%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% This file is the source code, in Latex, of the manuscript %% %% ON THE WULFF CONSTRUCTION AS A PROBLEM OF %% EQUIVALENCE OF STATISTICAL ENSEMBLES %% %% by: Salvador Miracle-Sole, Jean Ruiz %% %% after the \end{document} is an encaspuslated postscript file created %% by Cricket Draw %% on Macintosh corresponding to the figure. %% To print it, make the postscript part into a new file and send it % to the printer. %% If you have a problem, send your postal address to %% ruiz@cptsu2.univ-mrs.fr BODY \documentstyle[12pt]{article} %% special macros \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\beqn{\begin{eqnarray}} \def\eeqn{\end{eqnarray}} \def\naturel{ {\bf N} } \def\relatif{ {\bf Z} } \def\reel{ {\bf N} } \def\confh{ {\bf h} } \def\Reel{I\!\!R} \def\t{\thinspace} \def\acknowledgements {\vskip13pt plus12pt {\raggedright \noindent{ \normalsize\bf Acknowledgements\par}} \vskip13pt} \begin{document} \bibliographystyle{unsrt} \parskip=3pt plus 1pt minus 1pt \baselineskip=14pt \begin{center} {\Large \bf On the Wulff Construction as a Problem \\[2mm] of Equivalence of Statistical Ensembles}\\[5mm] \renewcommand{\thefootnote}{\fnsymbol{footnote}} \setcounter{footnote}{-1} \footnote{Preprint CPT-93/P.2952} \renewcommand{\thefootnote}{\arabic{footnote}} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \setcounter{footnote}{-1} \footnote{ To appear in ``Micro, Meso and Macroscopic Approaches in Physics'', edited by A. Verbeure, Plenum Press, New York (1993)} \renewcommand{\thefootnote}{\arabic{footnote}} {\bf Salvador MIRACLE-SOLE}\footnote{ {\it Email address\/}: miracle@cptsu2.univ-mrs.fr} {\bf and Jean RUIZ}\footnote{ {\it Email address\/}: ruiz@cptsu2.univ-mrs.fr}\\[4mm] {\small \sc Centre de Physique Th\'eorique, CNRS Luminy Case 907,\\ F-13288 Marseille Cedex 9, France} \end{center} \vspace{35pt plus 12pt} In this note, the statistical mechanics of SOS (solid-on-solid) 1-dimensio- \hfill\break nal models under the global constraint of having a specified area between the interface and the horizontal axis, is studied. We prove the existence of the thermodynamic limits and the equivalence of the corresponding statistical ensembles. This gives a simple alternative microscopic proof of the validity of Wulff construction for such models, first established in \cite{DDR}. We consider the SOS model defined as follows: To each site $i$ of the lattice $\relatif$ an integer variable $h_i$ is assigned which represents the height of the interface at this site. The energy $ H_N ({ \confh }) $ of a configuration ${\confh} = \{ h_0,h_1,...,h_N \} $, in the box $0 \le i \le N$, of length $N$, is equal to the length of the corresponding interface \beq H_N ({\confh}) = \sum _{i=1}^N ( 1 + |h_i - h_{i-1}|) \label{1} \eeq Its weight, at the inverse temperature $\beta $, is proportional to the Boltzmann factor \ $ \exp [ - \beta H({\confh})] $. More general Hamiltonians of the form $H=\sum P(|h_i - h_{i-1}|)$, where $P$ is a strictly increasing function such that $P(x)\geq |x|$, when $x\to \infty$, can be treated in the same way. The proofs extend also to the case of continuous height variables. We introduce the Gibbs ensemble which consits of all configurations, in the box of length $N$, with specified boundary conditions $h_0=0$ and $h_N=Y$. The associated partition function is given by \beq Z_1 (N,Y) = \sum _{\confh} e^{- \beta H({\confh})} \ \delta (h_0) \delta (h_N - Y) \label{2} \eeq where the sum runs over all configurations in the box and $\delta (t) $ is the dicrete Dirac delta ($\delta (t) = 1 $ if $t=0$ and $\delta (t) = 0 $ otherwise). We define the corresponding free energy per site as the limit \beq \tau _p (y) = \lim _{N \to \infty } -{ 1 \over {\beta N} } \ln Z_1 (N,yN) \label{3} \eeq where $y = - \tan \theta $, the slope of the interface, is a real number. This free energy is called the projected surface tension. The surface tension, which represents the interfacial free energy per unit length of the mean interface, is \beq \tau (\theta ) = \cos \theta \ \tau _p (- \tan \theta ) \label{4} \eeq We introduce also a second Gibbs ensemble, which is conjugate to the previous ensemble, and whose partition function, in the box of length $N$, is given by \beq Z_2 (N,x) = \sum _{\confh} e^{- \beta H({\confh})} e^{\beta x h_N} \ \delta (h_0) \label{5} \eeq where $x \in \reel $ replaces as a thermodynamic parameter the slope $y$ and $h_0=0$. We define the associated free energy as \beq \varphi (x) = \lim _{N \to \infty } -{ 1 \over {\beta N} } \ln Z_2 (N,x) \label{6} \eeq \medskip {\bf Theorem 1.} \it Limits (\ref{3}) and (\ref{6}), which define the above free energies, exist. The first, $\tau _p$, is a convex even function of $y$. The second, $\varphi $, is a concave even function of $x$. Moreover, $\tau _p$ and $ -\varphi $ are conjugate convex functions, i.e., they are related by the Legendre transformations \beq \begin{array}{rcl} - \varphi (x) &=& \displaystyle \sup_y \ [ x y - \tau_p (y) ] \\ \tau _p (y) &=& \displaystyle \sup_x \ [ x y + \varphi (x) ] \end{array} \label{7} \eeq \rm \noindent {\it Proof.\/} The validity of the above statements is well known. See for instance \cite{DKS,MMR} for a proof of these results in a more general setting. $\Box$ The convexity of $\tau _p$ is equivalent to the fact that the surface tension $\tau $ satisfies a stability condition called the triangular inequality \cite{DKS,MMR}. Relations (\ref{7}) between the free energies express the thermodynamic equivalence of the two ensembles (\ref{2}) and (\ref{5}). These relations imply that the curve $z = \varphi (x)$ gives, according to the Wulff construction or its modern equivalent the Andreev construction, the equilibrium shape of the crystal associated to our system \cite{MMR}. The function $\varphi (x)$ defined by (\ref{6}) is easily computed by summing a geometrical serie. One introduces the difference variables \beq n_i = h_{i-1} - h_i \label{8} \eeq for $i = 1,...,N $, so that the partition function factorizes and one obtains \beq \varphi (x) = 1 - \beta ^{-1} \ln \sum _{n\in \relatif } e^{ - \beta |n| + \beta x n} \label{9} \eeq The explicit form of this function is \beq \varphi (x) = 1 - \beta^{-1} \ln { {\sinh \beta} \over {\cosh \beta - \cosh \beta x} } \label{10} \eeq if $-1 < x < 1$ and $ \varphi (x) = - \infty $ otherwise. We next define two new Gibbs ensembles for the system under consideration. In the first of these ensembles we consider the configurations such that $h_N = 0$, which have a specified height at the origin $h_0 = M$ and which have a specified volume $V$ between the interface and the horizontal axis, this volume being counted negatively for negative heights: \beq V = V({\confh}) \equiv \sum _{i=0}^N h_i \label{11} \eeq The corresponding partition function is \beq Z_3 (N,V,M) = \sum _{\confh} e^{- \beta H({\confh})} \ \delta (h_N) \delta (V({\confh}) - V) \delta (h_0 - M) \label{12} \eeq The second of these ensembles is the conjugate ensemble of the first one. Its partition function is given by \beq Z_4 (N,u,\mu) = \sum _{\confh} e^{- \beta H({\confh})} \ e^{\beta u (V({\confh})/N) + \beta \mu h_0 } \ \delta (h_N) \label{13} \eeq where $u \in \reel $ and $\mu \in \reel $ are the conjugate variables. Our next step will be to prove the existence of the thermodynamic limit for these ensembles and their equivalence in this limit. We shall take $N=2^n$, $n \in \naturel$. In (\ref{12}), $V$ and $M$ must be understood as their integer part when they do not belong to $\relatif$. \medskip {\bf Theorem 2.} \it The following limits exist \beqn \psi _3 (v,m) &=& \lim _{N \to \infty } -{ 1 \over {\beta N} } \ \ln Z_3 (N, v N^2 , mN) \label{14} \\ \psi _4 (u,\mu) &=& \lim _{N \to \infty } -{ 1 \over {\beta N} } \ \ln Z_4 (N, u,\mu ) \label{15} \eeqn They define the free energies per site associated to the considered ensembles as, respectively, convex and concave functions of their variables. Moreover, $\psi _3$ and $- \psi _4$ are conjugate convex functions: \beq \begin{array}{rcl} - \psi_4 (u,\mu) &=& \displaystyle \sup_{v,m} \ [ u v + \mu m - \psi_3 (v,m) ] \\ \psi_3 (v, m) &=& \displaystyle \sup_{u,\mu} \ [ u v + \mu m + \psi_4 (u,\mu) ] \end{array} \label{16} \eeq \rm \noindent {\it Proof.\/} The crucial observation is the subadditivity property given in Lemma 1 below. Then we addapt known arguments \cite{R,GMS} in the theory of the thermodynamic limit. A more detailed discussion is given in the Appendix. $\Box$ \medskip {\bf Lemma 1.} \it The partition function $Z_3$ satisfies the subadditivity property \beq Z_3 ( 2N, 2(V' + V''), M' + M'' ) \ge Z_3 ( N, V', M')\ Z_3 ( N, V'', M'') \ e^{- 2 \beta |M''| / (2N-1) } \label{17} \eeq \rm \noindent {\it Proof.\/} In order to prove this property we associate a configuration ${\confh}$ of the first system in the box of length $2N$, to a pair of configurations ${\confh}'$ and ${\confh}''$ of the system in a box of length $N$, as follows \beq \begin{array}{rclr} h_{2i} &=& \displaystyle h'_{i} + h''_{i} \; , & i=0,\dots , N \\ h_{2i-1} &=& \displaystyle h'_{i-1 } + h''_{i} \; , & i=1,\dots , N \end{array} \label{18} \eeq Then $h_{2N} = h'_{N} + h''_{N} = 0$, $h_{0} = h'_{0} + h''_{0} = M' + M''$ and \beqn V({\confh}) &=& 2 \sum_{i=1}^N h'_i + \sum_{i=0}^N h''_i + \sum_{i=1}^N h''_i \nonumber \\ & =& 2\ [ V({\confh'}) + V({\confh''}) ] - M'' \nonumber \eeqn This shows that the configuration $\confh$ belongs to $ Z_3 ( 2N, 2(V' + V'') - M'', M' + M'') $. Since $ H_{2N}({\confh}) = H_N ({\confh'}) + H_N ({\confh''}) $, because $n_{2i} =n'_i$ and $n_{2i-1} =n''_i$ for $i=1,...,N-1 $, as follows from (\ref{18}), we get $$ Z_3 ( N, V', M')\ Z_3 ( N, V'', M'') \leq Z_3 ( 2N, 2(V' + V'') - M'', M' + M'' ) $$ Then we use the change of variables $\tilde{h}_i = h_i + [M''/(2N-1)]$ for $i=1,\dots,2N-1$, $\tilde{h}_0 = h_0$, $\tilde{h}_{2N} = h_{2N}=0$ which gives $$ Z_3 ( 2N, V - M'', M ) \leq e^{ 2 \beta |M''| / (2N-1) } \ Z_3 ( 2N, V , M ) $$ to conclude the proof. $\Box$ The subadditivity property and thus also Theorem 2 are satisfied when the height variables $h_i$ are restricted to take non negative values. \medskip \bf Theorem 3. \it The functions $\psi _3$ and $\psi _4$ can be expressed in terms of the functions $\varphi$ and $\tau_p$ as follows \beqn \psi _3 (v,m) &=& \frac{1}{u_0} \int _{\mu_0} ^{\mu_0 + u_0} \tau_p (\varphi' (x)) dx \label{19} \\ \psi _4 (u,\mu) &=& {1 \over u} \int _0 ^u \varphi (x + \mu) dx \label{20} \eeqn if $u_0$ and $\mu_0$ satisfy \beqn {1 \over {u_0^2} } \int _0 ^{u_0} \varphi (x+ \mu_0) dx - {1 \over {u_0} } \varphi ( \mu_0 + u_0 ) &=& v \label{21} \\ {1 \over u_0 } [\varphi (\mu_0 ) - \varphi ( \mu_0 + u_0) ] &=& m \label{22} \eeqn \rm \noindent {\it Proof.\/} We consider again the difference variables (\ref{8}) and observe that $$ V({\confh}) = \sum _{i=0}^N h_i = \sum _{i=1}^N \; i n _i $$ therefore $$ Z_4 (N,u,\mu) = \prod _{i=1} ^N \Big( \sum _{n_i \in \relatif } e^{ - \beta |n_i| + \beta ( u / N ) i n_i + \beta \mu n_i} \Big) $$ Taking expression (\ref{9}) into account it follows $$ Z_4 (N,u, \mu ) = \exp \Big( - \beta \sum _{i=1} ^N \varphi \big( { u \over N} i + \mu \big) \Big) $$ and $$ \psi _4 (u,\mu) = \lim _{N \to \infty} {1 \over N} \sum _{i=1}^N \varphi \big( { u \over N} i + \mu \big) = \lim _{N \to \infty} {1 \over u} \sum _{i=1}^N {u \over N} \varphi \big( { u \over N} i + \mu \big) $$ which implies expression (\ref{20}) in the Theorem. The function $\psi _3$ is determined by the Legendre transform (\ref{16}). The supremum over $u,\mu$ is obtained for the value $ u_0, \mu_0 $ for which the partial derivatives of the right hand side are zero: $ v + (\partial \psi _4/ \partial u) ( u_0,\mu_0 ) = 0, \; m + (\partial \psi _4/ \partial \mu) ( u_0,\mu_0 ) = 0 $. That is, for $ u_0, \mu_0 $ which satisfy (\ref{21}) and (\ref{22}). Then, from (\ref{16}), (\ref{20}), (\ref{21}) and (\ref{22}), we get \beq \psi _3 (v,m) = 2 \psi_4 (u_0, \mu_0) - \frac{1}{u_0} [(\mu_0 + u_0) \varphi(\mu_0 + u_0)- \mu_0 \varphi(\mu_0)] \label{23} \eeq The right hand side of (\ref{23}) represents twice the area of the sector $OBC$ in Fig.\thinspace 1 divided by $u_0$. But, it is a known property in the Wulff construction, that twice this area is equal to the integral in (\ref{19}). Indeed, by using the relation (\ref{7}) under the form $$ \varphi (x) = x \varphi'(x) + \tau _p (\varphi' (x)) $$ in (\ref{20}) and integrating by parts $x \varphi'(x)$, we get $$ 2 \psi _4 (u_0,\mu_0) = \frac{1}{u_0} \int _{\mu_0} ^{\mu_0 + u_0} \tau_p (\varphi' (x)) dx + \frac{1}{u_0} [(\mu_0 + u_0) \varphi(\mu_0 + u_0)- \mu_0 \varphi(\mu_0)] $$ which together with (\ref{23}) implies the expression (\ref{19}) in the Theorem. $\Box$ To interpret these relations, let us observe that the right hand side of (\ref{21}) represents the area $ABC$, in Fig.\t 1, divided by $\overline{AC}^2$. Therefore, the values $u_0$, $\mu_0$, which solve (\ref{21}) and (\ref{22}), are obtained when this area is equal to $v$, with the condition coming from (\ref{22}), that the slope $\overline{AB}/\overline{AC}$, is equal to $m$. Then, according to (\ref{19}), the free energy $\psi_3(v,m)$ is equal to the integral of the surface tension along the arc $BC$, of the curve $z=\varphi(x)$, divided by the same scaling factor $\overline{BC}=u_0$. We conclude that, for large $N$, the configurations of the SOS model, with a prescribed area $vN^2$, follow a well defined mean profile, the macroscopic profile given by the Wulff construction, with very small fluctuations. This follows from the fact that the probability of the configurations which deviate macroscopically from the mean profile is zero in the thermodynamic limit. The free energy associated to the configurations which satisfy the conditions above, and moreover, are constrained to pass through a given point not belonging to the mean profile, can be computed with the help of Theorem~3. The corresponding probabilities decay exponentially as $N\to \infty$, as a concequence of the usual large deviations theory in statistical mechanics \cite{RL}. \vskip35pt plus12pt \noindent { \Large\bf Appendix \par } \vskip13pt We give here a more detailed discussion of the proof of Theorem 2. We define: $$ f_n(v,m) = -{ 1 \over {\beta 2^n} } \ln Z_3 (2^n, 2^{2n} v, 2^n m ) $$ For $v$ and $m$ of the form $2^{-q}p$, the subadditivity property with $N=2^n$, $V'=V''=vN^2$ and $M'=M''=mN$ implies that $f_n$ is a decreasing sequence: $f_{n+1}(v,m) \leq f_n(v,m)$. Since this sequence is bounded from below its limit exits when n tends to infinity and coincide with \beq \psi_3(v,m) = \inf _N \left[ -{ 1 \over {\beta N} } \ln Z_3 (N, v N^2,mN )\right] \label{A.1} \eeq Indeed $$ Z_3 (N,V,M) \leq \sum _{\confh} e^{- \beta H({\confh})} \delta (h_N) $$ The right hand side of the above expression is easily computed by inroducing the difference variables (\ref{8}) and we get that $f_n$ is bounded from below by $1 + (1/\beta) \ln \tanh (\beta / 2)$. Let us notice that one can obtain a lower bound to $Z_3(N,vN^2,mN)$ by restricting the summation to the configuration such that $(N-1)h_i = v N^2 - mN$ for $i=1,\dots,N-1$. This gives that $f_n$ is bounded from above by $1 +|v-m| + |v|$. To prove that $\psi_3$ is convex, we notice that the subadditivity inequality (\ref{17}) with $N=2^n$, $V'=v_1 N^2$, $V''=v_2 N^2$ and $M'=M''=mN^2$ gives: $$ \psi_3 \Big( {1\over 2} (v_1 +v_2),m \Big)\leq {1\over 2} \psi_3(v_1,m) + {1\over 2} \psi_3(v_2,m) $$ which applied iteratively implies: $$ \psi_3 \Big( \alpha v_1 + (1- \alpha) v_2,m \Big)\leq \alpha \psi_3(v_1,m) + (1 -\alpha) \psi_3(v_2,m) $$ for $\alpha$ of the form $2^{-q}p$ and $0\leq \alpha \leq 1$. For such $\alpha$ we obtain analogously $$ \psi_3 \Big( v , \alpha m_1 + (1- \alpha) m_2 \Big)\leq \alpha \psi_3(v,m_1) + (1 -\alpha) \psi_3(v,m_2) $$ by applying the subadditivity inequality (\ref{17}) with $N=2^n$, $M'=m_1 N^2$, $M''=m_2 N^2$ and $V'=V''=vN^2$. Since $\psi _3$ is bounded, it follows, cf. \cite{R}, that $\psi _3$ can be extended to a convex Lipshitz continuous function of the real variables $v$ and $m$. To prove the the existence of the limit (\ref{15}) and relations (\ref{16}), we introduce $$ Z_4^+(N,u,\mu) =\sup_{V,M \in \relatif } \left[ e^{\beta u (V/N) + \beta \mu M } \ Z_3(N,V,M) \right] $$ and proceed, as in the Appendix of \cite{GMS}, to study the thermodynamic limit for this quantity. We introduce the convex function $$ \psi^{\ast}_4 (u,\mu) = \sup_{v,m} \ [uv+\mu m -\psi_3 (v,m)] $$ According to (\ref{A.1}), we have $$ e^{\beta u (V/N) + \beta \mu M} \ Z_3(N,V,M) \leq e^{\beta N [uv+\mu m - \psi_3(v,m)]} $$ for all $ V, M \in \relatif$, so that \beq Z_4^+(N,u,\mu) \; \leq \; e^{\beta N \psi_4^{\ast}(u,\mu)} \label{A.2} \eeq On the other hand for any $\delta>0$ and sufficiently large $N$ one can find, $V = v N^2$ and $M=mN$ such that \beqn e^{\beta u (V/N)+\beta \mu M} \; Z_3(N,V,M) &=& e^{\beta N [uv+\mu m - \psi_3(v,m)]} \; e^{\beta N \psi_3(v,m)} \; Z_3(N,V,M) \nonumber \\ &\geq & e^{\beta N [\psi_4^{\ast}(u,\mu)- \delta]} \nonumber \eeqn and therefore \beq Z_4^+(N,u,\mu) \; \geq \; e^{\beta N [\psi_4^{\ast}(u,\mu)- \delta]} \label{A.3} \eeq Inequalities (\ref{A.2}) and (\ref{A.3}) imply that \beq \lim_{N \rightarrow \infty}\; -{1\over \beta N} \; \ln Z_4^+(N,u,\mu) \; =\; - \psi_4^{\ast}(u,\mu) \label{A.4} \eeq We shall now prove that the thermodynamic limit (\ref{15}) exits and gives the same quantity following the argument of Theorem 2 in \cite{GMS}. First, we notice that $$ Z_4(N,u,\mu) =\sum_{V,M \in \relatif } e^{\beta u (V/N) + \beta \mu M } \ Z_3(N,V,M) $$ which implies \beq Z_4^+(N,u,\mu) \leq Z_4(N,u,\mu) \label{A.5} \eeq Moreover, the inequality $$ Z_3(N,V,M) \leq e^{-\beta {\bar u} (V/N)-\beta {\bar {\mu}} M} \; Z_4^+(N,{\bar u},{\bar {\mu}}) $$ used with ${\bar u}=u'$ and ${\bar u}=u''$, ${\bar {\mu}}=\mu'$ and ${\bar {\mu }}=\mu''$ gives for any $u''0 \nonumber \\ \Delta u &=& \min \{ \mu'-\mu,\mu-\mu'' \} >0 \nonumber \eeqn By referring to (\ref{A.4}), and to the continuity of $\psi^{\ast}_4$, the inequalities (\ref{A.5}) and (\ref{A.6}) imply (\ref{15}) and (\ref{16}). \acknowledgements The authors thank Mons University, where part of this work was done, for warm hospitality and acknowledge the NATO and the ERASMUS project for financial support. \begin{thebibliography}{9} \bibitem{DDR} J.\t De Coninck, F.\t Dunlop, and V.\t Rivasseau, On the microscopic validity of the Wulff construction and of the generalized Young equation, {\em Commun.\t Math.\t Phys\t.} {\bf 121}, 401--419 (1989). \bibitem{DKS} R.\t L.\t Dobrushin, R.\t Koteck\'y and S.\t B.\t Shlosman, ``Wulff Construction: A Global Shape from Local Interactions,'' Am.\t Math.\t Soc.\t, Providence, RI (1992). \bibitem{MMR} A.\t Messager, S.\t Miracle-Sol\'e and J.\t Ruiz, Convexity properties of the surface tension and equilibrium crystals, {\em J.\t Stat.\t Phys.\t} {\bf 67}, 449--470 (1992). \bibitem{R} D.\t Ruelle, `` Statistical Mechanics: Rigorous Results,'' Benjamin, New York Amsterdam (1969). \bibitem{GMS} L.\t Galgani, L.\t Manzoni, and A.\t Scotti, Asymptotic equivalence of equilibrium ensembles of classical statistical mechanics, {\em J.\t Math.\t Phys\t.} {\bf12}, 933--935 (1971). \bibitem{RL} D.\t Ruelle, ``Hasard et Chaos'' (chap.\t 19), Odile Jacob, Paris (1991). O.\t Landford, Entropy and equilibrium states in classical statistical mechanics, {\it in\/}: ``Statistical Mechanics and Mathematical Problems,'' A.\t Lenard, ed.\t , Springer, Berlin (1973). \end{thebibliography} \newpage \centerline{\bf Figure Captions} Figure 1. 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reflecty {yreflect} if shearangle dup 0 ne {shear}{pop} ifelse }bdef /$arc 6 dict def $arc /mtrx matrix put /doarc %Copyright 1986 Cricket Software {$arc begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /savematrix mtrx currentmatrix def xrad yrad scale xreflect 0 0 1 startangle endangle startangle endangle gt {arcn} {arc} ifelse savematrix setmatrix end}bdef /$rect 6 dict def /dorect %Copyright 1986 Cricket Software {$rect begin /yrad exch 2 div def /xrad exch 2 div def /left xrad neg def /right xrad def /bottom yrad neg def /top 0 yrad add def newpath left bottom moveto left top lineto right top lineto right bottom lineto closepath end}bdef /$corner 8 dict def $corner /mtrx matrix put /docorner %Copyright 1986 Cricket Software {$corner begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix end}bdef /$rrect 4 dict def /doroundrect {$rrect begin /ovalyrad exch def /ovalxrad exch def /yrad exch 2 div def /xrad exch 2 div def newpath xrad ovalxrad sub yrad ovalyrad sub ovalxrad ovalyrad 0 90 docorner xrad neg ovalxrad add yrad ovalyrad sub ovalxrad ovalyrad 90 180 docorner xrad neg ovalxrad add yrad neg ovalyrad add ovalxrad ovalyrad 180 270 docorner xrad ovalxrad sub yrad neg ovalyrad add ovalxrad ovalyrad 270 360 docorner closepath end}bdef /$diamond 6 dict def %Copyright 1986 Cricket Software /dodiamond {$diamond begin /yrad exch 2 div def /xrad exch 2 div def /left xrad neg def /right xrad def /bottom yrad neg def /top yrad def newpath left 0 moveto 0 top lineto right 0 lineto 0 bottom lineto closepath end}bdef /$grate 25 dict def /dograte %Copyright 1986 Cricket Software {$grate begin /dolog exch def /nlines exch def /y4 exch def /x4 exch def /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def /minx x1 def /maxx x1 def /miny y1 def /maxy y1 def x2 minx lt {/minx x2 def} if x2 maxx gt {/maxx x2 def} if x3 minx lt {/minx x3 def} if x3 maxx gt {/maxx x3 def} if x4 minx lt {/minx x4 def} if x4 maxx gt {/maxx x4 def} if y2 miny lt {/miny y2 def} if y2 maxy gt {/maxy y2 def} if y3 miny lt {/miny y3 def} if y3 maxy gt {/maxy y3 def} if y4 miny lt {/miny y4 def} if y4 maxy gt {/maxy y4 def} if /dx maxx minx sub def /dy maxy miny sub def /pen2 currentlinewidth 2 div def newpath x1 x2 lt {x1 pen2 sub} {x1 pen2 add} ifelse y1 pen2 sub moveto x2 x1 gt {x2 pen2 add}{x2 pen2 sub} ifelse y2 pen2 sub lineto x3 x4 lt {x3 pen2 sub}{x3 pen2 add} ifelse y3 pen2 add lineto x4 x3 gt {x4 pen2 add}{x4 pen2 sub} ifelse y4 pen2 add lineto closepath clip minx maxy translate newpath dolog {1 10 nlines div 10 {log dy mul 0 exch neg moveto dx 0 rlineto }for} {0 1 nlines 1 sub div 1 {dy mul 0 exch neg moveto dx 0 rlineto }for} ifelse end}bdef /$rgrate 25 dict def /doradialgrate %Copyright 1986 Cricket Software {$rgrate begin /nlines exch def /y4 exch def /x4 exch def /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def /minx x1 def /maxx x1 def /miny y1 def /maxy y1 def x2 minx lt {/minx x2 def} if x2 maxx gt {/maxx x2 def} if x3 minx lt {/minx x3 def} if x3 maxx gt {/maxx x3 def} if x4 minx lt {/minx x4 def} if x4 maxx gt {/maxx x4 def} if y2 miny lt {/miny y2 def} if y2 maxy gt {/maxy y2 def} if y3 miny lt {/miny y3 def} if y3 maxy gt {/maxy y3 def} if y4 miny lt {/miny y4 def} if y4 maxy gt {/maxy y4 def} if /centerx minx maxx add 2 div def /centery miny maxy add 2 div def /xradius maxx minx sub 2 div def /yradius maxy miny sub 2 div def /pen2 currentlinewidth def newpath x1 x2 lt {x1 pen2 sub} {x1 pen2 add} ifelse y1 pen2 sub moveto x2 x1 gt {x2 pen2 add}{x2 pen2 sub} ifelse y2 pen2 sub lineto x3 x4 lt {x3 pen2 sub}{x3 pen2 add} ifelse y3 pen2 add lineto x4 x3 gt {x4 pen2 add}{x4 pen2 sub} ifelse y4 pen2 add lineto closepath clip centerx centery translate xradius yradius gt {/step xradius nlines div def} {/step yradius nlines div def} ifelse /stepradius step def 1 1 nlines {newpath 0 0 stepradius 0 360 arc closepath stroke /stepradius stepradius step add def }for end}bdef /$star 8 dict def /dostarburst %Copyright 1986 Cricket Software {$star begin /endtheta exch def /inctheta exch def /begintheta exch def /yradius exch def /xradius exch def newpath begintheta inctheta endtheta {dup /x exch cos xradius mul def /y exch sin yradius mul neg def x y moveto 0 0 lineto }for end}bdef /$poly 2 dict def /dopoly %Copyright 1986 Cricket Software {$poly begin /polyArray exch def polyArray 0 get polyArray 1 get moveto 2 2 polyArray length 1 sub {/i exch def polyArray i get polyArray i 1 add get lineto }for end}bdef /$smoothpoly 20 dict def /dosmoothpoly %Copyright 1986 Cricket Software {$smoothpoly begin /list exch def /count list length 2 div cvi def /passes count 2 sub def /coeff1 .166667 def /coeff2 .833333 def /coeff3 .5 def /firstx list 0 get def /firsty list 1 get def /lastx list dup length 2 sub get def /lasty list dup length 1 sub get def /closed firstx lastx eq firsty lasty eq and def firstx firsty moveto firstx coeff3 mul list 2 get coeff3 mul add firsty coeff3 mul list 3 get coeff3 mul add closed {moveto} {lineto} ifelse 1 1 passes {/i exch 1 sub 2 mul def /x0 list 0 i add get def /x1 list 1 i add get def /x2 list 2 i add get def /x3 list 3 i add get def /x4 list 4 i add get def /x5 list 5 i add get def x0 coeff1 mul x2 coeff2 mul add x1 coeff1 mul x3 coeff2 mul add x2 coeff2 mul x4 coeff1 mul add x3 coeff2 mul x5 coeff1 mul add x2 coeff3 mul x4 coeff3 mul add x3 coeff3 mul x5 coeff3 mul add curveto } for closed {/x0 list dup length 4 sub get def /x1 list dup length 3 sub get def /x2 lastx def /x3 lasty def /x4 list 2 get def /x5 list 3 get def x0 coeff1 mul x2 coeff2 mul add x1 coeff1 mul x3 coeff2 mul add x2 coeff2 mul x4 coeff1 mul add x3 coeff2 mul x5 coeff1 mul add x2 coeff3 mul x4 coeff3 mul add x3 coeff3 mul x5 coeff3 mul add curveto}{lastx lasty lineto} ifelse end} def /$bitmap 20 dict def /dobitmap %Copyright 1986 Cricket Software {$bitmap begin /height exch def /width exch def bitgray setgray width height true [width 0 0 height 0 0] {currentfile picstring readhexstring pop} imagemask end}bdef /$head 15 dict def /arrowhead %Copyright 1986 Cricket Software {$head begin /thickness currentlinewidth def /taily exch def /tailx exch def /tipy exch def /tipx exch def /headlength 10 thickness mul neg def /halfwidth thickness dup add def /dx tipx tailx sub def /dy tipy taily sub def /priorpath [{/moveto load} {/lineto load} {/curveto load} {} pathforall] cvx def gsave .2 setlinewidth 0 setlinejoin tipx tipy translate dy dx atan rotate /offset headlength 2 div neg def newpath offset 0 moveto headlength halfwidth rlineto 0 halfwidth 2 mul neg rlineto closepath fill grestore priorpath stroke end} bdef /$arrow 15 dict def /strokearrow {$arrow begin /beginpath exch def /endpath exch def /counter 0 def /pointList 500 array def flattenpath {addpt} {addpt} {addpt} {} pathforall /pt1x pointList 0 get def /pt1y pointList 1 get def /pt2x pointList 2 get def /pt2y pointList 3 get def beginpath {pt1x pt1y pt2x pt2y arrowhead} if /pt1x pointList counter 2 sub get def /pt1y pointList counter 1 sub get def /pt2x pointList counter 4 sub get def /pt2y pointList counter 3 sub get def endpath {pt1x pt1y pt2x pt2y arrowhead} if end} bdef /$shadow 15 dict def $shadow begin /MoveProc { Xform transform moveto } def /LineProc { Xform transform lineto } def /CurveProc { 3 { Xform transform 6 2 roll} repeat curveto } def /CloseProc {closepath} def end /shadow %Copyright 1986 Cricket Software {save 6 1 roll $shadow begin /theStep dpi 4 div def setgray /endgray exch def /startgray exch def /ystep 1 index theStep div neg def /xstep 2 index theStep div neg def /Xform [ 1 0 0 1 8 -2 roll ] def /incgray startgray endgray sub theStep div neg def gsave {MoveProc}{LineProc}{CurveProc}{CloseProc} pathforall stroke grestore startgray incgray add incgray endgray {gsave setgray Xform aload pop ystep add exch xstep add exch Xform astore pop {MoveProc}{LineProc}{CurveProc}{CloseProc} pathforall stroke grestore } for end restore } bdef /$charshadow 20 dict def /charshadow %Copyright 1986 Cricket Software {$charshadow begin /xspace exch def /bordergray exch def /endgray exch def /startgray exch def /yoffset exch def /xoffset exch def /theString exch def /priorgray currentgray def /theStep dpi 4 div def /ystep yoffset theStep div def /xstep xoffset neg theStep div def endgray startgray eq {/startgray startgray .001 sub def} if /incgray endgray startgray sub theStep div neg def currentpoint /y exch def /x exch def x y moveto xoffset yoffset neg translate bordergray setgray xspace 0 32 theString widthshow endgray incgray add incgray startgray {setgray x y moveto xstep ystep translate xspace 0 32 theString widthshow }for x y moveto xstep ystep translate priorgray setgray xspace 0 32 theString widthshow end}bdef /makefountainstring %Copyright 1986 Cricket Software {/trfn exch def /endcolor exch 255 mul def /begcolor exch 255 mul def endcolor begcolor eq {/endcolor begcolor .001 sub def} if /range endcolor begcolor sub def 0 1 255 {fountainstring exch dup 255 div trfn range mul begcolor add round cvi put }for fountainstring }bdef /$fountain 10 dict def /fountain %Copyright 1986 Cricket Software {$fountain begin /theangle exch def gsave eoclip theangle 180 sub rotate pathbbox /ury exch def /urx exch def /lly exch def /llx exch def urx ury translate 180 rotate urx llx sub ury lly sub scale 1 256 8 [1 0 0 256 0 0] {makefountainstring} image grestore end} bdef /$radialfountain 12 dict def /radialfountain %Copyright 1986 Cricket Software {$radialfountain begin /endcolor exch def /begcolor exch def endcolor begcolor eq {/endcolor begcolor .001 sub def} if /range endcolor begcolor sub def gsave eoclip pathbbox /ury exch def /urx exch def /lly exch def /llx exch def /centerx urx llx add 2 div def /centery lly ury add 2 div def centerx centery translate /radius urx centerx sub dup mul lly centery sub dup mul add sqrt def /step radius 255 div def 0 1 255 {255 div range mul begcolor add setgray newpath 0 0 radius 0 360 arc closepath fill /radius radius step sub def }for grestore end}bdef /$offset 15 dict def /offsetcalc %Copyright 1986 Cricket Software {$offset begin /textlength exch def /just exch def /dist 0 def gsave flattenpath {/priory exch def /priorx exch def} {/newy exch def /newx exch def /dx newx priorx sub def /dy newy priory sub def /dist dx dup mul dy dup mul add sqrt dist add def /priorx newx def /priory newy def} {} {/newy exch def /newx exch def /dx newx priorx sub def /dy newy priory sub def /dist dx dup mul dy dup mul add sqrt def} pathforall just 0 eq {/pathoffset 0 store} if just 1 eq {/pathoffset dist textlength sub 2 div store} if just 2 eq {/pathoffset dist textlength sub store} if grestore end} def /$pathtext 35 dict def /pathtext {$pathtext begin %Copyright 1986 Cricket Software %Based on routines from Adobe /FirstMoveto true def /offset exch def /str exch def /offset offset str 0 1 getinterval stringwidth pop 2 div add def /pathdist 0 def /setdist offset def /charcount 0 def gsave flattenpath {movetoproc}{linetoproc} {curvetoproc}{closepathproc} pathforall grestore end}bdef $pathtext begin /movetoproc {/newy exch def /newx exch def /firstx newx def /firsty newy def /ovr FirstMoveto { offset /FirstMoveto false def } { 0 } ifelse def newx newy transform /cpy exch def /cpx exch def }bdef /linetoproc {/oldx newx def /oldy newy def /newy exch def /newx exch def /dx newx oldx sub def /dy newy oldy sub def /dist dx dup mul dy dup mul add sqrt def /dist 0 ne {/dsx dx dist div ovr mul def /dsy dy dist div ovr mul def oldx dsx add oldy dsy add transform /cpy exch def /cpx exch def /pathdist pathdist dist add def {setdist pathdist le {charcount str length lt {setchar }{ exit } ifelse } {/ovr setdist pathdist sub def exit } ifelse} loop }if }bdef /curvetoproc { }bdef /closepathproc {firstx firsty linetoproc firstx firsty movetoproc}bdef /setchar {/char str charcount 1 getinterval def /charcount charcount 1 add def /charwidth char stringwidth pop 2 div def gsave cpx cpy itransform translate dy dx atan rotate charwidth neg 0 moveto char show charcount str length lt { str charcount 1 getinterval stringwidth pop 2 div 0 rmoveto }if currentpoint transform /cpy exch def /cpx exch def grestore /setdist setdist charwidth add def /pathoffset pathoffset charwidth 2 mul add store charcount str length lt { /setdist setdist str charcount 1 getinterval stringwidth pop 2 div add def}if }bdef end /$makeoutline 20 dict def /MakeOutlineFont %Copyright 1986 Cricket Software %Based on routines from Adobe {$makeoutline begin /uniqueid exch def /strokewidth 1000 24 div def%.25 /newfontname exch def /basefontname exch def uniqueid 0 eq {/uniqueid basefontname findfont dup /UniqueID known {/UniqueID get 1 add}{pop 1}ifelse def} if /basefontdict basefontname findfont def /numentries basefontdict maxlength 1 add def basefontdict /UniqueID known not {/numentries numentries 1 add def} if /outfontdict numentries dict def basefontdict { exch dup /FID ne {exch outfontdict 3 1 roll put} {pop pop} ifelse }forall outfontdict /FontName newfontname put outfontdict /PaintType 2 put outfontdict /StrokeWidth strokewidth put outfontdict /UniqueID uniqueid put newfontname outfontdict definefont pop end} bdef /MakeUnderlineFont %Copyright 1986 Cricket Software {20 dict begin /uniqueid exch def exch /BaseFont exch findfont def uniqueid 0 eq {/uniqueid BaseFont findfont dup /UniqueID known {/UniqueID get 1 add}{pop 1}ifelse def} if /FontType 3 def /Upos BaseFont /FontInfo 2 copy known {get /UnderlinePosition 2 copy known {get }{ pop pop -100 } ifelse} { pop pop -100 } ifelse def /Uwid BaseFont /FontInfo 2 copy known {get /UnderlineThickness 2 copy known {get }{ pop pop 50 } ifelse} { pop pop 50 } ifelse def /FontMatrix BaseFont /FontMatrix get def /FontBBox BaseFont /FontBBox get dup 1 get Upos gt { dup 1 Upos put } if def /Encoding BaseFont /Encoding get def /theChar 1 string def /BuildChar {exch begin theChar 0 3 -1 roll put BaseFont 1000 scalefont setfont theChar stringwidth FontBBox setcachedevice 0 0 moveto theChar show 0 Upos rmoveto 0 Upos lineto Uwid setlinewidth stroke end}def currentdict end definefont pop }bdef /$lshow 20 dict def /leftshow {$lshow begin /theText exch def /spaceplus 0 def gsave currentpoint translate 0 0 moveto /show /myshow load def theText grestore end} bdef /$cshow 20 dict def $cshow /mtrx matrix put /centershow {$cshow begin /theText exch def /spaceplus 0 def gsave currentpoint translate /savematrix mtrx currentmatrix def /beforejustify save def nulldevice savematrix setmatrix 0 0 moveto theText currentpoint beforejustify restore pop 0 0 moveto neg 2 div 0 rmoveto /show /myshow load def theText grestore end} bdef /$rshow 20 dict def $rshow /mtrx matrix put /rightshow {$rshow begin /theText exch def /spaceplus 0 def gsave currentpoint translate /savematrix mtrx currentmatrix def /beforejustify save def nulldevice savematrix setmatrix 0 0 moveto theText currentpoint beforejustify restore pop 0 0 moveto neg 0 rmoveto /show /myshow load def theText grestore end} bdef /$fshow 20 dict def /fullshow {$fshow begin /priorstate save def /theText exch def /linewidth exch currentpoint pop sub def /numspaces 0 def currentpoint translate /savematrix mtrx currentmatrix def /beforejustify save def nulldevice savematrix setmatrix /textwidth 0 0 moveto theText currentpoint pop def /whitespace linewidth textwidth sub def /show /countspaces load def theText numspaces 0 gt {whitespace numspaces div} {0} ifelse beforejustify restore /spaceplus exch def 0 0 moveto /show /justifyshow load def theText priorstate restore end} bdef $fshow begin /countspaces {/theString exch def theString {32 eq {/numspaces numspaces 1 add def} if } forall } def /printChar {/charcode exch def /theChar (x) dup 0 charcode put def theChar myshow charcode 32 eq {spaceplus 0 rmoveto} if } def /justifyshow {/theString exch def theString {printChar} forall } def end /MacEncoding 256 array def MacEncoding 0 /Times-Roman findfont /Encoding get 0 128 getinterval putinterval MacEncoding 16#27 /quotesingle put MacEncoding 16#60 /grave put /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /degree /cent /sterling /section /bullet /paragraph /germandbls /registersans /copyrightsans /trademarksans /acute /dieresis /notequal /AE /Oslash /infinity /plusminus /lessequal /greaterequal /yen /mu /partialdiff /summation /product /pi /integral /ordfeminine /ordmasculine /Omega /ae /oslash /questiondown /exclamdown /logicalnot /radical /florin /approxequal /Delta /guillemotleft /guillemotright /ellipsis /space /Aacute /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /divide /lozenge /ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright /fi /fl /daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand /Acircumflex /Ecircumflex /Agrave /Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute /Ocircumflex /apple /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde /macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron MacEncoding 128 128 getinterval astore pop /local [/exch load /def load] cvx def /copydict {/newfontdict exch def {exch dup /FID ne {exch newfontdict 3 1 roll put} {pop pop} ifelse} forall newfontdict }bdef /makename {1 index length /prefixlength exch def dup length prefixlength add string dup prefixlength 4 -1 roll putinterval dup 0 4 -1 roll putinterval }bdef /coordinatefont {dup (|______) exch makename cvn /newname exch def FontDirectory newname known {pop} {cvn findfont dup maxlength dict copydict dup /fontname known {dup /fontname newname put} if dup /Encoding MacEncoding put newname exch definefont pop } ifelse } bdef %----------------------------------- %Encode PS Fonts to match Mac Fonts (Symbol) coordinatefont (Times-Roman) coordinatefont %----------------------------------- %----- Begin Main Program -----% gsave %Line 216 234.5 0 false false 0 fixcoordinates newpath 0 -125.5 moveto 0 125.5 lineto 1 setlinewidth 0 setgray stroke grestore gsave %Line 216 234.5 0 false false 0 fixcoordinates newpath 0 -125.5 moveto 0 125.5 lineto 1 setlinewidth 0 setgray stroke grestore gsave %Bezier Curve 307.237 271.248 0 false false 0 fixcoordinates newpath -91.259 -90.751 moveto -18.655 -85.709 40.839 -36.3 91.259 90.751 curveto 1 setlinewidth 0 setgray stroke grestore gsave %Bezier Curve 125.75 270.5 0 false true 0 fixcoordinates newpath -90.5 -90 moveto -18.5 -85 40.5 -36 90.5 90 curveto 1 setlinewidth 0 setgray stroke grestore gsave %Text Block 264.999 346.327 0 true false 0 fixcoordinates /myshow /show load def 0 setgray -19 -1 moveto { /|______Symbol findfont 14 scalefont setfont (m) show /|______Times-Roman findfont 11 scalefont setfont 0 -4 rmoveto (0) show 0 4 rmoveto } leftshow grestore gsave %Text Block 363 348 0 true false 0 fixcoordinates /myshow /show load def 0 setgray -23 0 moveto { /|______Symbol findfont 14 scalefont setfont (m) show /|______Times-Roman findfont 11 scalefont setfont 0 -4 rmoveto (0) show 0 4 rmoveto /|______Times-Roman findfont 14 scalefont setfont (+u) show /|______Times-Roman findfont 11 scalefont setfont 0 -4 rmoveto (0) show 0 4 rmoveto } leftshow grestore gsave %Text Block 199.941 117.999 0 true false 0 fixcoordinates /myshow /show load def 0 setgray -27 -3 moveto { /|______Symbol findfont 14 scalefont setfont (j) show /|______Times-Roman findfont 14 scalefont setfont (\(x\)) show } leftshow grestore gsave %Grouped Object 295 324 0 false false 0 fixcoordinates gsave %Polygon 135.5 0 0 false false 0 fixcoordinates /counter 0 def /pointList 10 array def 7.5 -0.5 addpt -7.5 3.5 addpt -7.5 -0.5 addpt -7.5 -3.5 addpt 7.5 -0.5 addpt newpath pointList dopoly closepath gsave 0 setgray eofill grestore 1 setlinewidth 0 setgray stroke grestore gsave %Line -7.5 -0.5 0 false false 0 fixcoordinates newpath -135.5 0 moveto 135.5 0 lineto 1 setlinewidth 0 setgray 0 setlinecap stroke grestore grestore gsave %Grouped Object 215.75 260.75 0 false false 0 fixcoordinates gsave %Polygon 0 -147 0 false false 0 fixcoordinates /counter 0 def /pointList 10 array def 0 -7 addpt 4 7 addpt 0 7 addpt -4 7 addpt 0 -7 addpt newpath pointList dopoly closepath gsave 0 setgray eofill grestore 1 setlinewidth 0 setgray stroke grestore gsave %Line 0 7 0 false false 0 fixcoordinates newpath 0 147 moveto 0 -147 lineto 1 setlinewidth 0 setgray 0 setlinecap stroke grestore grestore gsave %Line 93.75 323.5 0 false false 0 fixcoordinates newpath 57.75 0 moveto -57.75 0 lineto 1 setlinewidth 0 setgray stroke grestore gsave %Text Block 201.501 341.737 0 true false 0 fixcoordinates /myshow /show load def 0 setgray -8 -5 moveto { /|______Times-Roman findfont 14 scalefont setfont (O) show } leftshow grestore gsave %Text Block 282 178 0 true false 0 fixcoordinates /myshow /show load def 0 setgray -23 -5 moveto { /|______Times-Roman findfont 14 scalefont setfont (B) show } leftshow grestore gsave %Text Block 396 279 0 true false 0 fixcoordinates /myshow /show load def 0 setgray -23 -5 moveto { /|______Times-Roman findfont 14 scalefont setfont (C) show } leftshow grestore gsave %Text Block 245.562 287.499 0 true false 0 fixcoordinates /myshow /show load def 0 setgray -13 -7 moveto { /|______Times-Roman findfont 14 scalefont setfont (A) show } leftshow grestore gsave %Line 360 303 0 false false 0 fixcoordinates newpath 0 -21 moveto 0 21 lineto 1 setlinewidth 0 setgray 0 setlinecap [4 2] 0 setdash stroke grestore gsave %Line 306 282 0 false false 0 fixcoordinates newpath 54 0 moveto -54 0 lineto 1 setlinewidth 0 setgray stroke grestore gsave %Line 252 234.5 0 false false 0 fixcoordinates newpath 0 46.5 moveto 0 -46.5 lineto 1 setlinewidth 0 setgray stroke grestore gsave %Line 252 303 0 false false 0 fixcoordinates newpath 0 -21 moveto 0 21 lineto 1 setlinewidth 0 setgray 0 setlinecap [4 2] 0 setdash stroke grestore gsave %Line 234 255.5 0 false false 0 fixcoordinates newpath 18 -67.5 moveto -18 67.5 lineto 1 setlinewidth 0 setgray 0 setlinecap [9 3] 0 setdash stroke grestore gsave %Line 288 302.5 0 false false 0 fixcoordinates newpath 72 -20.5 moveto -72 20.5 lineto 1 setlinewidth 0 setgray 0 setlinecap [9 3] 0 setdash stroke grestore %------ End Main Program ------% showpage end vmstate restore %%Trailer %%DocumentFonts: Symbol %%+ Times-Roman %%Pages:1