv
1043 662 mv
1049 662 lv
1043 646 mv
1049 646 lv
1046 662 mv
1046 646 lv
1046 649 mv
1051 654 lv
1046 659 lv
1041 654 lv
1046 649 lv
1027 647 mv
1033 647 lv
1027 631 mv
1033 631 lv
1030 647 mv
1030 631 lv
1030 634 mv
1035 639 lv
1030 644 lv
1024 639 lv
1030 634 lv
1011 643 mv
1016 643 lv
1011 630 mv
1016 630 lv
1014 643 mv
1014 630 lv
1014 631 mv
1019 636 lv
1014 641 lv
1009 636 lv
1014 631 lv
995 657 mv
1001 657 lv
995 641 mv
1001 641 lv
998 657 mv
998 641 lv
998 644 mv
1003 649 lv
998 654 lv
993 649 lv
998 644 lv
979 664 mv
985 664 lv
979 649 mv
985 649 lv
982 664 mv
982 649 lv
982 652 mv
987 657 lv
982 662 lv
977 657 lv
982 652 lv
963 674 mv
969 674 lv
963 658 mv
969 658 lv
966 674 mv
966 658 lv
966 660 mv
971 666 lv
966 671 lv
961 666 lv
966 660 lv
946 699 mv
954 699 lv
946 678 mv
954 678 lv
950 699 mv
950 678 lv
950 683 mv
956 688 lv
950 693 lv
945 688 lv
950 683 lv
932 708 mv
937 708 lv
932 695 mv
937 695 lv
935 708 mv
935 695 lv
935 697 mv
940 702 lv
935 707 lv
930 702 lv
935 697 lv
916 722 mv
922 722 lv
916 708 mv
922 708 lv
919 722 mv
919 708 lv
919 710 mv
924 715 lv
919 720 lv
914 715 lv
919 710 lv
901 768 mv
905 768 lv
901 756 mv
905 756 lv
903 768 mv
903 756 lv
903 757 mv
908 762 lv
903 767 lv
898 762 lv
903 757 lv
885 794 mv
890 794 lv
885 781 mv
890 781 lv
887 794 mv
887 781 lv
887 782 mv
892 787 lv
887 792 lv
882 787 lv
887 782 lv
869 822 mv
874 822 lv
869 811 mv
874 811 lv
871 822 mv
871 811 lv
871 812 mv
876 817 lv
871 822 lv
866 817 lv
871 812 lv
854 862 mv
857 862 lv
854 852 mv
857 852 lv
856 862 mv
856 852 lv
856 852 mv
861 857 lv
856 863 lv
850 857 lv
856 852 lv
838 885 mv
841 885 lv
838 876 mv
841 876 lv
840 885 mv
840 876 lv
840 876 mv
845 881 lv
840 886 lv
835 881 lv
840 876 lv
823 934 mv
825 934 lv
823 928 mv
825 928 lv
824 934 mv
824 928 lv
824 926 mv
829 931 lv
824 936 lv
819 931 lv
824 926 lv
807 967 mv
809 967 lv
807 962 mv
809 962 lv
808 967 mv
808 962 lv
808 959 mv
813 964 lv
808 970 lv
803 964 lv
808 959 lv
791 1001 mv
793 1001 lv
791 994 mv
793 994 lv
792 1001 mv
792 994 lv
792 993 mv
797 998 lv
792 1003 lv
787 998 lv
792 993 lv
775 1052 mv
778 1052 lv
775 1046 mv
778 1046 lv
776 1052 mv
776 1046 lv
776 1044 mv
782 1049 lv
776 1054 lv
771 1049 lv
776 1044 lv
759 1094 mv
762 1094 lv
759 1086 mv
762 1086 lv
760 1094 mv
760 1086 lv
760 1085 mv
765 1090 lv
760 1095 lv
755 1090 lv
760 1085 lv
744 1133 mv
746 1133 lv
744 1127 mv
746 1127 lv
745 1133 mv
745 1127 lv
745 1125 mv
750 1130 lv
745 1135 lv
739 1130 lv
745 1125 lv
728 1174 mv
730 1174 lv
728 1169 mv
730 1169 lv
729 1174 mv
729 1169 lv
729 1167 mv
734 1172 lv
729 1177 lv
724 1172 lv
729 1167 lv
712 1213 mv
714 1213 lv
712 1206 mv
714 1206 lv
713 1213 mv
713 1206 lv
713 1204 mv
718 1209 lv
713 1215 lv
708 1209 lv
713 1204 lv
696 1244 mv
698 1244 lv
696 1240 mv
698 1240 lv
697 1244 mv
697 1240 lv
697 1237 mv
702 1242 lv
697 1247 lv
692 1242 lv
697 1237 lv
681 1271 mv
682 1271 lv
681 1267 mv
682 1267 lv
681 1271 mv
681 1267 lv
681 1263 mv
686 1268 lv
681 1274 lv
676 1268 lv
681 1263 lv
665 1291 mv
666 1291 lv
665 1286 mv
666 1286 lv
666 1291 mv
666 1286 lv
666 1283 mv
671 1289 lv
666 1294 lv
660 1289 lv
666 1283 lv
649 1300 mv
650 1300 lv
649 1295 mv
650 1295 lv
650 1300 mv
650 1295 lv
650 1293 mv
655 1298 lv
650 1303 lv
645 1298 lv
650 1293 lv
633 1298 mv
634 1298 lv
633 1293 mv
634 1293 lv
634 1298 mv
634 1293 lv
634 1290 mv
639 1295 lv
634 1301 lv
629 1295 lv
634 1290 lv
617 1279 mv
619 1279 lv
617 1275 mv
619 1275 lv
618 1279 mv
618 1275 lv
618 1272 mv
623 1277 lv
618 1282 lv
613 1277 lv
618 1272 lv
601 1237 mv
603 1237 lv
601 1231 mv
603 1231 lv
602 1237 mv
602 1231 lv
602 1229 mv
607 1234 lv
602 1239 lv
597 1234 lv
602 1229 lv
585 1180 mv
587 1180 lv
585 1174 mv
587 1174 lv
586 1180 mv
586 1174 lv
586 1172 mv
592 1177 lv
586 1182 lv
581 1177 lv
586 1172 lv
569 1094 mv
572 1094 lv
569 1086 mv
572 1086 lv
571 1094 mv
571 1086 lv
571 1085 mv
576 1090 lv
571 1096 lv
566 1090 lv
571 1085 lv
553 974 mv
556 974 lv
553 964 mv
556 964 lv
555 974 mv
555 964 lv
555 964 mv
560 969 lv
555 974 lv
549 969 lv
555 964 lv
534 808 mv
544 808 lv
534 784 mv
544 784 lv
539 808 mv
539 784 lv
539 791 mv
544 796 lv
539 801 lv
534 796 lv
539 791 lv
[] sd
394 183 mv
1584 183 lv
1584 1373 lv
394 1373 lv
394 183 lv
476 183 mv
476 201 lv
476 1355 mv
476 1373 lv
558 183 mv
558 201 lv
558 1355 mv
558 1373 lv
640 183 mv
640 201 lv
640 1355 mv
640 1373 lv
722 183 mv
722 201 lv
722 1355 mv
722 1373 lv
804 183 mv
804 220 lv
804 1337 mv
804 1373 lv
886 183 mv
886 201 lv
886 1355 mv
886 1373 lv
968 183 mv
968 201 lv
968 1355 mv
968 1373 lv
1050 183 mv
1050 201 lv
1050 1355 mv
1050 1373 lv
1132 183 mv
1132 201 lv
1132 1355 mv
1132 1373 lv
1215 183 mv
1215 220 lv
1215 1337 mv
1215 1373 lv
1297 183 mv
1297 201 lv
1297 1355 mv
1297 1373 lv
1379 183 mv
1379 201 lv
1379 1355 mv
1379 1373 lv
1461 183 mv
1461 201 lv
1461 1355 mv
1461 1373 lv
1543 183 mv
1543 201 lv
1543 1355 mv
1543 1373 lv
394 273 mv
412 273 lv
1566 273 mv
1584 273 lv
394 325 mv
412 325 lv
1566 325 mv
1584 325 lv
394 362 mv
412 362 lv
1566 362 mv
1584 362 lv
394 391 mv
412 391 lv
1566 391 mv
1584 391 lv
394 415 mv
412 415 lv
1566 415 mv
1584 415 lv
394 435 mv
412 435 lv
1566 435 mv
1584 435 lv
394 452 mv
412 452 lv
1566 452 mv
1584 452 lv
394 467 mv
412 467 lv
1566 467 mv
1584 467 lv
394 481 mv
430 481 lv
1547 481 mv
1584 481 lv
394 570 mv
412 570 lv
1566 570 mv
1584 570 lv
394 623 mv
412 623 lv
1566 623 mv
1584 623 lv
394 660 mv
412 660 lv
1566 660 mv
1584 660 lv
394 689 mv
412 689 lv
1566 689 mv
1584 689 lv
394 712 mv
412 712 lv
1566 712 mv
1584 712 lv
394 732 mv
412 732 lv
1566 732 mv
1584 732 lv
394 749 mv
412 749 lv
1566 749 mv
1584 749 lv
394 765 mv
412 765 lv
1566 765 mv
1584 765 lv
394 778 mv
430 778 lv
1547 778 mv
1584 778 lv
394 868 mv
412 868 lv
1566 868 mv
1584 868 lv
394 920 mv
412 920 lv
1566 920 mv
1584 920 lv
394 957 mv
412 957 lv
1566 957 mv
1584 957 lv
394 986 mv
412 986 lv
1566 986 mv
1584 986 lv
394 1010 mv
412 1010 lv
1566 1010 mv
1584 1010 lv
394 1030 mv
412 1030 lv
1566 1030 mv
1584 1030 lv
394 1047 mv
412 1047 lv
1566 1047 mv
1584 1047 lv
394 1062 mv
412 1062 lv
1566 1062 mv
1584 1062 lv
394 1076 mv
430 1076 lv
1547 1076 mv
1584 1076 lv
394 1165 mv
412 1165 lv
1566 1165 mv
1584 1165 lv
394 1218 mv
412 1218 lv
1566 1218 mv
1584 1218 lv
394 1255 mv
412 1255 lv
1566 1255 mv
1584 1255 lv
394 1284 mv
412 1284 lv
1566 1284 mv
1584 1284 lv
394 1308 mv
412 1308 lv
1566 1308 mv
1584 1308 lv
394 1327 mv
412 1327 lv
1566 1327 mv
1584 1327 lv
394 1345 mv
412 1345 lv
1566 1345 mv
1584 1345 lv
394 1360 mv
412 1360 lv
1566 1360 mv
1584 1360 lv
874 1297 mv
874 1264 lv
876 1297 mv
876 1264 lv
870 1297 mv
881 1297 lv
870 1264 mv
893 1264 lv
893 1274 lv
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900 1283 mv
928 1283 lv
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939 1290 lv
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986 1283 lv
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972 1280 lv
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974 1292 lv
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975 1266 lv
979 1264 lv
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1039 1292 lv
1026 1280 lv
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1102 1297 mv
1102 1264 lv
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1085 1274 lv
1109 1274 lv
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1106 1264 lv
1131 1293 mv
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1132 1264 lv
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1116 1274 lv
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1198 1286 lv
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1234 1281 mv
1237 1284 lv
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1251 1284 lv
1254 1281 lv
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1256 1274 lv
1254 1269 lv
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1234 1272 lv
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1246 1286 mv
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es
eop end
%%EndDocument
 @endspecial -14 1528 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
-14 1528 a Fg(\000)p Ff(2)17 b
 currentpoint grestore moveto
449 1528 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
449 1528 a Fg(\000)p Ff(1)p Fe(:)p Ff(5)g
 currentpoint grestore moveto
972 1528 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
972 1528 a Fg(\000)p Ff(1)f
 currentpoint grestore moveto
1435 1528 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
1435 1528 a Fg(\000)p Ff(0)p Fe(:)p Ff(5)g
 currentpoint grestore moveto
-186 1452 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
-186 1452 a Ff(0)p Fe(:)p Ff(001)g
 currentpoint grestore moveto
-148 1095 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
-148 1095 a Ff(0)p Fe(:)p Ff(01)g
 currentpoint grestore moveto
-111 738 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
-111 738 a Ff(0)p Fe(:)p Ff(1)g
 currentpoint grestore moveto
-53 380 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
-53 380 a Ff(1)h
 currentpoint grestore moveto
-90 24 a

 gsave currentpoint currentpoint translate
 0 neg rotate 0.9375 0.9375 scale neg exch neg exch translate
-90 24 a Ff(10)f
 currentpoint grestore moveto
592 1581 a

 gsave currentpoint currentpoint translate
 0 neg rotate 1.0 1.0 scale neg exch neg exch translate
592 1581 a Fd(Energy)g(/)g(Site)33 b
 currentpoint grestore moveto
692 1658 a Fc(Fig.)26 b(3)0 1718 y Fb(Ener)m(gy)c(Densities)h(for)g
Fa(L)e Fd(=)h Fb(36,)i(44)f(and)g(50.)37 b(Curves)22 b(ar)m(e)h(p)m(ar)m
(ameter)g(fr)m(e)m(e)0 1778 y(pr)m(e)m(dictions)d(Eqs.)k(\(13-14\).)p
eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF



