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%From: gidi@vesta.TAU.AC.IL (Gideon Lana)
%Date: Tue, 27 Jul 93 15:47:28 IDT


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June, 1993
\hfill
TAUP   2059-93
\vspace{1.5 cm}

\centerline{\Large \bf
Bubble-Bubble Interaction in the}
%\vspace{0.3 cm}
\centerline{\Large \bf
Quark-Gluon Plasma}
\vspace{0.5 cm}
\centerline{by}
\vspace{0.5 cm}
\centerline{\bf Gideon Lana}


\bigskip\centerline{School of Physics and Astronomy}
\centerline{Raymond and Beverly Sackler Faculty of Exact Sciences}
\centerline{Tel Aviv University, Ramat-Aviv 69978, Israel}
\vspace{2.0 cm}

\centerline{\bf ABSTRACT}


We calculate the two-body interaction of spherical hadronic bubbles immersed in
a gluon plasma at temperatures above the phase transition.
Modeling the bubbles with the MIT bag we find that the two body potential is
repulsive for all bubble-bubble distances considered.
This implies that a static configuration of spherical hadronic bubbles in a
gluon background is consistent with the bag model of QCD.
\vfil\eject


\section{Introduction}

Considerable theoretical work \cite{PW,FJ,SY} and many computer simulations
\cite{BP,HP} of high temperature QCD have focused on the order of the phase
transition between quark-gluon plasma and hadronic matter. Quenched simulations
convincingly showed the transition is first order but weak i.e., with a
relatively small latent heat. When dynamical quarks are incorporated, the
transition weakens as smaller values of the quark mass are being used, and
recent calculations suggest that in the limit $m_q\rightarrow 0$ the transition
becomes second order \cite{SG}.

Most investigations modeling this transition in the context of the Universe
about 10-20 $\mu$sec after the big bang have assumed that the QCD transition
proceeds by mechanisms similar to those encountered in first order phase
transitions in condensed-matter systems, such as the condensation of water
vapor.
The common picture is of a homogeneous medium of QGP which cools adiabatically
as the universe expands; as the temperature drops just below $T_0$, which is of
the order of 100-200 MeV, the QGP becomes metastable, and hadronic bubbles
nucleate by local fluctuations. These seeds expand slowly while exchanging heat
and baryon number with the rest of the universe \cite{EW,KH,FA}.

Recently, however, an exact bag-model calculation suggested that the high
temperature phase of QCD is not a simple homogeneous medium of quarks and
gluons \cite{MS,BS}. It was claimed that at temperatures just above $T_0$ a
single bubble of finite radius of hadronic matter in the plasma has lower free
energy than the plasma with no bubbles at all. This means that the high
temperature phase of QCD may be composed of a plasma ``soup'' with floating
bubbles filled mostly by an ideal gas of pions. However, the evidence for this
phenomenon from lattice Monte-Carlo calculations  \cite{HPR,KPR,SO} is not
clear and the interpretation of the results is not too convincing.
Without addressing to the issue of how well the bag-model describes QCD physics
at high temperatures, we try to understand whether within the arena of the
bag-model such mixed-phase structure is indeed possible.

The bag model calculation \cite{MS} was performed for a single spherical
hadronic  bubble immersed in a homogeneous medium of plasma. The first question
which arose was whether this spherical configuration is indeed a minimum of the
free-energy functional and not a saddle point. In particular, is a spherical
bubble stable against small deformations of shape?
Using rather straightforward analysis it was shown that perturbations of its
boundary indeed tend to grow, suggesting that a spherical bubble may be
unstable \cite{LS}.

The aim of this work is twofold: to see if bubble-bubble interactions may, on
the one hand, support a finite density of bubbles, and, on the other hand,
stabilize the walls of the bubbles, such that the spherical bubbles discussed
in \cite{MS,LS} are indeed stable objects within the scope of MIT bag model.

The next section will be devoted to the explanation of the strategy of
obtaining an estimate of the bubble-bubble interaction.
In section \ref{secmre} we shall show how the multiple reflection expansion may
be applied to calculating the density of energy levels for the simplest
two-bubble geometry, and some details of the calculation will be given in
section \ref{secgeo}.
In section \ref{secres} the results of the calculation will be displayed and in
section \ref{secdis} we shall discuss the implications of the results and the
validity of the calculation.


\section{The Method}

We assume that the high temperature phase of QCD is composed of identical
spherical domains (bubbles) where the fields are manifested as colorless
excitations (mainly pions); the bubbles are surrounded by freely propagating
gauge fields.

In the early universe, the rate of expansion is much slower than the time
scales of QCD. This suggests that the field configuration is, at any time, in
complete thermal equilibrium and, specifically, that there are no temperature
gradients in the medium. Accordingly, we view it as a static configuration. We
further assume that the fluctuations in bubble-bubble distances are small.

Notice that in the absence of baryon number, the bubbles' radii and the  mean
distance between them completely specify the configuration. The structure we
have in mind is rather similar to that of a type II superconductor; the
hadronic domains form in our case a three dimensional lattice within the
quark-gluon plasma, whereas in the type II superconducting phase the geometry
is essentially two dimensional.

Since bubbles are color singlet objects, their interaction should stem from a
mechanism similar to the one producing the Casimir effect at $T=0$.
Such forces usually fall off very quickly as the relevant distance of the
problem is increased.
This leads us to assume that as long as the distance $d$ between the bubbles is
not too small, the interaction between bubbles is dominated by a two body
interaction, i.e., that three body and higher order contributions are
negligible in all configurations which are important to the partition function.
Upon completion of the calculation, a consistency check for this assumption
will be made.
The smallness of the interaction enables us to express the free energy of a
two-bubble configuration as
\BE
F_{bb}(R,d) = 2 F_b(R) + \delta\! F_{bb}(R,d) \, ,
\label{fbbdef}
\EE
where $F_b(R,d)$ is the free energy of a single bubble of radius $R$ in
infinite space, and $\delta\! F_{bb}(R,d)$ is a small correction to be derived.
For brevity of notation we omit the explicit $T$ dependence.
We first calculate the shift of energy level density $\delta\! \rho(k)$ for
each gluon, from which the free energy shift  $\delta\! F_{bb}(R,d)$ is derived
via
\BE
\delta\! F_{bb}(R,d)=n_g T\int_0^\infty\! d\!k \,\delta \! \rho(k)
\ln\left(1-e^{-k/T}\right)
\label{deltaf}
\EE
where $n_g=8$ is the number of gluons.

In the next section we shall briefly expound the multiple reflection expansion,
the tool with which $\delta \! \rho(k)$ is isolated and estimated.


\section{The Multiple Reflection Expansion}
\label{secmre}

The multiple  reflection expansion is a method of calculating the distribution
of eigenfrequencies for the wave equation in a connected finite domain of
arbitrary shape. It was developed for scalar and vector fields \cite{BBs,BBv}
and later for the much more involved case of Dirac fields \cite{HJ}.
It uses the time independent Green function formalism for obtaining an
expansion in $1/k$ where $k$ is the wavenumber.
The general idea is to extract the density of eigenfrequencies $\rho(k)$ from
the discontinuity of the propagator along the cut in the complex $k$ plane,
\BE
\rho_\gamma(k)= {2\over\pi} \left[ {\rm Im} \int_V \, d^3\! r\,  {\rm
Tr}\,\BG_\perp (r r^\prime,k) \right]_{r^\prime=r}\, ,
\label{rhok}
\EE
where $V$ is the volume in which the fields propagate. $\gamma$ is an imaginary
part added to $k$ which measures the width of a Lorentzian smearing function,
chosen in order to turn the discrete eigenvalue distributions that pertains to
finite domains into continuous ones. $\gamma$ must be chosen to be as small as
possible so that $k$ lies as close as possible to the real axis.

$\BG_\perp (r r^\prime,k)$ is the physical propagator which satisfies the
differential equation together with the boundary conditions. In our model the
fields satisfy the Helmholtz equation inside $V$ ,
\BE
(\nabla^2 + k^2)\vec A =  0 \, , \qquad {\rm div}\vec A=0 \, ,
\EE
and dual-superconductor boundary conditions on $S$, the boundary of the domain,
\BE
\hat n \cdot \vec E = \hat n \times \vec B = 0 \, .
\EE
As no confusion may arise, we shall henceforth omit the explicit $k$ dependence
of $\BG$.

For deriving expressions for the energy level density it is technically
convenient to augment the transverse propagator with a longitudinal part which
propagates longitudinal modes,
\BE
\BG=\BG_\perp+\BG_\parallel \, .
\EE
Using $\BG$ for calculating $\rho_\gamma(k)$ via Eq. (\ref{rhok}), this
additional piece yields an extra contribution which will be subtracted at the
end.
Proceeding along the lines of \cite{BBv}, we define for each point $\alpha$ on
the boundary of the domain the operators
\begin{eqnarray}
\DL{\alpha}& = & {\bf P}_n(\alpha) + {\stackrel{\leftarrow}{\partial}
  \over \partial n_\alpha} {\bf P}_t(\alpha) \, , \nonumber \\
\DR{\alpha} & = & {\bf P}_n(\alpha) {\stackrel{\rightarrow}{\partial}
  \over \partial n_\alpha} - {\bf P}_t(\alpha) \, ,
\end{eqnarray}
where ${\bf P}_n(\alpha)$ and ${\bf P}_t(\alpha)$ are projection operators on
the normal $\vec n_\alpha$ and on the tangent plane to the surface at the point
$\alpha$ respectively.
Augmenting also the free transverse propagator with a longitudinal component,
\BE
\BGO=\BGO_\perp+\BGO_\parallel={\mbox{\bf 1}} \, G_0 \,  ,
\qquad
G_0={\exp(ik|r-r^\prime|) \over 4\pi\, |r-r^\prime|} \,  ,
\EE
enables us to expand $\BG$ in terms of $\BGO$, yielding a multiple reflection
expansion:

\begin{eqnarray}
\BG(r r^\prime)& = &\BGO(r r^\prime) + 2 \int_S d\sigma_{\!\alpha} \, \BGO (r
\alpha) \DL{\alpha}\DR{\alpha} \BGO (\alpha\, r^\prime) \nonumber \\
& &  + 2^2 \int_S d\sigma_{\!\alpha}\, d\sigma_{\!\beta} \,
\BGO
(r \alpha) \DL{\alpha}\DR{\alpha} \BGO (\alpha\beta)
 \DL{\beta} \DR{\beta} \BGO (\beta r^\prime)
+ \cdots \, .
\label{gmre}
\end{eqnarray}

The first term gives for the density of states the usual contribution
proportional to the (infinite) volume $V$, and is therefore irrelevant to
$\delta \! \rho(k)$ .
When computing the free energy of the system it is subtracted off.
The second term, which corresponds to a single reflection, vanishes identically
upon performing the trace and subtracting the contribution of the longitudinal
modes.

For the two-reflection term the integrand is
\begin{eqnarray}
& & G_0(r \alpha) {\partial G_0(\alpha \beta)\over\partial n_\alpha}
{\partial G_0(\beta r^\prime) \over\partial n_\beta} (1-\sin^2 \theta)
\nonumber \\
&& - \Biggl[G_0(r \alpha) {\partial^2 G_0(\alpha \beta)\over\partial n_\alpha
\, \partial n_\beta} G_0(\beta r^\prime)
+ {\partial G_0(r \alpha) \over\partial n_\alpha}  G_0(\alpha \beta) {\partial
G_0(\beta r^\prime) \over\partial n_\beta}
\Biggr]\sin^2 \theta \nonumber \\
&&+ {\partial G_0(r \alpha) \over\partial n_\alpha}
 {\partial G_0(\alpha \beta)\over\partial n_\beta} G_0(\beta r^\prime)
(2-\sin^2 \theta) \, ,
\label{g_two_refl}
\end{eqnarray}
where $\theta$ denotes the angle between the normal vectors to the surface $S$
at the points $\alpha$ and $\beta$, namely $\cos\theta=n_\alpha\cdot n_\beta$.
Upon integrating over $\alpha$, $\beta$ and taking $r=r^\prime$ the triple
products of propagators in the first and last terms give identical results.
Collecting all terms, subtracting the contribution of the longitudinal modes,
and performing the trace, we finally get for the two-reflection shift in the
density of states
\begin{eqnarray}
 \delta\! \rho_\gamma(k) & = & {8k\over \pi} {\rm Im} \int_V d^3\!r \int_S
d\sigma_{\!\alpha}\, d\sigma_{\!\beta} \,
\Biggl[2 G_0(r \alpha) {\partial G_0(\alpha \beta)\over\partial n_\alpha}
{\partial G_0(\beta r) \over\partial n_\beta}  \nonumber \\
& & -\sin^2 \theta {\partial^2\over\partial n_\alpha \, \partial n_\beta}
\left( G_0(r \alpha)  G_0(\alpha \beta) G_0(\beta r) \right)\Biggr] \, .
\label{trg}
\end{eqnarray}


\section{The Geometrical Setting}
\label{secgeo}

In order to calculate the two-bubble interaction alone, it is sufficient to
consider a simple geometry of two spherical bubbles in infinite space.
This is in practice similar to confining the two bubbles in a cavity, and
taking the walls of the cavity to infinity.
The actual application of this limiting procedure is considerably simplified by
the fact that only the bubble-bubble interaction is of interest and not the
artificial interactions of the bubbles with the walls of the cavity. Therefore,
we do not have to take into account reflections from the walls, i.e., the
integrations over the points $\alpha$ and $\beta$ are performed only over the
two spherical surfaces of the bubbles.

Figure \ref{fig_twoterms}a shows schematically the relevant contribution to the
propagator $\BG$, where the limit $r\rightarrow r^\prime$ has already been
taken. The points $\alpha$ and $\beta$ are on different spheres and $r$ is in
$V$.

For the free energy shift $\delta\! F_{bb}$ of the two-bubble configuration
relative to the free energy of two completely separate bubbles, one has to
subtract an excluded-volume contribution.
Had we computed the free energy of a single bubble in infinite space by using
the multiple reflection expansion, the two-reflection term would be as depicted
in Fig. \ref{fig_twoterms}b: The points $\alpha$ and $\beta$ lie on the surface
of the bubble, and the coordinate $r$ runs over all space, the interior of the
single bubble excluded.
When introducing another bubble, its interior is no longer included in the
region of integration, and therefore the excluded-volume terms must be
subtracted off.

\begin{figure}
%\centerline{\psfig{figure=bisphere.ps,height=6.0cm}}
\vspace{6.0 cm}
\caption{Schematic representation of the two kinds of contributions to the two
reflection term. See text.}
\label{fig_twoterms}
\end{figure}

\section{Results}
\label{secres}

We have computed the shift in the density of states $\delta\!\rho_\gamma(k)$ as
a function of the common radius $R$
of the bubbles and the distance $d$ between their centers.
This enables us to estimate the free energy shift associated with the
interaction of bubbles by Eq. (\ref{deltaf}).

Since for the equilibrium bubble radius (see Refs. \cite{MS,BS}) $RT\simeq 1$ ,
it was sufficient to compute $\delta\!\rho_\gamma(k)$ in the range $0.2\leq
kR\leq 10$. Lower or higher values of $k$ are expected to yield too small
contributions in Eq. (\ref{deltaf}).
The convergence parameter $\gamma$ was chosen as small as the numerics allowed.
The results which follow correspond to the choice of $\gamma=0.4/R$.
Calculations for lower values of $\gamma$ require much more iterations to
achieve similar accuracy, but lead substantially to the same results.
Physically, such a high value of $\gamma$ may be viewed as effectively
introducing a gluon mass into the calculation.

Integrating Eq. (\ref{trg}) we encounter seven-dimensional integrals.
Exploiting the azimuthal symmetry reduces them to six-dimensional integrals. We
were, therefore, forced to resort to numerical routines which sample the huge
space. We used the computer code VEGAS \cite{V}, which we found to be very
efficient for the task..
We ran the numerical program on a Cyber 920 Silicon-Graphics machine. For each
value of $d$ the program ran for about 30 hours, sampling about $2\cdot 10^6$
points for each value of $k$.

The integration was performed using the bispherical coordinate system (see e.g.
\cite{MF}) in terms of which $S$ is trivially expressed as a union of the
surfaces $\mu=-\mu_0$ and  $\mu=\mu_0$ where  $\mu$ is the radial coordinate;
$\mu_0$ fixes the ratio between the bubbles' radii and separation via
$d/2R=\cosh \mu_0$. The volume integration is then performed over the range
$-\mu_0< \mu< \mu_0$ for (a) type terms, and $\mu_0< \mu$ for (b) type terms
(see Fig. \ref{fig_twoterms}).


In general, the limit $r\rightarrow r^\prime$ must be taken with caution, since
for a simply connected boundary all four points $\alpha$, $\beta$, $r$ and
$r^\prime$ may become arbitrarily close.
In our case, however, the boundary is not simply connected and for both (a) and
(b) type terms, the four points may not coincide.
Therefore, the superficial singularities of the integrand of (\ref{trg}),
$\alpha\rightarrow r$ or $\beta\rightarrow r$ for (a) type terms, and
$\alpha\rightarrow \beta$ for (b) type terms, are not actual singularities due
to the vanishing of the measure.
 From the numerical point of view, it is then sufficient to avoid sampling the
space of integration too close to subspaces where the value of the integrand of
(\ref{trg}) becomes unbounded.

Figure \ref{figrho} is a typical plot of the shift in the density of levels as
a function of $k$ for $R=1$ fm and $d=2.59$ fm. As expected, for small values
of $k$,  $\delta\! \rho(k)$ approaches zero. This feature is common to all
combinations of $R$ and $d$ we explored.
The numerical integration, however, fails to yield the correct decrease of
$\delta\! \rho(k)$ for large values of $k$, where the integrand is highly
oscillatory. The insufficiency of the sampling mesh for large values of $k$ is
also reflected in the large error bars, which are of the order of the estimate
itself. This does not prevent us from obtaining a reliable estimate for
$\delta\! F(k)$ since for large values of $k$ the logarithmic factor in Eq.
\ref{deltaf} is exponentially small.

\begin{figure}
%\centerline{\psfig{figure=rho_vs_k.ps,height=8.0cm}}
\vspace{8.0 cm}
\caption{Typical behavior of $\delta\! \rho(k)$ for fixed values of $R$ and
$d$. }
\label{figrho}
\end{figure}

The free energy shift as a function of the bubbles' radius and their distance
is shown in Fig. \ref{figfdr}. The relative errors, which are not shown here,
are typically around 10 to 30 percent, though for the largest values of $d/R$
where $\delta\! F$ goes to zero the relative errors are higher.
The most important feature is the increase of $\delta\! F$ for small values of
$d/R$ which is most pronounced for $0.5\, {\rm fm}\lsim R\lsim 2\, {\rm fm}$.
The meaning of this is that bubbles tend to repel each other, i.e., bubbles do
not tend to coalesce. This result supports the picture suggested in Refs.
\cite{MS,BS} of a phase of quark-gluon plasma populated with hadronic bubbles
arranged in a lattice-like structure.

Another important result is that the largest values obtained for $\delta\! F$
are of the order of $100$ MeV, and typical values are of the order of $10$ MeV.
This is to be compared with the free energy shift $F_b$ due to a single bubble
[see  Eq. (\ref{fbbdef})], which is typically about $500$ MeV at the
equilibrium radius. This means that the presence of neighboring bubbles should
have little effect on the equilibrium radius.
The mean separation between bubbles, however, is determined both by $F_b$ and
$\delta\! F_{bb}$: On the one hand, the tendency of $F_b$ is to populate the
plasma with bubbles as densely as possible. On the other hand, $\delta \! F$
pushes them away from each other as far as possible. The net result of these
two mechanisms may be a stable lattice of bubbles.

\begin{figure}
%\centerline{\psfig{figure=f_drtex.ps,height=16.0cm}}
\vspace{16.0 cm}
\caption{$\delta\! F$ for as a function of $R$ and $d/R$ for a temperature
$T=155 MeV$.}
\label{figfdr}
\end{figure}


Three limiting regimes are worth mentioning.
 For $d-2R\ll R$ the results of the calculation are not shown, since for this
case the distance between the surfaces of the bubbles is very small, and the
propagator $\BGO$ will decrease considerably only after many reflections. In
other words, we expect that many reflections should be important in such
situation and the two-reflection term badly approximates  $\delta\! F$.
 For the limit $R\rightarrow 0$ with $d$ fixed, the shift in the level density
tends to zero due to the vanishing of the surface $S$ in the integral
(\ref{gmre}).
 For $d/R\rightarrow \infty$ the vanishing of $\delta\!$ is clearly exhibited;
this is to be expected due to the $1/R$ behavior of the propagator.

We are particularly interested in the behavior of $\delta \! F(R,d)$ as a
function of $d$ for fixed $R$.
For each temperature we fix the value of $R$ to the radius that minimizes the
free energy for a single bubble geometry (see Ref. \cite{MS}). The cuts through
the free-energy surfaces yield the curves shown in Fig. \ref{figfd}.

\begin{figure}
%\centerline{\psfig{figure=f_vs_d1.ps,height=10.0cm}}
\vspace{10.0 cm}
\caption{$\delta\! F$ for plotted as a function of $d$, the distance between
the bubbles for various temperatures. For each temperature $R$ is fixed to the
value which minimizes $F_b(R,t)$.}
\label{figfd}
\end{figure}


\section{Discussion}
\label{secdis}

We have estimated the shift of free energy related to the interaction between
two spherical bubbles of hadronic phase immersed in a gluon plasma. In the
relevant regime of parameters, a repulsion between the bubbles was found.
If quarks play only a minor role in this phenomenon, as was found in the single
bubble calculation in Ref. \cite{MS}, this finding supports the picture of the
Swiss cheese instability \cite{BS} of the early Universe.
As an approximation we have used the first non-trivial contribution of the
multiple reflection expansion. Since this is essentially an asymptotic
expansion in $1/k$, it is not easy to predict a priori the validity of such
truncation.

At first glance, truncating all higher-order reflections has some similarity to
the dilute gas approximation: For large $d$ the free propagator $\BGO$
decreases strongly for every trip from one bubble to another. In other words,
when the bubbles are dilute only terms which correspond to a minimal number of
trips between bubbles are important.
This may be misleading, since such terms arise at any higher order of the
expansion.

The correction we have computed is indeed considerably smaller than $F_b$, the
free energy of a single bubble in infinite space. But an explicit comparison
with terms stemming from larger number of reflections is completely
impractical, because of technical problems in calculating integrals of more
than nine dimensions. One may, however, be able to compute successive terms by
using completely different ways of performing this calculation: For example,
one might iterate the integral equation II.29 of Ref. \cite{BBv} on a grid, and
use this as an input for equation II.23.

Our results are only qualitative in nature; The error bars are considerable,
and we have only a limited window in $k$ from which we may get reliable
numbers. Therefore it seems to us premature even to use them to predict the
structure of the lattice of bubbles we suggest.

The repulsion we found, is intuitively exerted by a positive outside pressure
on the walls of the bubbles. However, by regarding the free energy as a
function of the bubbles' radii for fixed $d$, it may be immediately noticed
that it contains not only a pure pressure term, cubic in $R$, but other terms
as well. These terms are crucial to the issue of the surface instability of the
bubble, namely its tendency to grow fingerlike structures, as suggested in
\cite{LS}.
Again, much more precise calculation has to be performed in order to answer
whether the two-bubble interaction may eliminate or reduce the surface
instability.

\section*{Acknowledgments}
We thank Benjamin Svetitsky for many helpful discussions and for thorough
reading of the manuscript.
This work was supported by a Wolfson Research Award administered by the
Israel Academy of Sciences and Humanities.


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\bibitem{V} G. P. Lepage, J. Comp. Phys. {\bf 27} (1978) 192.  .
\bibitem{MF} P.~Morse and H.~Feshbach, {\it Methods of Theoretical Physics}
(McGraw-Hill, New York, 1953), p. 665.

\end{thebibliography}
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1453 178 L
1445 180 L
1443 183 L
1440 188 L
1440 190 L
1443 193 L
1445 190 L
1443 188 L
1471 203 M 1474 195 L
1474 188 L
1471 183 L
1468 180 L
1463 178 L
1834 178 M 1864 226 M 1864 178 L
1867 231 M 1867 178 L
1867 231 M 1839 193 L
1880 193 L
1857 178 M 1875 178 L
2235 178 M 2248 231 M 2242 206 L
2248 211 L
2255 213 L
2263 213 L
2271 211 L
2276 206 L
2278 198 L
2278 193 L
2276 185 L
2271 180 L
2263 178 L
2255 178 L
2248 180 L
2245 183 L
2242 188 L
2242 190 L
2245 193 L
2248 190 L
2245 188 L
2263 213 M 2268 211 L
2273 206 L
2276 198 L
2276 193 L
2273 185 L
2268 180 L
2263 178 L
2248 231 M 2273 231 L
2248 229 M 2260 229 L
2273 231 L
255 2261 M 2261 2261 L
255 2261 M 255 2206 L
335 2261 M 335 2233 L
415 2261 M 415 2233 L
495 2261 M 495 2233 L
576 2261 M 576 2233 L
656 2261 M 656 2206 L
736 2261 M 736 2233 L
816 2261 M 816 2233 L
897 2261 M 897 2233 L
977 2261 M 977 2233 L
1057 2261 M 1057 2206 L
1137 2261 M 1137 2233 L
1218 2261 M 1218 2233 L
1298 2261 M 1298 2233 L
1378 2261 M 1378 2233 L
1458 2261 M 1458 2206 L
1538 2261 M 1538 2233 L
1619 2261 M 1619 2233 L
1699 2261 M 1699 2233 L
1779 2261 M 1779 2233 L
1859 2261 M 1859 2206 L
1940 2261 M 1940 2233 L
2020 2261 M 2020 2233 L
2100 2261 M 2100 2233 L
2180 2261 M 2180 2233 L
2261 2261 M 2261 2206 L
255 255 M 255 2261 L
255 255 M 309 255 L
255 422 M 282 422 L
255 589 M 282 589 L
255 756 M 282 756 L
255 923 M 309 923 L
255 1090 M 282 1090 L
255 1258 M 282 1258 L
255 1425 M 282 1425 L
255 1592 M 309 1592 L
255 1759 M 282 1759 L
255 1926 M 282 1926 L
255 2093 M 282 2093 L
255 2261 M 309 2261 L
180 228 M 203 282 M 195 279 L
190 271 L
188 259 L
188 251 L
190 238 L
195 230 L
203 228 L
208 228 L
216 230 L
221 238 L
224 251 L
224 259 L
221 271 L
216 279 L
208 282 L
203 282 L
198 279 L
195 277 L
193 271 L
190 259 L
190 251 L
193 238 L
195 233 L
198 230 L
203 228 L
208 228 M 213 230 L
216 233 L
219 238 L
221 251 L
221 259 L
219 271 L
216 277 L
213 279 L
208 282 L
129 896 M 139 940 M 142 937 L
139 935 L
136 937 L
136 940 L
139 945 L
142 948 L
149 950 L
160 950 L
167 948 L
170 945 L
172 940 L
172 935 L
170 930 L
162 925 L
149 919 L
144 917 L
139 912 L
136 904 L
136 896 L
160 950 M 165 948 L
167 945 L
170 940 L
170 935 L
167 930 L
160 925 L
149 919 L
136 902 M 139 904 L
144 904 L
157 899 L
165 899 L
170 902 L
172 904 L
144 904 M 157 896 L
167 896 L
170 899 L
172 904 L
172 909 L
203 950 M 195 948 L
190 940 L
188 927 L
188 919 L
190 907 L
195 899 L
203 896 L
208 896 L
216 899 L
221 907 L
224 919 L
224 927 L
CS M
221 940 L
216 948 L
208 950 L
203 950 L
198 948 L
195 945 L
193 940 L
190 927 L
190 919 L
193 907 L
195 902 L
198 899 L
203 896 L
208 896 M 213 899 L
216 902 L
219 907 L
221 919 L
221 927 L
219 940 L
216 945 L
213 948 L
208 950 L
129 1565 M 160 1614 M 160 1565 L
162 1619 M 162 1565 L
162 1619 M 134 1580 L
175 1580 L
152 1565 M 170 1565 L
203 1619 M 195 1616 L
190 1609 L
188 1596 L
188 1588 L
190 1575 L
195 1567 L
203 1565 L
208 1565 L
216 1567 L
221 1575 L
224 1588 L
224 1596 L
221 1609 L
216 1616 L
208 1619 L
203 1619 L
198 1616 L
195 1614 L
193 1609 L
190 1596 L
190 1588 L
193 1575 L
195 1570 L
198 1567 L
203 1565 L
208 1565 M 213 1567 L
216 1570 L
219 1575 L
221 1588 L
221 1596 L
219 1609 L
216 1614 L
213 1616 L
208 1619 L
129 2234 M 167 2280 M 165 2277 L
167 2275 L
170 2277 L
170 2280 L
167 2285 L
162 2287 L
154 2287 L
147 2285 L
142 2280 L
139 2275 L
136 2264 L
136 2249 L
139 2241 L
144 2236 L
152 2234 L
157 2234 L
165 2236 L
170 2241 L
172 2249 L
172 2252 L
170 2259 L
165 2264 L
157 2267 L
154 2267 L
147 2264 L
142 2259 L
139 2252 L
154 2287 M 149 2285 L
144 2280 L
142 2275 L
139 2264 L
139 2249 L
142 2241 L
147 2236 L
152 2234 L
157 2234 M 162 2236 L
167 2241 L
170 2249 L
170 2252 L
167 2259 L
162 2264 L
157 2267 L
203 2287 M 195 2285 L
190 2277 L
188 2264 L
188 2257 L
190 2244 L
195 2236 L
203 2234 L
208 2234 L
216 2236 L
221 2244 L
224 2257 L
224 2264 L
221 2277 L
216 2285 L
208 2287 L
203 2287 L
198 2285 L
195 2282 L
193 2277 L
190 2264 L
190 2257 L
193 2244 L
195 2239 L
198 2236 L
203 2234 L
208 2234 M 213 2236 L
216 2239 L
219 2244 L
221 2257 L
221 2264 L
219 2277 L
216 2282 L
213 2285 L
208 2287 L
2261 255 M 2261 2261 L
2261 255 M 2206 255 L
2261 422 M 2233 422 L
2261 589 M 2233 589 L
2261 756 M 2233 756 L
2261 923 M 2206 923 L
2261 1090 M 2233 1090 L
2261 1258 M 2233 1258 L
2261 1425 M 2233 1425 L
2261 1592 M 2206 1592 L
2261 1759 M 2233 1759 L
2261 1926 M 2233 1926 L
2261 2093 M 2233 2093 L
2261 2261 M 2206 2261 L
CS [] 0 setdash M
CS [] 0 setdash M
1017 37 M 1082 128 M 1082 37 L
1086 128 M 1086 37 L
1082 85 M 1073 93 L
1065 98 L
1056 98 L
1043 93 L
1034 85 L
1030 72 L
1030 63 L
1034 50 L
1043 41 L
1056 37 L
1065 37 L
1073 41 L
1082 50 L
1056 98 M 1047 93 L
1039 85 L
1034 72 L
1034 63 L
1039 50 L
1047 41 L
1056 37 L
1069 128 M 1086 128 L
1082 37 M 1099 37 L
1225 145 M 1216 137 L
1208 124 L
1199 106 L
1195 85 L
1195 67 L
1199 46 L
1208 28 L
1216 15 L
1225 7 L
1216 137 M 1208 119 L
1203 106 L
1199 85 L
1199 67 L
1203 46 L
1208 33 L
1216 15 L
1281 124 M 1277 119 L
1281 115 L
1286 119 L
1286 124 L
1281 128 L
1273 128 L
1264 124 L
1260 115 L
1260 37 L
1273 128 M 1268 124 L
1264 115 L
1264 37 L
1247 98 M 1281 98 L
1247 37 M 1277 37 L
1316 98 M 1316 37 L
1320 98 M 1320 37 L
1320 85 M 1329 93 L
1342 98 L
1351 98 L
1364 93 L
1368 85 L
1368 37 L
1351 98 M 1359 93 L
1364 85 L
1364 37 L
1368 85 M 1377 93 L
1390 98 L
1398 98 L
1411 93 L
1416 85 L
1416 37 L
1398 98 M 1407 93 L
1411 85 L
1411 37 L
1303 98 M 1320 98 L
1303 37 M 1333 37 L
1351 37 M 1381 37 L
1398 37 M 1429 37 L
1450 145 M 1459 137 L
1468 124 L
1476 106 L
1481 85 L
1481 67 L
1476 46 L
1468 28 L
1459 15 L
1450 7 L
1459 137 M 1468 119 L
1472 106 L
1476 85 L
1476 67 L
1472 46 L
1468 33 L
1459 15 L
CS [] 0 setdash M
CS [] 0 setdash M
115 939 M 59 996 M 54 987 L
54 978 L
59 965 L
72 957 L
85 952 L
98 952 L
106 957 L
111 961 L
115 969 L
115 978 L
111 991 L
98 1000 L
85 1004 L
72 1004 L
63 1000 L
41 983 L
33 978 L
24 978 L
20 983 L
20 991 L
24 1000 L
33 1009 L
54 978 M 59 969 L
72 961 L
85 957 L
102 957 L
111 961 L
115 978 M 111 987 L
98 996 L
85 1000 L
67 1000 L
59 996 L
46 987 L
37 983 L
28 983 L
24 987 L
24 996 L
33 1009 L
24 1043 M 115 1043 L
24 1047 M 115 1047 L
50 1073 M 85 1073 L
24 1030 M 24 1099 L
50 1099 L
24 1095 L
67 1047 M 67 1073 L
115 1030 M 115 1060 L
7 1225 M 15 1216 L
28 1208 L
46 1199 L
67 1195 L
85 1195 L
106 1199 L
124 1208 L
137 1216 L
145 1225 L
CS M
15 1216 M 33 1208 L
46 1203 L
67 1199 L
85 1199 L
106 1203 L
119 1208 L
137 1216 L
24 1260 M 115 1260 L
24 1264 M 102 1290 L
24 1260 M 115 1290 L
24 1320 M 115 1290 L
24 1320 M 115 1320 L
24 1325 M 115 1325 L
24 1247 M 24 1264 L
24 1320 M 24 1338 L
115 1247 M 115 1273 L
115 1307 M 115 1338 L
80 1364 M 80 1416 L
72 1416 L
63 1411 L
59 1407 L
54 1398 L
54 1385 L
59 1372 L
67 1364 L
80 1359 L
89 1359 L
102 1364 L
111 1372 L
115 1385 L
115 1394 L
111 1407 L
102 1416 L
80 1411 M 67 1411 L
59 1407 L
54 1385 M 59 1377 L
67 1368 L
80 1364 L
89 1364 L
102 1368 L
111 1377 L
115 1385 L
24 1442 M 115 1472 L
24 1446 M 102 1472 L
24 1502 M 115 1472 L
24 1433 M 24 1459 L
24 1485 M 24 1511 L
7 1528 M 15 1537 L
28 1546 L
46 1554 L
67 1559 L
85 1559 L
106 1554 L
124 1546 L
137 1537 L
145 1528 L
15 1537 M 33 1546 L
46 1550 L
67 1554 L
85 1554 L
106 1550 L
119 1546 L
137 1537 L
CS [] 0 setdash M
1476 2048 M 1476 2090 L
1501 2090 M 1451 2090 L
1501 2090 L
1657 1323 M 1657 1381 L
1682 1381 M 1632 1381 L
1682 1381 L
1926 622 M 1926 679 L
1951 679 M 1901 679 L
1951 679 L
1476 2048 M 1476 2005 L
1501 2005 M 1451 2005 L
1501 2005 L
1657 1323 M 1657 1265 L
1682 1265 M 1632 1265 L
1682 1265 L
1926 622 M 1926 566 L
1951 566 M 1901 565 L
1951 566 L
CS [] 0 setdash M
1476 2048 M 1657 1323 L
1926 622 L
2261 337 L
CS [] 0 setdash M
1114 1533 M 1114 1558 L
1139 1558 M 1089 1558 L
1139 1558 L
1242 1150 M 1242 1182 L
1267 1182 M 1216 1182 L
1267 1182 L
1431 572 M 1431 600 L
1456 600 M 1406 600 L
1456 600 L
1694 322 M 1694 343 L
1719 343 M 1669 343 L
1719 343 L
2048 255 M 2048 264 L
2073 264 M 2022 264 L
2073 264 L
1114 1533 M 1114 1509 L
1139 1509 M 1089 1509 L
1139 1509 L
1242 1150 M 1242 1118 L
1267 1118 M 1216 1118 L
1267 1118 L
1431 572 M 1431 544 L
1456 544 M 1406 544 L
1456 544 L
1694 322 M 1694 302 L
1719 302 M 1669 302 L
1719 302 L
CS [6 12] 0 setdash M
1114 1533 M 1242 1150 L
1431 572 L
1694 322 L
2028 255 L
CS [] 0 setdash M
843 1128 M 843 1144 L
868 1144 M 818 1144 L
868 1144 L
930 958 M 930 978 L
955 978 M 905 978 L
955 978 L
1059 542 M 1059 557 L
1085 557 M 1034 557 L
1085 557 L
1240 331 M 1240 342 L
1265 342 M 1214 342 L
1265 342 L
1482 262 M 1482 268 L
1507 268 M 1456 268 L
1507 268 L
843 1128 M 843 1113 L
868 1113 M 818 1113 L
868 1113 L
930 958 M 930 938 L
955 938 M 905 938 L
955 938 L
1059 542 M 1059 527 L
1085 527 M 1034 527 L
1085 527 L
1240 331 M 1240 320 L
1265 320 M 1214 320 L
1265 320 L
1482 262 M 1482 255 L
1507 255 M 1456 255 L
1507 255 L
CS [32 24] 0 setdash M
843 1128 M 930 958 L
1059 542 L
1240 331 L
1482 262 L
CS [] 0 setdash M
662 1000 M 662 1014 L
687 1014 M 637 1014 L
687 1014 L
722 899 M 722 916 L
747 916 M 697 916 L
747 916 L
812 551 M 812 564 L
837 564 M 787 564 L
837 564 L
937 342 M 937 351 L
962 351 M 911 351 L
962 351 L
1104 268 M 1104 273 L
1129 273 M 1079 273 L
1129 273 L
662 1000 M 662 986 L
687 986 M 637 986 L
687 986 L
722 899 M 722 883 L
747 883 M 697 882 L
747 883 L
812 551 M 812 539 L
837 539 M 787 539 L
837 539 L
937 342 M 937 334 L
962 334 M 911 334 L
962 334 L
1104 268 M 1104 263 L
1129 263 M 1079 263 L
1129 263 L
CS [6 12 32 12] 0 setdash M
662 1000 M 722 899 L
812 551 L
937 342 L
1104 268 L
CS [] 0 setdash M
1218 2194 M CS [] 0 setdash M
1010 2170 M 1030 2217 M 1030 2170 L
1032 2217 M 1032 2170 L
1016 2217 M 1014 2204 L
1014 2217 L
1048 2217 L
1048 2204 L
1046 2217 L
1023 2170 M 1039 2170 L
1061 2197 M 1102 2197 L
1061 2184 M 1102 2184 L
1124 2208 M 1129 2211 L
1135 2217 L
1135 2170 L
1133 2215 M 1133 2170 L
1124 2170 M 1144 2170 L
1167 2217 M 1162 2195 L
1167 2199 L
1174 2202 L
1180 2202 L
1187 2199 L
1192 2195 L
1194 2188 L
1194 2184 L
1192 2177 L
1187 2172 L
1180 2170 L
1174 2170 L
1167 2172 L
1165 2175 L
1162 2179 L
1162 2181 L
1165 2184 L
1167 2181 L
1165 2179 L
1180 2202 M 1185 2199 L
1189 2195 L
1192 2188 L
1192 2184 L
1189 2177 L
1185 2172 L
1180 2170 L
1167 2217 M 1189 2217 L
1167 2215 M 1178 2215 L
1189 2217 L
1212 2217 M 1207 2195 L
1212 2199 L
1218 2202 L
1225 2202 L
1232 2199 L
1236 2195 L
1239 2188 L
1239 2184 L
1236 2177 L
1232 2172 L
1225 2170 L
1218 2170 L
1212 2172 L
1209 2175 L
1207 2179 L
1207 2181 L
1209 2184 L
1212 2181 L
1209 2179 L
1225 2202 M 1230 2199 L
1234 2195 L
1236 2188 L
1236 2184 L
1234 2177 L
1230 2172 L
1225 2170 L
1212 2217 M 1234 2217 L
1212 2215 M 1223 2215 L
1234 2217 L
1292 2217 M 1292 2170 L
1295 2217 M 1308 2177 L
1292 2217 M 1308 2170 L
1324 2217 M 1308 2170 L
1324 2217 M 1324 2170 L
1326 2217 M 1326 2170 L
1286 2217 M 1295 2217 L
1324 2217 M 1333 2217 L
1286 2170 M 1299 2170 L
1317 2170 M 1333 2170 L
1346 2188 M 1373 2188 L
1373 2192 L
1371 2197 L
1369 2199 L
1364 2202 L
1358 2202 L
1351 2199 L
1346 2195 L
1344 2188 L
1344 2184 L
1346 2177 L
1351 2172 L
1358 2170 L
1362 2170 L
1369 2172 L
1373 2177 L
1371 2188 M 1371 2195 L
1369 2199 L
1358 2202 M 1353 2199 L
1348 2195 L
1346 2188 L
1346 2184 L
1348 2177 L
1353 2172 L
1358 2170 L
1387 2217 M 1402 2170 L
1389 2217 M 1402 2177 L
1418 2217 M 1402 2170 L
1382 2217 M 1396 2217 L
1409 2217 M 1423 2217 L
CS [] 0 setdash M
1218 2117 M CS [] 0 setdash M
1015 2093 M 1027 2140 M 1027 2093 L
1029 2140 M 1029 2093 L
1020 2140 M 1047 2140 L
1054 2138 L
CS M
1056 2136 L
1058 2131 L
1058 2127 L
1056 2122 L
1054 2120 L
1047 2118 L
1029 2118 L
1047 2140 M 1051 2138 L
1054 2136 L
1056 2131 L
1056 2127 L
1054 2122 L
1051 2120 L
1047 2118 L
1020 2093 M 1036 2093 L
1040 2118 M 1045 2116 L
1047 2113 L
1054 2098 L
1056 2095 L
1058 2095 L
1060 2098 L
1045 2116 M 1047 2111 L
1051 2095 L
1054 2093 L
1058 2093 L
1060 2098 L
1060 2100 L
1074 2120 M 1114 2120 L
1074 2107 M 1114 2107 L
1137 2131 M 1141 2134 L
1148 2140 L
1148 2093 L
1145 2138 M 1145 2093 L
1137 2093 M 1157 2093 L
1179 2098 M 1177 2095 L
1179 2093 L
1181 2095 L
1179 2098 L
1199 2131 M 1202 2129 L
1199 2127 L
1197 2129 L
1197 2131 L
1199 2136 L
1202 2138 L
1208 2140 L
1217 2140 L
1224 2138 L
1226 2134 L
1226 2127 L
1224 2122 L
1217 2120 L
1211 2120 L
1217 2140 M 1222 2138 L
1224 2134 L
1224 2127 L
1222 2122 L
1217 2120 L
1222 2118 L
1226 2113 L
1228 2109 L
1228 2102 L
1226 2098 L
1224 2095 L
1217 2093 L
1208 2093 L
1202 2095 L
1199 2098 L
1197 2102 L
1197 2104 L
1199 2107 L
1202 2104 L
1199 2102 L
1224 2116 M 1226 2109 L
1226 2102 L
1224 2098 L
1222 2095 L
1217 2093 L
1246 2140 M 1242 2118 L
1246 2122 L
1253 2125 L
1260 2125 L
1267 2122 L
1271 2118 L
1273 2111 L
1273 2107 L
1271 2100 L
1267 2095 L
1260 2093 L
1253 2093 L
1246 2095 L
1244 2098 L
1242 2102 L
1242 2104 L
1244 2107 L
1246 2104 L
1244 2102 L
1260 2125 M 1264 2122 L
1269 2118 L
1271 2111 L
1271 2107 L
1269 2100 L
1264 2095 L
1260 2093 L
1246 2140 M 1269 2140 L
1246 2138 M 1258 2138 L
1269 2140 L
1338 2138 M 1336 2136 L
1338 2134 L
1341 2136 L
1341 2138 L
1338 2140 L
1334 2140 L
1329 2138 L
1327 2134 L
1327 2093 L
1334 2140 M 1332 2138 L
1329 2134 L
1329 2093 L
1320 2125 M 1338 2125 L
1320 2093 M 1336 2093 L
1356 2125 M 1356 2093 L
1359 2125 M 1359 2093 L
1359 2118 M 1363 2122 L
1370 2125 L
1374 2125 L
1381 2122 L
1383 2118 L
1383 2093 L
1374 2125 M 1379 2122 L
1381 2118 L
1381 2093 L
1383 2118 M 1388 2122 L
1394 2125 L
1399 2125 L
1406 2122 L
1408 2118 L
1408 2093 L
1399 2125 M 1403 2122 L
1406 2118 L
1406 2093 L
1350 2125 M 1359 2125 L
1350 2093 M 1365 2093 L
1374 2093 M 1390 2093 L
1399 2093 M 1415 2093 L
CS [] 0 setdash M
897 1759 M CS [] 0 setdash M
689 1736 M 709 1783 M 709 1736 L
711 1783 M 711 1736 L
696 1783 M 693 1769 L
693 1783 L
727 1783 L
727 1769 L
725 1783 L
702 1736 M 718 1736 L
740 1762 M 781 1762 L
740 1749 M 781 1749 L
803 1774 M 808 1776 L
814 1783 L
814 1736 L
812 1780 M 812 1736 L
803 1736 M 823 1736 L
868 1776 M 866 1774 L
868 1771 L
871 1774 L
871 1776 L
868 1780 L
864 1783 L
857 1783 L
850 1780 L
846 1776 L
844 1771 L
841 1762 L
841 1749 L
844 1742 L
848 1738 L
855 1736 L
859 1736 L
866 1738 L
871 1742 L
873 1749 L
873 1751 L
871 1758 L
866 1762 L
859 1765 L
857 1765 L
850 1762 L
846 1758 L
844 1751 L
857 1783 M 853 1780 L
848 1776 L
846 1771 L
844 1762 L
844 1749 L
846 1742 L
850 1738 L
855 1736 L
859 1736 M 864 1738 L
868 1742 L
871 1749 L
871 1751 L
868 1758 L
864 1762 L
859 1765 L
900 1783 M 893 1780 L
889 1774 L
886 1762 L
886 1756 L
889 1745 L
893 1738 L
900 1736 L
904 1736 L
911 1738 L
915 1745 L
918 1756 L
918 1762 L
915 1774 L
911 1780 L
904 1783 L
900 1783 L
895 1780 L
893 1778 L
891 1774 L
889 1762 L
889 1756 L
891 1745 L
893 1740 L
895 1738 L
900 1736 L
904 1736 M 909 1738 L
911 1740 L
913 1745 L
915 1756 L
915 1762 L
913 1774 L
911 1778 L
909 1780 L
904 1783 L
971 1783 M 971 1736 L
974 1783 M 987 1742 L
971 1783 M 987 1736 L
1003 1783 M 987 1736 L
1003 1783 M 1003 1736 L
1005 1783 M 1005 1736 L
965 1783 M 974 1783 L
1003 1783 M 1012 1783 L
965 1736 M 978 1736 L
996 1736 M 1012 1736 L
1025 1753 M 1052 1753 L
1052 1758 L
1050 1762 L
1048 1765 L
1043 1767 L
1037 1767 L
1030 1765 L
1025 1760 L
1023 1753 L
1023 1749 L
1025 1742 L
1030 1738 L
1037 1736 L
1041 1736 L
1048 1738 L
1052 1742 L
1050 1753 M 1050 1760 L
1048 1765 L
1037 1767 M 1032 1765 L
1028 1760 L
1025 1753 L
1025 1749 L
1028 1742 L
1032 1738 L
1037 1736 L
1066 1783 M 1081 1736 L
1068 1783 M 1081 1742 L
1097 1783 M 1081 1736 L
1061 1783 M 1075 1783 L
1088 1783 M 1102 1783 L
CS [] 0 setdash M
897 1682 M CS [] 0 setdash M
694 1659 M 706 1706 M 706 1659 L
708 1706 M 708 1659 L
699 1706 M 726 1706 L
733 1703 L
735 1701 L
737 1697 L
737 1692 L
735 1688 L
733 1686 L
726 1683 L
708 1683 L
726 1706 M 730 1703 L
733 1701 L
735 1697 L
735 1692 L
733 1688 L
730 1686 L
726 1683 L
699 1659 M 715 1659 L
719 1683 M 724 1681 L
726 1679 L
733 1663 L
735 1661 L
737 1661 L
739 1663 L
CS M
724 1681 M 726 1677 L
730 1661 L
733 1659 L
737 1659 L
739 1663 L
739 1665 L
753 1686 M 793 1686 L
753 1672 M 793 1672 L
822 1706 M 816 1703 L
811 1697 L
809 1686 L
809 1679 L
811 1668 L
816 1661 L
822 1659 L
827 1659 L
834 1661 L
838 1668 L
840 1679 L
840 1686 L
838 1697 L
834 1703 L
827 1706 L
822 1706 L
818 1703 L
816 1701 L
813 1697 L
811 1686 L
811 1679 L
813 1668 L
816 1663 L
818 1661 L
822 1659 L
827 1659 M 831 1661 L
834 1663 L
836 1668 L
838 1679 L
838 1686 L
836 1697 L
834 1701 L
831 1703 L
827 1706 L
858 1663 M 856 1661 L
858 1659 L
860 1661 L
858 1663 L
905 1690 M 903 1683 L
899 1679 L
892 1677 L
890 1677 L
883 1679 L
878 1683 L
876 1690 L
876 1692 L
878 1699 L
883 1703 L
890 1706 L
894 1706 L
901 1703 L
905 1699 L
908 1692 L
908 1679 L
905 1670 L
903 1665 L
899 1661 L
892 1659 L
885 1659 L
881 1661 L
878 1665 L
878 1668 L
881 1670 L
883 1668 L
881 1665 L
890 1677 M 885 1679 L
881 1683 L
878 1690 L
878 1692 L
881 1699 L
885 1703 L
890 1706 L
894 1706 M 899 1703 L
903 1699 L
905 1692 L
905 1679 L
903 1670 L
901 1665 L
896 1661 L
892 1659 L
925 1706 M 921 1683 L
925 1688 L
932 1690 L
939 1690 L
946 1688 L
950 1683 L
952 1677 L
952 1672 L
950 1665 L
946 1661 L
939 1659 L
932 1659 L
925 1661 L
923 1663 L
921 1668 L
921 1670 L
923 1672 L
925 1670 L
923 1668 L
939 1690 M 943 1688 L
948 1683 L
950 1677 L
950 1672 L
948 1665 L
943 1661 L
939 1659 L
925 1706 M 948 1706 L
925 1703 M 937 1703 L
948 1706 L
1017 1703 M 1015 1701 L
1017 1699 L
1020 1701 L
1020 1703 L
1017 1706 L
1013 1706 L
1008 1703 L
1006 1699 L
1006 1659 L
1013 1706 M 1011 1703 L
1008 1699 L
1008 1659 L
999 1690 M 1017 1690 L
999 1659 M 1015 1659 L
1035 1690 M 1035 1659 L
1038 1690 M 1038 1659 L
1038 1683 M 1042 1688 L
1049 1690 L
1053 1690 L
1060 1688 L
1062 1683 L
1062 1659 L
1053 1690 M 1058 1688 L
1060 1683 L
1060 1659 L
1062 1683 M 1067 1688 L
1074 1690 L
1078 1690 L
1085 1688 L
1087 1683 L
1087 1659 L
1078 1690 M 1083 1688 L
1085 1683 L
1085 1659 L
1029 1690 M 1038 1690 L
1029 1659 M 1044 1659 L
1053 1659 M 1069 1659 L
1078 1659 M 1094 1659 L
CS [] 0 setdash M
656 1358 M CS [] 0 setdash M
448 1334 M 468 1382 M 468 1334 L
471 1382 M 471 1334 L
455 1382 M 453 1368 L
453 1382 L
486 1382 L
486 1368 L
484 1382 L
462 1334 M 477 1334 L
500 1361 M 540 1361 L
500 1348 M 540 1348 L
563 1373 M 567 1375 L
574 1382 L
574 1334 L
572 1379 M 572 1334 L
563 1334 M 583 1334 L
601 1382 M 601 1368 L
601 1373 M 603 1377 L
607 1382 L
612 1382 L
623 1375 L
628 1375 L
630 1377 L
632 1382 L
603 1377 M 607 1379 L
612 1379 L
623 1375 L
632 1382 M 632 1375 L
630 1368 L
621 1357 L
619 1352 L
616 1346 L
616 1334 L
630 1368 M 619 1357 L
616 1352 L
614 1346 L
614 1334 L
659 1382 M 652 1379 L
648 1373 L
646 1361 L
646 1355 L
648 1343 L
652 1337 L
659 1334 L
664 1334 L
670 1337 L
675 1343 L
677 1355 L
677 1361 L
675 1373 L
670 1379 L
664 1382 L
659 1382 L
655 1379 L
652 1377 L
650 1373 L
648 1361 L
648 1355 L
650 1343 L
652 1339 L
655 1337 L
659 1334 L
664 1334 M 668 1337 L
670 1339 L
672 1343 L
675 1355 L
675 1361 L
672 1373 L
670 1377 L
668 1379 L
664 1382 L
731 1382 M 731 1334 L
733 1382 M 747 1341 L
731 1382 M 747 1334 L
762 1382 M 747 1334 L
762 1382 M 762 1334 L
764 1382 M 764 1334 L
724 1382 M 733 1382 L
762 1382 M 771 1382 L
724 1334 M 738 1334 L
755 1334 M 771 1334 L
785 1352 M 812 1352 L
812 1357 L
809 1361 L
807 1364 L
803 1366 L
796 1366 L
789 1364 L
785 1359 L
782 1352 L
782 1348 L
785 1341 L
789 1337 L
796 1334 L
800 1334 L
807 1337 L
812 1341 L
809 1352 M 809 1359 L
807 1364 L
796 1366 M 791 1364 L
787 1359 L
785 1352 L
785 1348 L
787 1341 L
791 1337 L
796 1334 L
825 1382 M 841 1334 L
827 1382 M 841 1341 L
857 1382 M 841 1334 L
821 1382 M 834 1382 L
848 1382 M 861 1382 L
CS [] 0 setdash M
656 1281 M CS [] 0 setdash M
454 1257 M 465 1305 M 465 1257 L
467 1305 M 467 1257 L
458 1305 M 485 1305 L
492 1302 L
494 1300 L
496 1296 L
496 1291 L
494 1287 L
492 1284 L
485 1282 L
467 1282 L
485 1305 M 490 1302 L
492 1300 L
494 1296 L
494 1291 L
492 1287 L
490 1284 L
485 1282 L
458 1257 M 474 1257 L
478 1282 M 483 1280 L
485 1278 L
492 1262 L
494 1260 L
496 1260 L
499 1262 L
483 1280 M 485 1275 L
490 1260 L
492 1257 L
496 1257 L
499 1262 L
499 1264 L
512 1284 M 553 1284 L
512 1271 M 553 1271 L
582 1305 M 575 1302 L
570 1296 L
CS M
568 1284 L
568 1278 L
570 1266 L
575 1260 L
582 1257 L
586 1257 L
593 1260 L
597 1266 L
600 1278 L
600 1284 L
597 1296 L
593 1302 L
586 1305 L
582 1305 L
577 1302 L
575 1300 L
573 1296 L
570 1284 L
570 1278 L
573 1266 L
575 1262 L
577 1260 L
582 1257 L
586 1257 M 591 1260 L
593 1262 L
595 1266 L
597 1278 L
597 1284 L
595 1296 L
593 1300 L
591 1302 L
586 1305 L
617 1262 M 615 1260 L
617 1257 L
620 1260 L
617 1262 L
662 1298 M 660 1296 L
662 1293 L
665 1296 L
665 1298 L
662 1302 L
658 1305 L
651 1305 L
644 1302 L
640 1298 L
638 1293 L
635 1284 L
635 1271 L
638 1264 L
642 1260 L
649 1257 L
653 1257 L
660 1260 L
665 1264 L
667 1271 L
667 1273 L
665 1280 L
660 1284 L
653 1287 L
651 1287 L
644 1284 L
640 1280 L
638 1273 L
651 1305 M 647 1302 L
642 1298 L
640 1293 L
638 1284 L
638 1271 L
640 1264 L
644 1260 L
649 1257 L
653 1257 M 658 1260 L
662 1264 L
665 1271 L
665 1273 L
662 1280 L
658 1284 L
653 1287 L
685 1305 M 680 1282 L
685 1287 L
692 1289 L
698 1289 L
705 1287 L
709 1282 L
712 1275 L
712 1271 L
709 1264 L
705 1260 L
698 1257 L
692 1257 L
685 1260 L
683 1262 L
680 1266 L
680 1269 L
683 1271 L
685 1269 L
683 1266 L
698 1289 M 703 1287 L
707 1282 L
709 1275 L
709 1271 L
707 1264 L
703 1260 L
698 1257 L
685 1305 M 707 1305 L
685 1302 M 696 1302 L
707 1305 L
777 1302 M 775 1300 L
777 1298 L
779 1300 L
779 1302 L
777 1305 L
772 1305 L
768 1302 L
765 1298 L
765 1257 L
772 1305 M 770 1302 L
768 1298 L
768 1257 L
759 1289 M 777 1289 L
759 1257 M 775 1257 L
795 1289 M 795 1257 L
797 1289 M 797 1257 L
797 1282 M 801 1287 L
808 1289 L
813 1289 L
819 1287 L
822 1282 L
822 1257 L
813 1289 M 817 1287 L
819 1282 L
819 1257 L
822 1282 M 826 1287 L
833 1289 L
837 1289 L
844 1287 L
846 1282 L
846 1257 L
837 1289 M 842 1287 L
844 1282 L
844 1257 L
788 1289 M 797 1289 L
788 1257 M 804 1257 L
813 1257 M 828 1257 L
837 1257 M 853 1257 L
CS [] 0 setdash M
415 1124 M CS [] 0 setdash M
207 1100 M 228 1147 M 228 1100 L
230 1147 M 230 1100 L
214 1147 M 212 1134 L
212 1147 L
246 1147 L
246 1134 L
243 1147 L
221 1100 M 237 1100 L
259 1127 M 299 1127 L
259 1114 M 299 1114 L
317 1138 M 320 1136 L
317 1134 L
315 1136 L
315 1138 L
317 1143 L
320 1145 L
326 1147 L
335 1147 L
342 1145 L
344 1143 L
347 1138 L
347 1134 L
344 1130 L
338 1125 L
326 1121 L
322 1118 L
317 1114 L
315 1107 L
315 1100 L
335 1147 M 340 1145 L
342 1143 L
344 1138 L
344 1134 L
342 1130 L
335 1125 L
326 1121 L
315 1105 M 317 1107 L
322 1107 L
333 1103 L
340 1103 L
344 1105 L
347 1107 L
322 1107 M 333 1100 L
342 1100 L
344 1103 L
347 1107 L
347 1111 L
374 1147 M 367 1145 L
362 1138 L
360 1127 L
360 1121 L
362 1109 L
367 1103 L
374 1100 L
378 1100 L
385 1103 L
389 1109 L
391 1121 L
391 1127 L
389 1138 L
385 1145 L
378 1147 L
374 1147 L
369 1145 L
367 1143 L
364 1138 L
362 1127 L
362 1121 L
364 1109 L
367 1105 L
369 1103 L
374 1100 L
378 1100 M 382 1103 L
385 1105 L
387 1109 L
389 1121 L
389 1127 L
387 1138 L
385 1143 L
382 1145 L
378 1147 L
418 1147 M 412 1145 L
407 1138 L
405 1127 L
405 1121 L
407 1109 L
412 1103 L
418 1100 L
423 1100 L
430 1103 L
434 1109 L
436 1121 L
436 1127 L
434 1138 L
430 1145 L
423 1147 L
418 1147 L
414 1145 L
412 1143 L
409 1138 L
407 1127 L
407 1121 L
409 1109 L
412 1105 L
414 1103 L
418 1100 L
423 1100 M 427 1103 L
430 1105 L
432 1109 L
434 1121 L
434 1127 L
432 1138 L
430 1143 L
427 1145 L
423 1147 L
490 1147 M 490 1100 L
492 1147 M 506 1107 L
490 1147 M 506 1100 L
522 1147 M 506 1100 L
522 1147 M 522 1100 L
524 1147 M 524 1100 L
483 1147 M 492 1147 L
522 1147 M 531 1147 L
483 1100 M 497 1100 L
515 1100 M 531 1100 L
544 1118 M 571 1118 L
571 1123 L
569 1127 L
566 1130 L
562 1132 L
555 1132 L
548 1130 L
544 1125 L
542 1118 L
542 1114 L
544 1107 L
548 1103 L
555 1100 L
560 1100 L
566 1103 L
571 1107 L
569 1118 M 569 1125 L
566 1130 L
555 1132 M 551 1130 L
546 1125 L
544 1118 L
544 1114 L
546 1107 L
551 1103 L
555 1100 L
584 1147 M 600 1100 L
587 1147 M 600 1107 L
616 1147 M 600 1100 L
580 1147 M 593 1147 L
607 1147 M 620 1147 L
CS M
CS [] 0 setdash M
415 1047 M CS [] 0 setdash M
213 1023 M 224 1071 M 224 1023 L
226 1071 M 226 1023 L
218 1071 M 244 1071 L
251 1068 L
253 1066 L
256 1062 L
256 1057 L
253 1053 L
251 1050 L
244 1048 L
226 1048 L
244 1071 M 249 1068 L
251 1066 L
253 1062 L
253 1057 L
251 1053 L
249 1050 L
244 1048 L
218 1023 M 233 1023 L
238 1048 M 242 1046 L
244 1044 L
251 1028 L
253 1026 L
256 1026 L
258 1028 L
242 1046 M 244 1041 L
249 1026 L
251 1023 L
256 1023 L
258 1028 L
258 1030 L
271 1050 M 312 1050 L
271 1037 M 312 1037 L
341 1071 M 334 1068 L
330 1062 L
327 1050 L
327 1044 L
330 1032 L
334 1026 L
341 1023 L
345 1023 L
352 1026 L
357 1032 L
359 1044 L
359 1050 L
357 1062 L
352 1068 L
345 1071 L
341 1071 L
336 1068 L
334 1066 L
332 1062 L
330 1050 L
330 1044 L
332 1032 L
334 1028 L
336 1026 L
341 1023 L
345 1023 M 350 1026 L
352 1028 L
354 1032 L
357 1044 L
357 1050 L
354 1062 L
352 1066 L
350 1068 L
345 1071 L
377 1028 M 375 1026 L
377 1023 L
379 1026 L
377 1028 L
415 1066 M 415 1023 L
417 1071 M 417 1023 L
417 1071 M 392 1037 L
428 1037 L
408 1023 M 424 1023 L
444 1071 M 440 1048 L
444 1053 L
451 1055 L
458 1055 L
464 1053 L
469 1048 L
471 1041 L
471 1037 L
469 1030 L
464 1026 L
458 1023 L
451 1023 L
444 1026 L
442 1028 L
440 1032 L
440 1035 L
442 1037 L
444 1035 L
442 1032 L
458 1055 M 462 1053 L
467 1048 L
469 1041 L
469 1037 L
467 1030 L
462 1026 L
458 1023 L
444 1071 M 467 1071 L
444 1068 M 455 1068 L
467 1071 L
536 1068 M 534 1066 L
536 1064 L
538 1066 L
538 1068 L
536 1071 L
532 1071 L
527 1068 L
525 1064 L
525 1023 L
532 1071 M 529 1068 L
527 1064 L
527 1023 L
518 1055 M 536 1055 L
518 1023 M 534 1023 L
554 1055 M 554 1023 L
556 1055 M 556 1023 L
556 1048 M 561 1053 L
567 1055 L
572 1055 L
579 1053 L
581 1048 L
581 1023 L
572 1055 M 576 1053 L
579 1048 L
579 1023 L
581 1048 M 585 1053 L
592 1055 L
597 1055 L
603 1053 L
606 1048 L
606 1023 L
597 1055 M 601 1053 L
603 1048 L
603 1023 L
547 1055 M 556 1055 L
547 1023 M 563 1023 L
572 1023 M 588 1023 L
597 1023 M 612 1023 L
CS [] 0 setdash M
stroke
grestore
showpage
end


