Crossingless matchings and the cohomology of $(n,n)$ Springer varieties
Mikhail Khovanov
center.tex

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\title{\bf Crossingless matchings and the cohomology of $(n,n)$ 
                 Springer varieties}
\author{Mikhail Khovanov} 
\date{February 12, 2002}

\newtheorem{prop}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newtheorem{definition}{Definition}
\newtheorem{conjecture}{Conjecture} 
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\begin{document}
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\tableofcontents

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\def\R{\mathbb R}
\def\N{\mathbb N}
\def\Z{\mathbb Z}
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\def\F{\mathbb F}
\def\H{\mathbb H} 
\def\P{\mathbb P}
\def\l{\lbrace}
\def\r{\rbrace}
\def\o{\otimes}
\def\D{\Delta}
\def\lra{\longrightarrow}
\def\slt{\mathfrak{sl}_2}

\newcommand{\sumop}[1]{{\mathop{\sum}\limits_{#1}}}
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\newcommand{\bigcupop}[1]{{\mathop{\bigcup}\limits_{#1}}}
\newcommand{\cupop}[1]{{\mathop{\cup}\limits_{#1}}}

\def\Hom{\mathrm{Hom}}

\def\sbinom#1#2{\left( \hspace{-0.06in}\begin{array}{c} #1 \\ #2 \end{array}
 \hspace{-0.06in} \right)}

\def\mc{\mathcal} 
\def\mf{\mathfrak}

\def\sl{\mathfrak{sl}}

\newcommand{\vsp}{\vspace{0.1in}}

\def\yesnocases#1#2#3#4{\left\{
\begin{array}{ll} #1 & #2 \\ #3 & #4 
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\def\hsm{\hspace{0.05in}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\Id}{\mathrm{Id}}
\newcommand{\cA}{\mc{A}}

\newcommand{\define}{\stackrel{\mbox{\scriptsize{def}}}{=}}
\newcommand{\spring}{\mc{B}_{n,n}}  % springer variety for (n,n)
\newcommand{\Stotal}{\widetilde{S}}  % main configuration space 
\newcommand{\Ttotal}{\widetilde{T}}  % Springer variety 
\newcommand{\cohom}{\mathrm{H}} % cohomology with integral coefficients
\newcommand{\hsmall}{\hspace{0.05in}} % break for _a  


\section{Introduction}

We constructed in \cite{me:tangles} a family of rings $H^n, n\ge 0,$ a new 
invariant of tangles, and a conjectural invariant of tangle cobordisms. The 
invariant of a tangle is a complex of $(H^n,H^m)$-bimodules 
(up to chain homotopy equivalences). 
This paper relates $H^n$ to the Springer variety $\spring$ 
of complete flags in $\C^{2n}$ stabilized by a fixed nilpotent operator 
with two Jordan blocks of size $n.$

\begin{theorem} \label{center-spring} The center of $H^n$ is isomorphic to 
the cohomology ring of $\spring$:   
\begin{equation*}
 Z(H^n)\cong \mathrm{H}^{\ast}(\spring,\Z). 
\end{equation*}
\end{theorem} 
Both rings have natural gradings and the isomorphism is grading-preserving. 

Theorem~\ref{center-spring} is proved in a roundabout way, by finding 
generators and defining relations for both rings. Cohomology ring of 
the Springer variety $\mc{B}_{\lambda},$ for a partition $\lambda$ of 
$n,$ is well-understood. In Section~\ref{spring-gen-rel} we use a 
presentation of $\mc{B}_{\lambda}$ via generators and relations obtained 
by de Concini and Procesi \cite{ConciniProcesi} to prove

\begin{theorem}\label{springer-present} 
The cohomology ring of $\mc{B}_{n,n}$ is isomorphic, as a 
graded ring, to the quotient of the 
polynomial ring $R=\Z[X_1, \dots, X_{2n}], \mathrm{deg} X_i =2,$ by the ideal 
$R_1$ with generators 
 \begin{eqnarray}
  \label{sq-zero}           X_i^2,  &  \hspace{0.1in} & i\in [1,2n]; \\
  \label{el-sym} \sum_{|I|=k} X_I,  &  \hspace{0.1in} & k\in [1,2n];
 \end{eqnarray}
where $X_I = X_{i_1}\dots X_{i_k}$ for $I=\{i_1, \dots, i_k\}$ and 
the sum is over all cardinality $k$ subsets of $[1,2n].$  
\end{theorem} 


Generators (\ref{el-sym}) are elementary symmetric polynomials in 
$X_1, \dots ,X_{2n}.$ The quotient of $R$ by the ideal generated by 
(\ref{el-sym}) is isomorphic to the cohomology ring of the variety $\mc{B}$ of 
complete flags in $\C^{2n}.$ The inclusion $\spring\subset \mc{B}$ induces 
a surjection of cohomology rings $\cohom^{\ast}(\mc{B},\Z)\longrightarrow 
\cohom^{\ast}(\spring, \Z).$ It turns out that by adding relations 
$X_i^2=0$ we get a presentation for the cohomology ring of $\spring.$ 

\vspace{0.07in}
 
We recall several notations and definitions from \cite{me:tangles}, including 
that of $H^n.$ Let $\mc{A} = \cohom^{\ast}(S^2,\Z)$ be the cohomology ring of 
the 2-sphere, $\mc{A}\cong \Z[X]/(X^2).$ The trace form 
\begin{equation*}
  \mathrm{tr}:\mc{A}\to \Z, \hspace{0.2in} \mathrm{tr}(1)=0, \hspace{0.1in}
  \mathrm{tr}(X)=1,
\end{equation*}
makes $\mc{A}$ into a commutative Frobenius algebra. We assign to $\mc{A}$ a 
2-dimensional topological quantum field theory $\mc{F},$ a functor 
from the category of oriented cobordisms between 1-manifolds to the 
category of abelian groups. $\mc{F}$ associates 
\begin{itemize}
\item $\mc{A}^{\otimes k}$ to the disjoint union of $k$ circles,
\item the multiplication map $\mc{A}^{\otimes 2}\to \mc{A}$ to the "pants" 
   cobordism (three-holed sphere viewed as a cobordism from two circles to 
   one circle),
\item the comultiplication 
 \begin{equation*} 
  \Delta: \mc{A}\to \mc{A}^{\otimes 2}, \hspace{0.2in}
  \Delta(1)= 1\otimes X + X\otimes 1, \hspace{0.1in}
  \Delta(X)= X \otimes X 
 \end{equation*}
to the "inverted pants" cobordism,
\item either the trace or the unit map to the disk (depending on whether 
 we consider the disk as a cobordism from one circle to the empty manifold 
 or vice versa). 
\end{itemize}

Let $B^n$ be the set of crossingless matchings of $2n$ points. Equivalently, 
$B^n$ is all pairings of integers from $1$ to $2n$ such that there 
is no quadruple $i<j<k<l$ with $(i,k)$ and $(j,l)$ paired. Most of the 
time $n$ is fixed, and we denote $B^n$ simply by $B.$ Figure~\ref{pic-match2} 
depicts elements of $B^2.$ 

 \begin{figure} [htb] \drawing{b2.eps}\caption{crossingless matchings 
   $\{(12),(34)\}$   and $\{ (14), (23) \}$} \label{pic-match2} 
 \end{figure}

For $a,b\in B$ denote by $W(b)$ the reflection of $b$ about the horizontal 
axis, and by $W(b)a$ the closed 1-manifold obtained by gluing $W(b)$ 
and $a$ along their boundaries, see figure~\ref{pic-glue}. 

 \begin{figure} [htb] \drawing{glue.eps}\caption{} \label{pic-glue} 
 \end{figure}

$\mc{F}(W(b)a)$ is an abelian group isomorphic to $\mc{A}^{\otimes I},$ 
where $I$ is the set of connected components of $W(b)a.$ 
For $a,b,c\in B$ there is a canonical cobordism from $W(c)bW(b)a$ to 
$W(c)a$ given by "contracting" $b$ with $W(b),$ see figure~\ref{pic-contr}. 

 \begin{figure} [htb] \drawing{contr.eps}\caption{The contraction cobordism} 
 \label{pic-contr} 
 \end{figure}

This cobordism induces a homomorphism of abelian groups 
 \begin{equation}\label{ind-hom}
   \mc{F}(W(c)b)\otimes \mc{F}(W(b)a) \lra \mc{F}(W(c)a).
 \end{equation}
Let
 \begin{equation*}
   H^n\define \oplusop{a,b\in B} \hspace{0.05in} {_b(H^n)_a}, 
  \hspace{0.2in} {_b(H^n)_a}\define \mc{F}(W(b)a).  
  \end{equation*} 
Homomorphisms (\ref{ind-hom}), over all $a,b,c,$ define an associative 
multiplication in $H^n$ (we let the products 
 ${_d(H^n)_c}\otimes\hspace{0.05in} {_b(H^n)_a}\to\hspace{0.05in}{_d(H^n)_a}$ 
 be zero if $b\not= c$). 

$_a(H^n)_a$ is a subring of $H^n,$ isomorphic to $\mc{A}^{\otimes n}.$ 
Its element $1_a\define 1^{\otimes n}\in \mc{A}^{\otimes n}$ is an 
idempotent in $H^n.$ The sum $\sum_a 1_a$ is the unit element of $H^n.$ 
Notice that $_b(H^n)_a= 1_b H^n 1_a.$ 

Cohomological grading of $\mc{A}$ ($\mbox{deg}(1)=0, \mbox{deg}(X)=2$) 
  gives rise to a grading of $H^n,$ see \cite{me:tangles} 
for details. 

\vspace{0.1in} 

Denote by $S$ the 2-sphere $\mathbb{S}^2.$ Consider the direct product 
 \begin{equation*}
   S^{\times 2n} \define S\times S \times \dots \times S \hspace{0.2in}
   \mbox{(2n terms)}
 \end{equation*}
and a submanifold $S_a\in S^{\times 2n},$ for $a\in B,$ which consists of 
sequences \newline 
$(x_1, \dots , x_{2n}), x_i\in S$ such that $x_i=x_j$ 
whenever $(i,j)$ is a pair in $a.$ This submanifold is diffeomorphic to 
$S^{\times n}.$ Let $\Stotal = \cup_{a\in B} S_a,$ a subspace 
in $S^{\times 2n}.$ For example, if $n=2$ then $\Stotal$ is homeomorphic to 
two copies of $S\times S$ glued together along their diagonals. 

$H^n$ and $\Stotal$ are constructed along similar lines:  
the cohomology ring of $S_a$ is canonically isomorphic to the ring 
$_a(H^n)_a,$  while the abelian group $_a(H^n)_b$ is canonically isomorphic to 
$\cohom^{\ast}(S_a\cap S_b, \Z).$ These isomorphisms are our starting point 
in the proof (Section~\ref{proof-3}) of  

\begin{theorem} \label{center-config} 
The center of $H^n$ is isomorphic to the cohomology ring of $\Stotal$: 
\begin{equation*}
  Z(H^n) \cong \mathrm{H}^{\ast} (\Stotal, \Z). 
\end{equation*} 
\end{theorem}


In Section~\ref{ZH-g-r} we prove 

\begin{theorem}\label{gen-and-rel} 
Cohomology ring of $\Stotal$ is isomorphic, as a graded ring, to the quotient 
of the polynomial ring $R=\Z[X_1, \dots, X_{2n}], \mathrm{deg} X_i =2,$ 
by relations
 \begin{eqnarray}
  \label{sq-zero2}           X_i^2 & = & 0, \hspace{0.2in} i\in [1,2n]; \\
  \label{el-sym2} \sum_{|I|=k} X_I & = & 0, \hspace{0.2in} k\in [1,2n].
 \end{eqnarray}
\end{theorem} 


Theorems~\ref{springer-present}, \ref{center-config} and 
\ref{gen-and-rel} imply Theorem~\ref{center-spring}.  They also show that
spaces $\Stotal$ and $\spring$ have isomorphic cohomology rings. 
These spaces have similar combinatorial structure. Irreducible components 
of $\spring,$ just like those of $\Stotal,$ are enumerated by crossingless 
matchings. Each component $K_a\subset \mc{B}_{n,n}$ is an 
iterated $\mathbb{P}^1$-bundle over $\mathbb{P}^1$ (see \cite{Fung}),  
and homeomorphic to $S_a\subset \Stotal.$ Moreover, $K_a\cap K_b$ and 
  $S_a\cap S_b$ are homeomorphic. We expect that there is a compatible family 
of homeomorphisms and suggest  

\begin{conjecture} $\spring$ and $\Stotal$ are homeomorphic. 
\end{conjecture}

\emph{Warning:}
$\Stotal$ can be naively upgraded to an algebraic variety, by changing the 
2-sphere $S$ to $\mathbb{P}^1$ in the definition of $\Stotal.$ With this 
structure, however, $\Stotal$ 
is not isomorphic to $\spring$ as an algebraic variety, since irreducible 
components of $\spring$ are nontrivial iterated $\mathbb{P}^1$ bundles over 
$\mathbb{P}^1,$ while those of $\Stotal$ are just direct 
products of $\mathbb{P}^1.$ 

\vspace{0.1in}

Let $\mc{B}_{\kappa}$ be the Springer variety of complete flags in $\C^n$ 
stabilized by a fixed nilpotent operator with Jordan decomposition 
$\kappa=(k_1, \dots, k_m).$ 
The cohomology ring of $\mc{B}_{\kappa}$ admits a natural action 
of the symmetric group, see \cite[Section 3.6]{ChrissGinzburg} and 
references therein.
In particular, $S_{2n}$ acts on the cohomology ring of $\spring.$ In view 
of Theorem~\ref{center-spring} it therefore acts on the center 
of $H^n.$ Explicitly, the action is by permutations of $X_i$'s. 
It does not come from any action of $S_{2n}$ on $H^n.$ 

In Section~\ref{symm-group-action} we present an intrinsic construction of 
this action. The $2n$-stranded braid group acts on the category 
$\mc{K}$ of complexes of $H^n$-modules modulo chain homotopies, as 
follows from \cite{me:tangles}. This action 
descends to the braid group action on centers of $\mc{K}$ and 
$H^n$ and factors through to the symmetric group action on the center 
of $H^n.$ 

In Section~\ref{conj-highest-weight} we discuss conjectural 
isomorphisms between centers of 
parabolic blocks of the highest weight category for the Lie algebra
$\mf{sl}_n$ and cohomology algebras of Springer varieties. 

\vspace{0.1in}

\emph{Acknowledgments:} Ideas relating rings $H^n$ and $(n,n)$ 
Springer 
varieties appeared during discussions between Paul Seidel and the author, 
and we are planning a joint paper on various aspects of this correspondence 
\cite{me:Paul2}. The present work can be viewed as a side result 
of \cite{me:Paul2}. I am grateful to Ragnar-Olaf Buchweitz and 
Ivan Mirkovic for useful consultations. This work was partially 
supported by NSF grant DMS-0104139. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GENERATORS AND RELATIONS FOR SPRINGER FIBER 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of Theorem~\ref{springer-present}} \label{spring-gen-rel}

De Concini and Procesi \cite{ConciniProcesi} found a presentation for 
the cohomology ring of the Springer variety associated to a partition.
We describe their result specialized to the $(n,n)$ partition. 

Start with the ring $R=\Z[X_1, \dots, X_{2n}].$ 
For $I\subset [1,2n]$ let $X_I=\prod_{i\in I}X_i$ and let
$e_k(I)$ be the elementary symmetric polynomial of order $k$ in variables 
$X_i,$ for $i\in I$: 
 \begin{equation*}
  e_k(I) = \sum_{|J|=k, J\subset I} X_J. 
 \end{equation*}

 \begin{prop} (see \cite{ConciniProcesi}) The cohomology ring of $\spring$ is 
  isomorphic to the quotient ring of $R=\Z[X_1,\dots, X_{2n}]$ by the ideal 
  $R_2$ generated by $e_k(I)$ for all $k+|I|=2n+1,$  $X_I$ for all $|I|=n+1,$ 
  and $X_i^2$ for $i\in [1,2n].$ 
 \end{prop} 

\emph{Remark:} Defining relations are expressed in \cite{ConciniProcesi}
in terms of complete symmetric functions. Complete and elementary symmetric 
functions coincide modulo the ideal generated by $X_i^2, i\in [1,2n].$ 

We want to show the equality of ideals $R_1=R_2,$ where $R_1$ was defined 
in Theorem~\ref{springer-present}. Let $R_3$ be the ideal of $R$ generated by  
$X_i^2, i\in [1,2n].$ Let $e_k=e_k([1,2n])\in R_1.$ 

\emph{Claim:} $R_2\subset R_1.$ Since both $R_1$ and $R_2$ are stable under 
the permutation action of $S_{2n}$ on $R,$ it suffices to show that 
$e_k([1,2n-k+1])$ and $X_{[1,n+1]}$ lie in  $R_1.$ Indeed, 
 \begin{eqnarray}
  \label{equality4} e_k([1,2n-k+1])& \equiv & \sum_{i=0}^{k-1} (-1)^i e_i
  ([2n-k+2, 2n])e_{k-i}\hspace{0.2in}(\mathrm{mod }\hspace{0.1in}R_3), \\
   X_{[1,n+1]} & \equiv & \sum_{i=0}^{n-1} (-1)^i e_i([n+2,2n]) e_{n+1-i} 
  \hspace{0.2in}(\mathrm{mod }\hspace{0.1in}R_3).
  \end{eqnarray} 
The right hand sides lie in $R_1/R_3$ and the claim follows. $\square$ 

\emph{Claim:} $R_1\subset R_2.$ Equalities (\ref{equality4}) and induction 
on $k$ imply that $e_k\in R_2$ for all $k\le n.$ Moreover, $e_k\in R_2$ 
for $k>n$ since $X_I\in R_2$ for any $|I|>n.$ $\square$ 

Therefore, $R_1=R_2$ and the cohomology ring of $\spring$ is isomorphic 
to $R/R_1.$ 

$\square$





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ISOMORPHISM BETWEEN CENTER AND COHOMOLOGY 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of Theorem~\ref{center-config}}
\label{proof-3}

$\cohom(Y)$ will denote the cohomology ring of a topological space $Y$ 
with integer coefficients. 

There is a canonical isomorphism of rings 
$\cohom(S_a) \cong \cA^{\otimes I}\cong \hspace{0.03in} _aH_a,$ where $I$ is 
the set of arcs of $a.$ Similarly, there are natural abelian group 
isomorphisms $\cohom(S_a\cap S_b) \cong \cA^{\otimes I}\cong 
  \hspace{0.03in} _a H_b,$
where $I$ is the set of connected components of $W(a)b.$   These isomorphisms 
allow us to make $_a H_b$ into a ring. $_a 1_b \define 1^{\otimes k} \in 
\mc{A}^{\otimes k}$ is the unit of this ring.

Inclusions $ S_b\supset (S_a\cap S_b) \subset S_a$ induce ring homomorphisms 
 \begin{equation*}
  \psi_{a;a,b}: \cohom(S_a) \longrightarrow \cohom(S_a\cap S_b),
   \hspace{0.2in} 
  \psi_{b;a,b}: \cohom(S_b) \longrightarrow \cohom(S_a\cap S_b). 
 \end{equation*}
Likewise, maps 
 \begin{equation*}
 \gamma_{a;a,b}:\hspace{0.06in} _a(H^n)_a \longrightarrow \hsmall _a(H^n)_b, 
  \hspace{0.2in}
 \gamma_{b;a,b}:\hspace{0.06in} _b(H^n)_b \longrightarrow  \hsmall _a(H^n)_b, 
 \end{equation*}
  given by $x \longmapsto \hsmall x\hspace{0.03in} _a 1_b $ and 
  $x \longmapsto \hspace{0.03in} _a 1_b x$ are ring homomorphisms.  
The following diagram made out of these homomorphisms and automorphisms 
 commutes: 
 \begin{equation} \label{psi-diag}
    \begin{CD}   
     \cohom(S_a) @>>>  \cohom(S_a\cap S_b) @<<<
     \cohom(S_b)    \\ 
     @VV{\cong}V                @VV{\cong}V   @VV{\cong}V   \\
     _a(H^n)_a   @>>>  _a(H^n)_b           
      @<<< _b(H^n)_b  
    \end{CD} 
 \end{equation} 


Suppose given finite sets $I$ and $J,$ rings $A_i,i\in I$ and $B_j, j\in J,$ 
and ring homomorphisms $\beta_{i,j}: A_i\to B_j$ for some pairs $(i,j)$. 
Let 
\begin{equation*}
  \beta= \sum \beta_{i,j}, \hspace{0.2in} \beta: \prod_{i\in I} A_i \lra 
  \prod_{j\in J} B_j.  
\end{equation*}
Define the \emph{equalizer} of $\beta$ (denoted $\mathrm{Eq}(\beta)$) 
as the subring of ${\mathop{\prod}\limits_{i\in I}A_i}$ which consist 
of ${\mathop{\times}\limits_{i\in I}a_i}$ such that 
$\beta_{i,j} a_i = \beta_{k,j}a_k$ whenever 
$\psi_{i,j}$ and $\psi_{k,j}$ are defined. 

\vspace{0.1in}

Diagrams (\ref{psi-diag}) give rise to a commutative diagram 
of ring homomorphisms

 \begin{equation} \label{larger-diag}
    \begin{CD}   
      \mathrm{Eq}(\psi)    @>>>  {\mathop{\prod}\limits_{a}} \hsmall 
     \cohom(S_a) @>{\psi}>>
    {\mathop{\prod}\limits_{a\not= b}} \cohom(S_a\cap S_b) \\ 
      @VV{\cong}V     @VV{\cong}V                @VV{\cong}V     \\
     \mathrm{Eq}(\gamma) @>>> {\mathop{\prod}\limits_{a}} 
     \hsmall _a(H^n)_a @>{\gamma}>> 
    {\mathop{\prod}\limits_{a\not= b}} \hsmall _a(H^n)_b     
    \end{CD} 
   \end{equation} 
where 
\begin{equation*}
\psi = \sum_{a\not= b}(\psi_{a;a,b}+ \psi_{b;a,b})\hspace{0.1in}
 \mathrm{and} \hspace{0.1in}
 \gamma = \sum_{a\not= b}(\gamma_{a;a,b}+ \gamma_{b;a,b}).
\end{equation*}

For an element $z\in H$ write $z= \sum_{a,b}\hsmall _b z_a$ where $_b z_a\in 
\hspace{0.03in} _b H_a.$ If $z$ is central, $_b z_a=0$ if $a\not= b,$ since 
$0 = z 1_b 1_a = 1_b z 1_a =\hsmall _b z_a.$ Thus, 
$z= \sum_a \hsmall _a z_a. $ Denote $_az_a$ by $z_a.$ 
Clearly, $z= \sum_a z_a$ is central iff 
$z_a \hspace{0.02in} _a1_b = 
 \hsmall _a1_b z_b$ for all $a,b$ such that $a\not= b.$  
 Therefore, $Z(H^n)\cong \mathrm{Eq}(\gamma).$ 

\vspace{0.1in}

Inclusions $S_a\subset \Stotal$ induce ring homomorphisms 
$\cohom(\Stotal) \lra {\mathop{\prod}\limits_{a}} S_a$ which factor 
through $\mathrm{Eq}(\psi).$ Putting everything together, we obtain the 
following diagram

 \begin{figure}[htb]
 \begin{pspicture}(0,0)(2.5,3)
 \begin{psmatrix}
   $\cohom(\Stotal)\hsmall$  &  \hsmall $\mathrm{Eq}(\psi)$ \hsmall &  
   \hsmall  ${\mathop{\prod}\limits_{a}} \hsmall \cohom(S_a)\hsmall $ &   
   \hsmall ${\mathop{\prod}\limits_{a\not= b}} \cohom(S_a\cap S_b)$ \hsmall \\
   \hsmall  $Z(H^n)$ \hsmall  & \hsmall $\mathrm{Eq}(\gamma)$ \hsmall &  
   \hsmall  ${\mathop{\prod}\limits_{a}}^{\mathrm{}} 
    \hsmall _a(H^n)_a$ \hsmall  &  
   \hsmall ${\mathop{\prod}\limits_{a\not= b}} \hsmall _a(H^n)_b$ \hsmall  
   \psset{arrows=->}
   \ncline{1,1}{1,2}^{$\tau$} 
   \ncline{1,2}{1,3}
   \ncline{1,3}{1,4}^{$\psi$}
   \ncline{2,1}{2,2}^{$\cong$} 
   \ncline{2,2}{2,3}
   \ncline{2,3}{2,4}^{$\gamma$}
   \ncline{1,2}{2,2}>{$\cong$}
   \ncline{1,3}{2,3}>{$\cong$}
   \ncline{1,4}{2,4}>{$\cong$}
 \end{psmatrix}
 \end{pspicture} 
 \end{figure} 


\vspace{0.2in}

Theorem~\ref{center-config} will follow from 

\begin{prop} \label{ta-isom} $\tau$ is an isomorphism. 
\end{prop}

Proof of this proposition occupies the rest of this section.  

\vspace{0.1in}

For $a,b\in B$ we will write $a\rightarrow b$ 
if there is a quadruple $i<j<k<l$ such that $(i,j)$ and $(k,l)$ are 
pairs in $a,$ $(i,l)$ and $(j,k)$ are pairs in $b,$ and otherwise $a$ and 
$b$ are identical (see Figure~\ref{pic-arrow}). 
Figure~\ref{pic-allarrow} depicts all arrow relations for $n=3.$ 

 \begin{figure}[htb]\drawing{arrow.eps} 
  \caption{$a\rightarrow b$} \label{pic-arrow} \end{figure}

Introduce a partial order on $B$ by $a\prec b$ iff there is 
a chain of arrows $a\rightarrow a_1 \rightarrow \dots \rightarrow a_m 
\rightarrow b.$ 
We extend the partial order $\prec$ to a total order on $B$ in an 
arbitrary way and denote it by $<.$ 
 

 \begin{figure}[htb]\drawing{altarrow.eps}\caption{Arrow relations for $n=3$}
 \label{pic-allarrow} \end{figure}

\vspace{0.2in}

Define the distance $d(a,b)$ between $a$ and $b$ as the minimal length $m$  
of a sequence $(a=a_0,a_1, \dots, a_m=b)$ such that for each $i$ either 
$a_i\rightarrow a_{i+1}$ or $a_{i+1} \rightarrow a_i.$ One geometric 
interpretation of the distance: the diagram $W(b)a$ has 
$n-d(a,b)$ circles. 

\begin{lemma} \label{lemma-sink}
For any $a,b\in B$ there is $c$ such that 
$d(a,b)= d(a,c) + d(c,b)$ and $a \succ  c \prec b.$  
\end{lemma}

Proof is left to the reader. $\square$ 


\begin{lemma} If $d(a,c) = d(a,b) + d(b,c)$ then 
  \begin{equation*} S_a \cap S_c = S_a \cap S_b \cap S_c. \end{equation*}
\end{lemma}

$\square$ 

\vspace{0.1in}

Let $S_{<a} = \bigcupop{b<a} S_b$ and $S_{\le a} = \bigcupop{b\le a} S_b.$ 
Note that if $c$ is the next element after $a$ in the total order $<$ on $B$
then $S_{<c} = S_{\le a}.$ 

\begin{lemma} 
\begin{equation*}
S_{<a}\cap S_a = \cup_{b\rightarrow a} (S_b\cap S_a).
\end{equation*}
\end{lemma}

Follows from the previous lemma and lemma~\ref{lemma-sink}.  $\square$

\begin{lemma} \label{even-only} $S_{<a}\cap S_a$ has cohomology in even 
degrees only. The inclusion $(S_{<a}\cap S_a) \subset S_a$ induces a 
\emph{surjective} homomorphism of cohomology rings 
$\cohom(S_a)\lra \cohom(S_{<a}\cap S_a).$
\end{lemma} 

\emph{Proof:} We construct a cell decomposition of $S_a.$ Let $I$ be the 
set of arcs of $a.$ There is a canonical homeomorphism 
 $S_a\cong S^{\times I}.$ 
Let $\Gamma$ be the graph with $I$ as the set of vertices and $y,z\in I$ 
are connected by an edge iff there exist $b\rightarrow a$ such that $b$ is 
obtained from $a$ by erazing $y,z$ and reconnecting their endpoints in 
a different way. See Figure~\ref{pic-graph} for an example. 

\begin{figure} [htb] \drawing{graph.eps} \caption{A crossingless 
 matching and its graph}  
 \label{pic-graph} \end{figure}


$\Gamma$ is a forest (a disjoint union of trees). Let $E$ be the set of 
edges of $\Gamma.$ Mark a vertex in each connected component of $\Gamma$ 
and denote by $M$ the set of marked vertices. Note that $|E|+|M|=n.$ 

Fix a point $p\in S.$ For each $J\subset (E\sqcup M)$ let $c(J)$ be the 
subset of $S^{\times I}$ consisting of points $\{ x_i\}_{i\in I}, x_i\in S$ 
such that 
 \[ 
 \begin{array}{ll}
  x_i = x_j       & \mbox{if $(i,j)\in J$}, \\
  x_i \not= x_j   & \mbox{if $(i,j)\notin J$}, \\
  x_i = p         & \mbox{if $i\in M\cap J$}, \\
  x_i \not= p     & \mbox{if $i\in M, i\notin J$}. 
 \end{array}
 \]
Clearly, $S^{\times I}= \sqcup_{J} c(J)$ and $c(J)$ is homeomorphic to 
$\R^{2(n-|J|)}.$ We obtain a decomposition of $S_a\cong S^{\times I}$ 
into even dimensional cells. It restricts to a cell decomposition of 
$S_{<a}\cap S_a,$ the latter a union of cells $c(J)$ such 
that $J\cap E \not= \emptyset.$ The lemma follows, since these 
decompositions give us cochain complexes with zero differentials that 
describe cohomology groups of $S_a$ and $S_{<a}\cap S_a.$ $\square$ 

\begin{lemma} \label{lemma-injective} Homomorphism 
 \begin{equation*}
   \cohom(S_{< a} \cap S_a) \lra \oplusop{b<a} \cohom(S_b\cap S_a) 
 \end{equation*}
induced by inclusions $(S_b\cap S_a)\subset (S_{<a}\cap S_a)$ is injective. 
\end{lemma}

\emph{Proof:} It suffices to check that 
 \begin{equation*}
   \cohom(S_{< a} \cap S_a) \lra \oplusop{b\rightarrow a} \cohom(S_b\cap S_a) 
 \end{equation*}
is injective. The cell decomposition of $S_{<a}\cap S_a$ constructed above 
restricts to a cell decomposition of $S_b\cap S_a,$ for each $b\rightarrow a.$
Since $S_{<a}\cap S_a = \cup_{b\rightarrow a} (S_b\cap S_a),$ the lemma 
follows. $\square$

Note that $S_{\le a} = S_{<a} \cup S_a.$ Consider the Mayer-Vietoris 
sequence for $(S_{<a},S_a):$ 
 \begin{equation*}
  \lra \cohom^m(S_{\le a}) \lra \cohom^m(S_{<a})\oplus \cohom^m(S_a) 
  \lra \cohom^m(S_{<a}\cap S_a) \lra 
 \end{equation*}
 
\vspace{0.1in}

\begin{prop} \label{prop-exact} $S_{\le a}$ has cohomology in even degrees
only. The Mayer-Vietoris sequence for $(S_{<a},S_a)$ breaks down into 
short exact sequences
\begin{equation}\label{seq-less}
0 \lra \cohom^{2m}(S_{\le a}) \lra \cohom^{2m}(S_{<a})\oplus \cohom^{2m}(S_a) 
\lra \cohom^{2m}(S_{<a} \cap S_a)\lra 0
\end{equation}
for $0\le m \le n.$  
\end{prop}
  

\emph{Proof:} Induction on $a$ with respect to the total order $<.$ 
Induction base is obvious. Induction step: let $e$ be the element before $a$ 
relative to $<$ and assume the proposition holds for $e.$ Then spaces 
$S_{<a} = S_{\le e}, S_a,$ and $S_{<a}\cap S_a$ have cohomology 
in even degrees only (the last one by lemma~\ref{even-only}) and the 
Mayer-Vietoris sequence degenerates into exact sequences 
 \begin{eqnarray*}
  & &  0 \lra \cohom^{2m}(S_{\le a}) \lra \cohom^{2m}(S_{<a})\oplus 
   \cohom^{2m}(S_a)\lra \cohom^{2m}(S_{<a} \cap S_a)\lra \\
  & & \lra   \cohom^{2m+1}(S_{\le a}) \lra 0. 
 \end{eqnarray*}
By lemma~\ref{even-only} the map $\cohom(S_a)\lra \cohom(S_{<a}\cap S_a)$ is 
surjective, so that the last term of the sequence is zero. $\square$ 


\begin{prop} \label{prop-exact2} The following sequence is exact
\begin{equation}\label{another-exact}
 0 \lra \cohom(S_{\le a}) \stackrel{\phi}{\lra} \oplusop{b\le a} \cohom(S_b) 
  \stackrel{\psi^-}{\lra}  
 \oplusop{b< c\le a} \cohom(S_b\cap S_c), 
\end{equation}
\end{prop} 
where $\phi$ is induced by inclusions $S_b\subset S_{\le a},$ while 
\begin{equation*}
\psi^- \define \sum_{b<c\le a} (\psi_{b,c}-\psi_{c,b}), 
\end{equation*}
where 
\begin{equation*}
\psi_{b,c}: \cohom(S_b) \lra \cohom(S_b\cap S_c)
\end{equation*}
is induced by the inclusion $(S_b\cap S_c )\subset S_b.$ 

\emph{Proof:} Induction on $a.$ The induction base, $a$ is minimal relative 
to $<,$  is obvious. 
Induction step: assume $e$ preceeds $a$ relative to $<$ and the claim is 
true for $e.$ 
Lemma~\ref{lemma-injective} allows us to substitute 
$\oplusop{b<a}\cohom(S_b\cap S_a)$ 
for $\cohom(S_{<a}\cap S_a)$ in the sequence (\ref{seq-less}) while 
maintaining exactness everywhere but in the last term. Thus, 
\begin{equation}\label{seq-plus}
0 \lra \cohom(S_{\le a}) \lra \cohom(S_{<a})\oplus \cohom(S_a) 
\lra \oplusop{b<a} \cohom(S_b \cap S_a)
\end{equation}
is exact. Moreover, $S_{<a}=S_{\le e}.$ By induction hypothesis 
\begin{equation*}
 0 \lra \cohom(S_{\le e}) \lra \oplusop{f\le e} \cohom(S_f) 
  {\lra}  
 \oplusop{f < g\le e} \cohom(S_f\cap S_e) 
\end{equation*}
is exact. Substituting in (\ref{seq-plus}), and using standard properties 
of complexes, we conclude that (\ref{another-exact}) is exact. $\square$ 

When $a$ is the maximal element of $B,$ 
Proposition~\ref{prop-exact2} tells us that the sequence
 \begin{equation}
 0 \lra \cohom(\Stotal) \stackrel{\phi}{\lra} \oplusop{b} \cohom(S_b) 
  \stackrel{\psi^-}{\lra}  
 \oplusop{b< c} \cohom(S_b\cap S_c) 
\end{equation}
is exact. This is equivalent to Proposition~\ref{ta-isom}. $\square$

\vspace{0.1in}

\emph{Remark:} A similar method establishes an isomorphism between the quotient
of $H^n$ by its commutant subspace and homology of $\Stotal$ with 
integer coefficients: 
\begin{equation*}
H^n/[H^n,H^n]\cong \cohom_{\ast}(\Stotal, \Z).
\end{equation*}
If $\Lambda$ is a symmetric ring, the center of $\Lambda$ is 
dual to $\Lambda/[\Lambda,\Lambda].$ Ring $H^n$ is symmetric 
\cite[Section 6.7]{me:tangles}.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GENERATORS AND RELATIONS FOR COHOMOLOGY OF CROSSINGLESS MATCHINGS SPACE 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Proof of Theorem~\ref{gen-and-rel}} \label{ZH-g-r}

\begin{lemma} \label{free} $\cohom(\Stotal)$ is a free abelian group of rank 
{\scriptsize $\sbinom{2n}{n}.$}  
\end{lemma} 

\emph{Proof}
The cell decomposition of $S_a$ defined in the proof of 
Lemma~\ref{even-only} restricts to a cell decomposition of $S_a\cap S_{<a}.$ 
Hence, a cell \emph{partition} of $\Stotal$ can be obtained starting 
with the cell decomposition of $S_a,$ for the minimal $a\in B,$ and 
then adding the cells of $S_a\setminus S_{<a},$ over all $a$ in 
$B$ following the total order $<.$ Note that this is a 
cell \emph{partition} of $\Stotal,$ not a cell decomposition, since the 
closure of a cell is not, in general, a union of cells. Nevertheless, since 
all cells are even-dimensional and the boundary of each cell has 
codimension 2 relative to the cell, $\cohom(\Stotal)$ is a free 
abelian group with a basis consisting of delta functions of these cells. 

For $a\in B$ let $t(a)$ be the number of "bottom" arcs of $a,$ that is, arcs  
with no arcs below them. $t(a)$ is also the number of connected components 
of the graph $\Gamma,$ defined in the proof of Lemma~\ref{even-only}. 
For instance, Figure~\ref{pic-match2} diagrams have two and one bottom 
arcs (the left diagram has two). Our decomposition of $S_a\setminus S_{<a}$ 
has $2^{t(a)}$ cells. Therefore, the cell partition of $\Stotal$ has 
$\sum_{a\in B} 2^{t(a)}$ cells. It is easy to see that this sum 
equals {\scriptsize $\sbinom{2n}{n}.$} Lemma follows. $\square$ 




$\square$ 


Recall that $X$ denotes a generator of $\cohom^2(S).$ The inclusion 
$\iota:\Stotal \subset S^{\times 2n}$ induces a homomorphism 
$\iota^{\ast}: \cohom(S^{\times 2n}) \to \cohom(\Stotal).$ Let 
$\phi_i: S^{\times 2n}\to S$ be the projection on the $i$-th component. 
Consider the composition $\phi_i\circ \iota $ and let 
\begin{equation*}
X_i \define (-1)^i \iota^{\ast} \circ \phi_i^{\ast} (X), \hspace{0.2in}
  X_i \in \cohom(\Stotal). 
\end{equation*}

We denote by $[1,2n]$ the set of integers from $1$ to $2n.$ For 
$I\subset [1,2n]$ let $X_I = \prod_{i\in I} X_i.$ 


\begin{prop} \label{ring-gen-def} The cohomology ring of $\Stotal$ is 
generated by $X_i, i\in [1,2n]$ and has defining relations 
 \begin{eqnarray}
  \label{sq-zero-s}           X_i^2 & = & 0, \hspace{0.2in} i\in [1,2n]; \\
  \label{el-sym-s} \sum_{|I|=k} X_I & = & 0, \hspace{0.2in} k\in [1,2n].
 \end{eqnarray}
\end{prop}

\emph{Proof:} First we show that these relations hold. (\ref{sq-zero-s}) 
is obvious. Let $j_a: S_a\subset \Stotal$ and $j_a^{\ast}$ be the induced 
map on cohomology. (\ref{el-sym-s}) will follow if we check that 
\begin{equation}\label{eq-for-a} 
 \sum_{|I|=k} j_a^{\ast} (X_I) =0
\end{equation}
for all $a\in B,$ since 
 \begin{equation*}
  \sum_{a\in B} j_a^{\ast}: \cohom(\Stotal) \lra \oplusop{a\in B} 
  \cohom(S_a)
 \end{equation*}
is an inclusion. If $(i,i')$ is a pair in $a$ then $j_a^{\ast}(X_i X_{i'})=0$ 
and $j_a^{\ast}(X_i + X_{i'})=0$ (because of the term $(-1)^i$ in the 
definition of $X_i,$ and since $i+i'\equiv 1 (\mbox{mod }2)$). Therefore, 
\begin{equation*}
  \sum_{|I|=k, \{i,i'\}\cap I \not= \emptyset} j_a^{\ast} (X_I) =0
\end{equation*}
where the sum is over all subsets of cardinality $k$ that intersect 
$\{i,i'\}$ nontrivially. To take care of the remaining terms in the 
L.H.S. of (\ref{eq-for-a}),  
 \begin{equation*}
  \sum_{|I|=k, \{i,i'\}\cap I = \emptyset} j_a^{\ast} (X_I),
 \end{equation*}
pick another pair $(r, r')$ in $a$ and apply the same reduction to 
it. After $\frac{n-k}{2}+1$ iterations all cardinality $k$ subsets will be 
accounted for. (\ref{eq-for-a}) follows. 

\vspace{0.07in}

We say that a subset $I$ of $[1,2n]$ is \emph{admissible} if $I\cap [1,m]$ 
has at most $\frac{m}{2}$ elements for each $m\in [1,2n].$ 


 \begin{lemma} \label{number-adm}
   There are {\scriptsize $\sbinom{2n}{n}$} admissible subsets. 
 \end{lemma}

\emph{Proof} is left to the reader. $\square$ 

 \begin{lemma} \label{lin-combin} $X_J,$ for any $J\subset [1,2n],$ is a 
 linear combination of $X_I,$ over admissible $I.$  
 \end{lemma}

 \emph{Proof:} let $y(J)= \sum_{j\in J} j.$ Assume the lemma is false, 
 and find such a non-admissible $J$ with the minimal possible $y(J).$ 
 Take the smallest possible $m$ such that $|J\cap [1,m]| > \frac{m}{2}.$
 Then $m$ is odd, $m=2r+1$ and $|J\cap [1,m]|=r+1.$ 
 Arguments in the proof of Theorem~\ref{springer-present} in 
 Section~\ref{spring-gen-rel} imply that 
 $e_{r+1}((J\cap [1,m])\cup [m+1,2n]) =0.$ Therefore, $X_{J\cap [1,m]}$ 
 is a linear combination of $X_K$ with $K\subset (J\cap [1,m])\cup [m+1,2n]$ 
 and $y(K)> y(J).$ Since $X_J= X_{J\cap [1,m]} X_{J\cap [m+1,2n]},$ 
  this contradicts minimality of $y(J).$   $\square$

 \begin{lemma} \label{linear-ind} $\{ X_I\},$ over all admissible $I,$ are 
   linearly independent in $\cohom(\Stotal).$
 \end{lemma}

\emph{Sketch of proof:} Induction on $n,$ use homomorphism  
 $\cohom(\Stotal_n) \lra \cohom(\Stotal_{n-1})$ induced by the inclusion 
 $\Stotal_{n-1}\subset \Stotal_n$ (where $\Stotal_n$ is what we usually 
 call $\Stotal$). Details are left to the reader. $\square$ 

Lemmas~\ref{free}, \ref{number-adm}, \ref{lin-combin} and 
\ref{linear-ind} imply Proposition~\ref{ring-gen-def} and the following 
results.  

 \begin{corollary} $\cohom(\Stotal)$ has a basis $\{ X_I\},$ over all 
  admissible $I.$ 
 \end{corollary}

 \begin{corollary} The inclusion $\Stotal \subset S^{\times 2n}$ induces a 
   \emph{surjective} ring homomorphism 
   $\cohom(S^{\times 2n}) \lra \cohom(\Stotal).$ 
 \end{corollary} 






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   SYMMETRIC GROUP ACTION 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Symmetric group action on the center of $H^n$} 
\label{symm-group-action}

\vspace{0.1in}

{\bf The center of a category}

\vspace{0.1in}

The center of a category is defined as the commutative monoid of natural 
transformations of the identity functor. If the category is pre-additive 
($\mbox{Hom}(X,Y)$ is an abelian group for any objects $X,Y,$ and the 
composition of morphisms is bilinear) then the center is a commutative ring. A 
down-to-earth example: the center of the category of modules 
over a ring $A$ is isomorphic to the center of $A.$ 

Let $F$ be a functor in a category $\mc{C}.$ The center of $\mc{C}$ 
acts in two ways on the set $\mbox{End}(F)$ of endomorphisms of $F,$ since
we can compose $\alpha\in Z(\mc{C})$ and $\beta\in \mbox{End}(F)$
on the left or on the right: 
\begin{eqnarray*}
 \alpha\circ\beta & : & F \cong \mathrm{Id}\circ F 
  \stackrel{\alpha\circ\beta}{\lra} \mathrm{Id}\circ F \cong F, \\
 \beta\circ\alpha & : & F \cong F\circ\mathrm{Id}  
  \stackrel{\beta\circ\alpha}{\lra} F\circ\mathrm{Id} \cong F. 
\end{eqnarray*}
Assume that $F$ is invertible. Then any endomorphism of $F$ has the form 
$\mbox{Id}_F\circ \alpha,$ for a unique $\alpha\in Z(\mc{C}),$ as well as
$\alpha'\circ \mbox{Id}_F,$ for a unique $\alpha'\in Z(\mc{C}).$   
Thus, $F$ defines an automorphism of the monoid $Z(\mc{C})$ which takes 
$\alpha$ to $\alpha'.$  

Suppose group $G$ acts weakly on $\mc{C},$ meaning that there are functors 
$F_g: \mc{C}\to \mc{C}$ for each $g\in G,$ such that $F_1\cong \mbox{Id}$ and 
$F_gF_h\cong F_{gh}$ (we do not impose compatibility conditions on these 
isomorphisms). Each $F_g$ is invertible and gives rise to an automorphism 
of the 
center of $\mc{C}.$ We get an action of $G$ on $Z(\mc{C}).$ Therefore, a weak 
group action on a category descends to an action on the center of the 
category. 

\vspace{0.1in}

{\bf Centers of triangulated and derived categories} 

\vspace{0.1in}

Define the center of a triangulated category $\mc{D}$
as the set of natural transformations of the identity functor that 
commute with the shift functor $[1].$ 
Let $\mc{C}$ be an abelian category and $\widehat{\mc{C}}$ one of the 
triangulated categories associated to $\mc{C}$ (for instance, the 
bounded derived category of $\mc{C},$ or the category of bounded 
complexes of objects of $\mc{C}$ modulo chain homotopies). 
There are ring homomorphisms 
\begin{equation}\label{center-hom} 
   Z(\mc{C})\stackrel{f}{\lra}Z(\widehat{\mc{C}})\stackrel{g}{\lra}Z(\mc{C})  
\end{equation}
whose composition is the identity. $f$ extends $\alpha\in Z(\mc{C})$ 
termwise to complexes of objects of $\mc{C}.$ Homomorphism $g$ is induced 
by the inclusion of categories $\mc{C} \subset \widehat{\mc{C}}.$ 

$f$ and $g$ are not always isomorphisms, as observed by Jeremy Rickard. 
If $\mc{C}$ is the category of modules over the exterior 
algebra in one generator, then $f,$ respectively $g,$ has a nontrivial 
cokernel, respectively kernel. 

\emph{Remark:} Ragnar-Olaf Buchweitz pointed out to me that a triangulated 
category $\mc{D}$ also has \emph{extended center} (or "Hochschild cohomology"),
\begin{equation*}
 \oplusop{m\in \Z} \mbox{Hom}_{\mathrm{Fun}(\mc{D})}(\mbox{Id},\mbox{Id}[m]), 
\end{equation*}
(only natural transformations that supercommute with the shift functor 
are included). The extended center of the derived category of 
$\Lambda\mbox{-mod},$ for an algebra $\Lambda,$  
contains the Hochschild cohomology algebra $\mbox{Ext}_{\Lambda^{e}}(\Lambda,
 \Lambda).$  

An action of $G$ on $\widehat{\mc{C}}$ induces an action on
$Z(\widehat{\mc{C}}).$ If $\mathrm{ker}(g)$ is $G$-stable, the 
action descends to $Z(\mc{C}).$ 
Let $\Lambda$ be a ring and $D(\Lambda)$ the bounded derived category 
of $\Lambda\mbox{-mod}.$ If a  self-equivalence $F$ of $D(\Lambda)$ 
is given by tensoring with a bounded complex of left and 
right projective $\Lambda$-bimodules, then it descends to an automorphism of 
$Z(\Lambda)$ (compare with \cite[Proposition 9.2]{Rickard3}).  


\vspace{0.1in}

{\bf Symmetric group action}

\vspace{0.1in}

The symmetric group action on $Z(H^n)$ can be described intrinsically as 
follows. Let $\mc{K}$ be one of triangulated categories associated to 
 $H^n$ (say, the category of bounded complexes of left 
$H^n$-modules up to chain homotopies).  
The structures described in \cite{me:tangles} lead to a weak action 
of the braid group with $2n$-strands
 on $\mc{K}.$ Diagram $U_i$ (see Figure~\ref{Uid-diag}) defines an 
$H^n$-bimodule $\mc{F}(U_i)$ together with bimodule homomorphisms 
\begin{equation*}
  \mc{F}(U_i) \stackrel{\alpha}{\lra} H^n, \hspace{0.2in}
   H^n \stackrel{\beta}{\lra} \mc{F}(U_i),
\end{equation*} 
induced by elementary cobordisms between $U_i$ and $Vert_{2n}$ 
(we use notations from \cite{me:tangles}, note 
that $H^n\cong \mc{F}(Vert_{2n})$).  

 \begin{figure} [htb] \drawing{Uid.eps}\caption{Diagrams $U_i$ and 
 $Vert_{2n}$} \label{Uid-diag} 
 \end{figure}

Let $\mc{R}_i: \mc{K}\to \mc{K}$ be the functor of tensoring with 
the complex of bimodules 
\begin{equation} \label{inv-complex}
 0 \lra \mc{F}(U_i) \stackrel{\alpha}{\lra} H^n \lra 0. 
\end{equation} 
$\mc{R}_i$'s are invertible and satisfy the braid group relations. 

\vspace{0.05in}

In Section~\ref{ZH-g-r} we defined generators $X_i$ of $\cohom(\Stotal).$ 
We now describe the image of $X_i$ (also denoted $X_i$) in $Z(H^n)$ under 
the isomorphism $\cohom(\Stotal) \cong Z(H^n)$ established in 
Section~\ref{proof-3}: 
\begin{eqnarray*} 
X_i & = & {\mathop{\sum}\limits_{a\in B}}\hsmall _a(X_i)_a, \hspace{0.2in} 
  _a(X_i)_a \in \hsmall _a(H^n)_a \cong \mc{A}^{\o n}, \\
  _a(X^i)_a & = & (-1)^i 1^{\otimes (n-1)} \otimes X \in \mc{A}^{\otimes (n-1)}
 \otimes \mc{A}, 
\end{eqnarray*}
where the separated $\mc{A}$ corresponds to the circle in $\mc{F}(W(a)a)$ 
that contains the $i$-th endpoint of $a,$ counting from the left. 

For a complex $V$ of $H^n$-bimodules and $z\in Z(H^n)$ let $l_z,$ respectively 
$r_z,$ be the endomorphism of $V$ given by left, respectively right, 
multiplication by $z.$   
Endomorphisms $l_{X_i} - r_{X_{i+1}}$ and $l_{X_{i+1}} - r_{X_i}$ of 
the complex (\ref{inv-complex}) are homotopic to $0,$ via the homotopy 
$\pm \beta: H^n \lra \mc{F}(U_i).$ Therefore, the braid group 
action on $\mc{K}$ descends to the action of the symmetric group 
$S_{2n}$ on $Z(H^n)$ by permutations of $X_i$'s. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  CONJECTURES ABOUT CENTERS 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Conjectures on centers of highest weight categories} 
\label{conj-highest-weight}


If $e$ is an idempotent in a ring $\Lambda,$ there is a homomorphism 
$Z(\Lambda)\lra Z(e\Lambda e)$ which takes $z\in Z(\Lambda)$ to $ze.$ 

Let $\mc{O}^{n,n}$ be the full subcategory of a regular block of 
the highest weight category fof $\mf{sl}_{2n}$ which consists of 
locally $U\mf{p}(n,n)$-finite modules, where $\mf{p}(n,n)$ is the 
parabolic subalgebra in $\mf{sl}_{2n}$ of $(n,n)$ block-uppertriangular 
matrices. $\mc{O}^{n,n}$ is equivalent to the category of perverse sheaves 
on the Grassmannian of $n$-planes in $\C^{2n},$ constructible relative 
to the Schubert stratification. There is a unique finite-dimensional 
$\C$-algebra $A_{n,n}$ such that 

 (i) $\mc{O}^{n,n}$ is equivalent to the category of finite-dimensional 
  $A_{n,n}$-modules, 

 (ii) every irreducible $A_{n,n}$-module is one-dimensional. 

$A_{n,n}$ was explicitly described by Tom Braden \cite{Braden}. 
In \cite{me:withTom} we'll construct an idempotent $e$ in $A_{n,n}$ and 
an isomorphism $h: H^n\otimes_{\Z}\C\cong eA_{n,n}e.$ 

 \begin{conjecture} $Z(A_{n,n})\cong Z(eA_{n,n}e),$ and $h$ induces an 
  isomorphism of centers of $H^n\otimes_{\Z}\C$ and $\mc{O}^{n,n}.$ 
 \end{conjecture}

We would like to suggest a more general conjecture relating parabolic highest 
weight categories and Springer varieties. 
Let $\kappa=(k_1, \dots , k_m)$ be a decomposition of $n,$ 
$k_1+ \dots +k_m=n$ and denote by $\mf{p}(\kappa)$ the corresponding 
Lie algebra of block upper-triangular $n$-by-$n$ matrices. Let 
$\mc{O}^{\kappa}$ be the full subcategory of locally 
$U\mf{p}(\kappa)$-finite modules in a regular block $\mc{O}_{reg}$ 
of the highest weight category  $\mc{O}$ for $\mf{sl}_n.$ 
Let $Y^{\kappa}$ be the partial flag variety associated to $\kappa.$ 
It is known that $\mc{O}^{\kappa}$ is equivalent to the category of 
perverse sheaves on $Y^{\kappa},$ smooth along Schubert cells. 

Let $\mc{B}_{\kappa}$ be the Springer variety of complete flags in $\C^n$ 
stabilized by a fixed nilpotent operator with Jordan decomposition 
$(k_1, \dots, k_m).$ 

\begin{conjecture} The center of $\mc{O}^{\kappa}$ is isomorphic 
to the cohomology algebra of $\mc{B}_{\kappa}$: 
\begin{equation}\label{conj-iso1}
Z(\mc{O}^{\kappa}) \cong \mathrm{H}^{\ast}(\mc{B}_{\kappa}, \C).
\end{equation}
\end{conjecture}


Note that the right hand side of the isomorphism (\ref{conj-iso1}) 
depends only on the partition type of $\kappa,$ i.e. preserved by 
permutations of terms 
$k_1, \dots, k_m$ of $\kappa.$ The category $\mc{O}^{\kappa},$ 
featured in the left hand side, generally does not possess the same kind of 
invariance: 

 \begin{prop} \label{inequivalent}
   Categories $\mc{O}^{2,1,1}$ and $\mc{O}^{1,2,1}$ are inequivalent. 
 \end{prop}
 
However, we have

 \begin{prop} \label{der-equiv} If decompositions $\kappa$ and $\kappa'$   
  differ only by a permutation of terms, the categories 
  $\mc{O}^{\kappa}$ and $\mc{O}^{\kappa'}$ are derived equivalent. 
 \end{prop}

Propositions~\ref{inequivalent} and \ref{der-equiv} are proved at the end 
of this section. 
  
If two rings are derived equivalent, their centers
are isomorphic (\cite[Proposition 9.2]{Rickard3}). Since 
$\mc{O}^{\kappa}$ and $\mc{O}^{\kappa'}$ are equivalent to categories 
of modules over finite-dimensional algebras, and these algebras are derived 
equivalent, centers of  $\mc{O}^{\kappa}$ and $\mc{O}^{\kappa'}$ are
isomorphic. Thus, the left hand side of $(\ref{conj-iso1})$ also depends only 
on the partition type of $\kappa.$ 
 

We would like conjectural isomorphisms (\ref{conj-iso1}) to be compatible 
with the inclusions of categories $\mc{O}^{\kappa}\subset \mc{O}_{reg}$ and 
topological spaces $\mc{B}_{\kappa}\subset \mc{B},$ where $\mc{B}$ is the 
variety 
of complete flags in $\C^n.$ These inclusion induce ring homomorphisms 
$Z(\mc{O}_{reg})\lra Z(\mc{O}^{\kappa})$ and 
$\cohom^{\ast}(\mc{B},\C)\lra \cohom^{\ast}(\mc{B}_{\kappa},\C)$ which should 
be a part of the following commutative diagram 


  \begin{equation} 
    \begin{CD}   
     \cohom^{\ast}(\mc{B},\C) @>>>  \cohom^{\ast}(\mc{B}_{\kappa},\C) \\
     @VV{\cong}V                @VV{\cong}V     \\
     Z(\mc{O}_{reg}) @>>>    Z(\mc{O}^{\kappa})    
    \end{CD} 
   \end{equation} 

\vspace{0.3in}


Let $\Theta_i: \mc{O}^{\kappa}\lra \mc{O}^{\kappa}$ be the functor of 
translation across the $i$-th wall. $\Theta_i$ is the product of 
two biadjoint functors (translations on and off the $i$-th wall). 
Let $\alpha_i : \Theta_i \lra \mathrm{Id}$ be one of the natural 
transformations 
coming from biadjointness. $\mc{R}_i,$ the cone of $\alpha_i,$ is 
a functor in the derived category of $\mc{O}^{\kappa}.$ Functors 
$\mc{R}_i$ are invertible and generate a braid group action in 
$D^b(\mc{O}^{\kappa}).$ We conjecture that this action descends to 
a symmetric group action on $Z(\mc{O}^{\kappa}),$ and the ring isomorphism 
(\ref{conj-iso1}) can be made $S_n$-equivariant.

\vspace{0.1in}

Let $A_{\kappa}$ be a finite-dimensional algebra such that 
$A_{\kappa}\mbox{-mod}\cong \mc{O}^{\kappa}.$ Let $e\in A_{\kappa}$ be 
the maximal idempotent such that the left $A_{\kappa}$-module 
$A_{\kappa}e$ is injective. 

\begin{conjecture} Inclusion $eA_{\kappa}e\subset A_{\kappa}$ induces an 
isomorphism of centers $Z(A_{\kappa}) \cong  Z(e A_{\kappa}e).$ 
\end{conjecture} 

\vspace{0.1in}

\emph{Proof of Proposition~\ref{inequivalent}:} 
Assume the two categories are equivalent. Any intrinsic homological 
information about them is identical. The equivalence restricts to a bijection 
between isomorphism classes of simple objects. The bijection induces 
isomorphisms between Ext rings of simple objects of these categories. 
                        
Simple objects are in a one-to-one correspondence with Schubert cells in 
partial flag varieties $X_{2,1,1}$ and $X_{1,2,1}.$ For a simple object $L$ 
let $IC(L)$ be the intersection cohomology sheaf on the closure of the 
Schubert cell associated to $L.$ Then $\mbox{Ext}(L,L)\cong 
\mbox{Ext}(IC(L),IC(L)).$ 
                        
Let us count the number of simple objects $L$ in each category with 
$\mbox{dim}(\mbox{Ext}(L,L))=3.$ The Schubert cell of such an object is 
necessarily 2-dimensional and its closure is diffeomorphic to 
$\C\mathbb{P}^2$ (use that the cohomology of the closure of the cell 
is a direct summand of $\mbox{Ext}(IC(L),IC(L)).$
$X_{2,1,1}$ has only one such cell, while $X_{1,2,1}$ has two. 
Contradiction. $\square$           
                 
\vspace{0.07in}
                 
\emph{Proof of Proposition~\ref{der-equiv}:}
To construct an equivalence 
between $D^b(\mc{O}^{\kappa})$ and $D^b(\mc{O}^{\kappa'})$
note that these categories are isomorphic to the derived categories 
of sheaves on partial flag varieties $Y_{\kappa}$ and $Y_{\kappa'},$
smooth along Schubert stratifications. It suffices to treat the case when 
$\kappa$ and $\kappa'$ differ by a transposition of adjacent terms, 
$\kappa=(k_1, \dots, k_m), \kappa'=(k_1, \dots, k_{i-1},k_i, \dots, k_m).$  
Let $U\subset Y_{\kappa}\times Y_{\kappa'}$ be the set 
\begin{equation*}
 ((F_1, \dots F_m)\in Y_{\kappa}, (F_1', \dots, F_m')\in Y_{\kappa'})| 
 F_j=F_j' \hspace{0.1in}\mathrm{ for }\hspace{0.1in}
 j\not=i, \hsmall F_i\cap F_i' = F_{i-1},
\end{equation*}
of pairs of partial flags.
Let $\mc{G}$ be sheaf on $Y_{\kappa}\times Y_{\kappa'}$ which is the 
continuation by $0$ of the constant sheaf on $U.$ 
Convolution with $\mc{G}$ is an equivalence of derived categories of sheaves 
on $Y_{\kappa}$ and $Y_{\kappa'},$ and restricts to an equivalence of 
subcategories of cohomologically constructible (relative to the 
Schubert startification) complexes of sheaves. The latter categories are 
equivalent to the derived categories of $\mc{O}^{\kappa}$ and 
$\mc{O}^{\kappa'}.$     $\square$ 





\bibliographystyle{abbrv} 
\bibliography{$HOME/bibl/me,$HOME/bibl/all}  


\end{document}







pst-node.tex
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%% See the betadoc documentation for usage. 
%% This uses the header file `pst-node.pro'.
%%
\def\fileversion{97 patch 9}
\def\filedate{1999/04/14}
%%
%% COPYRIGHT 1993, 1994, 1999 by Timothy Van Zandt, tvz@nwu.edu.
%%
%% This program can be redistributed and/or modified under the terms
%% of the LaTeX Project Public License Distributed from CTAN
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\def\pst@getnode#1#2{%
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\def\pst@@getnode#1,#2,#3\@nil#4{%
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\def\pnode{\@ifnextchar({\pnode@}{\pnode@(0,0)}}
\def\pnode@(#1)#2{%
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\@fixedradiusfalse
\def\pst@nodehook{\cnodeput@iv{#2}}%
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\def\cnodeput@iv#1{%
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\def\Cnodeput@iii(#1)#2{%
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\@fixedradiustrue
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\pst@makebox{\cput@iv{#1}}}
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\def\psset@nodesep#1{%
\pst@getlength{#1}\psk@nodesepA
\let\psk@nodesepB\psk@nodesepA
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\def\psk@nodeseptypeB{0 }}
\psset@nodesep{0pt}
\def\psset@XnodesepA#1{%
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\def\psset@XnodesepB#1{%
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\def\psk@nodeseptypeB{1 }}
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\pst@getlength{#1}\psk@nodesepA
\let\psk@nodesepB\psk@nodesepA
\def\psk@nodeseptypeA{1 }%
\def\psk@nodeseptypeB{1 }}
\def\psset@YnodesepA#1{%
\pst@getlength{#1}\psk@nodesepA
\def\psk@nodeseptypeA{2 }}
\def\psset@YnodesepB#1{%
\pst@getlength{#1}\psk@nodesepB
\def\psk@nodeseptypeB{2 }}
\def\psset@Ynodesep#1{%
\pst@getlength{#1}\psk@nodesepA
\let\psk@nodesepB\psk@nodesepA
\def\psk@nodeseptypeA{2 }%
\def\psk@nodeseptypeB{2 }}
\def\psset@armA#1{%
\pst@getlength{#1}\psk@armA
\def\psk@armtypeA{0 }}
\def\psset@armB#1{%
\pst@getlength{#1}\psk@armB
\def\psk@armtypeB{0 }}
\def\psset@arm#1{%
\pst@getlength{#1}\psk@armA
\let\psk@armB\psk@armA
\def\psk@armtypeA{0 }%
\def\psk@armtypeB{0 }}
\psset@arm{10pt}
\def\psset@XarmA#1{%
\pst@getlength{#1}\psk@armA
\def\psk@armtypeA{1 }}
\def\psset@XarmB#1{%
\pst@getlength{#1}\psk@armB
\def\psk@armtypeB{1 }}
\def\psset@Xarm#1{%
\pst@getlength{#1}\psk@armA
\let\psk@armB\psk@armA
\def\psk@armtypeA{1 }%
\def\psk@armtypeB{1 }}
\def\psset@YarmA#1{%
\pst@getlength{#1}\psk@armA
\def\psk@armtypeA{2 }}
\def\psset@YarmB#1{%
\pst@getlength{#1}\psk@armB
\def\psk@armtypeB{2 }}
\def\psset@Yarm#1{%
\pst@getlength{#1}\psk@armA
\let\psk@armB\psk@armA
\def\psk@armtypeA{2 }%
\def\psk@armtypeB{2 }}
\def\psset@offsetA#1{\pst@getlength{#1}\psk@offsetA}
\def\psset@offsetB#1{\pst@getlength{#1}\psk@offsetB}
\def\psset@offset#1{\psset@offsetA{#1}\let\psk@offsetB\psk@offsetA}
\psset@offset{0pt}
\def\psset@angleA#1{\pst@getangle{#1}\psk@angleA}
\def\psset@angleB#1{\pst@getangle{#1}\psk@angleB}%
\def\psset@angle#1{%
\pst@getangle{#1}\psk@angleA
\let\psk@angleB\psk@angleA}
\psset@angle{0}
\def\psset@arcangleA#1{\pst@getangle{#1}\psk@arcangleA}
\def\psset@arcangleB#1{\pst@getangle{#1}\psk@arcangleB}%
\def\psset@arcangle#1{%
\pst@getangle{#1}\psk@arcangleA
\let\psk@arcangleB\psk@arcangleA}
\psset@arcangle{8}
\def\psset@ncurvA#1{\pst@checknum{#1}\psk@ncurvA}
\def\psset@ncurvB#1{\pst@checknum{#1}\psk@ncurvB}%
\def\psset@ncurv#1{\psset@ncurvA{#1}\let\psk@ncurvB\psk@ncurvA}
\psset@ncurv{.67}
\def\tx@GetCenter{GetCenter }
\def\tx@XYPos{XYPos }
\def\tx@GetEdge{GetEdge }
\def\tx@AddOffset{AddOffset }
\def\tx@GetEdgeA{GetEdgeA }
\def\tx@GetEdgeB{GetEdgeB }
\def\tx@GetArmA{GetArmA }
\def\tx@GetArmB{GetArmB }
\def\check@arrow#1#2{%
\check@@arrow#2-\@nil
\if@pst
\addto@par{arrows=#2}%
\def\next{#1}%
\else
\def\next{#1{#2}}%
\fi
\next}
\def\check@@arrow#1-#2\@nil{%
\ifx\@nil#2\@nil\@pstfalse\else\@psttrue\fi}
\def\tx@InitNC{InitNC }
\def\nc@object#1#2#3#4#5{%
\csname begin@#1Obj\endcsname
\showpointsfalse
\pst@getnode{#2}\pst@tempa
\pst@getnode{#3}\pst@tempb
\gdef\npos@default{#4 }%
\addto@pscode{%
/NCLW CLW def
\pst@nodedict
\psk@offsetA
\psk@offsetB neg
\psk@nodesepA
\psk@nodesepB
\psk@nodeseptypeA
\psk@nodeseptypeB
\pst@tempa
\pst@tempb
\tx@InitNC { #5 } if
end}%
\def\use@pscode{%
\pst@Verb{gsave \tx@STV newpath \pst@code\space grestore}%
\gdef\pst@code{}}%
\csname end@#1Obj\endcsname
\pst@shortput}
\def\npos@default{.5 }
\def\pc@object#1{%
\@ifnextchar({\pc@@object#1}{\pst@getarrows{\pc@@object#1}}}
\def\pc@@object#1(#2)(#3){%
\pnode(#2){@@A}\pnode(#3){@@B}%
#1{@@A}{@@B}}
\def\tx@LPutLine{LPutLine }
\def\tx@LPutLines{LPutLines }
\def\tx@BezierMidpoint{BezierMidpoint }
\def\tx@HPosBegin{HPosBegin }
\def\tx@HPosEnd{HPosEnd }
\def\tx@HPutLine{HPutLine }
\def\tx@HPutLines{HPutLines }
\def\tx@VPosBegin{VPosBegin }
\def\tx@VPosEnd{VPosEnd }
\def\tx@VPutLine{VPutLine }
\def\tx@VPutLines{VPutLines }
\def\tx@HPutCurve{HPutCurve }
\def\tx@NCCoor{NCCoor }
\def\tx@NCLine{NCLine }
\def\ncline{\pst@object{ncline}}
\def\ncline@i{\check@arrow{\ncline@ii}}
\def\ncline@ii#1#2{\nc@object{Open}{#1}{#2}{.5}{\tx@NCLine}}
\def\pcline{\pst@object{pcline}}
\def\pcline@i{\pc@object\ncline@ii}
\def\ncLine{\pst@object{ncLine}}
\def\ncLine@i{\check@arrow{\ncLine@ii}}
\def\ncLine@ii#1#2{\nc@object{Open}{#1}{#2}{.5}%
% DG/SR modification begin - Apr. 14, 1999 - Patch 9
%{\tx@NCLine /LPutPos { xB xA yB yA \tx@LPutLine } def}}
{\tx@NCLine /LPutPos { xB yB xA yA \tx@LPutLine } def}}
% DG/SR modification end
\def\tx@NCLines{NCLines }
\def\nclines{\pst@object{nclines}}
\def\nclines@i{\check@arrow\nclines@ii}
\def\nclines@ii#1#2{%
\begingroup
\use@par
\def\pst@aftercoors{\nclines@iii{#1}{#2}}%
\def\pst@coors{}%
\pst@@getcoors}
\def\nclines@iii#1#2{%
\nc@object{Open}{#1}{#2}{.5}{%
tx@Dict begin \psline@iii pop end
mark \pst@coors \tx@NCLines}%
\endgroup
\ignorespaces}
\def\tx@NCCurve{NCCurve }
\def\nccurve{\pst@object{nccurve}}
\def\nccurve@i{\check@arrow{\nccurve@ii}}
\def\nccurve@ii#1#2{\nc@object{Open}{#1}{#2}{.5}{%
/AngleA \psk@angleA\space def /AngleB \psk@angleB\space def
\psk@ncurvB\space \psk@ncurvA\space
\tx@NCCurve}}
\def\pccurve{\pst@object{pccurve}}
\def\pccurve@i{\pc@object\nccurve@ii}
\def\ncarc{\pst@object{ncarc}}
\def\ncarc@i{\check@arrow{\ncarc@ii}}
\def\ncarc@ii#1#2{\nc@object{Open}{#1}{#2}{.5}{%
yB yA sub xB xA sub \tx@Atan dup
\psk@arcangleA\space add /AngleA ED
\psk@arcangleB\space sub 180 add /AngleB ED
\psk@ncurvB\space \psk@ncurvA\space
\tx@NCCurve}}
\def\pcarc{\pst@object{pcarc}}
\def\pcarc@i{\pc@object\ncarc@ii}
\def\tx@NCAngles{NCAngles }
\def\ncangles{\pst@object{ncangles}}
\def\ncangles@i{\check@arrow{\ncangles@ii}}
\def\ncangles@ii#1#2{%
\nc@object{Open}{#1}{#2}{1.5}{\ncangles@iii \tx@NCAngles}}
\def\ncangles@iii{%
tx@Dict begin \psline@iii pop end
/AngleA \psk@angleA def
/AngleB \psk@angleB def
/ArmA \psk@armA def
/ArmB \psk@armB def
/ArmTypeA \psk@armtypeA def
/ArmTypeB \psk@armtypeB def }
\def\pcangles{\pst@object{pcangles}}
\def\pcangles@i{\pc@object\ncangles@ii}
\def\tx@NCAngle{NCAngle }
\def\ncangle{\pst@object{ncangle}}
\def\ncangle@i{\check@arrow{\ncangle@ii}}
\def\ncangle@ii#1#2{%
\nc@object{Open}{#1}{#2}{1.5}{\ncangles@iii \tx@NCAngle}}
\def\pcangle{\pst@object{pcangle}}
\def\pcangle@i{\pc@object\ncangle@ii}
\def\tx@NCBar{NCBar }
\def\ncbar{\pst@object{ncbar}}
\def\ncbar@i{\check@arrow{\ncbar@ii}}
\def\ncbar@ii#1#2{\nc@object{Open}{#1}{#2}{1.5}{%
\ncangles@iii /AngleB \psk@angleA def \tx@NCBar}}
\def\pcbar{\pst@object{pcbar}}
\def\pcbar@i{\pc@object\ncbar@ii}
\def\tx@NCDiag{NCDiag }
\def\ncdiag{\pst@object{ncdiag}}
\def\ncdiag@i{\check@arrow{\ncdiag@ii}}
\def\ncdiag@ii#1#2{%
\nc@object{Open}{#1}{#2}{1.5}{\ncangles@iii \tx@NCDiag}}
\def\pcdiag{\pst@object{pcdiag}}
\def\pcdiag@i{\pc@object\ncdiag@ii}
\def\tx@NCDiagg{NCDiagg }
\def\ncdiagg{\pst@object{ncdiagg}}
\def\ncdiagg@i{\check@arrow{\ncdiagg@ii}}
\def\ncdiagg@ii#1#2{%
\nc@object{Open}{#1}{#2}{.5}{\ncangles@iii \tx@NCDiagg}}
\def\pcdiagg{\pst@object{pcdiagg}}
\def\pcdiagg@i{\pc@object\ncdiagg@ii}
\def\tx@NCLoop{NCLoop }
\def\psset@loopsize#1{\pst@getlength{#1}\psk@loopsize}
\psset@loopsize{1cm}
\def\ncloop{\pst@object{ncloop}}
\def\ncloop@i{\check@arrow{\ncloop@ii}}
\def\ncloop@ii#1#2{%
\nc@object{Open}{#1}{#2}{2.5}%
{\ncangles@iii /loopsize \psk@loopsize def \tx@NCLoop}}
\def\pcloop{\pst@object{pcloop}}
\def\pcloop@i{\pc@object\ncloop@ii}
\def\tx@NCCircle{NCCircle }
\def\nccircle{\pst@object{nccircle}}
\def\nccircle@i{\check@arrow{\nccircle@ii}}
\def\nccircle@ii#1#2{%
\pssetlength\pst@dima{#2}%
\nc@object{Open}{#1}{#1}{.5}{%
/AngleA \psk@angleA def
/r \pst@number\pst@dima def
\tx@NCCircle \psarc@v end}}
\def\tx@NCBox{NCBox }
\def\ncbox{\pst@object{ncbox}}
\def\ncbox@i{\check@arrow{\ncbox@ii}}
\def\ncbox@ii#1#2{%
\def\pst@linetype{2}%
\nc@object{Closed}{#1}{#2}{.5}{%
tx@Dict begin \psline@iii pop end
\psk@boxheight \psk@boxdepth
\tx@NCBox}}
\def\pcbox{\pst@object{pcbox}}
\def\pcbox@i{\pc@object\ncbox@ii}
\def\tx@NCArcBox{NCArcBox }
\def\psset@boxheight#1{\pst@getlength{#1}\psk@boxheight}
\def\psset@boxdepth#1{\pst@getlength{#1}\psk@boxdepth}
\def\psset@boxsize#1{%
\psset@boxheight{#1}%
\let\psk@boxdepth\psk@boxheight}
\psset@boxsize{.4cm}
\def\ncarcbox{\pst@object{ncarcbox}}
\def\ncarcbox@i{\check@arrow{\ncarcbox@ii}}
\def\ncarcbox@ii#1#2{%
\def\pst@linetype{1}%
\nc@object{Closed}{#1}{#2}{.5}{%
\psk@arcangleA \psk@boxheight \psk@boxdepth \pst@number\pslinearc
\tx@NCArcBox}}
\def\pcarcbox{\pst@object{pcarcbox}}
\def\pcarcbox@i{\pc@object\ncarcbox@ii}
\def\tx@Tfan{Tfan }
% Changed according pst-beta.bug December 3, 1993
% nrot=:<angle> does not work when : is active.
\begingroup
\catcode`\:=13
\gdef\pst@activerot{\def:{\string:}}
\endgroup
\def\psset@nrot#1{%
\begingroup
\pst@activerot
\pst@expandafter{\@ifnextchar:{\psset@@nrot}{\psset@@rot}}{#1}\@nil
\global\let\pst@tempg\psk@rot
\endgroup
\let\psk@nrot\pst@tempg}
\def\psset@@nrot:#1\@nil{%
\psset@@rot#1\@nil
\edef\psk@rot{NAngle \ifx\psk@rot\@empty\else\psk@rot add \fi}}
\psset@nrot{0}
\def\tx@LPutCoor{LPutCoor }
\def\tx@LPut{LPut }
\def\psset@npos#1{%
\def\pst@tempa{#1}%
\ifx\pst@tempa\@empty
\def\psk@npos{\npos@default}%
\else
\pst@checknum{#1}\psk@npos
\fi}
\psset@npos{}
\def\ncput{\pst@object{ncput}}
\def\ncput@i{\pst@killglue\pst@makebox{\ncput@ii}}
\def\ncput@ii{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\pst@makesmall\pst@hbox
\pst@rotate\psk@nrot\pst@hbox
\ncput@iii
\endgroup
\pst@shortput}
\def\ncput@iii{%
\leavevmode
\hbox{%
\pst@Verb{%
\pst@nodedict
/t \psk@npos def
\tx@LPut
end
\tx@PutBegin}%
\box\pst@hbox
\pst@Verb{\tx@PutEnd}}}
\def\naput{\pst@object{naput}}
\def\naput@i{\pst@killglue\pst@makebox{\naput@ii{NAngle 90 add}}}
\def\naput@ii#1{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\def\psk@refangle{#1 }%
\let\psk@rot\psk@nrot
\uput@vii
{exch pop add a \tx@PtoC h1 add exch w1 add exch }%
{tx@Dict /NCLW known { NCLW add } if }%
\ncput@iii
\endgroup
\pst@shortput}
\def\nbput{\pst@object{nbput}}
\def\nbput@i{\pst@killglue\pst@makebox{\naput@ii{NAngle 90 sub}}}
\def\psset@tpos#1{%
\pst@checknum{#1}\psk@tpos
\ifdim\psk@tpos \p@<\z@
\def\psk@tpos{.5}%
% DG/SR modification begin - Sep. 23, 1998 - Patch 7
%\@pstrickserr{Bad `tpos' value: `#1'. Must be 0<tpos<1}\@epha
\@pstrickserr{Bad `tpos' value: `#1'. Must be 0<tpos<1}\@ehpa
% DG/SR modification end
\else
\ifdim\psk@tpos \p@>\p@
\def\psk@tpos{.5}%
% DG/SR modification begin - Sep. 23, 1998 - Patch 7
%\@pstrickserr{Bad `tpos' value: `#1'. Must be 0<tpos<1}\@epha
\@pstrickserr{Bad `tpos' value: `#1'. Must be 0<tpos<1}\@ehpa
% DG/SR modification end
\fi
\fi}
\psset@tpos{.5}
\def\tvput{\pst@object{tvput}}
\def\tvput@i{\pst@makebox{\psput@tput{H}{1}}}
\def\tlput{\pst@object{tlput}}
\def\tlput@i{\pst@makebox{\psput@tput{H}{true}}}
\def\trput{\pst@object{trput}}
\def\trput@i{\pst@makebox{\psput@tput{H}{false}}}
\def\thput{\pst@object{thput}}
\def\thput@i{\pst@makebox{\psput@tput{V}{1}}}
\def\taput{\pst@object{taput}}
\def\taput@i{\pst@makebox{\psput@tput{V}{true}}}
\def\tbput{\pst@object{tbput}}
\def\tbput@i{\pst@makebox{\psput@tput{V}{false}}}
\def\tx@HPutAdjust{HPutAdjust }
\def\tx@VPutAdjust{VPutAdjust }
\def\psput@tput#1#2{%
\begingroup
\use@par
\pst@tputmakesmall
\leavevmode
\hbox{%
\pst@Verb{%
\pst@nodedict
/t \psk@tpos \pst@tposflip def
tx@NodeDict /HPutPos known
{ #1PutPos }
{ CP /Y ED /X ED /NAngle 0 def /NCLW 0 def }
ifelse
/Sin NAngle sin def
/Cos NAngle cos def
/s \pst@number\pslabelsep NCLW add def
/l \pst@number\pst@dima def
/r \pst@number\pst@dimb def
/h \pst@number\pst@dimc def
/d \pst@number\pst@dimd def
% DG/SR modification begin - Sep. 26, 1997 - Patch 4
%\ifnum1=0#2\else
\ifnum1=0#2 \else
% DG/SR modification end
/flag #2 def
\csname tx@#1PutAdjust\endcsname
\fi
\tx@LPutCoor
end
\tx@PutBegin}%
\box\pst@hbox
\pst@Verb{\tx@PutEnd}}%
\endgroup
\pst@shortput}
\def\pst@tposflip{}
\def\pst@tputmakesmall{%
\pst@dima=\wd\pst@hbox
\divide\pst@dima 2
\pst@dimg=\psk@href\pst@dimg
\pst@dimb\pst@dima
\advance\pst@dima\pst@dimg % leftsize
\advance\pst@dimb-\pst@dimg % rightsize
\pst@dimd=\psk@vref\relax
\pst@dimc=\ht\pst@hbox
\advance\pst@dimc-\pst@dimd % height
\advance\pst@dimd\dp\pst@hbox % depth
\setbox\pst@hbox=\hbox to\z@{%
\kern-\pst@dima\vbox to\z@{\vss\box\pst@hbox\vskip-\pst@dimd}\hss}}
\def\MakeShortNab#1#2{%
  \def\pst@shortput@nab{%
    \def\pst@tempg{\next}%
    \ifx#1\next
      \let\pst@tempg\naput
    \else
      \ifx#2\next
        \let\pst@tempg\nbput
      \else
        \ifx\@sptoken\next
          \let\pst@tempg\pst@shortput
        \fi
      \fi
    \fi
    \pst@tempg}}
\MakeShortNab{^}{_}
\def\MakeShortTablr#1#2#3#4{%
  \def\pst@shortput@tablr{%
    \def\pst@tempg{\next}%
    \ifx#1\next
      \let\pst@tempg\taput
    \else
      \ifx#2\next
        \let\pst@tempg\tbput
      \else
        \ifx#3\next
          \let\pst@tempg\tlput
        \else
          \ifx#4\next
            \let\pst@tempg\trput
          \else
            \ifx\@sptoken\next
              \let\pst@tempg\pst@shortput
            \fi
          \fi
        \fi
      \fi
    \fi
    \pst@tempg}}
\MakeShortTablr{^}{_}{<}{>}
\def\MakeShortTab#1#2{%
  \def\pst@shortput@tab{%
    \def\pst@tempg{\next}%
    \ifx#1\next
      \def\pst@tempg{%
        \@nameuse{%
          t\ifodd\psk@treemode\ifpstreeflip b\else a\fi
          \else\ifpstreeflip r\else l\fi\fi put}}%
    \else
      \ifx#2\next
        \def\pst@tempg{%
          \@nameuse{%
            t\ifodd\psk@treemode\ifpstreeflip a\else b\fi
            \else\ifpstreeflip l\else r\fi\fi put}}%
      \else
        \ifx\@sptoken\next
          \let\pst@tempg\pst@shortput
        \fi
      \fi
    \fi
    \pst@tempg}}
\MakeShortTab{^}{_}
\def\psset@shortput#1{%
\def\pst@tempg{#1}%
\ifx\pst@tempg\@none
\let\pst@shortput\ignorespaces
\else
\@ifundefined{pst@shortput@#1}%
{\@pstrickserr{Bad short put: `#1'}\@ehpa}%
{\edef\pst@shortput{\noexpand\afterassignment\expandafter\noexpand
\csname pst@shortput@#1\endcsname\noexpand\let\noexpand\next}}%
\fi}
\psset@shortput{none}
\def\lput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\lput@i}{\lput@ii}}}
\def\lput@i[#1]{\addto@par{ref=#1}\lput@ii}
\def\lput@ii{\@ifnextchar({\lput@iv}{\lput@iii}}
\def\lput@iii#1{\addto@par{nrot=#1}\@ifnextchar({\lput@iv}{\ncput@i}}
\def\lput@iv(#1){\addto@par{npos=#1}\ncput@i}
\def\mput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\mput@i}{\ncput@i}}}
\def\mput@i[#1]{\addto@par{ref=#1}\ncput@i}
\def\Lput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Lput@ii}{\Lput@i}}}
\def\Lput@i#1{\addto@par{labelsep=#1}\Lput@ii}
\def\Lput@ii[#1]{\addto@par{ref={#1}}\@ifnextchar({\Lput@iv}{\Lput@iii}}
\def\Lput@iii#1{\addto@par{nrot={#1}}\@ifnextchar({\Lput@iv}{\Lput@v}}
\def\Lput@iv(#1){\addto@par{npos=#1}\Lput@v}
\def\Lput@v{\pst@killglue\pst@makebox{\Lput@vi}}
\def\Lput@vi{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\Rput@vi
\pst@makesmall\pst@hbox
\pst@rotate\psk@nrot\pst@hbox
\ncput@iii
\endgroup
\pst@shortput}
\def\Mput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Mput@ii}{\Mput@i}}}
\def\Mput@i#1{\addto@par{labelsep=#1}\Mput@ii}
\def\Mput@ii[#1]{\addto@par{ref={#1}}\Lput@v}
\def\aput@#1{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\aput@i#1}{\aput@ii#1}}}
\def\aput@i#1[#2]{\addto@par{labelsep=#2}\aput@ii#1}
\def\aput@ii#1{\@ifnextchar({\aput@iv#1}{\aput@iii#1}}
\def\aput@iii#1#2{\addto@par{nrot=#2}\@ifnextchar({\aput@iv#1}{#1}}
\def\aput@iv#1(#2){\addto@par{npos=#2}#1}
\def\aput{\aput@\naput@i}
\def\bput{\aput@\nbput@i}
\def\Aput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Aput@i}{\naput@i}}}
\def\Aput@i[#1]{\addto@par{labelsep=#1}\naput@i}
\def\Bput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Bput@i}{\nbput@i}}}
\def\Bput@i[#1]{\addto@par{labelsep=#1}\nbput@i}
\def\node@coor#1;#2\@nil{%
\pst@getnode{#1}\pst@tempg
\edef\pst@coor{%
\pst@nodedict
tx@NodeDict \pst@tempg known
{ \pst@tempg load \tx@GetCenter }
{ 0 0 }
ifelse
end }}
\def\Node@coor[#1]#2;#3\@nil{%
\begingroup
\psset{#1}%
\@ifnextchar\bgroup{\Node@@@coor}{\Node@@coor}#2\@nil
\endgroup
\let\pst@coor\pst@tempg}
\def\Node@@coor#1\@nil{%
\pst@getnode{#1}\pst@tempg
\xdef\pst@tempg{%
\pst@nodedict
tx@NodeDict \pst@tempg known
{ \psk@nodesepA \psk@angleA
\pst@tempg load \psk@nodeseptypeA \tx@GetEdge
\psk@offsetA \psk@angleA \tx@AddOffset
\pst@tempg load \tx@GetCenter
3 -1 roll add 3 1 roll add exch }
{ CP }
ifelse
end }}%
\def\Node@@@coor#1{%
\pst@@getcoor{#1}%
\def\psk@angleA{%
\pst@tempg load \tx@GetCenter \pst@coor
3 -1 roll sub 3 1 roll sub neg \tx@Atan}%
\Node@@coor}
\def\nput{\pst@object{nput}}
\def\nput@i#1#2{\pst@killglue\pst@makebox{\nput@ii{#1}{#2}}}
\def\nput@ii#1#2{%
\begingroup
\use@par
\psset@refangle{#1}%
\let\psk@angleA\psk@refangle
\edef\psk@nodesepA{\pst@number\pslabelsep}%
\def\psk@nodeseptypeA{0 }%
\pslabelsep\z@
\uput@vi
\Node@@coor#2\@nil
\let\pst@coor\pst@tempg
\leavevmode
\psput@special\pst@hbox
\endgroup
\ignorespaces}
\newcount\psrow
\newcount\pscol
\newcount\psmatrixcnt
\newskip\psrowsep
\newskip\pscolsep
\def\psset@colsep#1{\pssetlength\pscolsep{#1}}
\def\psset@rowsep#1{\pssetlength\psrowsep{#1}}
\psset@colsep{1.5cm}
\psset@rowsep{1.5cm}
\newif\ifpsmatrix
% DG/SR modification begin - Nov. 27, 1998 - Patch 8
%\let\mscount\@multicnt
\ifx\mscount\@undefined\let\mscount\@multicnt\fi
% DG/SR modification end
\def\psmatrix{%
  \begingroup
{\ifnum0=`}\fi % Don't want to expand any &.
\@ifnextchar[{\psmatrix@i}{\ifnum0=`{\fi}{}\psmatrix@ii}}
\def\psmatrix@i[#1]{%
\ifnum0=`{\fi}{}%
\psset{#1}%
\psmatrix@ii}
\def\psmatrix@ii{%
\KillGlue
\edef\psm@beginmath{%
\ifmmode$\m@th\ifinner\textstyle\else\displaystyle\fi\fi}%
\edef\psm@endmath{\ifmmode$\fi}%
\let\\\psm@cr
\advance\psmatrixcnt 1
\def\psm@thenode{M-\the\psmatrixcnt-\the\psrow-\the\pscol}%
\tabskip\z@
\psrow1
\pscol\z@
\psset@shortput{tablr}%
\leavevmode
\vbox\bgroup\halign\bgroup&%
\begingroup
\global\advance\pscol 1
\csname psrowhook\romannumeral\psrow\endcsname
\csname pscolhook\romannumeral\pscol\endcsname
\psm@beginnode##\psm@endnode\endgroup
\cr}
\def\endpsmatrix{%
\crcr\egroup\unskip\egroup
\endgroup}
\def\psm@cr{{\ifnum0=`}\fi\@ifnextchar[{\psm@@cr}{\psm@@@cr{}}}
\def\psm@@cr[#1]{\psm@@@cr{\vskip#1\relax}}
\def\psm@@@cr#1{%
\ifnum0=`{\fi}{}\cr
\noalign{%
\global\advance\psrow 1
\global\pscol\z@
\vskip\psrowsep
#1}}
\def\psm@beginnode{%
\@ifnextchar\psm@endnode
{\let\psm@endnode@i\relax\setbox\pst@hbox=\hbox{}}%
{\pst@object{psm@beginnode}}}
\def\psm@beginnode@i{%
\setbox\pst@hbox=\hbox\bgroup
\psm@beginmath
\begingroup
\ignorespaces}
\def\psm@endnode@i{%
\unskip
\endgroup
\psm@endmath
\egroup
\use@par
\@psttrue}
\def\psm@endnode{%
\@pstfalse
\psm@endnode@i
\ifnum\pscol>1 \hskip\pscolsep \fi
\psk@mnodesize
\hfil
\nodealigntrue
\if@pst
\csname mnode@\psk@mnode\endcsname
\else
\csname mnode@\psk@emnode\endcsname
\fi
\psk@mcol
\psk@@mnodesize}
\def\psspan#1{\mscount#1\relax\loop\ifnum\mscount>\@ne \sp@n\repeat}
\def\psset@name#1{\pst@getnode{#1}\psk@name}
\let\psk@name\relax
\def\psset@mcol#1{%
\ifx r#1\relax
\let\psk@mcol\relax
\else
\ifx l#1\relax
\let\psk@mcol\hfill
\else
\let\psk@mcol\hfil
\fi
\fi}
\psset@mcol{c}
\def\psset@mnodesize#1{%
\pssetlength\pst@dimg{#1}%
\ifdim\pst@dimg<\z@
\let\psk@mnodesize\relax
\let\psk@@mnodesize\relax
\else
\edef\psk@mnodesize{\noexpand\hbox to\number\pst@dimg sp\noexpand\bgroup}%
\let\psk@@mnodesize\egroup
\fi}
\psset@mnodesize{-1pt}
\def\mnode@R{\rnode@iii\Rnode@ii{\psm@thenode}}
\def\mnode@r{\rnode@iii\rnode@iv{\psm@thenode}}
\def\mnode@oval{\ovalnode@ii{\psm@thenode}}
\def\mnode@tri{\trinode@ii{\psm@thenode}}
\def\mnode@dia{\dianode@ii{\psm@thenode}}
\def\mnode@C{{\nodealigntrue\cnode@ii(\z@,\z@){\psk@radius}{\psm@thenode}}}
\def\mnode@f{{\nodealigntrue\fnode@ii(\z@,\z@){\psm@thenode}}}
\def\mnode@circle{\circlenode@ii{\psm@thenode}}
\def\mnode@p{\pnode(\z@,\z@){\psm@thenode}}
% DG/SR modification begin - Jul. 22, 1997 - Patch 1
\def\mnode@dot{\dotnode@ii(\z@,\z@){\psm@thenode}}
% DG/SR modification end
\def\mnode@none{\box\pst@hbox}
\def\psset@mnode#1{%
\@ifundefined{mnode@#1}%
{\@pstrickserr{\string\psmatrix\space node `#1' not defined.}\@ehpa}%
{\edef\psk@mnode{#1}}}
\def\psset@emnode#1{%
\@ifundefined{mnode@#1}%
{\@pstrickserr{\string\psmatrix\space node `#1' not defined.}\@ehpa}%
{\edef\psk@emnode{#1}}}
\psset@mnode{R}
\psset@emnode{none}
%%%% FROM pst-coil.tex
\def\nccoil{\pst@object{nccoil}}
\def\nccoil@i{\check@arrow{\nccoil@ii}}
\def\nccoil@ii#1#2{\nc@object{Open}{#1}{#2}{.5}{%
\tx@NCCoor
tx@Dict begin
% DG/SR modification begin - Mar. 19, 1998 - Patch 5
4 2 roll
% DG/SR modification end
\psk@coilwidth \pscoilheight
\psk@coilarmA \psk@coilarmB
\psk@coilaspect \psk@coilinc
\pst@coildict \tx@Coil end
end}}
\def\nczigzag{\pst@object{nczigzag}}
\def\nczigzag@i{\check@arrow{\nczigzag@ii}}
\def\nczigzag@ii#1#2{\nc@object{Open}{#1}{#2}{.5}{%
\tx@NCCoor
tx@Dict begin
% DG/SR modification begin - Mar. 19, 1998 - Patch 5
4 2 roll
% DG/SR modification end
\pscoilheight
\psk@coilwidth
\psk@coilarmA
\psk@coilarmB
\pst@coildict \tx@ZigZag end
\psline@iii
\tx@Line
end}}
\catcode`\@=\TheAtCode\relax
\endinput
%%
%% END pst-node.tex

pstricks.tex
%% BEGIN: pstricks.tex
%%
%% PostScript macros for Generic TeX: main input file for PSTricks 97.
%% See the PSTricks User's Guide for description.
%% This uses the header file `pstricks.pro'.
%%
\def\fileversion{97 patch 10}
\def\filedate{1999/03/24}
%%
%% COPYRIGHT 1993, 1994, 1999 by Timothy Van Zandt, tvz@nwu.edu.
%%
%% This program can be redistributed and/or modified under the terms
%% of the LaTeX Project Public License Distributed from CTAN
%% archives in directory macros/latex/base/lppl.txt.
%%
\csname PSTricksLoaded\endcsname
\let\PSTricksLoaded\endinput
\edef\PstAtCode{\the\catcode`\@}
\catcode`\@=11\relax
\expandafter\ifx\csname @latexerr\endcsname\relax
\long\def\@ifundefined#1#2#3{\expandafter\ifx\csname
#1\endcsname\relax#2\else#3\fi}
\def\@namedef#1{\expandafter\def\csname #1\endcsname}
\def\@nameuse#1{\csname #1\endcsname}
\def\@eha{%
Your command was ignored.^^J
Type \space I <command> <return> \space to replace
it with another command,^^J
or \space <return> \space to continue without it.}
\def\@spaces{\space\space\space\space}
\def\typeout#1{\immediate\write\@unused{#1}}
\alloc@7\write\chardef\sixt@@n\@unused
\def\@empty{}
\def\@gobble#1{}
\def\@nnil{\@nil}
\def\@ifnextchar#1#2#3{%
\let\@tempe#1\def\@tempa{#2}\def\@tempb{#3}\futurelet\@tempc\@ifnch}
\def\@ifnch{%
\ifx\@tempc\@sptoken
\let\@tempd\@xifnch
\else
\ifx\@tempc\@tempe \let\@tempd\@tempa \else \let\@tempd\@tempb \fi
\fi
\@tempd}
\begingroup
\def\:{\global\let\@sptoken= } \:
\def\:{\@xifnch} \expandafter\gdef\: {\futurelet\@tempc\@ifnch}
\endgroup
\fi
\typeout{`PSTricks' v\fileversion\space\space <\filedate> (tvz)}
\def\@pstrickserr#1#2{%
\begingroup
\newlinechar`\^^J
\edef\pst@tempc{#2}%
\expandafter\errhelp\expandafter{\pst@tempc}%
\typeout{%
PSTricks error. \space See User's Guide for further information.^^J
\@spaces\@spaces\@spaces\@spaces
Type \space H <return> \space for immediate help.}%
\errmessage{#1}%
\endgroup}
\def\@ehpa{%
Your command was ignored. Default value substituted.^^J
Type \space <return> \space to procede.}
\def\@ehpb{%
Your command was ignored. Will recover best I can.^^J
Type \space <return> \space to procede.}
\def\@ehpc{%
You better fix this before proceding.^^J
See the PSTricks User's Guide or ask your system administrator for help.^^J
Type \space X <return> \space to quit.}
\def\pst@misplaced#1{\@pstrickserr{Misplaced \string#1 command}\@ehpb}
\newdimen\pst@dima
\newdimen\pst@dimb
\newdimen\pst@dimc
\newdimen\pst@dimd
\newdimen\pst@dimg
\newdimen\pst@dimh
\newbox\pst@hbox
\newbox\pst@boxg
\newcount\pst@cnta
\newcount\pst@cntb
\newcount\pst@cntc
\newcount\pst@cntd
\newcount\pst@cntg
\newcount\pst@cnth
\newif\if@pst
\newtoks\pst@toks
\newif\if@star
\def\pst@ifstar#1{%
\@ifnextchar*{\@startrue\def\next*{#1}\next}{\@starfalse#1}}
\def\pst@expandafter#1#2{%
\def\next{#1}%
\edef\@tempa{#2}%
\ifx\@tempa\@empty
\@pstrickserr{Unexpected empty argument!}\@ehpb
\def\@tempa{\@empty}%
\fi
\expandafter\next\@tempa}
\def\pst@dimtonum#1#2{\edef#2{\pst@@dimtonum#1}}
\def\pst@@dimtonum#1{\expandafter\pst@@@dimtonum\the#1}
{\catcode`\p=12 \catcode`\t=12 \global\@namedef{pst@@@dimtonum}#1pt{#1}}
\def\pst@pyth#1#2#3{%
\ifdim#1>#2\pst@@pyth#1#2#3\else\pst@@pyth#2#1#3\fi}
\def\pst@@pyth#1#2#3{%
\ifdim4#1>9#2%
#3=#1\advance#3 .2122#2%
\else
#3=.8384#1\advance#3 .5758#2%
\fi}
\def\pst@divide#1#2#3{%
\pst@@divide{#1}{#2}%
\pst@dimtonum\pst@dimg{#3}}
\def\pst@@divide#1#2{%
\pst@dimg=#1\relax
\pst@dimh=#2\relax
\pst@cntg=\pst@dimh
\pst@cnth=67108863
\pst@@@divide\pst@@@divide\pst@@@divide\pst@@@divide
\divide\pst@dimg\pst@cntg}
\def\pst@@@divide{%
\ifnum
\ifnum\pst@dimg<\z@-\fi\pst@dimg<\pst@cnth
\multiply\pst@dimg\sixt@@n
\else
\divide\pst@cntg\sixt@@n
\fi}
\def\pst@configerr#1{%
\@pstrickserr{\string#1 not defined in pstricks.con}\@ehpc}
% % \begin{macrocode}
\def\pstVerb#1{\pst@configerr\pstVerb}
\def\pstverb#1{\pst@configerr\pstverb}
\def\pstverbscale{\pst@configerr\pstverbscale}
\def\pstrotate{\pst@configerr\pstrotate}
\def\pstheader#1{\pst@configerr\pstheader}
\def\pstdriver{\pst@configerr\pstdriver}
\@ifundefined{pstcustomize}%
{\def\pstcustomize{\endinput\let\pstcustomize\relax}}{}
\input pstricks.con
\newif\ifPSTricks
\PSTrickstrue
\def\PSTricksOff{%
\def\pstheader##1{}%
\def\pstverb##1{}%
\def\pstVerb##1{}%
\PSTricksfalse}
\@ifundefined{pst@def}{\def\pst@def#1<#2>{\@namedef{tx@#1}{#2 }}}{}
\@ifundefined{pst@ATH}{\def\pst@ATH<#1>{}}{}
\pstheader{pstricks.pro}
\def\pst@dict{tx@Dict begin }
\def\pst@theheaders{pstricks.pro}
\def\pst@Verb#1{\pstVerb{\pst@dict #1 end}}
\def\tx@Atan{Atan }
\def\tx@Div{Div }
\def\tx@NET{NET }
\def\tx@Pyth{Pyth }
\def\tx@PtoC{PtoC }
\def\tx@PathLength@{PathLength@ }
\def\tx@PathLength{PathLength }
\pst@dimg=\pstunit\relax
\ifdim\pst@dimg=1bp
\def\pst@stp{.996264 dup scale}
\else
\edef\pst@stp{1 \pst@@dimtonum\pst@dimg\space div dup scale}
\fi
\def\tx@STP{STP }
\def\tx@STV{STV }
\def\pst@number#1{\pst@@dimtonum#1\space}
\def\pst@checknum#1#2{%
\edef\next{#1}%
\ifx\next\@empty
\let\pst@num\z@
\else
\expandafter\pst@@checknum\next..\@nil
\fi
\ifnum\pst@num=\z@
\@pstrickserr{Bad number: `#1'. 0 substituted.}\@ehpa
\def#2{0 }%
\else
\edef#2{\ifnum\pst@num=\tw@-\fi\the\pst@cntg.%
\expandafter\@gobble\the\pst@cnth\space}%
\fi}
\def\pst@@checknum{%
\@ifnextchar-%
{\let\pst@num\tw@\expandafter\pst@@@checknum\@gobble}%
{\let\pst@num\@ne\pst@@@checknum}}
\def\pst@@@checknum#1.#2.#3\@nil{%
\afterassignment\pst@@@@checknum\pst@cntg=0#1\relax\@nil
\afterassignment\pst@@@@checknum\pst@cnth=1#2\relax\@nil}
\def\pst@@@@checknum#1\relax\@nil{%
\ifx\@nil#1\@nil\else\let\pst@num\z@\fi}
\def\pst@getnumii#1 #2 #3\@nil{%
\pst@checknum{#1}\pst@tempg
\pst@checknum{#2}\pst@temph}
\def\pst@getnumiii#1 #2 #3 #4\@nil{%
\pst@checknum{#1}\pst@tempg
\pst@checknum{#2}\pst@temph
\pst@checknum{#3}\pst@tempi}
\def\pst@getnumiv#1 #2 #3 #4 #5\@nil{%
\pst@checknum{#1}\pst@tempg
\pst@checknum{#2}\pst@temph
\pst@checknum{#3}\pst@tempi
\pst@checknum{#4}\pst@tempj}
\def\pst@getdimnum#1 #2 #3\@nil{%
\pssetlength\pst@dimg{#1}%
\pst@checknum{#2}\pst@tempg}
% DG/SR modification begin - Jan. 7, 1998 - Patch 9
% Missing from pstricks.bug 0.93
%\def\pst@getscale#1#2{%
%\pst@expandafter\pst@getnumii{#1 #1} {} {} {}\@nil
%\@psttrue
%\ifdim\pst@tempg\p@=\z@
%\@pstrickserr{Bad scaling argument `#1'}\@ehpa
%\def\pst@tempg{1 }%
%\@pstfalse
%\fi
%\ifdim\pst@temph\p@=\z@
%\if@pst\@pstrickserr{Bad scaling argument `#1'}\@ehpa\fi
%\def\pst@temph{1 }%
%\fi
%\edef#2{\pst@tempg\space \pst@temph\space scale }%
%\ifdim\pst@tempg\p@=\p@ \ifdim\pst@temph\p@=\p@
%\def#2{}%
%\fi\fi}
\def\pst@getscale#1#2{%
\edef\pst@tempg{#1}%
\ifx\pst@tempg\@none
\def#2{}%
\else
\pst@expandafter\pst@getnumii{#1 #1} {} {} {}\@nil
\ifdim\pst@tempg\p@=\z@
\@pstrickserr{Bad scaling argument `#1'}\@ehpa
\def#2{}%
\else
\ifdim\pst@temph\p@=\z@
\@pstrickserr{Bad scaling argument `#1'}\@ehpa
\def#2{}%
\else
\edef#2{\pst@tempg\space \pst@temph\space scale }%
\fi
\fi
\fi}
% DG/SR modification end
\def\pst@getint#1#2{%
\pst@cntg=#1\relax
\edef#2{\the\pst@cntg\space}}
\begingroup
\catcode`\{=12
\catcode`\}=12
\catcode`\[=1
\catcode`\]=2
\gdef\pslbrace[{ ]
\gdef\psrbrace[} ]
\endgroup
\def\@newcolor#1#2{%
\expandafter\edef\csname #1\endcsname{\noexpand\pst@color{#2}}%
\expandafter\edef\csname color@#1\endcsname{#2}%
\ignorespaces}
\def\pst@color#1{%
\def\pst@currentcolor{#1}\pstVerb{#1}\aftergroup\pst@endcolor}
\def\pst@endcolor{\pstVerb{\pst@currentcolor}}
\def\pst@currentcolor{0 setgray}
\def\altcolormode{%
\def\pst@color##1{%
\pstVerb{gsave ##1}\aftergroup\pst@endcolor}%
\def\pst@endcolor{\pstVerb{\pst@grestore}}}
\def\pst@grestore{%
currentpoint
matrix currentmatrix
currentfont
grestore
setfont
setmatrix
moveto}
\def\pst@usecolor#1{\csname color@#1\endcsname\space}
\def\newgray#1#2{%
\pst@checknum{#2}\pst@tempg
\@newcolor{#1}{\pst@tempg setgray}}
\def\newrgbcolor#1#2{%
\pst@expandafter\pst@getnumiii{#2} {} {} {} {}\@nil
\@newcolor{#1}{\pst@tempg \pst@temph \pst@tempi setrgbcolor}}
\def\newhsbcolor#1#2{%
\pst@expandafter\pst@getnumiii{#2} {} {} {} {}\@nil
\@newcolor{#1}{\pst@tempg \pst@temph \pst@tempi sethsbcolor}}
\def\newcmykcolor#1#2{%
\pst@expandafter\pst@getnumiv{#2} {} {} {} {} {}\@nil
\@newcolor{#1}{\pst@tempg \pst@temph \pst@tempi \pst@tempj setcmykcolor}}
\newgray{black}{0}
\newgray{darkgray}{.25}
\newgray{gray}{.5}
\newgray{lightgray}{.75}
\newgray{white}{1}
\newrgbcolor{red}{1 0 0}
\newrgbcolor{green}{0 1 0}
\newrgbcolor{blue}{0 0 1}
\newrgbcolor{yellow}{1 1 0}
\newrgbcolor{cyan}{0 1 1}
\newrgbcolor{magenta}{1 0 1}
\def\psset#1{\@psset#1,\@nil\ignorespaces}
\def\@psset#1,{%
\@@psset#1==\@nil
\@ifnextchar\@nil{\@gobble}{\@psset}}
\def\@@psset#1=#2=#3\@nil{%
\@ifundefined{psset@#1}%
{\@pstrickserr{Graphics parameter `#1' not defined.}\@ehpa}%
{\@nameuse{psset@#1}{#2}}}%
\def\psset@style#1{%
\@ifundefined{pscs@#1}%
{\@pstrickserr{Custom style `#1' undefined}\@ehpa}%
{\@nameuse{pscs@#1}}}
\def\newpsstyle#1#2{\@namedef{pscs@#1}{\psset{#2}}}
\def\@none{none}
\def\pst@getcolor#1#2{%
\@ifundefined{color@#1}%
{\@pstrickserr{Color `#1' not defined}\@eha}%
{\edef#2{#1}}}
\newdimen\psunit \psunit 1cm
\newdimen\psxunit \psxunit 1cm
\newdimen\psyunit \psyunit 1cm
\let\psrunit\psunit
\def\pstunit@off{\let\@psunit\ignorespaces\ignorespaces}
\def\pssetlength#1#2{%
\let\@psunit\psunit
\afterassignment\pstunit@off
#1 #2\@psunit}
\def\psaddtolength#1#2{%
\let\@psunit\psunit
\afterassignment\pstunit@off
\advance#1 #2\@psunit}
\def\pssetxlength#1#2{%
\let\@psunit\psxunit
\afterassignment\pstunit@off
#1 #2\@psunit}
\def\pssetylength#1#2{%
\let\@psunit\psyunit
\afterassignment\pstunit@off
#1 #2\@psunit}
\def\psset@unit#1{%
\pssetlength\psunit{#1}%
\psxunit=\psunit
\psyunit=\psunit}
\def\psset@runit#1{\pssetlength\psrunit{#1}}
\def\psset@xunit#1{\pssetxlength\psxunit{#1}}
\def\psset@yunit#1{\pssetylength\psyunit{#1}}
\def\pst@getlength#1#2{%
\pssetlength\pst@dimg{#1}%
\edef#2{\pst@number\pst@dimg}}
\def\pst@@getlength#1#2{%
\pssetlength\pst@dimg{#1}%
\edef#2{\number\pst@dimg sp}}
\def\pst@getcoor#1#2{\pst@@getcoor{#1}\let#2\pst@coor}
\def\pst@coor{0 0 }
\def\pst@getcoors#1#2{%
\def\pst@aftercoors{\addto@pscode{#1 \pst@coors }#2}%
\def\pst@coors{}%
\pst@@getcoors}
\def\pst@@getcoors(#1){%
\pst@@getcoor{#1}%
\edef\pst@coors{\pst@coor\pst@coors}%
\@ifnextchar({\pst@@getcoors}{\pst@aftercoors}}
\def\pst@getangle#1#2{\pst@@getangle{#1}\let#2\pst@angle}
\def\pst@angle{0 }
\def\cartesian@coor#1,#2,#3\@nil{%
\pssetxlength\pst@dimg{#1}%
\pssetylength\pst@dimh{#2}%
\edef\pst@coor{\pst@number\pst@dimg \pst@number\pst@dimh}}
\def\NormalCoor{%
\def\pst@@getcoor##1{\pst@expandafter\cartesian@coor{##1},\relax,\@nil}%
\def\pst@@getangle##1{%
\pst@checknum{##1}\pst@angle
\edef\pst@angle{\pst@angle \pst@angleunit}}%
\def\psput@##1{\pst@@getcoor{##1}\leavevmode\psput@cartesian}}
\NormalCoor
\def\degrees{\@ifnextchar[{\@degrees}{\def\pst@angleunit{}}}
\def\@degrees[#1]{%
\pst@checknum{#1}\pst@tempg
\edef\pst@angleunit{360 \pst@tempg div mul }%
\ignorespaces}
\def\radians{\def\pst@angleunit{57.2956 mul }}
\def\pst@angleunit{}
\def\SpecialCoor{%
\def\pst@@getcoor##1{%
\begingroup
\pst@activecoor
\xdef\pst@tempg{##1}%
\endgroup
\expandafter\special@coor\pst@tempg||\@nil}%
\def\pst@@getangle##1{%
\begingroup
\pst@activecoor
\xdef\pst@tempg{##1}%
\endgroup
\expandafter\special@angle\pst@tempg\@empty)\@nil}%
\def\psput@##1{\pst@@getcoor{##1}\leavevmode\psput@special}}
\begingroup
\catcode`\|=13
\catcode`\;=13
\catcode`\!=13
\gdef\pst@activecoor{%
\def|{\string|}%
\def;{\string;}%
\def!{\string!}}
\endgroup
\def\special@coor#1|#2|#3\@nil{%
\ifx#3|\relax
\mixed@coor{#1}{#2}%
\else
\special@@coor#1;;\@nil
\fi}
\def\special@@coor#1{%
\ifcat#1a\relax
\def\next{\node@coor#1}%
\else
\ifx#1[\relax
\def\next{\Node@coor[}%
\else
\ifx#1!\relax
\def\next{\raw@coor}%
\else
\def\next{\special@@@coor#1}%
\fi
\fi
\fi
\next}
\def\special@@@coor#1;#2;#3\@nil{%
\ifx#3;\relax
\polar@coor{#1}{#2}%
\else
\cartesian@coor#1,\relax,\@nil
\fi}
\def\mixed@coor#1#2{%
\begingroup
% DG/SR modification begin - Oct. 27, 1997 - Patch 6
%\specialcoor@ii#1;;\@nil
%\let\pst@tempa\pst@coor
%\specialcoor@ii#2;;\@nil
\special@@coor#1;;\@nil
\let\pst@tempa\pst@coor
\special@@coor#2;;\@nil
% DG/SR modification end
\xdef\pst@tempg{\pst@tempa pop \pst@coor exch pop }%
\endgroup
\let\pst@coor\pst@tempg}
\def\polar@coor#1#2{%
\pssetlength\pst@dimg{#1}%
\pst@@getangle{#2}%
\edef\pst@coor{\pst@number\pst@dimg \pst@angle \tx@PtoC}}
\def\raw@coor#1;#2\@nil{%
\edef\pst@coor{%
#1 \pst@number\psyunit mul exch \pst@number\psxunit mul exch }}
\def\node@coor#1\@nil{%
\@pstrickserr{You must load `pst-node.tex' to use node coordinates.}\@ehps
\def\pst@coor{0 0 }}
\def\Node@coor{\node@coor}
\def\special@angle#1#2)#3\@nil{%
\ifx!#1\relax
\edef\pst@angle{#2 \pst@angleunit}%
\else
\ifx(#1\relax
\pst@@getcoor{#2}%
\edef\pst@angle{\pst@coor exch \tx@Atan}%
\else
\pst@checknum{#1#2}\pst@angle
\edef\pst@angle{\pst@angle \pst@angleunit}%
\fi
\fi}
\def\Cartesian{%
\def\cartesian@coor##1,##2,##3\@nil{%
\pssetxlength\pst@dimg{##1}%
\pssetylength\pst@dimh{##2}%
\edef\pst@coor{\pst@number\pst@dimg \pst@number\pst@dimh}}%
\@ifnextchar({\Cartesian@}{}}
\def\Cartesian@(#1,#2){%
\pssetxlength\psxunit{#1}%
\pssetylength\psyunit{#2}%
\ignorespaces}
\def\Polar{%
\def\psput@cartesian{\psput@special}%
\def\cartesian@coor##1,##2,##3\@nil{\polar@coor{##1}{##2}}}%
\def\psset@origin#1{%
\pst@@getcoor{#1}%
\edef\psk@origin{\pst@coor \tx@NET }}
\def\psk@origin{}
\newif\ifswapaxes
\def\psset@swapaxes#1{%
\@nameuse{@pst#1}%
\if@pst
\def\psk@swapaxes{-90 rotate -1 1 scale }%
\else
\def\psk@swapaxes{}%
\fi}
\psset@swapaxes{false}
\newif\ifshowpoints
\def\psset@showpoints#1{\@nameuse{showpoints#1}}
\psset@showpoints{false}
\let\pst@setrepeatarrowsflag\relax
\def\psset@border#1{%
\pst@getlength{#1}\psk@border
\pst@setrepeatarrowsflag}
\psset@border{0pt}
\def\psset@bordercolor#1{\pst@getcolor{#1}\psbordercolor}
\psset@bordercolor{white}
\newif\ifpsdoubleline
\def\psset@doubleline#1{%
\@nameuse{psdoubleline#1}%
\pst@setrepeatarrowsflag}
\psset@doubleline{false}
\def\psset@doublesep#1{\def\psdoublesep{#1}}
\psset@doublesep{1.25\pslinewidth}
\def\psset@doublecolor#1{\pst@getcolor{#1}\psdoublecolor}
\psset@doublecolor{white}
\newif\ifpsshadow
\def\psset@shadow#1{%
\@nameuse{psshadow#1}%
\pst@setrepeatarrowsflag}
\psset@shadow{false}
\def\psset@shadowsize#1{\pst@getlength{#1}\psk@shadowsize}
\psset@shadowsize{3pt}
\def\psset@shadowangle#1{\pst@getangle{#1}\psk@shadowangle}
\psset@shadowangle{-45}
\def\psset@shadowcolor#1{\pst@getcolor{#1}\psshadowcolor}
\psset@shadowcolor{darkgray}
\def\pst@repeatarrowsflag{\z@}
\def\pst@setrepeatarrowsflag{%
\edef\pst@repeatarrowsflag{%
\ifdim\psk@border\p@>\z@ 1\else\ifpsdoubleline 1\else
\ifpsshadow 1\else \z@\fi\fi\fi}}
\def\psls@none{}
\newdimen\pslinewidth
\def\psset@linewidth#1{\pssetlength\pslinewidth{#1}}
\psset@linewidth{.8pt}
\def\psset@linecolor#1{\pst@getcolor{#1}\pslinecolor}
\psset@linecolor{black}
\def\psls@solid{0 setlinecap stroke }
\def\pst@missing{%
\z@
\@pstrickserr{Missing number or dimension. 0 substituted}\@ehpa}
\def\psset@dash#1{%
\pst@expandafter\psset@@dash{#1} {\pst@missing} {\pst@missing} {}\@nil
\edef\psk@dash{\pst@number\pst@dimg \pst@number\pst@dimh}}
\def\psset@@dash#1 #2 #3\@nil{%
\pssetlength\pst@dimg{#1}%
\pssetlength\pst@dimh{#2}}
\psset@dash{5pt 3pt}
\newif\ifpsdashadjust
\def\psset@dashadjust#1{\@nameuse{psdashadjust#1}}
\psset@dashadjust{true}
\def\psls@dashed{%
\ifpsdashadjust
\psk@dash \pst@linetype\space \tx@DashLine
\else
[ \psk@dash ] 0 setdash stroke
\fi}
\def\tx@DashLine{DashLine }
\def\psset@dotsep#1{\pst@getlength{#1}\psk@dotsep}
\psset@dotsep{3pt}
\def\psls@dotted{%
\ifpsdashadjust
\psk@dotsep \pst@linetype\space \tx@DotLine
\else
[ 0 \psk@dotsep CLW add ] 0 setdash 1 setlinecap stroke
\fi}
\def\tx@DotLine{DotLine }
\def\psset@linestyle#1{%
\@ifundefined{psls@#1}%
{\@pstrickserr{Line style `#1' not defined}\@eha}%
{\edef\pslinestyle{#1}}}
\psset@linestyle{solid}
\def\psfs@none{}
\def\psset@fillcolor#1{\pst@getcolor{#1}\psfillcolor}
\psset@fillcolor{white}
\def\psfs@solid{\pst@fill{\pst@usecolor\psfillcolor fill}}
\def\psset@hatchwidth#1{\pst@getlength{#1}\psk@hatchwidth}
\psset@hatchwidth{.8pt}
\def\psset@hatchsep#1{\pst@getlength{#1}\psk@hatchsep}
\psset@hatchsep{4pt}
\def\psset@hatchcolor#1{\pst@getcolor{#1}\pshatchcolor}
\psset@hatchcolor{black}
\def\psset@hatchangle#1{\pst@getangle{#1}\psk@hatchangle}
\psset@hatchangle{45}
\def\pst@linefill{%
\psk@hatchangle rotate
\psk@hatchwidth SLW
\pst@usecolor\pshatchcolor
\psk@hatchsep \tx@LineFill}
\def\psfs@vlines{\pst@fill\pst@linefill}
\@namedef{psfs@vlines*}{\psfs@solid \psfs@vlines}
\def\psfs@hlines{\pst@fill{90 rotate \pst@linefill}}
\@namedef{psfs@hlines*}{\psfs@solid \psfs@hlines}
\def\psfs@crosshatch{\psfs@vlines \psfs@hlines}
\@namedef{psfs@crosshatch*}{\psfs@solid \psfs@vlines \psfs@hlines}
\def\tx@LineFill{LineFill }
\def\psset@fillstyle#1{%
\edef\pst@tempg{#1}\def\pst@temph{none}%
\ifx\pst@tempg\pst@temph
\let\psk@fillstyle\relax
\else
\@ifundefined{psfs@#1}%
{\@pstrickserr{Undefined fill style: `#1'}\@eha}%
{\edef\psk@fillstyle{\expandafter\noexpand\csname psfs@#1\endcsname}}%
\fi}
\def\psset@addfillstyle#1{%
\@ifundefined{psfs@#1}%
{\@pstrickserr{Undefined fill style: `#1'}\@eha}%
{\edef\psk@fillstyle{%
\expandafter\noexpand\psk@fillstyle
\expandafter\noexpand\csname psfs@#1\endcsname}}}
\psset@fillstyle{none}
\def\psset@arrows#1{%
\begingroup
\pst@activearrows
\xdef\pst@tempg{#1}%
\endgroup
\expandafter\psset@@arrows\pst@tempg\@empty-\@empty\@nil
\if@pst\else
\@pstrickserr{Bad arrows specification: #1}\@ehpa
\fi}
\def\psset@@arrows#1-#2\@empty#3\@nil{%
\@psttrue
\def\next##1,#1-##2,##3\@nil{\def\pst@tempg{##2}}%
\expandafter\next\pst@arrowtable,#1-#1,\@nil
\@ifundefined{psas@\pst@tempg}%
{\@pstfalse\def\psk@arrowA{}}%
{\let\psk@arrowA\pst@tempg}%
\@ifundefined{psas@#2}%
{\@pstfalse\def\psk@arrowB{}}%
{\def\psk@arrowB{#2}}}
\def\psk@arrowA{}
\def\psk@arrowB{}
\def\pst@arrowtable{,<->,<<->>,>-<,>>-<<,(-),[-]}
\begingroup
\catcode`\<=13
\catcode`\>=13
\catcode`\|=13
\gdef\pst@activearrows{\def<{\string<}\def>{\string>}\def|{\string|}}
\endgroup
\def\tx@BeginArrow{BeginArrow }
\def\tx@EndArrow{EndArrow }
\def\psset@arrowscale#1{\pst@getscale{#1}\psk@arrowscale}
\psset@arrowscale{1}
\def\psset@arrowsize#1{%
\pst@expandafter\pst@getdimnum{#1} 0 {} {}\@nil
\edef\psk@arrowsize{\pst@number\pst@dimg \pst@tempg}}
\psset@arrowsize{1.5pt 2}
\def\psset@arrowlength#1{\pst@checknum{#1}\psk@arrowlength}
\psset@arrowlength{1.4}
\def\psset@arrowinset#1{\pst@checknum{#1}\psk@arrowinset}%
\psset@arrowinset{.4}
\def\tx@Arrow{Arrow }
\@namedef{psas@>}{%
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\@namedef{psas@>>}{%
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow
0 h T
gsave
newpath
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow
CP
grestore
CP newpath moveto
2 copy
L
stroke
moveto}
\@namedef{psas@<}{%
true \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\@namedef{psas@<<}{%
true \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow
CP newpath moveto 0 a neg L stroke 0 h neg T
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\def\psset@tbarsize#1{%
\pst@expandafter\pst@getdimnum{#1} 0 {} {}\@nil
\edef\psk@tbarsize{\pst@number\pst@dimg \pst@tempg}}
\psset@tbarsize{2pt 5}
\def\tx@Tbar{Tbar }
\@namedef{psas@|}{\psk@tbarsize \tx@Tbar}
\@namedef{psas@|*}{0 CLW -2 div T \psk@tbarsize \tx@Tbar}
\@namedef{psas@>|}{%
\psk@tbarsize \tx@Tbar
0 CLW 2 div T
newpath
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\@namedef{psas@>|*}{%
0 CLW -2 div T
\psk@tbarsize \tx@Tbar
0 CLW 2 div T
newpath
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\edef\pst@arrowtable{\pst@arrowtable,|<*->|*,|<->|}
\def\psset@bracketlength#1{\pst@checknum{#1}\psk@bracketlength}
\psset@bracketlength{.15}
\def\tx@Bracket{Bracket }
\@namedef{psas@]}{\psk@bracketlength \psk@tbarsize \tx@Bracket}
\def\psset@rbracketlength#1{\pst@checknum{#1}\psk@rbracketlength}
\psset@rbracketlength{.15}
\def\tx@RoundBracket{RoundBracket }
\@namedef{psas@)}{\psk@rbracketlength \psk@tbarsize \tx@RoundBracket}
\def\psas@c{1 \psas@@c}
\def\psas@cc{0 CLW 2 div T 1 \psas@@c}
\def\psas@C{2 \psas@@c}
\def\psas@@c{%
setlinecap
0 0 moveto
0 CLW 2 div L
stroke
0 0 moveto}
\def\psas@{}
\psset@arrows{-}
\def\tx@SD{SD }
\def\tx@EndDot{EndDot }
\def\psas@oo{{\pst@usecolor\psfillcolor true} true \psk@dotsize \tx@EndDot}
\def\psas@o{{\pst@usecolor\psfillcolor true} false \psk@dotsize \tx@EndDot}
\@namedef{psas@**}{{false} true \psk@dotsize \tx@EndDot}
\@namedef{psas@*}{{false} false \psk@dotsize \tx@EndDot}
\def\pst@par{}
\def\addto@par#1{%
\ifx\pst@par\@empty
\def\pst@par{#1}%
\else
\expandafter\def\expandafter\pst@par\expandafter{\pst@par,#1}%
\fi}
\def\addbefore@par#1{%
\ifx\pst@par\@empty
\def\pst@par{#1}%
\else
\toks@{#1}%
\pst@toks\expandafter{\pst@par}%
\edef\pst@par{\the\toks@,\the\pst@toks}%
\fi}
\def\use@par{%
\ifx\pst@par\@empty\else
\expandafter\@psset\pst@par,\@nil
\def\pst@par{}%
\fi}
\def\pst@object#1{%
\pst@ifstar{%
\@ifnextchar[%
{\pst@@object{#1}}%
{\def\pst@par{}\@nameuse{#1@i}}}}
\def\pst@@object#1[#2]{%
\def\pst@par{#2}%
\@ifnextchar+{\@nameuse{#1@i}}{\@nameuse{#1@i}}}
\def\newpsobject#1#2#3{%
\@ifundefined{#2@i}%
{\@pstrickserr{Graphics object `#2' not defined}\@eha}{%
\@namedef{#1}{\pst@object{#1}}%
\@namedef{#1@i}{\addbefore@par{#3}\@nameuse{#2@i}}}%
\ignorespaces}
\def\pst@getarrows#1{\@ifnextchar({#1}{\pst@@getarrows{#1}}}
\def\pst@@getarrows#1#2{\addto@par{arrows=#2}#1}
\def\begin@ClosedObj{%
\leavevmode
\pst@killglue
\begingroup
\use@par
\solid@star
\ifpsdoubleline \pst@setdoublesep \fi
\init@pscode}
\def\end@ClosedObj{%
\ifpsshadow \pst@closedshadow \fi
\ifdim\psk@border\p@>\z@ \pst@addborder \fi
\psk@fillstyle
\pst@stroke
\ifpsdoubleline \pst@doublestroke \fi
\ifshowpoints
% DG modification begin - Mar. 4, 1995
%\addto@pscode{Points aload length 2 div cvi /N ED \psdots@iii}%
\pst@OpenShowPoints
% DG modification end
\fi
\use@pscode
\endgroup
\ignorespaces}
\def\begin@OpenObj{%
\begin@ClosedObj
\let\pst@linetype\pst@arrowtype
\pst@addarrowdef}
\def\begin@AltOpenObj{%
\begin@ClosedObj
\def\pst@repeatarrowsflag{\z@}%
\def\pst@linetype{0}}
\def\end@OpenObj{%
\ifpsshadow \pst@openshadow \fi
\ifdim\psk@border\p@>\z@ \pst@addborder \fi
\psk@fillstyle
\pst@stroke
\ifpsdoubleline \pst@doublestroke \fi
\ifnum\pst@repeatarrowsflag>\z@ \pst@repeatarrows \fi
\ifshowpoints \pst@OpenShowPoints \fi
\use@pscode
\endgroup
\ignorespaces}
\def\begin@SpecialObj{%
\leavevmode
\pst@killglue
\begingroup
\use@par
\init@pscode}
\def\end@SpecialObj{%
\use@pscode
\endgroup
\ignorespaces}
\def\pst@code{}%
\def\init@pscode{%
\addto@pscode{%
\pst@number\pslinewidth SLW
\pst@usecolor\pslinecolor}}
\def\addto@pscode#1{\xdef\pst@code{\pst@code#1\space}}
\def\use@pscode{%
\pstverb{%
\pst@dict
\tx@STP
\pst@newpath
\psk@origin
\psk@swapaxes
\pst@code
end}%
\gdef\pst@code{}}
\def\pst@newpath{newpath }
\def\pst@@killglue{\unskip\ifdim\lastskip>\z@\expandafter\pst@@killglue\fi}
\def\KillGlue{\let\pst@killglue\pst@@killglue}
\def\DontKillGlue{\let\pst@killglue\relax}
\DontKillGlue
\def\solid@star{%
\if@star
\pslinewidth=\z@
\psdoublelinefalse
\def\pslinestyle{none}%
\def\psk@fillstyle{\psfs@solid}%
\let\psfillcolor\pslinecolor
\fi}
\def\pst@setdoublesep{%
\pst@getlength\psdoublesep\psdoublesep
\pslinewidth=2\pslinewidth
\advance\pslinewidth\psdoublesep\p@
\let\pst@setdoublesep\relax}
\def\tx@Shadow{Shadow }
\def\pst@closedshadow{%
\addto@pscode{%
gsave
\psk@shadowsize \psk@shadowangle \tx@PtoC
\tx@Shadow
\pst@usecolor\psshadowcolor
gsave fill grestore
stroke
grestore
gsave
\pst@usecolor\psfillcolor
gsave fill grestore
stroke
grestore}}
\def\pst@openshadow{%
\addto@pscode{%
gsave
\psk@shadowsize \psk@shadowangle \tx@PtoC
\tx@Shadow
\pst@usecolor\psshadowcolor
\ifx\psk@fillstyle\relax\else
gsave fill grestore
\fi
stroke}%
\pst@repeatarrows
\addto@pscode{grestore}
\ifx\psk@fillstyle\relax\else
\addto@pscode{%
gsave
\pst@usecolor\psfillcolor
gsave fill grestore
stroke
grestore}
\fi}
\def\pst@addborder{%
\addto@pscode{%
gsave
\psk@border 2 mul
CLW add SLW
\pst@usecolor\psbordercolor
stroke
grestore}}
\def\pst@stroke{%
\ifx\pslinestyle\@none\else
\addto@pscode{%
gsave
\pst@number\pslinewidth SLW
\pst@usecolor\pslinecolor
\@nameuse{psls@\pslinestyle}
grestore}%
\fi}
\def\pst@fill#1{\addto@pscode{gsave #1 grestore}}%
\def\pst@doublestroke{%
\addto@pscode{%
gsave
\psdoublesep SLW
\pst@usecolor\psdoublecolor
stroke
grestore}}
\def\pst@arrowtype{%
\ifx\psk@arrowB\@empty 0 \else -2 \fi
\ifx\psk@arrowA\@empty 0 \else -1 \fi
add}
\def\pst@addarrowdef{%
\addto@pscode{%
/ArrowA {
\ifx\psk@arrowA\@empty
\pst@oplineto
\else
\pst@arrowdef{A}
moveto
\fi
} def
/ArrowB {
\ifx\psk@arrowB\@empty \else \pst@arrowdef{B} \fi
} def}}
\def\pst@arrowdef#1{%
\ifnum\pst@repeatarrowsflag>\z@
/Arrow#1c [ 6 2 roll ] cvx def Arrow#1c
\fi
\tx@BeginArrow
\psk@arrowscale
\@nameuse{psas@\@nameuse{psk@arrow#1}}
\tx@EndArrow}
\def\pst@repeatarrows{%
\addto@pscode{%
gsave
\ifx\psk@arrowA\@empty\else
ArrowAc ArrowA pop pop
\fi
\ifx\psk@arrowB\@empty\else
ArrowBc ArrowB pop pop pop pop
\fi
grestore}}
\def\pst@OpenShowPoints{%
\addto@pscode{%
gsave
\psk@dotsize
\@nameuse{psds@\psk@dotstyle}
newpath
Points aload length 2 div 2 sub cvi /N ED
N 0 ge
{ \ifx\psk@arrowA\@empty
Dot
\else
pop pop
\fi
N { Dot } repeat
\ifx\psk@arrowB\@empty
Dot
\else
pop pop
\fi }
{ N 2 mul { pop } repeat }
ifelse
grestore}}
\def\pscustom{\pst@object{pscustom}}
\long\def\pscustom@i#1{%
\begin@SpecialObj
\solid@star
\let\pst@ifcustom\iftrue
\let\begin@ClosedObj\begin@CustomObj
\let\end@ClosedObj\endgroup
\def\begin@OpenObj{\begin@CustomObj\pst@addarrowdef}%
\let\end@OpenObj\endgroup
\let\begin@AltOpenObj\begin@CustomObj
\def\begin@SpecialObj{%
\begingroup
\pst@misplaced{special graphics object}%
\def\addto@pscode####1{}
\let\end@SpecialObj\endgroup}%
\def\@multips(##1)(##2)##3##4{\pst@misplaced\multips}%
\def\psclip##1{\pst@misplaced\psclip}%
\def\pst@repeatarrowsflag{\z@}%
\let\pst@setrepeatarrowsflag\relax
\showpointsfalse
\let\showpointstrue\relax
\def\pst@linetype{\pslinetype}%
\let\psset@liftpen\psset@@liftpen
\psset@liftpen{\z@}%
\def\pst@cp{/currentpoint load stopped pop }%
\def\pst@oplineto{/lineto load stopped { moveto } if }%
\def\pst@optcp##1##2{%
\ifnum##1=\z@\def##2{/currentpoint load stopped { 0 0 } if }\fi}%
\let\caddto@pscode\addto@pscode
\def\cuse@par##1{{\use@par##1}}%
\the\pst@customdefs
\setbox\pst@hbox=\hbox{#1}%
\psk@fillstyle
\pst@stroke
\end@SpecialObj}
\def\begin@CustomObj{%
\begingroup
\use@par
\addto@pscode{%
\pst@number\pslinewidth SLW
\pst@usecolor\pslinecolor}}
\def\pst@oplineto{moveto }
\def\pst@cp{}
\def\pst@optcp#1#2{}
\def\psset@liftpen#1{}
\def\psset@@liftpen#1{%
\ifcase#1\relax
\def\psk@liftpen{\z@}%
\def\pst@cp{/currentpoint load stopped pop }%
\def\pst@oplineto{/lineto load stopped { moveto } if }%
\or
\def\psk@liftpen{1}%
\def\pst@cp{}%
\def\pst@oplineto{/lineto load stopped { moveto } if }%
\or
\def\psk@liftpen{2}%
\def\pst@cp{}%
\def\pst@oplineto{moveto }%
\fi}
\psset@liftpen{0}
\def\psk@liftpen{-1}
\def\psset@linetype#1{%
\pst@getint{#1}\pslinetype
\ifnum\pst@dimg<-3
\@pstrickserr{linetype must be greater than -3}\@ehpa
\def\pslinetype{0}%
\fi}
\psset@linetype{0}
\def\caddto@pscode#1{%
\@pstrickserr{Command can only be used in \string\pscustom}\@ehpa}
\let\cuse@par\caddto@pscode
\def\tx@MSave{%
/msavemtrx
[ tx@Dict /msavemtrx known { msavemtrx aload pop } if CM ]
def }
\def\tx@MRestore{%
tx@Dict /msavemtrx known { length 0 gt } { false } ifelse
{ /msavematrx [ msavematrx aload pop setmatrix ] def }
if }
\newtoks\pst@customdefs
\pst@customdefs{%
\def\newpath{\addto@pscode{newpath}}%
\def\moveto(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor moveto}}%
\def\closepath{\addto@pscode{closepath}}%
\def\gsave{\begingroup\addto@pscode{gsave}}%
\def\grestore{\endgroup\addto@pscode{grestore}}%
% DG/SR modification begin - May 12, 1997 - Patch 2
%\def\translate(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor moveto}}%
\def\translate(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor translate}}%
% DG/SR modification end
\def\rotate#1{\pst@@getangle{#1}\addto@pscode{\pst@angle rotate}}%
\def\scale#1{\pst@getscale{#1}\pst@tempg\addto@pscode{\pst@tempg}}%
\def\msave{\addto@pscode{\tx@MSave}}%
\def\mrestore{\addto@pscode{\tx@MRestore}}%
\def\swapaxes{\addto@pscode{-90 rotate -1 1 scale}}%
\def\stroke{\pst@object{stroke}}%
\def\fill{\pst@object{fill}}%
\def\openshadow{\pst@object{openshadow}}%
\def\closedshadow{\pst@object{closedshadow}}%
% DG/SR modification begin - Jan. 7, 1998 - Patch 8
%\def\movepath(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor tx@Shadow}}%
\def\movepath(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor \tx@Shadow}}%
% DG/SR modification end
\def\lineto{\pst@onecoor{lineto}}%
\def\rlineto{\pst@onecoor{rlineto}}%
\def\curveto{\pst@threecoor{curveto}}%
\def\rcurveto{\pst@threecoor{rcurveto}}%
\def\code#1{\addto@pscode{#1}}%
\def\coor(#1){\pst@@getcoor{#1}\addto@pscode\pst@coor\@ifnextchar({\coor}{}}%
\def\rcoor{\pst@getcoors{}{}}%
\def\dim#1{\pssetlength\pst@dimg{#1}\addto@pscode{\pst@number\pst@dimg}}%
\def\setcolor#1{%
\@ifundefined{color@#1}{}{\addto@pscode{\use@color{#1}}}}%
\def\arrows#1{{\psset@arrows{#1}\pst@addarrowdef}}%
\let\file\pst@rawfile
} % END \pst@customdefs
\def\closedshadow@i{\cuse@par\pst@closedshadow}
\def\openshadow@i{\cuse@par\pst@openshadow}
\def\stroke@i{\cuse@par\pst@stroke}%
\def\fill@i{\cuse@par\psk@fillstyle}%
\def\pst@onecoor#1(#2){%
\pst@@getcoor{#2}%
\addto@pscode{\pst@coor #1}}
\def\pst@threecoor#1(#2)#3(#4)#5(#6){%
\begingroup
\pst@getcoor{#2}\pst@tempa
\pst@getcoor{#4}\pst@tempb
\pst@getcoor{#6}\pst@tembc
\addto@pscode{\pst@tempa \pst@tempb \pst@tempc #1}%
\endgroup}
\def\pst@rawfile#1{%
\begingroup
\def\do##1{\catcode`##1=12\relax}"
\dospecials
\catcode`\%=14
\pst@@rawfile{#1}%
\endgroup}
\def\pst@@rawfile#1{%
\immediate\openin1 #1
\ifeof1
\@pstrickserr{File `#1' not found}\@ehpa
\else
\immediate\read1 to \pst@tempg
\loop
\ifeof1 \@pstfalse\else\@psttrue\fi
\if@pst
\addto@pscode\pst@tempg
\immediate\read1 to \pst@tempg
\repeat
\fi
\immediate\closein1\relax}
\def\tx@NArray{NArray }
\def\tx@NArray{NArray }
\def\tx@Line{Line }
\def\tx@Arcto{Arcto }
\def\tx@CheckClosed{CheckClosed }
\def\tx@Polygon{Polygon }
\def\psset@gangle#1{\pst@getangle{#1}\psk@gangle}
\psset@gangle{0}
\def\tx@Diamond{Diamond }
\def\psdiamond{\pst@object{psdiamond}}
\def\psdiamond@i(#1){%
\@ifnextchar({\psdiamond@ii(#1)}{\psdiamond@ii(0,0)(#1)}}
\def\psdiamond@ii(#1)(#2){%
\begin@ClosedObj
\pst@getcoor{#1}\pst@tempa
\pst@getcoor{#2}\pst@tempb
\addto@pscode{%
\psline@iii
pop
\psk@dimen
\pst@tempb
\psk@gangle
\pst@tempa
\tx@Diamond}%
\def\pst@linetype{4}%
\end@ClosedObj}
\def\tx@Triangle{Triangle }
\def\pstriangle{\pst@object{pstriangle}}
\def\pstriangle@i(#1){%
\@ifnextchar({\pstriangle@ii(#1)}{\pstriangle@ii(0,0)(#1)}}
\def\pstriangle@ii(#1)(#2){%
\begin@ClosedObj
\pst@getcoor{#1}\pst@tempa
\pst@getcoor{#2}\pst@tempb
\addto@pscode{%
\psline@iii
pop
\psk@dimen
\pst@tempb
\psk@gangle
\pst@tempa
\tx@Triangle}%
\def\pst@linetype{2}%
\end@ClosedObj}
\def\tx@CCA{CCA }
\def\tx@CCA{CCA }
\def\tx@CC{CC }
\def\tx@IC{IC }
\def\tx@BOC{BOC }
\def\tx@NC{NC }
\def\tx@EOC{EOC }
\def\tx@BAC{BAC }
\def\tx@NAC{NAC }
\def\tx@EAC{EAC }
\def\tx@OpenCurve{OpenCurve }
\def\tx@AltCurve{AltCurve }
\def\tx@ClosedCurve{ClosedCurve }
\def\psset@curvature#1{%
\edef\pst@tempg{#1 }%
\expandafter\psset@@curvature\pst@tempg * * * \@nil}
\def\psset@@curvature#1 #2 #3 #4\@nil{%
\pst@checknum{#1}\pst@tempg
\pst@checknum{#2}\pst@temph
\pst@checknum{#3}\pst@tempi
\edef\psk@curvature{\pst@tempg \pst@temph \pst@tempi}}
\psset@curvature{1 .1 0}
\def\pscurve{\pst@object{pscurve}}
\def\pscurve@i{%
\pst@getarrows{%
\begin@OpenObj
\pst@getcoors[\pscurve@ii}}
\def\pscurve@ii{%
\addto@pscode{%
\pst@cp
\psk@curvature\space /c ED /b ED /a ED
\ifshowpoints true \else false \fi
\tx@OpenCurve}%
\end@OpenObj}
\def\psecurve{\pst@object{psecurve}}
\def\psecurve@i{%
\pst@getarrows{%
\begin@OpenObj
\pst@getcoors[\psecurve@ii}}
\def\psecurve@ii{%
\addto@pscode{%
\psk@curvature\space /c ED /b ED /a ED
\ifshowpoints true \else false \fi
\tx@AltCurve}%
\end@OpenObj}
\def\psccurve{\pst@object{psccurve}}
\def\psccurve@i{%
\begin@ClosedObj
\pst@getcoors[\psccurve@ii}
\def\psccurve@ii{%
\addto@pscode{%
\psk@curvature\space /c ED /b ED /a ED
\ifshowpoints true \else false \fi
\tx@ClosedCurve}%
\def\pst@linetype{1}%
\end@ClosedObj}
\def\psset@dotsize#1{%
\pst@expandafter\pst@getdimnum{#1} 0 {} {}\@nil
\edef\psk@@dotsize{\pst@number\pst@dimg}%
\let\psk@@@dotsize\pst@tempg
\edef\psk@dotsize{%
/DS \psk@@dotsize \psk@@@dotsize CLW mul add 2 div def }}
\psset@dotsize{2pt 2}
\def\psset@dotscale#1{%
\pst@getscale{#1}\psk@dotscale
\ifx\psk@dotscale\@empty
\def\psk@xdotscale{1 }%
\def\psk@ydotscale{1 }%
\else
\let\psk@xdotscale\pst@tempg
\let\psk@ydotscale\pst@temph
% DG/SR modification begin - Oct. 17, 1997 - Patch 5
%\fi}
\fi
\pst@getdotsize%
\psset@dotsize{\pst@dimg\space\pst@@dimtonum\pst@dimh}}
%\psset@dotscale{1}
% DG/SR modification end
\def\pst@Getangle#1#2{%
\pst@getangle{#1}\pst@tempg
\def\pst@temph{0. }%
\ifx\pst@tempg\pst@temph
\def#2{}%
\else
\edef#2{\pst@tempg\space rotate }%
\fi}
\def\psset@dotangle#1{%
\pst@getangle{#1}\psk@@dotangle
\ifdim\psk@@dotangle\p@=\z@
\let\psk@dotangle\@empty
\else
% DG/SR modification begin - Aug. 8, 1997 - Patch 4
%\edef\psk@dotangle{\psk@@dotangle rotate }
\edef\psk@dotangle{\psk@@dotangle rotate }%
% DG/SR modification end
\fi}
\psset@dotangle{0}
\def\pst@getdotsize{%
\pst@dimg=\psk@@@dotsize\pslinewidth
\advance\pst@dimg\psk@@dotsize\p@
\pst@dimh=\psk@ydotscale\pst@dimg
\pst@dimg=\psk@xdotscale\pst@dimg
\divide\pst@dimh 2
\divide\pst@dimg 2\relax}
% DG/SR modification begin - Oct. 17, 1997 - Patch 5
\psset@dotscale{1}
% DG/SR modification end
\def\psdot{\pst@object{psdot}}
\def\psdot@i{\@ifnextchar({\psdot@ii}{\psdot@ii(\z@,\z@)}}
\def\psdot@ii(#1){%
\begin@SpecialObj
\pst@@getcoor{#1}%
\addto@pscode{%
\psk@dotsize
\@nameuse{psds@\psk@dotstyle}%
\pst@coor Dot}%
\end@SpecialObj}
\def\psdots{\pst@object{psdots}}
\def\psdots@i{%
\begin@SpecialObj
\pst@getcoors[\psdots@ii}
\def\psdots@ii{%
\addto@pscode{false \tx@NArray \psdots@iii}%
\end@SpecialObj}
\def\psdots@iii{%
\psk@dotsize
\@nameuse{psds@\psk@dotstyle}
newpath
n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat}
% DG: dead code (to suppress until \psset@dotstyle) ? - Aug. 4, 1997
\def\tx@SQ{SQ }
\def\tx@ST{ST }
\def\tx@SP{SP }
\def\pst@gdot#1{/Dot { gsave T \psk@dotangle \psk@dotscale #1 grestore } def }
\@namedef{psds@*}{\pst@gdot{0 0 DS \tx@SD}}
\@namedef{psds@o}{%
/r2 DS CLW sub def
\pst@gdot{0 0 DS \tx@SD \pst@usecolor\psfillcolor 0 0 r2 \tx@SD}}
\@namedef{psds@square*}{%
/r1 DS .886 mul def
\pst@gdot{r1 \tx@SQ}}
\@namedef{psds@square}{%
/r1 DS .886 mul def /r2 r1 CLW sub def
\pst@gdot{r1 \tx@SQ \pst@usecolor\psfillcolor r2 \tx@SQ}}
\@namedef{psds@triangle*}{%
/y1 DS .778 mul neg def /x1 y1 1.732 mul neg def
\pst@gdot{x1 y1 \tx@ST}}
\@namedef{psds@triangle}{%
/y1 DS .778 mul neg def /x1 y1 1.732 mul neg def
/y2 y1 CLW add def /x2 y2 1.732 mul neg def
\pst@gdot{x1 y1 \tx@ST \pst@usecolor\psfillcolor x2 y2 \tx@ST}}
\@namedef{psds@pentagon*}{%
/r1 DS 1.149 mul def
\pst@gdot{r1 \tx@SP}}
\@namedef{psds@pentagon}{%
DS .93 mul dup 1.236 mul /r1 ED CLW sub 1.236 mul /r2 ED
\pst@gdot{r1 \tx@SP \pst@usecolor\psfillcolor r2 \tx@SP}}
\@namedef{psds@+}{%
/DS DS 1.253 mul def
\pst@gdot{DS 0 moveto DS neg 0 L stroke 0 DS moveto 0 DS neg L stroke}}
\@namedef{psds@|}{%
\psk@tbarsize CLW mul add 2 div /DS ED
\pst@gdot{0 DS moveto 0 DS neg L stroke}}
% DG: end dead code?
\def\psset@dotstyle#1{%
\@ifundefined{psds@#1}%
{\@pstrickserr{Dot style `#1' not defined}\@eha}%
{\edef\psk@dotstyle{#1}}}
\psset@dotstyle{*}
\def\tx@FontDot{FontDot }
\def\newpsfontdot#1[#2]#3#4{%
\@namedef{psds@#1}{%
/#3 \psk@@dotangle [#2] \tx@FontDot
/Dot { moveto #4 show } bind def }}
\def\newpsfontdotH#1[#2]#3#4#5{%
\@namedef{psds@#1}{%
/#3 \psk@@dotangle [#2] \tx@FontDot
/Dot {
moveto
\iftrue
gsave \pst@usecolor\psfillcolor #5 show grestore
\fi
#4 show
} bind def }}
\pstheader{pst-dots.pro}
\newpsfontdot{*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(b)}
\newpsfontdotH{o}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(c)}{(b)}
\newpsfontdotH{Bo}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(C)}{(b)}
\newpsfontdotH{triangle}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(t)}{(u)}
\newpsfontdotH{Btriangle}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(T)}{(u)}
\newpsfontdot{triangle*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(u)}
\newpsfontdotH{square}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(s)}{(r)}
\newpsfontdotH{Bsquare}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(S)}{(r)}
\newpsfontdot{square*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(r)}
\newpsfontdotH{pentagon}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(p)}{(q)}
\newpsfontdotH{Bpentagon}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(P)}{(q)}
\newpsfontdot{pentagon*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(q)}
% DG/SR modification begin - Mar. 18, 1997
%\newpsfontdot{diamond*}%
%[1.9 0.0 0.0 1.9 -0.4598 -0.70775]{Symbol}{<E0>}
%\newpsfontdot{diamond}%
%[2.3 0.0 0.0 2.3 -0.8533 -0.5336]{Symbol}{<A8>}
\newpsfontdot{diamond}%
[1.9 0.0 0.0 1.9 -0.4598 -0.70775]{Symbol}{<E0>}
\newpsfontdot{diamond*}%
[2.3 0.0 0.0 2.3 -0.8533 -0.5336]{Symbol}{<A8>}
% DG/SR modification end
\newpsfontdot{oplus}%
[1.44928 0.0 0.0 1.44928 -0.562319 -0.478261]{Symbol}{<C5>}
\newpsfontdot{otimes}%
[1.44928 0.0 0.0 1.44928 -0.562319 -0.475362]{Symbol}{<C4>}
\newpsfontdot{x}%
[1.8 0.0 0.0 1.8 -0.495 -0.4788]{Symbol}{<B4>}
\newpsfontdot{+}%
[2.3 0.0 0.0 2.3 -0.6486 -0.5819]{Times-Roman}{<2B>}
\newpsfontdot{asterisk}%
[2.43309 0.0 0.0 2.43309 -0.609489 -1.14477]{Times-Roman}{<2A>}
\newpsfontdot{B+}%
[2.3 0.0 0.0 2.3 -0.6555 -0.5819]{Times-Bold}{<2B>}
\newpsfontdot{Basterisk}%
[2.29358 0.0 0.0 2.29358 -0.576835 -1.08486]{Times-Bold}{<2A>}
\newpsfontdot{|}%
% DG/SR modification begin - Oct. 27, 1997 - Patch 7
%[1.98413 0.0 0.0 1.98413 -0.258929 -0.712302]{Helvetica}{(|)}
[1.98413 0.0 0.0 1.38 -0.258929 -0.5]{Helvetica}{(|)}
% DG/SR modification end
\newpsfontdot{B|}%
% DG/SR modification begin - Oct. 27, 1997 - Patch 7
%[1.98413 0.0 0.0 1.98413 -0.277778 -0.78302]{Helvetica-Bold}{(|)}
[1.98413 0.0 0.0 1.38 -0.277778 -0.5]{Helvetica-Bold}{(|)}
% DG/SR modification end
\iffalse
\newpsfontdot{*}%
[2.77778 0.0 0.0 2.77778 -0.638889 -0.813889]{Symbol}{<B7>}
\newpsfontdot{o}%
[3.33333 0.0 0.0 3.33333 -0.666667 -1.78167]{Symbol}{<B0>}
\newpsfontdot{Bo}%
[4.69484 0.0 0.0 4.69484 -0.78169 -2.97418]{Times-Bold}{<CA>}
\fi
\newdimen\pslinearc
\def\psset@linearc#1{\pssetlength\pslinearc{#1}}
\psset@linearc{0pt}
\def\psline{\pst@object{psline}}
\def\psline@i{%
\pst@getarrows{%
\begin@OpenObj
\pst@getcoors[\psline@ii}}
\def\psline@ii{%
\addto@pscode{\pst@cp \psline@iii \tx@Line}%
\end@OpenObj}
\def\psline@iii{%
\ifdim\pslinearc>\z@
/r \pst@number\pslinearc def
/Lineto { \tx@Arcto } def
\else
/Lineto /lineto load def
\fi
\ifshowpoints true \else false \fi}
\def\qline(#1)(#2){%
\def\pst@par{}%
\begin@SpecialObj
\def\pst@linetype{0}%
\pst@getcoor{#1}\pst@tempa
\pst@@getcoor{#2}%
\addto@pscode{%
\pst@tempa moveto \pst@coor L
\@nameuse{psls@\pslinestyle}}%
\end@SpecialObj}
\def\pspolygon{\pst@object{pspolygon}}
\def\pspolygon@i{%
\begin@ClosedObj
\def\pst@cp{}%
\pst@getcoors[\pspolygon@ii}
\def\pspolygon@ii{%
\addto@pscode{\psline@iii \tx@Polygon}%
\def\pst@linetype{1}%
\end@ClosedObj}
\def\psset@framearc#1{\pst@checknum{#1}\psk@framearc}
\psset@framearc{0}
\def\psset@cornersize#1{%
\pst@expandafter\psset@@cornersize{#1}\@nil}
\def\psset@@cornersize#1#2\@nil{%
\if #1a\relax
\def\psk@cornersize{\pst@number\pslinearc false }%
\else
\def\psk@cornersize{\psk@framearc true }%
\fi}
\psset@cornersize{relative}
\def\tx@Rect{Rect }
\def\tx@OvalFrame{OvalFrame }
\def\tx@Frame{Frame }
\def\psset@dimen#1{%
\pst@expandafter\psset@@dimen{#1}\@nil}
\def\psset@@dimen#1#2\@nil{%
\if #1o\relax
\def\psk@dimen{.5 }%
\else
\if #1m\relax
\def\psk@dimen{0 }%
\else
\if #1i\relax
\def\psk@dimen{-.5 }%
\fi
\fi
\fi}
\psset@dimen{outer}
\def\psframe{\pst@object{psframe}}
\def\psframe@i(#1){%
\@ifnextchar({\psframe@ii(#1)}{\psframe@ii(0,0)(#1)}}
\def\psframe@ii(#1)(#2){%
\begin@ClosedObj
\pst@getcoor{#1}\pst@tempa
\pst@@getcoor{#2}%
\addto@pscode{\psk@cornersize \pst@tempa \pst@coor \psk@dimen \tx@Frame}%
\def\pst@linetype{2}%
\showpointsfalse
\end@ClosedObj}
\def\tx@BezierNArray{BezierNArray }
\def\tx@OpenBezier{OpenBezier }
\def\tx@ClosedBezier{ClosedBezier }
\def\tx@BezierShowPoints{BezierShowPoints }
\def\psbezier{\pst@object{psbezier}}
\def\psbezier@i{%
\pst@getarrows{%
\begin@OpenObj
\pst@getcoors[\psbezier@ii}}
\def\psbezier@ii{%
\addto@pscode{%
% DG/SR modification begin - Apr. 28, 1997 - Patch 1
% \psbezier doesn't work inside \pscustom
%\pst@cp
% DG/SR modification end
\ifshowpoints true \else false \fi
\tx@OpenBezier
\ifshowpoints \tx@BezierShowPoints \fi}%
\end@OpenObj}
\def\pscbezier{\pst@object{pscbezier}}
\def\pscbezier@i{%
\begin@ClosedObj
\pst@getcoors[\pscbezier@ii}
\def\pscbezier@ii{%
\addto@pscode{%
\ifshowpoints true \else false \fi
\tx@ClosedBezier
\ifshowpoints \tx@BezierShowPoints \fi}%
\chardef\pst@linetype=1
\end@ClosedObj}
\def\tx@Parab{Parab }
\def\parabola{\pst@object{parabola}}
\def\parabola@i{\pst@getarrows\parabola@ii}
\def\parabola@ii#1(#2)#3(#4){%
\begin@OpenObj
\pst@getcoor{#2}\pst@tempa
\pst@@getcoor{#4}%
\addto@pscode{\pst@tempa \pst@coor \tx@Parab}%
\end@OpenObj}
\def\psset@gridwidth#1{\pst@getlength{#1}\psk@gridwidth}
\psset@gridwidth{.8pt}
\def\psset@griddots#1{%
\pst@cntg=#1\relax
\edef\psk@griddots{\the\pst@cntg}}
\psset@griddots{0}
\def\psset@gridcolor#1{\pst@getcolor{#1}\psgridcolor}
\psset@gridcolor{black}
\def\psset@subgridwidth#1{\pst@getlength{#1}\psk@subgridwidth}
\psset@subgridwidth{.4pt}
\def\psset@subgridcolor#1{\pst@getcolor{#1}\pssubgridcolor}
\psset@subgridcolor{gray}
\def\psset@subgriddots#1{%
\pst@cntg=#1\relax\edef\psk@subgriddots{\the\pst@cntg}}
\psset@subgriddots{0}
\def\psset@subgriddiv#1{%
\pst@cntg=#1\relax\edef\psk@subgriddiv{\the\pst@cntg}}
\psset@subgriddiv{5}
\def\psset@gridlabels#1{\pst@getlength{#1}\psk@gridlabels}
\psset@gridlabels{10pt}
\def\psset@gridlabelcolor#1{\pst@getcolor{#1}\psgridlabelcolor}
\psset@gridlabelcolor{black}
\def\tx@Grid{Grid }
\def\psgrid{\pst@object{psgrid}}
\def\psgrid@i{\@ifnextchar(%
{\psgrid@ii}{\expandafter\psgrid@iv\pic@coor}}
\def\psgrid@ii(#1){\@ifnextchar(%
{\psgrid@iii(#1)}{\psgrid@iv(0,0)(0,0)(#1)}}
\def\psgrid@iii(#1)(#2){\@ifnextchar(%
{\psgrid@iv(#1)(#2)}{\psgrid@iv(#1)(#1)(#2)}}
\def\psgrid@iv(#1)(#2)(#3){%
\begin@SpecialObj
\pst@getcoor{#1}\pst@tempa
\pst@getcoor{#2}\pst@tempb
\pst@@getcoor{#3}%
\ifnum\psk@subgriddiv>1
\addto@pscode{gsave
\psk@subgridwidth SLW \pst@usecolor\pssubgridcolor
\pst@tempb \pst@coor \pst@tempa
\pst@number\psxunit \pst@number\psyunit
\psk@subgriddiv\space \psk@subgriddots\space
{} 0 \tx@Grid grestore}%
\fi
\addto@pscode{gsave
\psk@gridwidth SLW \pst@usecolor\psgridcolor
\pst@tempb \pst@coor \pst@tempa
\pst@number\psxunit \pst@number\psyunit
1 \psk@griddots\space { \pst@usecolor\psgridlabelcolor }
\psk@gridlabels \tx@Grid grestore}%
\end@SpecialObj}
\newif\ifpsmathbox
\psmathboxtrue
\def\pst@mathflag{\z@}
\newtoks\everypsbox
\let\pst@thisbox\relax
\long\def\pst@makenotverbbox#1#2{%
\edef\pst@mathflag{%
\ifpsmathbox\ifmmode\ifinner 1\else 2\fi\else \z@\fi\else \z@\fi}%
\setbox\pst@hbox=\hbox{%
\ifcase\pst@mathflag\or$\m@th\textstyle\or$\m@th\displaystyle\fi
{\pst@thisbox\the\everypsbox#2}%
\ifnum\pst@mathflag>\z@$\fi}%
#1}
\def\pst@makeverbbox#1{%
\def\pst@afterbox{#1}%
\edef\pst@mathflag{%
\ifpsmathbox\ifmmode\ifinner 1\else 2\fi\else \z@\fi\else \z@\fi}%
\afterassignment\pst@beginbox
\setbox\pst@hbox\hbox}
\def\pst@beginbox{%
\ifcase\pst@mathflag\or$\m@th\or$\m@th\displaystyle\fi
\bgroup\aftergroup\pst@endbox
\pst@thisbox
\the\everypsbox}
\def\pst@endbox{%
\ifnum\pst@mathflag>\z@$\fi
\egroup
\pst@afterbox}
\def\pst@makebox{\pst@@makebox}
\def\psverbboxtrue{\def\pst@@makebox{\pst@makeverbbox}}
\def\psverbboxfalse{\def\pst@@makebox{\pst@makenotverbbox}}
\psverbboxfalse
\def\pst@longbox{%
\def\pst@makebox{%
\gdef\pst@makebox{\pst@@makebox}%
\pst@makelongbox}}
\def\pst@makelongbox#1{%
\def\pst@afterbox{#1}%
\edef\pst@mathflag{%
\ifpsmathbox\ifmmode\ifinner 1\else 2\fi\else \z@\fi\else \z@\fi}%
\setbox\pst@hbox\hbox\bgroup
\aftergroup\pst@afterbox
\ifcase\pst@mathflag\or$\m@th\or$\m@th\displaystyle\fi
\begingroup
\pst@thisbox
\the\everypsbox}
\def\pst@endlongbox{%
\endgroup
\ifnum\pst@mathflag>\z@$\fi
\egroup}
\def\pslongbox#1#2{%
\@namedef{#1}{\pst@longbox#2}%
\@namedef{end#1}{\pst@endlongbox}}
\newdimen\psframesep
\def\psset@framesep#1{\pssetlength\psframesep{#1}}
\psset@framesep{3pt}
\newif\ifpsboxsep
\def\psset@boxsep#1{\@nameuse{psboxsep#1}}
\psset@boxsep{true}
\def\pst@useboxpar{%
\use@par
\if@star
\let\pslinecolor\psfillcolor
\solid@star
\let\solid@star\relax
\fi
\ifpsdoubleline \pst@setdoublesep \fi}
\def\psframebox{\pst@object{psframebox}}
\def\psframebox@i{\pst@makebox\psframebox@ii}
\def\psframebox@ii{%
\begingroup
\pst@useboxpar
\pst@dima=\pslinewidth
\advance\pst@dima by \psframesep
\pst@dimc=\wd\pst@hbox\advance\pst@dimc by \pst@dima
\pst@dimb=\dp\pst@hbox\advance\pst@dimb by \pst@dima
\pst@dimd=\ht\pst@hbox\advance\pst@dimd by \pst@dima
\setbox\pst@hbox=\hbox{%
\ifpsboxsep\kern\pst@dima\fi
\begin@ClosedObj
\addto@pscode{%
\psk@cornersize
\pst@number\pst@dima neg
\pst@number\pst@dimb neg
\pst@number\pst@dimc
\pst@number\pst@dimd
.5
\tx@Frame}%
\def\pst@linetype{2}%
\showpointsfalse
\end@ClosedObj
\box\pst@hbox
\ifpsboxsep\kern\pst@dima\fi}%
\ifpsboxsep\dp\pst@hbox=\pst@dimb\ht\pst@hbox=\pst@dimd\fi
\leavevmode\box\pst@hbox
\endgroup}
\def\psdblframebox{\pst@object{psdblframebox}}
\def\psdblframebox@i{\addto@par{doubleline=true}\psframebox@i}
\def\psclip#1{%
\leavevmode
\begingroup
\begin@psclip
\begingroup
\def\use@pscode{%
\pstVerb{%
\pst@dict
/mtrxc CM def
CP CP T
\tx@STV
\psk@origin
\psk@swapaxes
newpath
\pst@code
clip
newpath
mtrxc setmatrix
moveto
0 setgray
end}%
\gdef\pst@code{}}%
\def\@multips(##1)(##2)##3##4{\pst@misplaced\multips}%
\def\nc@object##1##2##3##4{\pst@misplaced{node connection}}%
\hbox to\z@{#1}%
\endgroup
\def\endpsclip{%
\end@psclip
\endgroup}%
\ignorespaces}
\def\endpsclip{\pst@misplaced\endpsclip}
\let\begin@psclip\relax
\def\end@psclip{\pstVerb{currentpoint initclip moveto}}
\def\AltClipMode{%
\def\end@psclip{\pstVerb{\pst@grestore}}%
\def\begin@psclip{\pstVerb{gsave}}}
\def\clipbox{\@ifnextchar[{\clipbox@}{\clipbox@[\z@]}}
% DG modification begin - Apr. 3, 1997
% From paulus@immd5.informatik.uni-erlangen.de (Dietrich Paulus)
%\def\clipbox@[#1]{\pst@makebox\clipbox@@{#1}}
\def\clipbox@[#1]{\pst@makebox{\clipbox@@{#1}}}
% DG modification end
\def\clipbox@@#1{%
\pssetlength\pst@dimg{#1}%
\leavevmode\hbox{%
\begin@psclip
\pst@Verb{%
CM \tx@STV CP T newpath
/a \pst@number\pst@dimg def
/w \pst@number{\wd\pst@hbox}a add def
/d \pst@number{\dp\pst@hbox}a add neg def
/h \pst@number{\ht\pst@hbox}a add def
a neg d moveto
a neg h L
w h L
w d L
closepath
clip
newpath
0 0 moveto
setmatrix}%
\unhbox\pst@hbox
\end@psclip}}
\def\psshadowbox{\pst@object{psshadowbox}}
\def\psshadowbox@i{\pst@makebox\psshadowbox@ii}
\def\psshadowbox@ii{%
\begingroup
\pst@useboxpar
\psshadowtrue
\psboxseptrue
\def\psk@shadowangle{-45 }%
\setbox\pst@hbox=\hbox{\psframebox@ii}%
\pst@dimh=\psk@shadowsize\p@
\pst@dimh=.7071\pst@dimh
\pst@dimg=\dp\pst@hbox
\advance\pst@dimg\pst@dimh
\dp\pst@hbox=\pst@dimg
\pst@dimg=\wd\pst@hbox
\advance\pst@dimg\pst@dimh
\wd\pst@hbox=\pst@dimg
\leavevmode
\box\pst@hbox
\endgroup}
\def\pscirclebox{\pst@object{pscirclebox}}
\def\pscirclebox@i{\pst@makebox\pscirclebox@ii}
\def\pscirclebox@ii{%
\begingroup
\pst@useboxpar
\setbox\pst@hbox=\hbox{%
\pst@nodehook
\pscirclebox@iii
\box\pst@hbox}%
\ifpsboxsep \pscirclebox@sep \fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\pscirclebox@iii{%
\if@star
\pslinewidth\z@
\pstverb{\pst@dict \tx@STP \pst@usecolor\psfillcolor
newpath \pscirclebox@iv \tx@SD end}%
\else
\begin@ClosedObj
\def\pst@linetype{4}\showpointsfalse
\addto@pscode{%
\pscirclebox@iv CLW 2 div add 0 360 arc closepath}%
\end@ClosedObj
\fi}
\def\pscirclebox@iv{%
\pst@number{\wd\pst@hbox}2 div
\pst@number{\ht\pst@hbox}\pst@number{\dp\pst@hbox}add 2 div
2 copy \pst@number{\dp\pst@hbox}sub 4 2 roll
\tx@Pyth \pst@number\psframesep add }
\def\pscirclebox@sep{%
\pst@dimb=\ht\pst@hbox
\advance\pst@dimb\dp\pst@hbox
\divide\pst@dimb 2
\pst@dima=.5\wd\pst@hbox
\pst@pyth\pst@dima\pst@dimb\pst@dimc
\advance\pst@dimc\pslinewidth
\advance\pst@dimc\psframesep
\advance\pst@dimb-\pst@dimc
\setbox\pst@hbox=\hbox to2\pst@dimc{%
\hss
\vbox{\kern-\pst@dimb\box\pst@hbox}%
\hss}%
\advance\pst@dimb-\dp\pst@hbox
\dp\pst@hbox=-\pst@dimb}
\let\pst@nodehook\relax
\def\psCirclebox{\pst@object{psCirclebox}}
\def\psCirclebox@i{\pst@makebox\psCirclebox@ii}
\def\psCirclebox@ii{%
\begingroup
\pst@useboxpar
\pst@dima=\ht\pst@hbox
\advance\pst@dima\dp\pst@hbox
\divide\pst@dima\tw@
\pssetlength\pst@dimb\psk@radius
\setbox\pst@hbox=\hbox{%
\pst@nodehook
\pscircle(.5\wd\pst@hbox,\pst@dima){\pst@dimb}%
\box\pst@hbox}%
\ifpsboxsep \psCirclebox@sep \fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\psCirclebox@sep{%
\pst@dimc=\pst@dimb
\advance\pst@dimb-\pst@dima
\advance\pst@dima\pst@dimc
\setbox\pst@hbox=\hbox to\tw@\pst@dimc{%
\hss
\vrule width \z@ depth \pst@dimb height \pst@dima
\box\pst@hbox
\hss}}%
\def\psovalbox{\pst@object{psovalbox}}
\def\psovalbox@i{\pst@makebox{\psovalbox@ii}}
\def\psovalbox@ii{%
\begingroup
\pst@useboxpar
\psovalbox@iii
\ifpsboxsep\psovalbox@sep\fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\psovalbox@iii{%
\psovalbox@iv
\setbox\pst@hbox=\hbox{%
\begin@ClosedObj
\addto@pscode{%
0 360
\pst@number\pst@dimc CLW 2 div sub
\pst@number\pst@dimd CLW 2 div sub
\pst@number\pst@dima
\pst@number\pst@dimb
\tx@Ellipse
closepath}%
\def\pst@linetype{2}%
\end@ClosedObj
\unhbox\pst@hbox}}
\def\psovalbox@iv{%
\pst@dimc=\pslinewidth\advance\pst@dimc\psframesep
\pst@dimd=\ht\pst@hbox\advance\pst@dimd\dp\pst@hbox
\pst@dima=.5\wd\pst@hbox
\pst@dimb=.5\pst@dimd\advance\pst@dimb-\dp\pst@hbox
\pst@dimd=.707\pst@dimd
\advance\pst@dimd\pst@dimc
\advance\pst@dimc.707\wd\pst@hbox}
\def\psovalbox@sep{%
\setbox\pst@hbox\hbox to 2\pst@dimc{\hss\unhbox\pst@hbox\hss}%
\pst@dimg=\pst@dimd
\advance\pst@dimg-\pst@dimb
\dp\pst@hbox=\pst@dimg
\advance\pst@dimd\pst@dimb
\ht\pst@hbox=\pst@dimd}
\def\psdiabox{\pst@object{psdiabox}}
\def\psdiabox@i{\pst@makebox{\psdiabox@ii}}
\def\psdiabox@ii{%
\begingroup
\pst@useboxpar
\psdiabox@iii
\ifpsboxsep\psdiabox@sep\fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\psdiabox@iv{%
\pst@dimg=.707\pslinewidth
\advance\pst@dimg.707\psframesep
\pst@dima=\wd\pst@hbox
\divide\pst@dima 2
\pst@dimc=\pst@dima
\advance\pst@dimc\pst@dimg
\pst@dimd=\ht\pst@hbox
\advance\pst@dimd\dp\pst@hbox
\divide\pst@dimd 2
\pst@dimb=\pst@dimd
\advance\pst@dimb-\dp\pst@hbox
\advance\pst@dimd\pst@dimg}
\def\psdiabox@iii{%
\psdiabox@iv
\setbox\pst@hbox=\hbox{%
\begin@ClosedObj
\addto@pscode{%
\psline@iii
pop
.5
\pst@number\pst@dimc 2 mul \pst@number\pst@dimd 2 mul
0
\pst@number\pst@dima \pst@number\pst@dimb
\tx@Diamond}%
\def\pst@linetype{4}%
\end@ClosedObj
\box\pst@hbox}}
\def\psdiabox@sep{%
\setbox\pst@hbox\hbox to 4\pst@dimc{\hss\unhbox\pst@hbox\hss}%
\multiply\pst@dimd 2
\advance\pst@dimd\pst@dimb
\ht\pst@hbox\pst@dimd
\advance\pst@dimd-2\pst@dimb
\dp\pst@hbox\pst@dimd}
\def\psset@trimode#1{\pst@expandafter\psset@@trimode{#1}\@empty\@empty\@nil}
\def\psset@@trimode#1#2#3\@nil{%
\let\pst@tempg#1\relax
\ifx\pst@tempg*%
\let\psk@@trimode\@empty
\let\pst@tempg#2\relax
\else
\let\psk@@trimode\relax
\fi
\edef\psk@trimode{%
\ifx R\pst@tempg 1 \else\ifx D\pst@tempg 2
\else\ifx L\pst@tempg 3 \else 0 \fi\fi\fi}}
\psset@trimode{U}
\def\pstribox{\pst@object{pstribox}}
\def\pstribox@i{\pst@makebox{\pstribox@ii}}
\def\pstribox@ii{%
\begingroup
\pst@useboxpar
\pstribox@iii
\ifpsboxsep\pstribox@sep\fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\pstribox@iii{%
\pstribox@iv
\setbox\pst@hbox=\hbox{%
\begin@ClosedObj
\addto@pscode{%
\psline@iii
pop
.5
\pst@number\pst@dimc \pst@number\pst@dimd
\ifodd\psk@trimode exch \fi
\psk@trimode -90 mul
\pst@number\pst@dima \pst@number\pst@dimb
\tx@Triangle}%
\def\pst@linetype{2}%
\end@ClosedObj
\box\pst@hbox}}
\def\pstribox@iv{%
\pst@dimh=\pslinewidth
\advance\pst@dimh\psframesep
\pst@dimg=\ht\pst@hbox
\advance\pst@dimg-\dp\pst@hbox
\divide\pst@dimg 2
\edef\pst@tempa{\number\pst@dimg sp}% For use by nodes.
\ifodd\psk@trimode
\pst@dimb\pst@dimg
\else
\pst@dima=\wd\pst@hbox
\divide\pst@dima 2
\fi
\ifcase\psk@trimode
\pst@dimb=-\dp\pst@hbox
\advance\pst@dimb-\pst@dimh
\or
\pst@dima=-\pst@dimh
\or
\pst@dimb=\ht\pst@hbox
\advance\pst@dimb\pst@dimh
\or
\pst@dima=\wd\pst@hbox
\advance\pst@dima\pst@dimh
\fi
\pst@dimd=\dp\pst@hbox
\advance\pst@dimd\ht\pst@hbox
\ifx\psk@@trimode\relax
\pst@dimc=\wd\pst@hbox
\advance\pst@dimc\ifodd\psk@trimode 1.447\else 1.789\fi\pst@dimh
\multiply\pst@dimc 2
\advance\pst@dimd\ifodd\psk@trimode 1.789\else 1.447\fi\pst@dimh
\multiply\pst@dimd 2
\else
\ifodd\psk@trimode
\advance\pst@dimd 1.1547\wd\pst@hbox
\advance\pst@dimd 3.4641\pst@dimh
\pst@dimc=.866\pst@dimd
\else
\advance\pst@dimd .866\wd\pst@hbox %.866=(sqrt(3)/2)
\advance\pst@dimd 3\pst@dimh
\pst@dimc=1.1547\pst@dimd % 1.1547=(2/sqrt(3))
\fi
\fi}
\def\pstribox@sep{%
\ifodd\psk@trimode
\advance\pst@dimb.5\pst@dimd
\ht\pst@hbox=\pst@dimb
\advance\pst@dimd-\pst@dimb
\dp\pst@hbox=\pst@dimd
\else
\setbox\pst@hbox\hbox to \pst@dimc{\hss\unhbox\pst@hbox\hss}%
\global\pst@dimg=.5\pst@dimc
\fi
\ifcase\psk@trimode
\dp\pst@hbox-\pst@dimb
\advance\pst@dimd\pst@dimb
\ht\pst@hbox\pst@dimd
\or
\pst@dimg=.5\wd\pst@hbox
\global\advance\pst@dimg-\pst@dima
\setbox\pst@hbox\hbox to \pst@dimc{\kern-\pst@dima\box\pst@hbox\hss}%
\or
\ht\pst@hbox\pst@dimb
\advance\pst@dimd-\pst@dimb
\dp\pst@hbox\pst@dimd
\or
\pst@dimg=\pst@dimc
\advance\pst@dimg-\pst@dima
\global\advance\pst@dimg.5\wd\pst@hbox
\setbox\pst@hbox\hbox to \pst@dimc{%
\hss\box\pst@hbox\kern\psframesep\kern\pslinewidth}%
\fi}
\def\psset@arcsepA#1{\pst@getlength{#1}\psk@arcsepA}
\def\psset@arcsepB#1{\pst@getlength{#1}\psk@arcsepB}
\def\psset@arcsep#1{%
\psset@arcsepA{#1}\let\psk@arcsepB\psk@arcsepA}
\psset@arcsep{0}
\def\tx@ArcArrow{ArcArrow }
\def\psarc{\pst@object{psarc}}
\def\psarc@i{%
\@ifnextchar({\psarc@iii}{\psarc@ii}}
\def\psarc@ii#1{\addto@par{arrows=#1}%
\@ifnextchar({\psarc@iii}{\psarc@iii(0,0)}}
\def\psarc@iii(#1)#2#3#4{%
\begin@OpenObj
\pst@getangle{#3}\pst@tempa
\pst@getangle{#4}\pst@tempb
\pst@@getcoor{#1}%
\pssetlength\pst@dima{#2}%
\addto@pscode{\psarc@iv \psarc@v}%
\gdef\psarc@type{0}%
\showpointsfalse
\end@OpenObj}
\def\psarc@iv{%
\pst@coor /y ED /x ED
/r \pst@number\pst@dima def
/c 57.2957 r \tx@Div def
/angleA
\pst@tempa
\psk@arcsepA c mul 2 div
\ifcase \psarc@type add \or sub \fi
def
/angleB
\pst@tempb
\psk@arcsepB c mul 2 div
\ifcase \psarc@type sub \or add \fi
def
\ifshowpoints\psarc@showpoints\fi
\ifx\psk@arrowA\@empty
\ifnum\psk@liftpen=2
r angleA \tx@PtoC
y add exch x add exch
moveto
\fi
\fi}
\def\psarc@v{%
x y r
angleA
\ifx\psk@arrowA\@empty\else
{ ArrowA CP }
{ \ifcase\psarc@type add \or sub \fi }
\tx@ArcArrow
\fi
angleB
\ifx\psk@arrowB\@empty\else
{ ArrowB }
{ \ifcase\psarc@type sub \or add \fi }
\tx@ArcArrow
\fi
\ifcase\psarc@type arc \or arcn \fi}
\def\psarc@type{0}
\def\psarc@showpoints{%
gsave
newpath
x y moveto
x y r \pst@tempa \pst@tempb
\ifcase\psarc@type arc \or arcn \fi
closepath
CLW 2 div SLW
[ \psk@dash\space ] 0 setdash stroke
grestore }
\def\psarcn{\pst@object{psarcn}}
\def\psarcn@i{\def\psarc@type{1}\psarc@i}
\def\pscircle{\pst@object{pscircle}}
\def\pscircle@i{\@ifnextchar({\pscircle@do}{\pscircle@do(0,0)}}
\def\pscircle@do(#1)#2{%
\if@star
{\use@par\qdisk(#1){#2}}%
\else
\begin@ClosedObj
\pst@@getcoor{#1}%
\pssetlength\pst@dimc{#2}%
\def\pst@linetype{4}%
\addto@pscode{%
\pst@coor
\pst@number\pst@dimc
\psk@dimen CLW mul sub
0 360 arc
closepath}%
\showpointsfalse
\end@ClosedObj
\fi
\ignorespaces}
\def\qdisk(#1)#2{%
\def\pst@par{}%
\begin@SpecialObj
\pst@@getcoor{#1}%
\pssetlength\pst@dimg{#2}%
\addto@pscode{\pst@coor \pst@number\pst@dimg \tx@SD}%
\end@SpecialObj}
\def\psset@radius#1{\pst@@getlength{#1}\psk@radius}
\psset@radius{.25cm}
\def\psCircle{\pst@object{psCircle}}
\def\psCircle@i{\@ifnextchar({\psCircle@ii}{\psCircle@ii(0,0)}}
\def\psCircle@ii(#1){\pscircle@do(#1){\psk@radius}}
\def\pswedge{\pst@object{pswedge}}
\def\pswedge@i{\@ifnextchar({\pswedge@ii}{\pswedge@ii(0,0)}}
\def\pswedge@ii(#1)#2#3#4{%
\begin@ClosedObj
\pssetlength\pst@dimc{#2}
\pst@getangle{#3}\pst@tempa
\pst@getangle{#4}\pst@tempb
\pst@@getcoor{#1}%
\def\pst@linetype{1}%
\addto@pscode{%
\pst@coor
2 copy
moveto
\pst@number\pst@dimc \psk@dimen CLW mul sub % Adjusted radius
\pst@tempa \pst@tempb
arc
closepath}%
\showpointsfalse
\end@ClosedObj}
\def\tx@Ellipse{Ellipse }
\def\psellipse{\pst@object{psellipse}}
\def\psellipse@i(#1){\@ifnextchar(%
{\psellipse@ii(#1)}{\psellipse@ii(0,0)(#1)}}
\def\psellipse@ii(#1)(#2){%
\begin@ClosedObj
\pst@getcoor{#1}\pst@tempa
\pst@@getcoor{#2}%
\addto@pscode{%
0 360
\pst@coor
\ifdim\psk@dimen\p@=\z@\else
\psk@dimen CLW mul
dup 4 -1 roll sub neg 3 1 roll sub
\fi
\pst@tempa
\tx@Ellipse
closepath}%
\def\pst@linetype{2}%
\end@ClosedObj}
\def\multips{\@ifnextchar({\def\pst@par{}\multips@ii}{\multips@i}}
\def\multips@i#1{\def\pst@par{rot=#1}\multips@ii}
\def\multips@ii(#1){\@ifnextchar({\multips@iii(#1)}{\multips@iii(\z@,\z@)(#1)}}
\long\def\multips@iii(#1)(#2)#3#4{%
\begingroup
\use@par
\pst@getcoor{#1}\pst@tempa
\pst@@getcoor{#2}%
\pst@cnta=#3\relax
\init@pscode
\addto@pscode{%
\pst@tempa T \the\pst@cnta\space \pslbrace
gsave \ifx\psk@rot\@empty\else\psk@rot rotate \fi}%
\hbox to\z@{%
\def\init@pscode{%
\addto@pscode{%
gsave
\pst@number\pslinewidth SLW
\pst@usecolor\pslinecolor}}%
\def\use@pscode{\addto@pscode{grestore}}%
\def\psclip##1{\pst@misplaced\psclip}%
\def\nc@object##1##2##3##4{\pst@misplaced{node connection}}%
#4}%
\addto@pscode{grestore \pst@coor T \psrbrace repeat}%
\leavevmode
\use@pscode
\endgroup
\ignorespaces}
\def\scalebox#1{\pst@makebox{\@scalebox{#1}}}
\def\@scalebox#1{%
\begingroup
\pst@getscale{#1}\pst@tempa
\let\pst@tempc\pst@tempg
\let\pst@tempd\pst@temph
\@@scalebox
\endgroup}
\def\@@scalebox{%
\leavevmode
\hbox{%
\ifdim\pst@tempd\p@<\z@
\pst@dimg=\pst@tempd\ht\pst@hbox
\pst@dimh=\pst@tempd\dp\pst@hbox
\dp\pst@hbox=-\pst@dimg
\ht\pst@hbox=-\pst@dimh
\else
\ht\pst@hbox=\pst@tempd\ht\pst@hbox
\dp\pst@hbox=\pst@tempd\dp\pst@hbox
\fi
\pst@dima=\pst@tempc\wd\pst@hbox
\ifdim\pst@dima<\z@\kern-\pst@dima\fi
\pst@Verb{CP CP translate \pst@tempa \tx@NET}%
\hbox to \z@{\box\pst@hbox\hss}%
\pst@Verb{%
CP CP translate
1 \pst@tempc div 1 \pst@tempd div scale
\tx@NET}%
\ifdim\pst@dima>\z@\kern\pst@dima\fi}}
\pslongbox{Scalebox}{\scalebox}
\def\scaleboxto(#1,#2){\pst@makebox{\@scaleboxto(#1,#2)}}
\def\@scaleboxto(#1,#2){%
\begingroup
\pssetlength\pst@dima{#1}%
\pssetlength\pst@dimb{#2}%
\ifdim\pst@dima=\z@\else
\pst@divide{\pst@dima}{\wd\pst@hbox}\pst@tempc
\edef\pst@tempc{\pst@tempc\space}%
\fi
\ifdim\pst@dimb=\z@
\ifdim\pst@dima=\z@
\@pstrickserr{%
\string\scaleboxto\space dimensions cannot both be zero}\@ehpa
\def\pst@tempa{}%
\def\pst@tempc{1 }%
\def\pst@tempd{1 }%
\else
\let\pst@tempd\pst@tempc
\fi
\else
\pst@dimc=\ht\pst@hbox
\advance\pst@dimc\dp\pst@hbox
\pst@divide{\pst@dimb}{\pst@dimc}\pst@tempd
\edef\pst@tempd{\pst@tempd\space}%
\ifdim\pst@dima=\z@ \let\pst@tempc\pst@tempd \fi
\fi
\edef\pst@tempa{\pst@tempc \pst@tempd scale }%
\@@scalebox
\endgroup}
\pslongbox{Scaleboxto}{\scaleboxto}
\def\tx@Rot{Rot }
\def\rotateleft{\pst@makebox{\@rotateleft\pst@hbox}}
\def\@rotateleft#1{%
\leavevmode\hbox{\hskip\ht#1\hskip\dp#1\vbox{\vskip\wd#1%
\pst@Verb{90 \tx@Rot}
\vbox to \z@{\vss\hbox to \z@{\box#1\hss}\vskip\z@}%
\pst@Verb{-90 \tx@Rot}}}}
\def\rotateright{\pst@makebox{\@rotateright\pst@hbox}}
\def\@rotateright#1{%
\hbox{\hskip\ht#1\hskip\dp#1\vbox{\vskip\wd#1%
\pst@Verb{-90 \tx@Rot}
\vbox to \z@{\hbox to \z@{\hss\box#1}\vss}%
\pst@Verb{90 \tx@Rot}}}}
\def\rotatedown{\pst@makebox{\@rotatedown\pst@hbox}}
\def\@rotatedown#1{%
\hbox{\hskip\wd#1\vbox{\vskip\ht#1\vskip\dp#1%
\pst@Verb{180 \tx@Rot}%
\vbox to \z@{\hbox to \z@{\box#1\hss}\vss}%
\pst@Verb{-180 \tx@Rot}}}}
\pslongbox{Rotateleft}{\rotateleft}
\pslongbox{Rotateright}{\rotateright}
\pslongbox{Rotatedown}{\rotatedown}
\def\pst@starbox{%
\setbox\pst@hbox\hbox{\psframebox*[boxsep=false]{\unhbox\pst@hbox}}}
\def\pst@@makesmall#1{%
\setbox#1=\hbox to\z@{\hss\vbox to \z@{\vss\box#1\vss}\hss}}
\def\pst@@@makesmall#1{%
\pst@dimh=\psk@xref\wd#1%
\ifx\psk@yref\relax
\pst@dimg=\dp#1%
\else
\pst@dimg=\psk@yref\ht#1%
\advance\pst@dimg\psk@yref\dp#1%
\fi
\setbox#1=\hbox to\z@{%
\kern-\pst@dimh\vbox to\z@{\vss\box#1\kern-\pst@dimg}\hss}}
\def\psset@ref#1{\pst@expandafter\psset@@ref{#1}\@empty,,\@nil}
\def\psset@@ref#1#2,#3,#4\@nil{%
\def\psk@xref{.5}%
\def\psk@yref{.5}%
\let\pst@makesmall\pst@@@makesmall
\ifx\@empty#3\@empty
\@nameuse{getref@#1}%
\@nameuse{getref@#2}%
\else
\pst@checknum{#1#2}\psk@xref
\pst@checknum{#3}\psk@yref
\fi}
\def\getref@c{\let\pst@makesmall\pst@@makesmall}
\def\getref@t{\def\psk@yref{1}}
\def\getref@b{\def\psk@yref{0}}
\def\getref@B{\let\psk@yref\relax}
\def\getref@l{\def\psk@xref{0}}
\def\getref@r{\def\psk@xref{1}}
\psset@ref{c}
\def\psset@rot#1{%
\pst@expandafter{\@ifnextchar*{\psset@@@rot}{\psset@@rot}}{#1}\@nil}
\def\psset@@rot#1\@nil{%
\def\next##1@#1=##2@##3\@nil{%
\ifx\relax##2%
\pst@getangle{#1}\psk@rot
\else
\def\psk@rot{##2}%
\fi}%
\expandafter\next\pst@rottable @#1=\relax @\@nil}
\def\psset@@@rot#1#2\@nil{%
\psset@@rot#2\@nil
\edef\psk@rot{\pst@rotlist \ifx\psk@rot\@empty\else\psk@rot add \fi}}
\def\pst@rotlist{mark RAngle /a ED cleartomark a neg }
\def\pst@rottable{%
@0=%
@U=%
@L=90 %
@D=180 %
@R=-90 %
@N=\pst@rotlist
@W=\pst@rotlist 90 add %
@S=\pst@rotlist 180 add %
@E=\pst@rotlist 90 sub }
\psset@rot{0}
\def\tx@RotBegin{RotBegin }
\def\tx@RotEnd{RotEnd }
\def\pst@rotate#1#2{%
\ifx#1\@empty\else
\setbox#2=\hbox{\pst@Verb{#1 \tx@RotBegin}\box#2\pst@Verb{\tx@RotEnd}}%
\fi}
\def\psput@cartesian#1{%
\hbox to \z@{\kern\pst@dimg{\vbox to \z@{\vss\box#1\vskip\pst@dimh}\hss}}}
\def\psput@special#1{%
\hbox{%
\pst@Verb{{ \pst@coor } \tx@PutCoor \tx@PutBegin}%
\box#1%
\pst@Verb{\tx@PutEnd}}}
\def\tx@PutCoor{PutCoor }
\def\tx@PutBegin{PutBegin }
\def\tx@PutEnd{PutEnd }
\def\rput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\rput@i}{\rput@ii}}}
\def\rput@i[#1]{\addto@par{ref={#1}}\rput@ii}
\def\rput@ii{\@ifnextchar({\rput@iv}{\rput@iii}}
\def\rput@iii#1{\addto@par{rot={#1}}\@ifnextchar({\rput@iv}{\rput@iv(\z@,\z@)}}
\def\rput@iv(#1){\pst@killglue\pst@makebox{\rput@v{#1}}}
\def\rput@v#1{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\pst@makesmall\pst@hbox
\pst@rotate\psk@rot\pst@hbox
\psput@{#1}\pst@hbox
\endgroup
\ignorespaces}
\def\multirput{%
\def\pst@par{}%
\pst@ifstar{\@ifnextchar[{\multirput@i}{\multirput@ii}}}
\def\multirput@i[#1]{\addto@par{ref={#1}}\multirput@ii}
\def\multirput@ii{\@ifnextchar({\multirput@iv}{\multirput@iii}}
\def\multirput@iii#1{\addto@par{rot={#1}}\multirput@iv}
\def\multirput@iv(#1){%
\@ifnextchar({\multirput@v(#1)}{\multirput@v(\z@,\z@)(#1)}}
\def\multirput@v(#1,#2)(#3,#4)#5{%
\pst@makebox{\multirput@vi(#1,#2)(#3,#4){#5}}}
\def\multirput@vi(#1,#2)(#3,#4)#5{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\pst@makesmall\pst@hbox
\pst@rotate\psk@rot\pst@hbox
\pssetxlength\pst@dima{#1}%
\pssetylength\pst@dimb{#2}%
\pssetxlength\pst@dimc{#3}%
\pssetylength\pst@dimd{#4}%
\pst@cntg=#5\relax
\pst@cnth=\@ne
\leavevmode
\loop
\vbox to \z@{%
\vss
\hbox to \z@{\kern\pst@dima\copy\pst@hbox\hss}%
\vskip\pst@dimb}%
\ifnum\pst@cntg>\pst@cnth
\advance\pst@dima\pst@dimc
\advance\pst@dimb\pst@dimd
\advance\pst@cnth\@ne
\repeat
\endgroup
\ignorespaces}
\newif\if@fixedradius
\def\cput{\pst@object{cput}}
\def\cput@i{\@fixedradiusfalse\cput@ii}
\def\cput@ii{\pst@killglue\@ifnextchar({\cput@iv}{\cput@iii}}
\def\cput@iii#1{%
\addto@par{rot={#1}}%
\@ifnextchar({\cput@iv}{\cput@iv(\z@,\z@)}}
\def\cput@iv(#1){\pst@makebox{\cput@v{#1}}}
\def\cput@v#1{%
\begingroup
\use@par
\setbox\pst@hbox=\hbox{%
\psboxsepfalse
\if@fixedradius\psCirclebox@ii\else\pscirclebox@ii\fi}%
\pst@@makesmall\pst@hbox
\pst@rotate\psk@rot\pst@hbox
\psput@{#1}\pst@hbox
\endgroup
\ignorespaces}
\def\Cput{\pst@object{Cput}}
\def\Cput@i{\@fixedradiustrue\cput@ii}
\newdimen\pslabelsep
\def\psset@labelsep#1{\pssetlength\pslabelsep{#1}}
\psset@labelsep{5pt}
\def\psset@refangle#1{\pst@expandafter\psset@@refangle{#1}\@nil}
\def\psset@@refangle#1\@nil{%
\def\next##1@#1=##2"##3@##4\@nil{%
\ifx\relax##2%
\pst@getangle{#1}\psk@refangle
\def\psk@uputref{}%
\else
\def\psk@refangle{##2 }%
\def\psk@uputref{##3}%
\fi}%
\expandafter\next\pst@refangletable @#1=\relax"@\@nil}
\def\pst@refangletable{%
@r=0"20%
@u=90"02%
@l=180"10%
@d=-90"01%
@ur=45"22%
@ul=135"12%
@dr=-135"21%
@dl=-45"11}
\psset@refangle{0}
% DG/SR modification begin - Mar. 24, 1999 - Patch 10
%\def\uput{\def\pst@par{}\@ifnextchar[{\uput@ii}{\uput@i}}
\def\uput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\uput@ii}{\uput@i}}} 
% DG/SR modification end
\def\uput@i#1{\addto@par{labelsep=#1}\uput@ii}
\def\uput@ii[#1]{%
\addto@par{refangle={#1}}%
\@ifnextchar({\uput@iv}{\uput@iii}}
\def\uput@iii#1{%
\addto@par{rot={#1}}%
\@ifnextchar({\uput@iv}{\uput@iv(\z@,\z@)}}
\def\uput@iv(#1){\pst@killglue\pst@makebox{\uput@v{#1}}}
\def\uput@v#1{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\uput@vi
\psput@{#1}\pst@hbox
\endgroup
\ignorespaces}
\def\uput@vi{%
\ifx\psk@uputref\@empty
\uput@vii\tx@UUput{}%
\else
\ifx\psk@rot\@empty
\expandafter\uput@viii\psk@uputref
\else
\uput@vii\tx@UUput{}%
\fi
\fi}
\def\uput@vii#1#2{%
\edef\pst@coor{%
\pst@number\pslabelsep
#2%
\pst@number{\wd\pst@hbox}%
\pst@number{\ht\pst@hbox}%
\pst@number{\dp\pst@hbox}%
\psk@refangle\space \ifx\psk@rot\@empty\else\psk@rot\space sub \fi
\tx@Uput #1}%
\setbox\pst@hbox=\hbox to\z@{\hss\vbox to\z@{\vss\box\pst@hbox\vss}\hss}%
\setbox\pst@hbox=\psput@special\pst@hbox
\ifx\psk@rot\@empty\else\pst@rotate\psk@rot\pst@hbox\fi}
\def\uput@viii#1#2{%
\ifnum#1>\z@\ifnum#2>\z@\pslabelsep=.707\pslabelsep\fi\fi
\setbox\pst@hbox=\vbox to\z@{%
\ifnum#2=1 \vskip\pslabelsep\else\vss\fi
\hbox to\z@{%
\ifnum#1=2 \hskip\pslabelsep\else\hss\fi
\box\pst@hbox
\ifnum#1=1 \hskip\pslabelsep\else\hss\fi}%
\ifnum#2=2 \vskip\pslabelsep\else\vss\fi}}
\def\tx@Uput{Uput }
\def\tx@UUput{UUput }
\def\Rput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Rput@ii}{\Rput@i}}}
\def\Rput@i#1{\addto@par{labelsep=#1}\Rput@ii}
\def\Rput@ii[#1]{\addto@par{ref={#1}}\@ifnextchar({\Rput@iv}{\Rput@iii}}
\def\Rput@iii#1{\addto@par{rot={#1}}\@ifnextchar({\Rput@iv}{\Rput@iv(\z@,\z@)}}
\def\Rput@iv(#1){\pst@killglue\pst@makebox{\Rput@v{#1}}}
\def\Rput@v#1{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\Rput@vi
\pst@makesmall\pst@hbox
\pst@rotate\psk@rot\pst@hbox
\psput@{#1}\pst@hbox
\endgroup
\ignorespaces}
\def\Rput@vi{%
\pst@dimg=\dp\pst@hbox
\advance\pst@dimg\pslabelsep
\dp\pst@hbox=\pst@dimg
\pst@dimg=\ht\pst@hbox
\advance\pst@dimg\pslabelsep
\ht\pst@hbox=\pst@dimg
\setbox\pst@hbox\hbox{\kern\pslabelsep\box\pst@hbox\kern\pslabelsep}}%
\def\oldpsput{%
\def\pst@par{}\pst@ifstar{\@ifnextchar[{\oldpsput@i}{\oldpsput@ii}}}
\def\oldpsput@i[#1]{\addto@par{ref={#1}}\oldpsput@ii}
\def\oldpsput@ii{\@ifnextchar<{\oldpsput@iii}{\oldpsput@iv}}
\def\oldpsput@iii<#1>{\rput@iii{#1}}
\def\OldPsput{\let\psput\oldpsput}
\def\NewPsput{\let\psput\rput}
\def\pspicture{\begingroup\pst@ifstar\pst@picture}
\def\pst@picture{%
\@ifnextchar[{\pst@@picture}{\pst@@picture[0]}}
\def\pst@@picture[#1]#2(#3,#4){%
\@ifnextchar({\pst@@@picture[#1](#3,#4)}%
{\pst@@@picture[#1](0,0)(#3,#4)}}
\def\pst@@@picture[#1](#2,#3)(#4,#5){%
\pssetxlength\pst@dima{#2}%
\pssetylength\pst@dimb{#3}%
\pssetxlength\pst@dimc{#4}%
\pssetylength\pst@dimd{#5}%
\ifdim\pst@dima>\pst@dimc
\pst@dimg=\pst@dima
\pst@dima=\pst@dimc
\pst@dimc=\pst@dimg
\fi
\ifdim\pst@dimb>\pst@dimd
\pst@dimg=\pst@dimb
\pst@dimb=\pst@dimd
\pst@dimd=\pst@dimg
\fi
\def\pst@tempa{#1}%
\setbox\pst@hbox=\hbox\bgroup
\begingroup\KillGlue
\@ifundefined{@latexerr}{}{\let\unitlength\psunit}%
\edef\pic@coor{(#2,#3)(#2,#3)(#4,#5)}\ignorespaces}
\def\pic@coor{(0,0)(0,0)(10,10)}
\def\endpspicture{%
\pst@killglue
\endgroup
\egroup
\ifdim\wd\pst@hbox=\z@\else
%\@pstrickserr{Extraneous space in the pspicture environment}%
%{Type \space <return> \space to procede.}%
\fi
\ht\pst@hbox=\pst@dimd
\dp\pst@hbox=-\pst@dimb
\setbox\pst@hbox=\hbox{%
\kern-\pst@dima
\ifx\pst@tempa\@empty\else
\advance\pst@dimd-\pst@dimb
\pst@dimd=\pst@tempa\pst@dimd
\advance\pst@dimd\pst@dimb
\lower\pst@dimd
\fi
\box\pst@hbox
\kern\pst@dimc}%
\if@star\setbox\pst@hbox=\hbox{\clipbox@@\z@}\fi
\leavevmode\box\pst@hbox
\endgroup}
\@namedef{pspicture*}{\pspicture*}
\@namedef{endpspicture*}{\endpspicture}
\def\tx@BeginOL{BeginOL }
\def\tx@InitOL{InitOL }
\def\pst@initoverlay#1{\pst@Verb{\tx@InitOL /TheOL (#1) def}}
\def\AltOverlayMode{%
\def\pst@initoverlay##1{%
\pst@Verb{%
\tx@InitOL
/Visible { initclip } def
/Invisible {
CP newpath OLUnit itransform moveto clip newpath moveto
} def
/TheOL (##1) def}}}
\def\pst@overlay#1{%
\edef\curr@overlay{#1}%
\pst@Verb{(#1) BOL}%
\aftergroup\pst@endoverlay}
\def\pst@endoverlay{%
\pst@Verb{(\curr@overlay) BOL}}
\def\curr@overlay{all}
\newbox\theoverlaybox
\def\overlaybox{%
\global\setbox\theoverlaybox=\hbox\bgroup
\begingroup
\let\psoverlay\pst@overlay
\def\overlaybox{%
\@pstrickserr{Overlays cannot be nested}\@eha}%
\def\putoverlaybox{%
\@pstrickserr{You must end the overlay box
before using \string\putoverlaybox}}%
\psoverlay{main}%
\ignorespaces}
\def\endoverlaybox{\endgroup\egroup}
\def\putoverlaybox#1{%
\hbox{\pst@initoverlay{#1}\copy\theoverlaybox}}
\def\psoverlay{\@pstrickserr{\string\psoverlay\space
can only be used after \string\overlaybox}}
\ifx\pstcustomize\relax \input pstricks.con \fi
\catcode`\@=\PstAtCode\relax
%
\endinput
%%
%% END: pstricks.tex

center.bbl
\begin{thebibliography}{1}

\bibitem{Braden}
T.~Braden.
\newblock Perverse sheaves on {G}rassmannians.
\newblock To appear in {C}anadian {J}ournal of {Math.}, arXiv:math.AG/9907152.

\bibitem{me:withTom}
T.~Braden and M.~Khovanov.
\newblock In preparation.

\bibitem{ChrissGinzburg}
N.~Chriss and V.~Ginzburg.
\newblock {\em Representation theory and complex geometry}.
\newblock Birkh\"auser Boston, Boston, MA, 1997.

\bibitem{ConciniProcesi}
C.~de~Concini and C.~Procesi.
\newblock Symmetric functions, conjugacy classes and the flag variety.
\newblock {\em Invent. math.}, 64:203--219, 1981.

\bibitem{Fung}
F.~Y.~C. Fung.
\newblock {\em On the relation between {S}pringer fibers of the general linear
  group and {K}azhdan-{L}usztig theory}.
\newblock PhD thesis, Princeton University, June 1997.

\bibitem{me:tangles}
M.~Khovanov.
\newblock A functor-valued invariant of tangles.
\newblock arXiv math.QA/0103190.

\bibitem{me:Paul2}
M.~Khovanov and P.~Seidel. 
\newblock Work in progress. 

\bibitem{Rickard3}
J.~Rickard.
\newblock Morita theory for derived categories.
\newblock {\em J. London Math. Soc.}, 39:436--456, 1989.

\end{thebibliography}
