%Paper: hep-th/9401002
%From: Claudio Lucchesi <LUCCHESI@PIERRE.MIT.EDU>
%Date: Mon, 3 Jan 1994 10:46:27 -0500 (EST)
%Date (revised): Mon, 7 Mar 1994 17:51:44 -0500 (EST)


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% A Postscript file, appended after the "\end{document}",    %%%%%
%%%% contains one figure. This PS file should be cut out and    %%%%%
%%%% printed separately. Latexing this document without cutting %%%%%
%%%% out the figure will simply ignore it.                      %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentstyle[preprint,eqsecnum,aps]{revtex}

\def\footnoterule{\kern-3pt \hrule width \hsize \kern6.2pt}
\def\tr{~{\rm Tr}~}

\def\pmb#1{\setbox0=\hbox{$#1$}%
\kern-.025em\copy0\kern-\wd0%
\kern.05em\copy0\kern-\wd0%
\kern-.025em\raise.0433em\box0}%

\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\beq{\begin{eqnarray}}
\def\eeq{\end{eqnarray}}
\def\m{\mu}
\def\n{\nu}
\def\r{\rho}
\def\s{\sigma}
\def\z{\zeta}
\def\p{\psi}
\def\G{\Gamma}
\def\pa{\partial}
\def\d{\delta}
\def\g{\gamma}
\def\gg{g}
\def\t{\tau}
\def\l{\lambda}
\def\Gc{\Gamma_{\rm c}}
\def\mn{{\m\n}}
\def\mnr{{\m\n\r}}
\def\ab{{ab}}
\def\abc{{abc}}
\def\intd{\int d^{4}\!x\,}
\def\dqx{d^{4}\!x\,}
\def\dqs{d^{4}\!s\,}
\def\dqk{d^{4}\!k\,}
\def\ln{{\rm ln}\,}
\def\D{{\cal D}}
\def\hq{{\hat q}}
\def\bk{{\bf k}}
\def\disp{\displaystyle}
\def\al{\alpha}
\def\bet{\beta}

\begin{document}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\thispagestyle{empty}
\setcounter{page}{0}

\title{HARD THERMAL LOOPS, STATIC RESPONSE AND\\THE COMPOSITE EFFECTIVE
ACTION\footnotemark[1]}

\footnotetext[1]{This work is supported in part by funds
provided by the U.S.~Department of Energy (D.O.E.) under contract
\#DE-AC02-76ER03069.}

\author{R.~Jackiw,~Q.~Liu,~and~C.~Lucchesi\footnotemark[2]}
\footnotetext[2]{Supported by the Swiss National Science
Foundation.}


\address{Center for Theoretical Physics,
         Laboratory for Nuclear Science,
         and Department of Physics\\
         Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139}

\maketitle

\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\setcounter{page}{0}
\thispagestyle{empty}

\begin{abstract}
$\!\!$First, we investigate the static non-Abelian Kubo equation.
We prove that it does not possess finite energy solutions; thereby we
establish that gauge theories do not support hard thermal solitons.
This general result is verified by a numerical solution of the equations.
A similar argument shows that ``static" instantons are absent. In addition,
we note that the static equations reproduce the expected screening of the
non-Abelian electric field by a gauge invariant Debye mass
$m=\gg T\, \sqrt{{{N+N_F/2}\over 3}}$. Second, we derive the non-Abelian Kubo
equation from the composite effective action. This is achieved by showing
that the requirement of stationarity of the composite effective action
is equivalent, within a kinematical approximation scheme, to the
condition of gauge invariance for the generating functional of hard thermal
loops.
\end{abstract}

\vfill
\centerline{Submitted to: {\em Physical Review D}}
\vfill
\hbox to \hsize{CTP\#2261 \hfil November 1993}
\vskip-12pt
\eject

\section{Introduction}
\label{sec:1}

When it was realized \cite{ref1} that the gauge invariance condition
\cite{ref2} on the generating functional $\Gamma(A)$ for hard thermal loops in
a gauge theory \cite{ref3} (with or without fermions) coincides with a similar
requirement on the wave functional of Chern-Simons theory, one could use the
known solution for the latter, non-thermal problem \cite{ref4} to give a
construction of $\Gamma(A)$ relevant in the former, thermal context.
The expression for $\Gamma(A)$ is non-local and not very explicit: $\Gamma(A)$
can be presented either as a power series in the gauge field $A$ \cite{ref1}
[the $O(A^n)$ contribution determines the hard thermal gauge field
(and fermion) loop with
$n$ external gauge field lines] or as an explicit
functional of path ordered variables
$P \exp \int d x^\mu \, A_\mu$~ \cite{ref4}.

More accessible is the expression for the induced current
$- {\delta \Gamma(A) \over \delta A_\mu} \equiv
- T^a {\delta \Gamma(A) \over \delta A_\mu^a}$,
which enters (high-temperature) response theory, in a non-Abelian
generalization of Kubo's formula (in Minkowski space-time)
\cite{ref5}:
\be
D_\n \, F^{\n\m}(x)
= - {\delta \Gamma(A) \over \delta A_\m(x)}
\equiv {m^2\over 2} \, j^\m (x)\ .
\label{1.1}
\ee
$T^a$ is an anti-hermitian representation of the Lie algebra,
the gauge covariant derivative is defined as $D_\n =\pa_\n + \gg [A_\n,\ ]$,
and $m$ is the Debye mass determined by the matter
content: in an
$SU(N)$ gauge theory at temperature $T$, with fermions in the
representation
${\cal T}^a$, and $\tr ({\cal T}^a {\cal T}^b) = -{N_F \over 2}
\delta^{ab}$ where $N_F$ counts the number of flavors, the Debye mass
satisfies
\be
m^2 = {\gg^2 T^2\over 3} \, \left( N + {N_F \over 2} \right)\ .
\label{1.2}
\ee
Henceforth, through Section \ref{sec:2}, we scale the gauge coupling constant
to unity.
The functional form of $j^\mu$ can be given as \cite{ref5}
\be
j^\mu(x) = \int {d \hat{q} \over 4\pi}
\, \Biggl\{ Q^\mu_{+} \biggl( a_{-}(x) - A_{-}(x) \biggr) +
Q^\mu_{-} \biggl( a_{+}(x) - A_{+}(x) \biggr) \Biggr\}\ .
\label{1.3}
\ee
Here $Q^\mu_{\pm}$ are the light-like 4-vectors
${1\over\sqrt{2}} (1, \pm \hat{q})$, with $\hat{q}^2=1$,
$A_{\pm}$ are the light-like projections $A_\pm = Q^\mu_{\pm} \, A_\mu$,
while $a_\pm$ are given by \cite{ref4,ref5}
\be
a_+ = g^{-1} \, \partial_{+} \, g ~,~~~~~
a_- = h^{-1} \, \partial_{-} \, h ~~~~~~~
(\partial_\pm \equiv Q^\mu_\pm \, \partial_\mu)
\label{1.5}
\ee
when $A_\pm$ are parametrized as
\be
A_+ = h^{-1} \, \partial_+ \, h ~,~~~~~~
A_- = g^{-1} \, \partial_- \, g ~~.
\label{1.6}
\ee
In other words, $a_\pm$ satisfy the equations
\begin{mathletters}
\label{1.7}
\beq
\partial_+ a_- - \partial_- A_+ + [A_+, a_-] &=& 0\ , \\
\partial_+ A_- - \partial_- a_+ + [a_+, A_-] &=& 0\ ,
\eeq
\end{mathletters}
$\!\!$whose solution can be presented as in (\ref{1.5}) when $A_\pm$ are
parameterized as in (\ref{1.6}) ---
evidently $g$ and $h$ involve path ordered exponential integrals of
$A_\pm$.
(Alternatively $a_\pm$ may be given by a power series in $A_\mp$
\cite{ref1}.)
Finally (\ref{1.3}) requires averaging over the directions of
$\hat{q}$.

It is easy to verify that (\ref{1.7}) ensure covariant conservation of
$j^\mu$.
Moreover, gauge invariance is maintained: for (\ref{1.1}) to be gauge
covariant, it is necessary that $j^\mu$ transform gauge covariantly.
That the
expression in (\ref{1.3}) possesses this property is seen as follows.
When
$A_\pm$ transform by $U^{-1} \, A_\pm \, U + U^{-1} \partial_\pm
U$,
Eqs.~(\ref{1.5}) -- (\ref{1.7}) show that $a_\pm$ transform similarly,
hence the
differences $a_\pm - A_\pm$ transform covariantly. The manifest
gauge
covariance of (\ref{1.1}) ensures that $m$ is a gauge invariant
parameter;
that it also has the interpretation of an electric (Debye) mass will be
evident when we consider the static limit.

It is of obvious interest to discuss solutions of (\ref{1.1}).  In the
Abelian, electrodynamical case this is easy to do, since (\ref{1.7})
can be
readily solved for $a_\pm$, and the solutions of the linear problem
are the
well-known plasma waves \cite{ref6}.  The non-linear
problem of
finding non-Abelian plasma waves is much more formidable.  Also,
one inquires
whether the non-linear equations support soliton solutions, and
(after an
appropriate continuation to imaginary time) instanton solutions.
[The
time-dependent equation (\ref{1.1}) in Minkowski space-time must
be
supplemented with boundary conditions, which are determined by the
physical
context.  For example, response theory requires retarded boundary
conditions,
which in fact preclude deriving (\ref{1.1}) variationally \cite{ref5}.
Here we shall not be concerned with this issue.]

Our paper concerns the following two topics.  In Section~\ref{sec:2}, we
analyze
(\ref{1.1}) for static fields.  It turns out that in the time-independent
case
(\ref{1.7}) can be solved for $a_{\pm}$ and (\ref{1.1}) is presented
in closed form. We prove that the resulting equation does not possess
finite-energy solutions, thereby
establishing that gauge theories do not support hard thermal solitons.
Also some negative conclusions
about instantons are given. In Section~\ref{sec:3} we present an
alternative
derivation of (\ref{1.1}), which relies on the composite effective
action
\cite{ref7}, and makes use of approximations recently developed in an
analysis of hard thermal loops based on the Schwinger-Dyson
equations \cite{ref8}. The Appendix presents a numerical analysis of the
solutions to equation (\ref{1.1}) for $SU(2)$, which supports the
general result in Section~\ref{sec:2}.

\section{Static Response}
\label{sec:2}

When $A_\pm$ are time-independent, we seek solutions of
(\ref{1.7}) that are
also time-independent.  Acting on static fields, the derivatives
$\partial_\pm$
become $\pm {1\over\sqrt{2}} \hat{q} \cdot {\bf \nabla} \equiv
\pm\partial_\tau$,
and (\ref{1.7}) may be written as the equations
\be
\partial_\tau \, {\cal A}_\pm \pm [A_\pm, {\cal A}_\pm] = 0
\label{2.2}
\ee
for the unknowns ${\cal A}_\pm \equiv A_\pm + a_\mp$.
These are solved trivially by ${\cal A}_\pm = 0$, that is
\be
a_\mp = -A_\pm\ .
\label{2.3}
\ee
This solution is also the one that is deduced from the perturbative
series
expression for $a_\pm$, when restricted to static $A_\pm$.

[A non trivial solution can be constructed with the help of
representations
similar to (\ref{1.6}).  Upon defining in the static case
\beq
A_+ &=& h_0^{-1} \, \partial_\tau \, h_0 \ ,\nonumber\\
A_- &=& -g_0^{-1} \, \partial_\tau \, g_0 \nonumber
\eeq
($h_0$ and $g_0$ involve path-ordered exponentials along the path
${\bf r} + \hat{q}\tau$), we find
\beq
{\cal A}_+ = h_0^{-1} \, I_+ \, h_0 \ ,\nonumber\\
{\cal A}_- = g_0^{-1} \, I_- \, g_0 \ ,\nonumber
\eeq
where $I_{\pm}$ are arbitrary Lie algebra elements, independent of
$\tau$: $\hat{q} \cdot {\bf \nabla} I_\pm = 0$.
Since these solutions involve the arbitrary quantities $I_\pm$, and
since they
do not arise in the perturbative series, we do not consider them
further
and remain with the trivial solution (\ref{2.3}), which corresponds to
$I_\pm =0$.]

{}From (\ref{2.3}) it follows that the current for static fields is
\beq
j^\mu({\bf r})
&=& - \int {d\hat{q} \over 4\pi} \,
\biggl( Q^\mu_{+} + Q^\mu_{-} \biggr) \,
\biggl( A_+({\bf r}) + A_-({\bf r}) \biggr) \nonumber\\
&=& - \int {d\hat{q} \over 4\pi} \,
\biggl( Q^\mu_{+} + Q^\mu_{-} \biggr)
\biggl( Q^\nu_{+} + Q^\nu_{-} \biggr)
A_\nu({\bf r})\ .
\label{2.4}
\eeq
With ${\bf Q}_{+} + {\bf Q}_- = 0$ and $Q_{+}^0 + Q_{-}^0 = \sqrt{2}$,
we compute $j^\mu = - 2 \, \delta^{\mu 0}A_0 $.
The response equations (\ref{1.1}) then become, in the static limit:
\begin{mathletters}
\label{2.6}
\beq
{D}_i E^i + m^2 A_0 &=& 0 \ ,\label{2.6a} \\
\epsilon^{ijk} {D}_j B^k &=& [A_0, E^i]\ , \label{2.6b}
\eeq
\end{mathletters}
\noindent
$\!\!$where $E^i \equiv F^{i 0}$ and $F^{ij} \equiv -\epsilon^{ijk} B^k$.
Eqs. (\ref{2.6}) give clear indication that $m$ plays the role of a gauge
invariant,
electric mass.  The fact that the static current is linear in the vector
potential implies the vanishing of hard thermal loops with more that
two
external gauge-field lines, and zero energy --- a fact which can be
checked
from the relevant graphs.

Unfortunately, Eqs.~(\ref{2.6}) do not possess any finite energy
solutions.
This is established by a variant of the argument relevant to the $m^2
= 0$
case \cite{ref9}.

Consider the symmetric tensor
\be
\theta^{ij} = 2 \tr \left( E^i E^j + B^i B^j - {\delta^{ij} \over 2}
(E^2 + B^2 + m^2 A_0^2) \right)\ .
\label{2.7}
\ee
Using (\ref{2.6}) one verifies that for static fields
$\partial_j \, \theta^{ji} = 0$.  Therefore
\be
\int d^3 r \, \theta^{i i} =  \int d^3 r \, \partial_j (x^i \, \theta^{ji})
=  \int dS^j x^i \theta^{ji}\ .
\label{2.8}
\ee
Moreover, the energy of a massive gauge field (with no mass for the
spatial components) can be written as
\begin{mathletters}
\label{2.9}
\be
{\cal E} = \int d^3 r \left\{ - \tr
\left( E^2 + B^2  + {1 \over m^2} (D_i E^i)^2 \right)
+ \tr \left( m A_0 + {D_i E^i \over m} \right)^2 \right\}\ .
\label{2.9a}
\ee
The second trace in the integrand enforces the constraint (\ref{2.6a}).
Consequently, on the constrained surface the energy is a sum of
positive terms \cite{ref10}:
\be
{\cal E} = \int d^3 r
           \left\{ - \tr \left( E^2 + B^2 + m^2 A_0^2 \right) \right\}
\label{2.9b}
\ee
\end{mathletters}
$\!\!$and ${\bf E}$, ${\bf B}$ and $A_0$ must decrease at large distances
sufficiently rapidly so that each of them is square integrable. This in turn
ensures that the surface integral at infinity in (\ref{2.8}) vanishes, so that
static solutions require
\begin{mathletters}
\label{2.10}
\be
\int d^3 r \theta^{i i} = 0\ .
\label{2.10a}
\ee
On the other hand, from (\ref{2.7}), we see that $\theta^{ii}$ is a sum
of positive terms
\be
\theta^{ii} = -\tr ( E^2 + B^2 + 3 m^2 A_0^2 ) \ ,
\label{2.10b}
\ee
\end{mathletters}
$\!\!$hence (\ref{2.10}) imply the vanishing of
$\bf E$, $\bf B$ and $A_0$.

The absence of finite energy static solutions can also be understood
from the
differential equations (\ref{2.6}).  Eq.~(\ref{2.6a}) possesses solutions
for $A_0$ that are either exponentially increasing or decreasing at infinity.
Rejecting
the former removes the freedom of imposing further conditions at
the origin,
and necessarily the exponentially damped solution devolves into one
that is
singular (not integrable) at the origin; see  the Appendix.
(This situation can be contrasted
with, {\it e.g.}, the magnetic dyon solution \cite{ref11}, where
absence of
the mass term allows solutions for $A_0$ with unconstrained large-$r$
behavior, leaving the freedom to select the solution that is regular at the
origin.)

A similar argument shows that there are no ``static'' instanton
solutions.
These would be solutions for which $t$ is replaced by $- i x_4$, $A_0$
by $i
A_4$ and presumably one would seek solutions periodic in $x_4$
with period
$\beta ={1\over T}={1\over m}\sqrt{{{N+N_F/2}\over 3}}\,$.
An $x_4$-independent solution is necessarily
periodic; it would satisfy (\ref{2.6}) with $A_4$ replacing $A_0$
and
opposite sign in the right side of (\ref{2.6b}).
But analysis similar to the above shows that finite-action solutions
do not exist.

\section{HARD THERMAL LOOPS FROM THE COMPOSITE EFFECTIVE
ACTION}
\label{sec:3}

In this Section, we present a derivation of the non-Abelian Kubo
equation (\ref{1.1}) based on the composite effective action of
\cite{ref7}, a generalization of the usual effective action
(obtained by coupling local sources to the fields) in
which one additionally introduces bilocal sources. In the
QCD case, the composite effective action is given by
$S(A)+\Gc(A,G_{\phi})$, where $G_{\phi}(x,y)$ are (undetermined) two-point
functions, and the labels $\phi=A,\p,\z$ denote either gluons, or
fermions-antifermions, or
ghosts-antighosts, respectively (in the end, ghosts play no dynamical role,
beyond maintaining gauge covariance of the final result). $S(A)$ is the pure
Yang-Mills action, and
\beq
\Gc(A,G_{\phi})&=& {{i}\over{2}}
\biggl(\tr\ln G_A^{-1} + \tr\D_A^{-1}G_A\biggr)\nonumber\\
&&-i\biggl(\tr\ln G_\p^{-1}+\tr\D_\p^{-1}G_\p+\tr\ln G_\z^{-1}
+\tr\D_\z^{-1}G_\z\biggr)
\label{3.1}
\eeq
when 2PI contributions are omitted.
The trace is over
space-time arguments as well as over Lorentz and group indices.
The gauge coupling constant $\gg$, which was previously scaled to
unity, is here reinserted.
$\D_\phi^{-1}$ is computed from the usual QCD action $S_{QCD}$
({\it e.g.} in the Feynman gauge):
\be
i\D_{\phi}^{-1}(x,y)=
{\disp{\d^2 S_{QCD}}\over\disp{\d \phi (x)\,\d \phi (y)}}
\label{3.2}
\ee
The fields carry group and space-time
indices, which are symbolically subsumed into the space-time labels
$x,y$.

The truncated composite effective action (\ref{3.1})
comprises the first, dynamical approximation that we make and reflects
the known fact \cite{ref3} that hard thermal loops arise from one-loop graphs.
The full composite effective action of course coincides
with the ordinary effective
action when the two-point functions are evaluated by imposing stationarity
requirements, and the above truncation reproduces the standard
one-loop action involving $\tr\ln\D_\phi^{-1}$.
Nevertheless subsequent analysis is more transparently organized in the
composite effective action formalism.

As indicated in \cite{ref7}, $S+\Gc$ is stationary for physical
processes. This yields the conditions
\begin{mathletters}
\label{3.4}
\beq
D_\n F^{\n\m}  &=& J^{\m}\ , \label{3.4a} \\
G_\phi^{-1} &=& \D_\phi^{-1}\ ,\ \ \ \phi=A,\p,\z \ .\label{3.4b}
\eeq
\end{mathletters}
$\!\!$Computing the local induced current
$J^{\m}(x)=-{{\d\Gc}\over{\d A_\m(x)}}$
involves differentiating $\D_\phi^{-1}$ with respect to $A_\m$. Since
the $\D_\phi^{-1}$ depend locally on $A$, the resulting current is
the local limit of a bilocal expression constructed from the two-point
functions $G_\phi(x,y)$:
\be
J^\m (x)= \lim_{y\rightarrow x} J^\m (x,y)\ ,
\label{3.41}
\ee
where the bilocal current $J^\m(x,y)=T^a J^{\m}_a(x,y)$ is given by
\beq
J^{\m}(x,y)=g\left(\Gamma^{\m}_{\al\bet\g}\,D^\al_x
{\bf G}_{A}^{\ \,\bet\g} (x,y)
+ \pa^\m_y {\bf G}_{\zeta}(x,y)\right)
+ ig T^a \,{\rm tr}\, \gamma^\m{\cal T}^a G_{\psi}(x,y)
\label{3.411}
\eeq
with $\Gamma^\m_{\al\bet\g}\equiv 2g^\m_\bet g_{\al\g} - g^\m_\al g_{\bet\g}
- g^\m_\g g_{\bet\al}$. The trace ``${\rm tr}$" is taken over Dirac spinor
as well as internal symmetry indices, and we have defined
${\bf G}_{A,\zeta}(x,y)=[T^a,T^b] G_{A,\zeta\, ab}(x,y)$ with
$D_x{\bf G}_{A}(x,y)
=\pa_x{\bf G}_{A}(x,y)+g[[A(x),T^b],T^c]G_{A\,bc}(x,y)$.

We now use eqs. (\ref{3.4}) -- (\ref{3.411})
to study ``soft" plasma excitations. ``Soft'' means
that both the energy and the momentum carried by a particle are of
order
$\gg T$, for a coupling constant $\gg\ll 1$, while particles with energy or
momentum
of order $T$ are called hard (see {\it e.g.} \cite{ref3}).
The strategy is to solve the system of
coupled equations (\ref{3.4}), in order to derive from (\ref{3.411})
the expression (\ref{1.3}) for the local current $J^{\m}$.
We approximate eqs. (\ref{3.4}) by expanding them in powers of $\gg$.
The approximation scheme we use was first proposed in \cite{ref8} for
deriving hard thermal loops from the Schwinger-Dyson equations. It
represents an essential step in that derivation. Earlier work on the QCD
plasma (in which this approximation was not used) is reviewed in
\cite{elzeheinz}.
Following \cite{ref8}, we introduce
relative and center of mass coordinates, $s=x-y$ and $X={1\over
2}(x+y)$,
respectively. In these new variables the partial derivatives
carry different dependences on $\gg$: $\pa_s\sim T$ and $\pa_X \sim
\gg T$.
This comes from the fact that $\pa_s$ corresponds to hard loop
momenta,
whereas $\pa_X$ is related to soft external momenta. See \cite{ref8} for a
detailed account.

Next, motivated by the expression (\ref{3.41})--(\ref{3.411}) for the current,
we expand $G_\phi$ in powers of $\gg$:
\be
G_\phi=G_\phi^{(0)}+\gg G_\phi^{(1)}+\gg^2 G_\phi^{(2)}+ ...\ ,
\label{3.42}
\ee
where $G_\phi^{(0)}$ is just the free propagator at temperature $T$ and
$G_\phi^{(i)},\ i\geq 1$ are determined by (\ref{3.4b}).
At leading order in $\gg$ (to which we restrict ourselves in the sequel),
the bilocal current (\ref{3.411})
depends on $G_\phi^{(0)}$  and $G_\phi^{(1)}$:
\beq
J^{\m}_a(X,s) &=& g^2 f^\abc\Biggl[
\Gamma^{\m}_{\al\bet\g}\biggl(\pa^\al_s G_{A\,bc}^{(1) \,\bet\g}(X,s)
+f^{bde}A^\al_d(X)G_{A\,ec}^{(0)\,\bet\g}(X,s)\biggr)
- \pa^\m_s G_{\zeta\,bc}^{(1)}(X,s)\Biggr] \nonumber\\
&&+ ig^2\,{\rm tr}\,\gamma^\m{\cal T}^a G_{\psi}^{(1)}(X,s)
+ \delta J_a^\m(X,s)\ ,
\label{3.43}
\eeq
where $G_\phi(X,s)\equiv G_\phi(X+{s\over 2},X-{s\over 2})$ [and similarly for
$J(X,s)$]. We have added the term $\delta J^{\m}_a(X,s)$ in order to
compensate for the loss of gauge covariance due to non-locality:
\beq
\delta J^{\m}_a(X,s)= g^2 s\!\cdot\! A^b(X)\Biggl[ f^{ace}f^{bcd}
\biggl(3\,\pa_\n^s G_{A\,de}^{(0)\,\m\n}(s)
+ \pa^\m_s G_{\zeta}^{(0)\,de}(s)\biggr)
+i \,{\rm tr}\, {\cal T}^b{\cal T}^a \gamma^\m G_{\psi}^{(0)}(s)\Biggr] .
\label{3.44}
\eeq
Note that this term vanishes in the local limit.

Now, we derive from (\ref{3.4b}) a condition on $G_\phi^{(1)}$.
[It turns out to be convenient to expand,
instead of (\ref{3.4b}), the equivalent equations $\D^{-1}_\phi G_\phi=
G_\phi\D^{-1}_\phi=I$, in which we disregard temperature-independent
contributions.] The ${\cal O}(\gg)$-condition does not fix $G_\phi^{(1)}$
uniquely; hence we need to go to ${\cal O}(\gg^2)$. The condition
so obtained on $G_\phi^{(1)}$ can be used to derive a constraint on the
bilocal current.
The subsequent derivation of this constraint [eq. (\ref{3.10})] is
similar to the one given in \cite{ref8}, to which we refer the
reader for details. Momentum space is most convenient, {\it i.e.}
\be
G_\phi (X,k)=\int \dqs e^{ik\cdot s}G_\phi (X,s)\ ,
\label{3.5}
\ee
the explicit forms for the thermal parts of the free propagators
being ({\it e.g.} in Feynman gauge):
\begin{mathletters}
\label{3.501}
\beq
G_{A\,ab}^{(0)\,\m\n}(k)&=&-2\pi\,\delta^{ab}g^\mn\delta(k^2)\, n_B(k_0)\ ,\\
G_{\psi}^{(0)\,mn}(k)&=&-2\pi\,\delta^{mn} k\!\!\! / \delta(k^2)\, n_F(k_0)\
,\\
G_{\zeta}^{(0)\,ab}(k)&=&\phantom{-}2\pi\,\delta^{ab}\delta(k^2)\, n_B(k_0)\ ,
\eeq
\end{mathletters}
$\!\!$where $n_{B,F}(k_0)=1/(e^{\beta |k_0|} \mp 1)$ are the bosonic
and fermionic probability distributions.

Similarly, for the bilocal current in momentum space one writes
\be
J^\m (X,k)=\int \dqs e^{ik\cdot s}J^\m (X,s)\ .
\label{3.51}
\ee
In the limit $s\rightarrow 0$, or equivalently $y\rightarrow x$,
where $X=x$,
\be
J^\m (x)=J^\m (X) = \int {{\dqk}\over{(2\pi)^4}}\,J^{\m}(X,k)\ .
\label{3.6}
\ee

The resulting constraint on the bilocal current is \cite{ref8}:
\be
Q\cdot D_X\, {J}^{\m}(X,k)=4\pi \gg^2 Q^\m Q^\r k_0 \, F_{\rho 0} \,
\delta(k^2){d\over{dk_0}}[N\,n_B(k_0)+N_F\,n_F(k_0)]\ ,
\label{3.10}
\ee
where $Q^\m\equiv {{k^\m}\over{k_0}} = (1,{\bf Q})$.

Our next task is to make contact between (\ref{3.10}) and the gauge
invariance condition for the generating functional of hard thermal loops.
Our strategy consists in transforming (\ref{3.10}) into two distinct
conditions for positive and negative $k_0$'s, and in taking advantage of
the symmetry properties that arise.
We first integrate the equation (\ref{3.10})
over $|\bk|$ and $k_0\geq 0$. Due to the $\d (k^2)$ on
the right side, the bilocal current is non-vanishing only when
$k_0=|{\bk}|$; hence $\bf Q$ can be replaced by a unit vector
$\hat q\equiv {\bk\over |\bk|}$. The integration thus yields:
\be
Q_+\cdot D_X \, {\cal J}^{\m}_+(X,{\hat q})=- 2 \sqrt{2} \, \pi^3
 m^2 Q_+^\m Q_+^\r F_{\r 0}\ ,\label{3.103}
\ee
where we have defined
\be
{\cal J}^\m_+ (X,\hq) = \int|{\bf k}|^2 d|{\bf k}|\int_0^\infty dk_0\,
J^\m(X,k)\ .
\label{3.1031}
\ee
Similarly, upon introducing
\be
{\cal J}^\m_- (X,\hq) = \int|{\bf k}|^2 d|{\bf k}|\int_{-\infty}^0 dk_0\,
J^\m(X,k)\ ,\label{3.1033}
\ee
the integration of (\ref{3.10}) over $|\bk|$ and $k_0\leq 0$ gives:
\be
Q_-\cdot D_X \, {\cal J}^{\m}_-(X,{\hat q}) = -2 \sqrt{2} \, \pi^3
m^2 Q_-^\m Q_-^\r F_{\r 0}\ ,\label{3.104}
\ee
wherefrom one sees that ${\cal J}^\m_-(X,-\hq)$ satisfies the same
equation (\ref{3.103}) as ${\cal J}^\m_+(X,\hq)$.

Now, using $\int\dqk = \int d\Omega |{\bf k}|^2 d|{\bf k}|dk_0$, we
rewrite the expression (\ref{3.6}) for the local current as
$J^\m(X)=
\int{d{\hat q}\over (2\pi)^4}[{\cal J}^\m_+(X,\hq)+{\cal J}^\m_-(X,\hq)]$.
Here, $\hq$ can be replaced by $-\hq$ in each term of the integrand
separately, since $\hq$ spans the whole solid angle. Therefore, we can write
\be
J^\m (X)=\int{d{\hat q}\over (2\pi)^4}\, {\cal J}^\m(X,\hq)\ ,
\label{3.1041}
\ee
where ${\cal J}^\m(X,\hq)$ is defined as
\be
{\cal J}^\m(X,\hq)\equiv {\cal J}^\m_+(X,\hq) + {\cal J}^\m_-(X,-\hq)\ ,
\label{3.1042}
\ee
and satisfies, as a consequence of (\ref{3.103}) and (\ref{3.104}),
\be
Q_+\cdot D_X \, {\cal J}^{\m}(X,{\hat q})=-4 \sqrt{2} \, \pi^3
 m^2 Q_+^\m Q_+^\r F_{\r 0}\ .
\label{3.11}
\ee
{}From this, after decomposing
\be
{\cal J}^{\m}(X,{\hat q})={\tilde {\cal J}}^{\m}(X,{\hat q}) - 4 \sqrt{2}
\, \pi^3 m^2 Q_+^\m A_0\ ,
\label{3.13}
\ee
we get as our final condition on the bilocal current:
\be
Q_+\cdot D_X \, {\tilde {\cal J}}^{\m}(X,{\hat q})=4 \sqrt{2} \, \pi^3
m^2 Q_+^\m \pa^0_X (Q_+\cdot A)\ .
\label{3.14}
\ee

Let us now assume that
${\tilde {\cal J}}^{\m}(X,\hq)$ can be obtained from a functional
$W(A,\hq)$ as
\be
{\tilde {\cal J}}^{\m}(X,\hq)={{\d W(A,\hq)}\over{\d A_\m(X)}}\ .
\label{3.15}
\ee
Equation (\ref{3.14}) then implies that $W(A,\hq)$ depends only on
$A_+$, {\it i.e.} $W(A,\hq) =W(A_+)$,
and ${\tilde {\cal J}}^{\m}={{\d W( A_+ )}\over{\d A_+ }}Q_+^\m$.
In turn, $W( A_+ )$ satisfies, as a consequence of (\ref{3.14}),
\be
Q_+ \cdot D_X \, {{\d W( A_+ )}\over{\d  A_+ }}= 4 \sqrt{2} \, \pi^3
m^2 \pa^0_X  A_+  \ .
\label{3.16}
\ee
By introducing new coordinates $(x_+,x_-,{\bf x}_\bot )$,
\be
x_+= Q_-\cdot X,\qquad  x_-=Q_+ \cdot X,\qquad
{\bf x}_\bot \cdot {\hat q} =0\ ,
\label{3.17}
\ee
we can rewrite $Q_+ \cdot \pa_X$ as $\pa_+$ and (\ref{3.16})
becomes
\be
\pa_+ \, {{\d W( A_+ )}\over{\d  A_+ }}
+\gg \left[ A_+ \, ,{{\d W( A_+ )}\over{\d  A_+ }}\right]
=4 \sqrt{2} \, \pi^3 m^2 \pa^0_X A_+ \ .
\label{3.18}
\ee
This equation was first derived in \cite{ref2}, as an expression of
gauge invariance of the generating functional for hard
thermal loops, and has since then been studied by several authors.
Here, it is seen to be a consequence of the stationarity requirement on
the composite effective action.

It has been shown in \cite{ref1} that $W( A_+ )$ is given by the
eikonal of a Chern-Simons gauge theory. This observation is our last step
towards deriving the approximate expression for the local current $J^\m(x)$ in
eq. (\ref{3.4a}).
The subsequent development follows \cite{ref5} and the result is exactly the
non-Abelian Kubo equation (\ref{1.1}) with the form (\ref{1.3}) for the induced
current.


\section{CONCLUSIONS}
\label{sec:4}

The behavior of the quark-gluon plasma at high temperature is described by the
non-Abelian Kubo equation (\ref{1.1}) -- (\ref{1.3}). We have
studied the static response of such a plasma and proved that there are no
hard thermal solitons (this result is supported and illustrated by
numerical integration). The absence of ``static" instantons is established by
invoking a similar argument. In addition, the static non-Abelian Kubo
equation indicates that the non-Abelian electric field is screened by a
gauge invariant Debye mass $m=\gg T\, \sqrt{{{N+N_F/2}\over 3}}$.

Furthermore, we have derived  the non-Abelian Kubo equation from the
composite effective action formalism. Indeed, the requirement that the
composite effective action be stationary leads, within a kinematical
approximation scheme taken at the leading order, to the equation
obtained in \cite{ref2} by imposing gauge invariance on the
generating funtional of hard thermal loops.

Let us mention some problems deserving further investigation.
Finding non-static solutions to the non-Abelian Kubo equation is an
appealing --- if difficult --- task, since
such solutions would correspond to collective excitations of the
quark-gluon plasma at high temperature. Also, it would be interesting to
investigate the next-to-leading order effects in the kinematical
approximation and to see if they are gauge invariant; we hope that
our formalism is well suited for such an investigation. Furthermore,
it is clear that $\Gc(A,G_{\phi})$, when evaluated on the solution for
$G_{\phi}$ obtained from (\ref{3.4b}) and (\ref{3.42}), coincides with the
$\G(A)$ constructed from the Chern-Simons eikonal. While our derivation
establishes this
fact indirectly, an explicit evaluation of the relevant functional
determinants in the hard thermal limit would be welcome.

\vskip0.8truecm
\centerline{\bf NOTE ADDED}
\vskip0.7truecm
\setcounter{equation}{0}
%%\renewcommand{\theequation}{N\arabic{equation}}
%%\renewcommand{\theequation@prefix}{N}
\makeatletter
\def\theequation@prefix{N}
\makeatother

We have now seen recent papers \cite{BInew}
wherein the response equations
are also analyzed.  Moreover, local equations are found
for time-dependent, but space-independent gauge
fields, and for non-Abelian plane waves.
The starting point of these investigations is a non-local
expression for the induced current (see \cite{ref8,BInew}),
\be
j^{\rm ind}_{\mu\ a}(x)=3\,\omega_p^2\int
{{d\Omega}\over{4\pi}}\,v_\mu \int_0^\infty du\,U_{ab}(x,x-vu)
\,{\bf v}\cdot{\bf E}^b(x-vu)\ ,
\label{A.1}
\ee
which appears different from our local, but coupled, form
(\ref{1.3}) -- (\ref{1.7}). Here we exhibit the steps that
explicitly relate the two.

Beginning with our form for the induced current,
(\ref{1.3}) -- (\ref{1.7}), we observe that, owing to
the integration over the angles of $\hat q$, we may collapse these
expressions into
\be
{m^2\over 2}\,j^{\mu}(x) = m^2\,\int {d \hat{q} \over 4\pi}
\, Q^\mu_{+} \biggl( a_{-}(x) - A_{-}(x) \biggr) \ ,
\label{A.2}
\ee
where
\be
\partial_+ a_- + [A_+,a_-] = \partial_- A_+ \ .
\label{A.3}
\ee
Eq. (\ref{A.3}) may be integrated, yielding
\be
a^a_-(x) = \int_0^\infty du\,U_{ab}(x, x - Q_+ u)\,
\partial_-  A_+^b(x - Q_+ u) \ .
\label{A.4}
\ee
Here $U_{ab}$ satisfies
\beq
{\partial\over \partial u} U_{ab}(x,x-Q_+u)
&=&U_{ac}(x,x-Q_+u)\,f_{cbd}\,A_+^d(x-Q_+u)\ ,\nonumber\\
U_{ab}(x,x)&=&\delta_{ab}\ .
\eeq
Also $A_-^a$ may be presented as
\beq
A^a_-(x) &=& - \int_0^\infty du \, {d\over du} \biggl\{
U_{ab}(x, x-Q_+ u)\, A^b_-(x-Q_+ u)\biggr\}\nonumber\\
&=& \int_0^\infty du \, U_{ab}(x, x-Q_+ u)
\biggl\{ \partial_+ A_-^b(x-Q_+u)\nonumber\\
&&\phantom{\int_0^\infty}
- f^{bcd}\, A_-^c(x-Q_+u)\,A_+^d(x-Q_+u)\biggr\}.
\label{A.5}
\eeq
[We have assumed that no contributions
arise at infinity.] From (\ref{A.2}), (\ref{A.4}) and (\ref{A.5}),
it follows that the induced current can be written as
\be
{m^2\over 2}\,j^{\mu}_a(x) =  m^2 \int {d{\hat q}\over 4\pi}
\,Q^\mu_+ \int_0^\infty du\,U_{ab}(x, x-Q_+u)\
F_{-+}^b(x-Q_+u)\ ,
\label{A.6}
\ee
which coincides with the expression (\ref{A.1}) derived in \cite{BInew},
after the notational replacements
$m \rightarrow \sqrt{3}\,\omega_p$, $d{\hat q} \rightarrow d\Omega$,
$Q^\mu_+\rightarrow
v^\mu$ and $F_{-+}\rightarrow {\bf v}\cdot{\bf E}$ are performed.

The time-dependent, space-independent equation found in \cite{BInew}
is easily derived in
our formalism, also. When there is no space dependence, eqs. (\ref{1.7})
can be written as
\be
\partial_+(a_\mp - A_\pm)+[A_\pm,a_\mp -A_\pm] = 0
\label{A.7}
\ee
and are solved by $a_\mp = A_\pm$. Hence:
\be
{m^2\over 2}\,j^{\mu} = {{m^2}\over 2}\int {d{\hat q}\over 4\pi}\,
(Q_+ - Q_-)^\mu (Q_+ - Q_-)^\nu A_\nu \ ,
\label{A.8}
\ee
of which only the spatial component is non-vanishing:
\be
{m^2\over 2}\,j^i = {m^2} \int {d{\hat q}\over 4\pi}\,
{\hat q}^i {\hat q}^j A_j
=- {1\over 3}\, m^2 A^i\ .
\label{A.9}
\ee
This coincides with the result in \cite{BInew}.

Similarly, the induced current for the non-Abelian plane wave in
\cite{BInew} corresponds to:
\be
a_\pm = {Q_\pm\cdot p\over Q_\mp\cdot p}\, A_\mp (p\cdot x)
\ee
in our formalism, with $p=(\omega,{\vec k})$ being the corresponding
wave vector.

\vskip0.8truecm
\centerline{\bf APPENDIX}
\vskip0.7truecm
\setcounter{equation}{0}
%%\renewcommand{\theequation}{N\arabic{equation}}
%%\renewcommand{\theequation@prefix}{N}
\makeatletter
\def\theequation@prefix{A}
\makeatother

In this Appendix we analyze in greater detail and integrate numerically the
radially symmetric version of the static response equations (\ref{2.6}), in
the $SU(2)$ case. Radially symmetric $SU(2)$ gauge potentials
take the forms:
\beq
\label{eq:a1}
A^a_i &=& ( \delta^{ai} - \hat{r}^a \hat{r}^i ) \, {\phi_2(r) \over r} +
\varepsilon^{aij} \, \hat{r}^j \, {{1 - \phi_1(r)} \over r}\ , \nonumber \\
A_0^a &=& \hat{r}^a \, { g(r) \over r}\ ,
\eeq
where a residual gauge freedom has been used to eliminate
a term proportional to $ \hat{r}^a \hat{r}^i$.

We substitute the {\it Ansatz} (\ref{eq:a1}) into (\ref{2.6}).
The resulting equations
give us the freedom to set one of the two $\phi_i$'s to zero; we
obtain,
\beq
\label{eq:a3}
x^2 {d^2 \over dx^2}\, J &=& (x^2 + 2 K^2) \, J\ , \nonumber \\
x^2 {d^2 \over dx^2}\, K &=& (K^2 -J^2 -1) \, K\ ,
\eeq
where we have set $\phi_2$ to zero, rescaled $x=mr$ and defined
$J(x)=g(r)$, $K(x)=\phi_1(r)$.

We now investigate this system of coupled second-order differential
equations. First, we see
that they possess the following two exact solutions:
\begin{mathletters}
\beq
J&=&0,~K=\pm 1 \ , \label{eq:a4a} \\
J&=&J_0\,e^{-x},~K=0\ . \label{eq:a4b}
\eeq
\end{mathletters}
$\!\!$Eq.~(\ref{eq:a4a}) corresponds to the Yang-Mills vacuum, while
(\ref{eq:a4b}) is the celebrated Wu-Yang monopole plus a screened electric
field.

In the asymptotic region $x\rightarrow\infty$, the
regular solution of the system (\ref{eq:a3}) tends to (\ref{eq:a4a}), with
$J$ approaching its asymptote exponentially. (Of course there is also the
solution with $J$ growing exponentially, which we do not consider.)

Near the origin, $J$ and $K$ behave either like the vacuum (\ref{eq:a4a}) or
approach the monopole solution (\ref{eq:a4b}) as follows,
\beq
J(x) &\rightarrow& J_0 + ... \ ,\nonumber\\
K(x) &\rightarrow& K_0\sqrt{x} \, cos\biggl( {2\pi\over \tau} ln {x \over x_0}
\biggr)+ ...\ ,
\label{eq:a44}
\eeq
where $\tau$ is correlated with $J_0$ as
\be
\tau={4\pi\over\sqrt{4J_0^2+3}}\ .
\label{eq:a45}
\ee
Only the vacuum alternative at the origin leads to finite energy.
However, since we must choose one of two possible solutions at infinity
(obviously we pick the regular one), the behavior at
the origin is determined and can be exhibited explicitly by integrating the
equations (\ref{eq:a3}) numerically. Starting
with regular boundary conditions at infinity, we find the profiles
presented in Figure 1. They show that the monopole solution (\ref{eq:a4b})
is reached at the origin, with $K$ vanishing as in (\ref{eq:a44})
-- (\ref{eq:a45}),
a result consistent with our analytic proof that there are no
finite energy static solutions in hard thermal gauge theories.

\begin{references}

\bibitem{ref1}
R.~Efraty and V.~P.~Nair,
{\it Phys.~Rev.~Lett.}~{\bf 68}, 2891 (1992);
{\it Phys.~Rev.~D}~{\bf 47}, 5601 (1993).

\bibitem{ref2}
J.~C.~Taylor and S.~Wong,
{\it Nucl.~Phys.}~{\bf B346}, 115 (1990).

\bibitem{ref3}
E.~Braaten and R.~Pisarski,
{\it Phys.~Rev.~D}~{\bf 42}, 2156 (1990), {\bf 45}, 1827 (1992);
{\it Nucl.~Phys.}~{\bf B337}, 569 (1990), {\bf B339}, 310 (1992).

\bibitem{ref4}
D.~Gonzales and A.~Redlich, {\it Ann.~Phys.} (NY) {\bf 169}, 104
(1986);
G.~Dunne, R.~Jackiw and C.~Trugenberger, {\it Ann.~Phys.} (NY) {\bf
149}, 197
(1989).

\bibitem{ref5}
R.~Jackiw and V.~P.~Nair, {\it Phys.~Rev.~D} {\bf 48}, 4991 (1993).

\bibitem{ref6}
V.~P.~Silin, {\it Zh.~Eksp.~Teor.~Fiz.}~{\bf 38}, 1577 (1960)
[Engl.~trans: {\it Sov.~Phys.~JETP}~{\bf 11}, 1136 (1960)];
E.~Lifshitz and L.~Pitaevskii, {\it Physical Kinetics}
(Pergamon, Oxford, 1981).

\bibitem{ref7}
J.~M.~Cornwall, R.~Jackiw and E.~Tomboulis,
{\it Phys.~Rev.~D}~{\bf 10}, 2428 (1974).

\bibitem{ref8}
J.~Blaizot and E.~Iancu, {\it Phys.~Rev.~Lett.}~{\bf 70}, 3376 (1993);
Saclay preprint SPhT/93-064.

\bibitem{ref9}
S.~Deser, {\it Phys.~Lett.} {\bf 64B}, 463 (1976).

\bibitem{ref10}
That the energy is positive on the constrained surface even for non-static
fields has been shown by V.~P.~Nair, {\it Phys.~Rev.~D}~{\bf 48}, 3432
(1993).

\bibitem{ref11}
B.~Julia and A.~Zee, {\it Phys.~Rev.~D}~{\bf 11}, 2227 (1975).

\bibitem{elzeheinz}
H.-Th. Elze and U.~Heinz, {\it Phys.~Rep.} {\bf 183}, 81 (1989).

\bibitem{BInew}
J.~Blaizot and E.~Iancu, Saclay preprints T94/02, T94/03, January
1994, and T94/013, February 1994.

\end{references}
\end{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% PS file containing 1 figure & caption   %%%%%%%%%%%%%%%%%%%%%%%%
%%% Cut out here and print separately       %%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!PS-Adobe-2.0 EPSF-2.0
%%BoundingBox: 72.0 72.0 540.0 720.0
%%Creator: Mathematica
%%CreationDate: Wed Feb  2 12:20:41 EST 1994
%%EndComments

gsave
-24 83 translate

/Mathdict 100 dict def
Mathdict begin
/Mlmarg		1.0 72 mul def
/Mrmarg		1.0 72 mul def
/Mbmarg		1.0 72 mul def
/Mtmarg		1.0 72 mul def
/Mwidth		8.5 72 mul def
/Mheight	11 72 mul def
/Mtransform	{  } bind def
/Mnodistort	true def
/Mfixwid	false	def
/Mfixdash	false def
/Mrot		0	def
/Mpstart {
MathPictureStart
} bind def
/Mpend {
MathPictureEnd
} bind def
/Mscale {
0 1 0 1
5 -1 roll
MathScale
} bind def
/ISOLatin1Encoding dup where
{ pop pop }
{
[
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright
/parenleft /parenright /asterisk /plus /comma /minus /period /slash
/zero /one /two /three /four /five /six /seven
/eight /nine /colon /semicolon /less /equal /greater /question
/at /A /B /C /D /E /F /G
/H /I /J /K /L /M /N /O
/P /Q /R /S /T /U /V /W
/X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore
/quoteleft /a /b /c /d /e /f /g
/h /i /j /k /l /m /n /o
/p /q /r /s /t /u /v /w
/x /y /z /braceleft /bar /braceright /asciitilde /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent
/dieresis /.notdef /ring /cedilla /.notdef /hungarumlaut /ogonek /caron
/space /exclamdown /cent /sterling /currency /yen /brokenbar /section
/dieresis /copyright /ordfeminine /guillemotleft
/logicalnot /hyphen /registered /macron
/degree /plusminus /twosuperior /threesuperior
/acute /mu /paragraph /periodcentered
/cedilla /onesuperior /ordmasculine /guillemotright
/onequarter /onehalf /threequarters /questiondown
/Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla
/Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis
/Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply
/Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls
/agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla
/egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis
/eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide
/oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis
] def
} ifelse
/MFontDict 50 dict def
/MStrCat
{
exch
dup length
2 index length add
string
dup 3 1 roll
copy
length
exch dup
4 2 roll exch
putinterval
} def
/MCreateEncoding
{
1 index
255 string cvs
(-) MStrCat
1 index MStrCat
cvn exch
(Encoding) MStrCat
cvn dup where
{
exch get
}
{
pop
StandardEncoding
} ifelse
3 1 roll
dup MFontDict exch known not
{
1 index findfont
dup length dict
begin
{1 index /FID ne
{def}
{pop pop}
ifelse} forall
/Encoding 3 index
def
currentdict
end
1 index exch definefont pop
MFontDict 1 index
null put
}
if
exch pop
exch pop
} def
/ISOLatin1 { (ISOLatin1) MCreateEncoding } def
/ISO8859 { (ISOLatin1) MCreateEncoding } def
/Mcopyfont {
dup
maxlength
dict
exch
{
1 index
/FID
eq
{
pop pop
}
{
2 index
3 1 roll
put
}
ifelse
}
forall
} def
/Plain	/Courier findfont Mcopyfont definefont pop
/Bold	/Courier-Bold findfont Mcopyfont definefont pop
/Italic /Courier-Oblique findfont Mcopyfont definefont pop
/MathPictureStart {
gsave
Mtransform
Mlmarg
Mbmarg
translate
/Mtmatrix
matrix currentmatrix
def
/Mgmatrix
matrix currentmatrix
def
} bind def
/MathPictureEnd {
grestore
} bind def
/MathSubStart {
Momatrix
Mgmatrix Mtmatrix
Mlmarg Mrmarg
Mbmarg Mtmarg
Mwidth Mheight
11 -2 roll
moveto
Mtmatrix setmatrix
currentpoint
Mgmatrix setmatrix
13 -2 roll
moveto
Mtmatrix setmatrix
currentpoint
2 copy translate
/Mtmatrix matrix currentmatrix def
/Mlmarg 0 def
/Mrmarg 0 def
/Mbmarg 0 def
/Mtmarg 0 def
3 -1 roll
exch sub
/Mheight exch def
sub
/Mwidth exch def
} bind def
/MathSubEnd {
/Mheight exch def
/Mwidth exch def
/Mtmarg exch def
/Mbmarg exch def
/Mrmarg exch def
/Mlmarg exch def
/Mtmatrix exch def
dup setmatrix
/Mgmatrix exch def
/Momatrix exch def
} bind def
/Mdot {
moveto
0 0 rlineto
stroke
} bind def
/Mtetra {
moveto
lineto
lineto
lineto
fill
} bind def
/Metetra {
moveto
lineto
lineto
lineto
closepath
gsave
fill
grestore
0 setgray
stroke
} bind def
/Mistroke {
flattenpath
0 0 0
{
4 2 roll
pop pop
}
{
4 -1 roll
2 index
sub dup mul
4 -1 roll
2 index
sub dup mul
add sqrt
4 -1 roll
add
3 1 roll
}
{
stop
}
{
stop
}
pathforall
pop pop
currentpoint
stroke
moveto
currentdash
3 -1 roll
add
setdash
} bind def
/Mfstroke {
stroke
currentdash
pop 0
setdash
} bind def
/Mrotsboxa {
gsave
dup
/Mrot
exch def
Mrotcheck
Mtmatrix
dup
setmatrix
7 1 roll
4 index
4 index
translate
rotate
3 index
-1 mul
3 index
-1 mul
translate
/Mtmatrix
matrix
currentmatrix
def
grestore
Msboxa
3  -1 roll
/Mtmatrix
exch def
/Mrot
0 def
} bind def
/Msboxa {
newpath
5 -1 roll
Mvboxa
pop
Mboxout
6 -1 roll
5 -1 roll
4 -1 roll
Msboxa1
5 -3 roll
Msboxa1
Mboxrot
[
7 -2 roll
2 copy
[
3 1 roll
10 -1 roll
9 -1 roll
]
6 1 roll
5 -2 roll
]
} bind def
/Msboxa1 {
sub
2 div
dup
2 index
1 add
mul
3 -1 roll
-1 add
3 -1 roll
mul
} bind def
/Mvboxa	{
Mfixwid
{
Mvboxa1
}
{
dup
Mwidthcal
0 exch
{
add
}
forall
exch
Mvboxa1
4 index
7 -1 roll
add
4 -1 roll
pop
3 1 roll
}
ifelse
} bind def
/Mvboxa1 {
gsave
newpath
[ true
3 -1 roll
{
Mbbox
5 -1 roll
{
0
5 1 roll
}
{
7 -1 roll
exch sub
(m) stringwidth pop
.3 mul
sub
7 1 roll
6 -1 roll
4 -1 roll
Mmin
3 -1 roll
5 index
add
5 -1 roll
4 -1 roll
Mmax
4 -1 roll
}
ifelse
false
}
forall
{ stop } if
counttomark
1 add
4 roll
]
grestore
} bind def
/Mbbox {
1 dict begin
0 0 moveto
/temp (T) def
{	gsave
currentpoint newpath moveto
temp 0 3 -1 roll put temp
false charpath flattenpath currentpoint
pathbbox
grestore moveto lineto moveto} forall
pathbbox
newpath
end
} bind def
/Mmin {
2 copy
gt
{ exch } if
pop
} bind def
/Mmax {
2 copy
lt
{ exch } if
pop
} bind def
/Mrotshowa {
dup
/Mrot
exch def
Mrotcheck
Mtmatrix
dup
setmatrix
7 1 roll
4 index
4 index
translate
rotate
3 index
-1 mul
3 index
-1 mul
translate
/Mtmatrix
matrix
currentmatrix
def
Mgmatrix setmatrix
Mshowa
/Mtmatrix
exch def
/Mrot 0 def
} bind def
/Mshowa {
4 -2 roll
moveto
2 index
Mtmatrix setmatrix
Mvboxa
7 1 roll
Mboxout
6 -1 roll
5 -1 roll
4 -1 roll
Mshowa1
4 1 roll
Mshowa1
rmoveto
currentpoint
Mfixwid
{
Mshowax
}
{
Mshoway
}
ifelse
pop pop pop pop
Mgmatrix setmatrix
} bind def
/Mshowax {
0 1
4 index length
-1 add
{
2 index
4 index
2 index
get
3 index
add
moveto
4 index
exch get
Mfixdash
{
Mfixdashp
}
if
show
} for
} bind def
/Mfixdashp {
dup
length
1
gt
1 index
true exch
{
45
eq
and
} forall
and
{
gsave
(--)
stringwidth pop
(-)
stringwidth pop
sub
2 div
0 rmoveto
dup
length
1 sub
{
(-)
show
}
repeat
grestore
}
if
} bind def
/Mshoway {
3 index
Mwidthcal
5 1 roll
0 1
4 index length
-1 add
{
2 index
4 index
2 index
get
3 index
add
moveto
4 index
exch get
[
6 index
aload
length
2 add
-1 roll
{
pop
Strform
stringwidth
pop
neg
exch
add
0 rmoveto
}
exch
kshow
cleartomark
} for
pop
} bind def
/Mwidthcal {
[
exch
{
Mwidthcal1
}
forall
]
[
exch
dup
Maxlen
-1 add
0 1
3 -1 roll
{
[
exch
2 index
{
1 index
Mget
exch
}
forall
pop
Maxget
exch
}
for
pop
]
Mreva
} bind def
/Mreva	{
[
exch
aload
length
-1 1
{1 roll}
for
]
} bind def
/Mget	{
1 index
length
-1 add
1 index
ge
{
get
}
{
pop pop
0
}
ifelse
} bind def
/Maxlen	{
[
exch
{
length
}
forall
Maxget
} bind def
/Maxget	{
counttomark
-1 add
1 1
3 -1 roll
{
pop
Mmax
}
for
exch
pop
} bind def
/Mwidthcal1 {
[
exch
{
Strform
stringwidth
pop
}
forall
]
} bind def
/Strform {
/tem (x) def
tem 0
3 -1 roll
put
tem
} bind def
/Mshowa1 {
2 copy
add
4 1 roll
sub
mul
sub
-2 div
} bind def
/MathScale {
Mwidth
Mlmarg Mrmarg add
sub
Mheight
Mbmarg Mtmarg add
sub
0 0 moveto
1 index 0
lineto
2 copy
lineto
0 1 index
lineto
clip newpath
Mlp
translate
dup
/Mathabs
exch def
scale
/yscale exch def
/ybias exch def
/xscale exch def
/xbias exch def
/Momatrix
xscale yscale matrix scale
xbias ybias matrix translate
matrix concatmatrix def
/Mgmatrix
matrix currentmatrix
def
} bind def
/Mlp {
3 copy
Mlpfirst
{
Mnodistort
{
Mmin
dup
} if
4 index
2 index
2 index
Mlprun
11 index
11 -1 roll
10 -4 roll
Mlp1
8 index
9 -5 roll
Mlp1
4 -1 roll
and
{ exit } if
3 -1 roll
pop pop
} loop
exch
3 1 roll
7 -3 roll
pop pop pop
} bind def
/Mlpfirst {
3 -1 roll
dup length
2 copy
-2 add
get
aload
pop pop pop
4 -2 roll
-1 add
get
aload
pop pop pop
6 -1 roll
3 -1 roll
5 -1 roll
sub
div
4 1 roll
exch sub
div
} bind def
/Mlprun {
2 copy
4 index
0 get
dup
4 1 roll
Mlprun1
3 copy
8 -2 roll
9 -1 roll
{
3 copy
Mlprun1
3 copy
11 -3 roll
/gt Mlpminmax
8 3 roll
11 -3 roll
/lt Mlpminmax
8 3 roll
} forall
pop pop pop pop
3 1 roll
pop pop
aload pop
5 -1 roll
aload pop
exch
6 -1 roll
Mlprun2
8 2 roll
4 -1 roll
Mlprun2
6 2 roll
3 -1 roll
Mlprun2
4 2 roll
exch
Mlprun2
6 2 roll
} bind def
/Mlprun1 {
aload pop
exch
6 -1 roll
5 -1 roll
mul add
4 -2 roll
mul
3 -1 roll
add
} bind def
/Mlprun2 {
2 copy
add 2 div
3 1 roll
exch sub
} bind def
/Mlpminmax {
cvx
2 index
6 index
2 index
exec
{
7 -3 roll
4 -1 roll
} if
1 index
5 index
3 -1 roll
exec
{
4 1 roll
pop
5 -1 roll
aload
pop pop
4 -1 roll
aload pop
[
8 -2 roll
pop
5 -2 roll
pop
6 -2 roll
pop
5 -1 roll
]
4 1 roll
pop
}
{
pop pop pop
} ifelse
} bind def
/Mlp1 {
5 index
3 index	sub
5 index
2 index mul
1 index
le
1 index
0 le
or
dup
not
{
1 index
3 index	div
.99999 mul
8 -1 roll
pop
7 1 roll
}
if
8 -1 roll
2 div
7 -2 roll
pop sub
5 index
6 -3 roll
pop pop
mul sub
exch
} bind def
/intop 0 def
/inrht 0 def
/inflag 0 def
/outflag 0 def
/xadrht 0 def
/xadlft 0 def
/yadtop 0 def
/yadbot 0 def
/Minner {
outflag
1
eq
{
/outflag 0 def
/intop 0 def
/inrht 0 def
} if
5 index
gsave
Mtmatrix setmatrix
Mvboxa pop
grestore
3 -1 roll
pop
dup
intop
gt
{
/intop
exch def
}
{ pop }
ifelse
dup
inrht
gt
{
/inrht
exch def
}
{ pop }
ifelse
pop
/inflag
1 def
} bind def
/Mouter {
/xadrht 0 def
/xadlft 0 def
/yadtop 0 def
/yadbot 0 def
inflag
1 eq
{
dup
0 lt
{
dup
intop
mul
neg
/yadtop
exch def
} if
dup
0 gt
{
dup
intop
mul
/yadbot
exch def
}
if
pop
dup
0 lt
{
dup
inrht
mul
neg
/xadrht
exch def
} if
dup
0 gt
{
dup
inrht
mul
/xadlft
exch def
} if
pop
/outflag 1 def
}
{ pop pop}
ifelse
/inflag 0 def
/inrht 0 def
/intop 0 def
} bind def
/Mboxout {
outflag
1
eq
{
4 -1
roll
xadlft
leadjust
add
sub
4 1 roll
3 -1
roll
yadbot
leadjust
add
sub
3 1
roll
exch
xadrht
leadjust
add
add
exch
yadtop
leadjust
add
add
/outflag 0 def
/xadlft 0 def
/yadbot 0 def
/xadrht 0 def
/yadtop 0 def
} if
} bind def
/leadjust {
(m) stringwidth pop
.5 mul
} bind def
/Mrotcheck {
dup
90
eq
{
yadbot
/yadbot
xadrht
def
/xadrht
yadtop
def
/yadtop
xadlft
def
/xadlft
exch
def
}
if
dup
cos
1 index
sin
Checkaux
dup
cos
1 index
sin neg
exch
Checkaux
3 1 roll
pop pop
} bind def
/Checkaux {
4 index
exch
4 index
mul
3 1 roll
mul add
4 1 roll
} bind def
/Mboxrot {
Mrot
90 eq
{
brotaux
4 2
roll
}
if
Mrot
180 eq
{
4 2
roll
brotaux
4 2
roll
brotaux
}
if
Mrot
270 eq
{
4 2
roll
brotaux
}
if
} bind def
/brotaux {
neg
exch
neg
} bind def
/Mabswid {
Mathabs
div
setlinewidth
} bind def
/Mabsdash {
exch
Mathabs
[
3 1 roll
exch
{
exch
dup
3 -1 roll
exch
div
exch
}
forall
pop ]
exch
setdash
} bind def
/MBeginOrig { Momatrix concat} bind def
/MEndOrig { Mgmatrix setmatrix} bind def
/colorimage where
{ pop }
{
/colorimage {
3 1 roll
pop pop
5 -1 roll
mul
4 1 roll
{
currentfile
1 index
readhexstring
pop }
image
} bind def
} ifelse
/sampledsound where
{ pop}
{ /sampledsound {
exch
pop
exch
5 1 roll
mul
4 idiv
mul
2 idiv
exch pop
exch
/Mtempproc exch def
{ Mtempproc pop}
repeat
} bind def
} ifelse
/setcmykcolor where
{ pop}
{ /setcmykcolor {
4 1
roll
[
4 1
roll
]
{
1 index
sub
1
sub neg
dup
0
lt
{
pop
0
}
if
dup
1
gt
{
pop
1
}
if
exch
} forall
pop
setrgbcolor
} bind def
} ifelse

%%AspectRatio: 0.67422
MathPictureStart
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.0178469 0.952381 [
[ 0 0 0 0 ]
[ 1 0.674219 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
%%Object: Graphics
[ ] 0 setdash
0 setgray
gsave
grestore
0 0 moveto
1 0 lineto
1 0.67422 lineto
0 0.67422 lineto
closepath
clip
newpath
gsave
% Start of sub-graphic
gsave
0.02381 0.39062 0.45671 0.65817 MathSubStart
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.31746 0 0.154508 [
[(1)] 0.34127 0 0 2 Msboxa
[(2)] 0.65873 0 0 2 Msboxa
[(3)] 0.97619 0 0 2 Msboxa
[(1)] 0.01131 0.15451 1 0 Msboxa
[(2)] 0.01131 0.30902 1 0 Msboxa
[(3)] 0.01131 0.46353 1 0 Msboxa
[(4)] 0.01131 0.61803 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 0.61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
%%Object: Graphics
[ ] 0 setdash
gsave
gsave
0.002 setlinewidth
0.34127 0 moveto
0.34127 0.00625 lineto
stroke
grestore
[(1)] 0.34127 0 0 2 Mshowa
gsave
0.002 setlinewidth
0.65873 0 moveto
0.65873 0.00625 lineto
stroke
grestore
[(2)] 0.65873 0 0 2 Mshowa
gsave
0.002 setlinewidth
0.97619 0 moveto
0.97619 0.00625 lineto
stroke
grestore
[(3)] 0.97619 0 0 2 Mshowa
gsave
0.002 setlinewidth
0 0 moveto
1 0 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.02381 0.15451 moveto
0.03006 0.15451 lineto
stroke
grestore
[(1)] 0.01131 0.15451 1 0 Mshowa
gsave
0.002 setlinewidth
0.02381 0.30902 moveto
0.03006 0.30902 lineto
stroke
grestore
[(2)] 0.01131 0.30902 1 0 Mshowa
gsave
0.002 setlinewidth
0.02381 0.46353 moveto
0.03006 0.46353 lineto
stroke
grestore
[(3)] 0.01131 0.46353 1 0 Mshowa
gsave
0.002 setlinewidth
0.02381 0.61803 moveto
0.03006 0.61803 lineto
stroke
grestore
[(4)] 0.01131 0.61803 1 0 Mshowa
gsave
0.002 setlinewidth
0.02381 0 moveto
0.02381 0.61803 lineto
stroke
grestore
grestore
0 0 moveto
1 0 lineto
1 0.61803 lineto
0 0.61803 lineto
closepath
clip
newpath
gsave
gsave
gsave
gsave
gsave
0.004 setlinewidth
0.02505 0.54961 moveto
0.02629 0.54402 lineto
0.02877 0.534 lineto
0.03373 0.51621 lineto
0.04365 0.48486 lineto
0.06349 0.43172 lineto
0.10317 0.35047 lineto
0.14286 0.28098 lineto
0.18254 0.22363 lineto
0.22222 0.17973 lineto
0.2619 0.14644 lineto
0.30159 0.12081 lineto
0.34127 0.10071 lineto
0.38095 0.08467 lineto
0.42063 0.07166 lineto
0.46032 0.06098 lineto
0.5 0.05213 lineto
0.53968 0.04473 lineto
0.57937 0.0385 lineto
0.61905 0.03323 lineto
0.65873 0.02874 lineto
0.69841 0.0249 lineto
0.7381 0.02161 lineto
0.77778 0.01879 lineto
0.81746 0.01635 lineto
0.85714 0.01425 lineto
0.89683 0.01242 lineto
0.93651 0.01084 lineto
0.97619 0.00947 lineto
stroke
grestore
grestore
grestore
grestore
gsave
gsave
gsave
[ 0.03 0.01 ] 0 setdash
gsave
0.004 setlinewidth
0.02505 0.48312 moveto
0.02629 0.45569 lineto
0.02877 0.41011 lineto
0.03373 0.34329 lineto
0.03869 0.28412 lineto
0.04365 0.23312 lineto
0.04861 0.19377 lineto
0.05357 0.16412 lineto
0.05853 0.14155 lineto
0.06349 0.12402 lineto
0.06845 0.11011 lineto
0.07341 0.09884 lineto
0.08333 0.08177 lineto
0.09325 0.06951 lineto
0.10317 0.06028 lineto
0.1131 0.05309 lineto
0.12302 0.04733 lineto
0.14286 0.03867 lineto
0.1627 0.03246 lineto
0.18254 0.02778 lineto
0.20238 0.02412 lineto
0.22222 0.02118 lineto
0.2619 0.01674 lineto
0.30159 0.01354 lineto
0.34127 0.01114 lineto
0.38095 0.00928 lineto
0.42063 0.0078 lineto
0.46032 0.0066 lineto
0.5 0.00562 lineto
0.53968 0.0048 lineto
0.57937 0.00412 lineto
0.61905 0.00355 lineto
0.65873 0.00306 lineto
0.69841 0.00265 lineto
0.7381 0.0023 lineto
0.77778 0.00199 lineto
0.81746 0.00173 lineto
0.85714 0.00151 lineto
0.89683 0.00131 lineto
0.93651 0.00115 lineto
0.97619 0.001 lineto
stroke
grestore
grestore
grestore
grestore
grestore
MathSubEnd
grestore
% End of sub-graphic
% Start of sub-graphic
gsave
0.54329 0.39062 0.97619 0.65817 MathSubStart
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.31746 0.181775 0.363549 [
[(1)] 0.34127 0.18177 0 2 Msboxa
[(2)] 0.65873 0.18177 0 2 Msboxa
[(3)] 0.97619 0.18177 0 2 Msboxa
[(-0.5)] 0.01131 0 1 0 Msboxa
[(0.5)] 0.01131 0.36355 1 0 Msboxa
[(1)] 0.01131 0.54532 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 0.61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
%%Object: Graphics
[ ] 0 setdash
gsave
gsave
0.002 setlinewidth
0.34127 0.18177 moveto
0.34127 0.18802 lineto
stroke
grestore
[(1)] 0.34127 0.18177 0 2 Mshowa
gsave
0.002 setlinewidth
0.65873 0.18177 moveto
0.65873 0.18802 lineto
stroke
grestore
[(2)] 0.65873 0.18177 0 2 Mshowa
gsave
0.002 setlinewidth
0.97619 0.18177 moveto
0.97619 0.18802 lineto
stroke
grestore
[(3)] 0.97619 0.18177 0 2 Mshowa
gsave
0.002 setlinewidth
0 0.18177 moveto
1 0.18177 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.02381 0 moveto
0.03006 0 lineto
stroke
grestore
[(-0.5)] 0.01131 0 1 0 Mshowa
gsave
0.002 setlinewidth
0.02381 0.36355 moveto
0.03006 0.36355 lineto
stroke
grestore
[(0.5)] 0.01131 0.36355 1 0 Mshowa
gsave
0.002 setlinewidth
0.02381 0.54532 moveto
0.03006 0.54532 lineto
stroke
grestore
[(1)] 0.01131 0.54532 1 0 Mshowa
gsave
0.002 setlinewidth
0.02381 0 moveto
0.02381 0.61803 lineto
stroke
grestore
grestore
0 0 moveto
1 0 lineto
1 0.61803 lineto
0 0.61803 lineto
closepath
clip
newpath
gsave
gsave
gsave
gsave
gsave
0.004 setlinewidth
0.02505 0.16868 moveto
0.02629 0.19994 lineto
0.02753 0.19212 lineto
0.02877 0.16671 lineto
0.03001 0.15148 lineto
0.03125 0.15105 lineto
0.03373 0.17676 lineto
0.03497 0.19334 lineto
0.03621 0.20832 lineto
0.03745 0.22023 lineto
0.03869 0.2285 lineto
0.03993 0.23307 lineto
0.04117 0.23424 lineto
0.04241 0.23243 lineto
0.04365 0.22816 lineto
0.04613 0.21419 lineto
0.04861 0.19592 lineto
0.05357 0.15673 lineto
0.05605 0.139 lineto
0.05853 0.12365 lineto
0.06101 0.11101 lineto
0.06349 0.10119 lineto
0.06473 0.09733 lineto
0.06597 0.09413 lineto
0.06721 0.09157 lineto
0.06845 0.08964 lineto
0.06969 0.0883 lineto
0.07093 0.08752 lineto
0.07217 0.08727 lineto
0.07341 0.08751 lineto
0.07465 0.08823 lineto
0.07589 0.08938 lineto
0.07837 0.09287 lineto
0.08085 0.09775 lineto
0.08333 0.10381 lineto
0.09325 0.13625 lineto
0.10317 0.17522 lineto
0.12302 0.25269 lineto
0.14286 0.31692 lineto
0.1627 0.36582 lineto
0.18254 0.40206 lineto
0.20238 0.42886 lineto
0.22222 0.44883 lineto
0.24206 0.46392 lineto
0.2619 0.47548 lineto
0.28175 0.48448 lineto
0.30159 0.49159 lineto
0.32143 0.49729 lineto
0.34127 0.50193 lineto
0.38095 0.50892 lineto
Mistroke
0.42063 0.51389 lineto
0.46032 0.51758 lineto
0.5 0.52041 lineto
0.53968 0.52265 lineto
0.57937 0.52448 lineto
0.61905 0.52599 lineto
0.65873 0.52728 lineto
0.69841 0.52839 lineto
0.7381 0.52936 lineto
0.77778 0.53022 lineto
0.81746 0.53098 lineto
0.85714 0.53166 lineto
0.89683 0.53228 lineto
0.93651 0.53284 lineto
0.97619 0.53336 lineto
Mfstroke
grestore
grestore
grestore
grestore
gsave
gsave
gsave
[ 0.03 0.01 ] 0 setdash
gsave
0.004 setlinewidth
0.02505 0.18745 moveto
0.02629 0.22876 lineto
0.02753 0.15019 lineto
0.02877 0.09197 lineto
0.03001 0.07656 lineto
0.03125 0.09202 lineto
0.03373 0.16525 lineto
0.03621 0.24807 lineto
0.03869 0.31877 lineto
0.04117 0.37374 lineto
0.04365 0.41508 lineto
0.04613 0.44594 lineto
0.04861 0.46906 lineto
0.05109 0.48653 lineto
0.05357 0.49988 lineto
0.05605 0.51019 lineto
0.05853 0.51824 lineto
0.06101 0.5246 lineto
0.06349 0.52966 lineto
0.06597 0.53372 lineto
0.06845 0.53701 lineto
0.07093 0.53969 lineto
0.07341 0.54189 lineto
0.07589 0.54369 lineto
0.07837 0.54519 lineto
0.08085 0.54643 lineto
0.08333 0.54747 lineto
0.08581 0.54833 lineto
0.08829 0.54905 lineto
0.09077 0.54965 lineto
0.09325 0.55016 lineto
0.09573 0.55058 lineto
0.09821 0.55094 lineto
0.10069 0.55123 lineto
0.10317 0.55147 lineto
0.10565 0.55167 lineto
0.10813 0.55184 lineto
0.11062 0.55197 lineto
0.1131 0.55208 lineto
0.11558 0.55216 lineto
0.11806 0.55222 lineto
0.1193 0.55225 lineto
0.12054 0.55227 lineto
0.12178 0.55229 lineto
0.12302 0.5523 lineto
0.12426 0.55231 lineto
0.1255 0.55232 lineto
0.12674 0.55233 lineto
0.12798 0.55233 lineto
0.12922 0.55233 lineto
Mistroke
0.13046 0.55233 lineto
0.1317 0.55232 lineto
0.13294 0.55232 lineto
0.13542 0.5523 lineto
0.1379 0.55228 lineto
0.14286 0.55221 lineto
0.15278 0.55205 lineto
0.1627 0.55185 lineto
0.18254 0.55141 lineto
0.22222 0.55057 lineto
0.2619 0.54988 lineto
0.30159 0.54932 lineto
0.34127 0.54887 lineto
0.38095 0.54851 lineto
0.42063 0.54821 lineto
0.46032 0.54796 lineto
0.5 0.54774 lineto
0.53968 0.54756 lineto
0.57937 0.5474 lineto
0.61905 0.54727 lineto
0.65873 0.54715 lineto
0.69841 0.54704 lineto
0.7381 0.54695 lineto
0.77778 0.54686 lineto
0.81746 0.54678 lineto
0.85714 0.54671 lineto
0.89683 0.54665 lineto
0.93651 0.54659 lineto
0.97619 0.54654 lineto
Mfstroke
grestore
grestore
grestore
grestore
grestore
MathSubEnd
grestore
% End of sub-graphic
% Start of sub-graphic
gsave
0.02381 0.01605 0.45671 0.2836 MathSubStart
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.97619 0.0680272 0.309017 0.103006 [
[(-10)] 0.29592 0.30902 0 2 Msboxa
[(-5)] 0.63605 0.30902 0 2 Msboxa
[(-3)] 0.96369 0 1 0 Msboxa
[(-2)] 0.96369 0.10301 1 0 Msboxa
[(-1)] 0.96369 0.20601 1 0 Msboxa
[(1)] 0.96369 0.41202 1 0 Msboxa
[(2)] 0.96369 0.51503 1 0 Msboxa
[(3)] 0.96369 0.61803 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 0.61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
%%Object: Graphics
[ ] 0 setdash
gsave
gsave
0.002 setlinewidth
0.02381 0.30902 moveto
0.02381 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.09184 0.30902 moveto
0.09184 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.15986 0.30902 moveto
0.15986 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.22789 0.30902 moveto
0.22789 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.29592 0.30902 moveto
0.29592 0.31527 lineto
stroke
grestore
[(-10)] 0.29592 0.30902 0 2 Mshowa
gsave
0.002 setlinewidth
0.36395 0.30902 moveto
0.36395 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.43197 0.30902 moveto
0.43197 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.5 0.30902 moveto
0.5 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.56803 0.30902 moveto
0.56803 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.63605 0.30902 moveto
0.63605 0.31527 lineto
stroke
grestore
[(-5)] 0.63605 0.30902 0 2 Mshowa
gsave
0.002 setlinewidth
0.70408 0.30902 moveto
0.70408 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.77211 0.30902 moveto
0.77211 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.84014 0.30902 moveto
0.84014 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.90816 0.30902 moveto
0.90816 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0 0.30902 moveto
1 0.30902 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.97619 0 moveto
0.98244 0 lineto
stroke
grestore
[(-3)] 0.96369 0 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.10301 moveto
0.98244 0.10301 lineto
stroke
grestore
[(-2)] 0.96369 0.10301 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.20601 moveto
0.98244 0.20601 lineto
stroke
grestore
[(-1)] 0.96369 0.20601 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.41202 moveto
0.98244 0.41202 lineto
stroke
grestore
[(1)] 0.96369 0.41202 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.51503 moveto
0.98244 0.51503 lineto
stroke
grestore
[(2)] 0.96369 0.51503 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.61803 moveto
0.98244 0.61803 lineto
stroke
grestore
[(3)] 0.96369 0.61803 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0 moveto
0.97619 0.61803 lineto
stroke
grestore
grestore
0 0 moveto
1 0 lineto
1 0.61803 lineto
0 0.61803 lineto
closepath
clip
newpath
gsave
gsave
gsave
0.004 setlinewidth
0.02381 0.25321 moveto
0.04365 0.30495 lineto
0.05357 0.33694 lineto
0.05853 0.35047 lineto
0.06101 0.35616 lineto
0.06349 0.36099 lineto
0.06597 0.36487 lineto
0.06845 0.36774 lineto
0.06969 0.36878 lineto
0.07093 0.36954 lineto
0.07217 0.37003 lineto
0.07341 0.37024 lineto
0.07465 0.37017 lineto
0.07589 0.36983 lineto
0.07713 0.3692 lineto
0.07837 0.3683 lineto
0.08085 0.3657 lineto
0.08333 0.36207 lineto
0.08581 0.35747 lineto
0.08829 0.35199 lineto
0.09325 0.3388 lineto
0.10317 0.30706 lineto
0.10813 0.2908 lineto
0.1131 0.27587 lineto
0.11806 0.26334 lineto
0.12054 0.25826 lineto
0.12302 0.25412 lineto
0.1255 0.25097 lineto
0.12674 0.24978 lineto
0.12798 0.24887 lineto
0.12922 0.24823 lineto
0.13046 0.24787 lineto
0.1317 0.24779 lineto
0.13294 0.24799 lineto
0.13418 0.24846 lineto
0.13542 0.24921 lineto
0.13666 0.25023 lineto
0.1379 0.25152 lineto
0.14038 0.25488 lineto
0.14286 0.25922 lineto
0.14782 0.27053 lineto
0.15278 0.28463 lineto
0.1627 0.31698 lineto
0.16766 0.33289 lineto
0.17262 0.34706 lineto
0.17758 0.35848 lineto
0.18006 0.36289 lineto
0.1813 0.36473 lineto
0.18254 0.36632 lineto
0.18378 0.36764 lineto
Mistroke
0.18502 0.3687 lineto
0.18626 0.36949 lineto
0.1875 0.37 lineto
0.18874 0.37023 lineto
0.18998 0.37019 lineto
0.19122 0.36987 lineto
0.19246 0.36927 lineto
0.1937 0.36839 lineto
0.19494 0.36725 lineto
0.19742 0.36417 lineto
0.1999 0.36008 lineto
0.20238 0.35507 lineto
0.2123 0.32777 lineto
0.22222 0.29513 lineto
0.22718 0.27973 lineto
0.23214 0.26645 lineto
0.23462 0.26091 lineto
0.2371 0.25625 lineto
0.23958 0.25255 lineto
0.24082 0.25108 lineto
0.24206 0.24987 lineto
0.2433 0.24894 lineto
0.24454 0.24828 lineto
0.24578 0.24789 lineto
0.24702 0.24778 lineto
0.24826 0.24796 lineto
0.2495 0.24841 lineto
0.25074 0.24913 lineto
0.25198 0.25013 lineto
0.25446 0.25293 lineto
0.25694 0.25674 lineto
0.25942 0.26151 lineto
0.2619 0.26714 lineto
0.27183 0.29607 lineto
0.28175 0.32868 lineto
0.28671 0.34343 lineto
0.29167 0.3557 lineto
0.29415 0.3606 lineto
0.29663 0.36458 lineto
0.29911 0.36754 lineto
0.30035 0.36862 lineto
0.30159 0.36943 lineto
0.30283 0.36997 lineto
0.30407 0.37023 lineto
0.30531 0.37021 lineto
0.30655 0.36991 lineto
0.30779 0.36934 lineto
0.30903 0.36849 lineto
0.31151 0.36598 lineto
0.31399 0.36244 lineto
Mistroke
0.31647 0.35793 lineto
0.32143 0.34633 lineto
0.33135 0.31606 lineto
0.34127 0.28379 lineto
0.34623 0.26982 lineto
0.34871 0.26385 lineto
0.35119 0.25869 lineto
0.35367 0.25446 lineto
0.35615 0.25121 lineto
0.35739 0.24998 lineto
0.35863 0.24901 lineto
0.35987 0.24832 lineto
0.36111 0.24791 lineto
0.36235 0.24777 lineto
0.36359 0.24791 lineto
0.36483 0.24833 lineto
0.36607 0.24903 lineto
0.36731 0.25 lineto
0.36855 0.25124 lineto
0.37103 0.2545 lineto
0.37351 0.25874 lineto
0.37599 0.26391 lineto
0.38095 0.27658 lineto
0.40079 0.33952 lineto
0.40575 0.35257 lineto
0.40823 0.35797 lineto
0.41071 0.36248 lineto
0.41319 0.36602 lineto
0.41443 0.3674 lineto
0.41567 0.36852 lineto
0.41691 0.36937 lineto
0.41815 0.36994 lineto
0.41939 0.37024 lineto
0.42063 0.37025 lineto
0.42188 0.36999 lineto
0.42312 0.36946 lineto
0.42436 0.36864 lineto
0.4256 0.36756 lineto
0.43056 0.36064 lineto
0.43304 0.35573 lineto
0.43552 0.34998 lineto
0.44048 0.33636 lineto
0.4504 0.30433 lineto
0.45536 0.28823 lineto
0.46032 0.27362 lineto
0.46528 0.26158 lineto
0.46776 0.25679 lineto
0.47024 0.25296 lineto
0.47148 0.25141 lineto
0.47272 0.25013 lineto
Mistroke
0.47396 0.24912 lineto
0.4752 0.24838 lineto
0.47644 0.24791 lineto
0.47768 0.24772 lineto
0.47892 0.24781 lineto
0.48016 0.24818 lineto
0.5 0.28678 lineto
0.50992 0.31923 lineto
0.51984 0.34879 lineto
0.52232 0.35469 lineto
0.5248 0.35978 lineto
0.52728 0.36394 lineto
0.52852 0.36566 lineto
0.52976 0.36712 lineto
0.531 0.36832 lineto
0.53224 0.36925 lineto
0.53348 0.3699 lineto
0.53472 0.37029 lineto
0.53596 0.3704 lineto
0.5372 0.37023 lineto
0.53844 0.36978 lineto
0.53968 0.36906 lineto
0.54216 0.36682 lineto
0.5434 0.3653 lineto
0.54464 0.36353 lineto
0.5496 0.35409 lineto
0.55456 0.34142 lineto
0.55952 0.32644 lineto
0.56944 0.29389 lineto
0.5744 0.27865 lineto
0.57937 0.26558 lineto
0.58433 0.2556 lineto
0.58681 0.252 lineto
0.58805 0.25058 lineto
0.58929 0.24942 lineto
0.59053 0.24853 lineto
0.59177 0.2479 lineto
0.59301 0.24755 lineto
0.59425 0.24747 lineto
0.59549 0.24767 lineto
0.59673 0.24814 lineto
0.59797 0.24888 lineto
0.59921 0.24989 lineto
0.60045 0.25116 lineto
0.60169 0.25269 lineto
0.60417 0.25649 lineto
0.60913 0.26681 lineto
0.61905 0.29545 lineto
0.62897 0.32785 lineto
0.63393 0.34263 lineto
Mistroke
0.63889 0.35506 lineto
0.64137 0.36012 lineto
0.64385 0.36428 lineto
0.64509 0.36601 lineto
0.64633 0.36748 lineto
0.64757 0.3687 lineto
0.64881 0.36966 lineto
0.65005 0.37035 lineto
0.65129 0.37078 lineto
0.65253 0.37094 lineto
0.65377 0.37083 lineto
0.65501 0.37045 lineto
0.65625 0.3698 lineto
0.65749 0.36889 lineto
0.65873 0.36772 lineto
0.66121 0.36463 lineto
0.66369 0.36057 lineto
0.66865 0.34987 lineto
0.67857 0.321 lineto
0.68849 0.28892 lineto
0.69345 0.27437 lineto
0.69841 0.26215 lineto
0.70089 0.25717 lineto
0.70337 0.25306 lineto
0.70585 0.24988 lineto
0.70709 0.24865 lineto
0.70833 0.24768 lineto
0.70957 0.24696 lineto
0.71081 0.24649 lineto
0.71205 0.24629 lineto
0.71329 0.24634 lineto
0.71453 0.24665 lineto
0.71577 0.24722 lineto
0.71701 0.24804 lineto
0.71825 0.24912 lineto
0.72073 0.25199 lineto
0.72321 0.25579 lineto
0.72817 0.26591 lineto
0.7381 0.29362 lineto
0.74802 0.32519 lineto
0.75794 0.35282 lineto
0.76042 0.3583 lineto
0.7629 0.36303 lineto
0.76538 0.36695 lineto
0.76662 0.36859 lineto
0.76786 0.37 lineto
0.7691 0.37119 lineto
0.77034 0.37214 lineto
0.77158 0.37287 lineto
0.77282 0.37336 lineto
Mistroke
0.77406 0.37361 lineto
0.7753 0.37362 lineto
0.77654 0.3734 lineto
0.77778 0.37295 lineto
0.77902 0.37227 lineto
0.78026 0.37135 lineto
0.78274 0.36886 lineto
0.78522 0.36551 lineto
0.7877 0.36137 lineto
0.79762 0.33821 lineto
0.80754 0.30878 lineto
0.81746 0.27935 lineto
0.82242 0.2665 lineto
0.82738 0.25571 lineto
0.82986 0.25124 lineto
0.83234 0.24744 lineto
0.83482 0.24434 lineto
0.8373 0.24198 lineto
0.83854 0.24107 lineto
0.83978 0.24036 lineto
0.84102 0.23983 lineto
0.84226 0.23949 lineto
0.8435 0.23933 lineto
0.84474 0.23936 lineto
0.84598 0.23958 lineto
0.84722 0.23998 lineto
0.84846 0.24055 lineto
0.8497 0.24131 lineto
0.85218 0.24333 lineto
0.85466 0.24601 lineto
0.85714 0.24931 lineto
0.86706 0.26782 lineto
0.87698 0.29221 lineto
0.89683 0.34438 lineto
0.90675 0.36635 lineto
0.91171 0.37553 lineto
0.91667 0.38338 lineto
0.92163 0.38991 lineto
0.92659 0.39515 lineto
0.93155 0.39918 lineto
0.93403 0.40077 lineto
0.93651 0.4021 lineto
0.93899 0.40318 lineto
0.94147 0.40404 lineto
0.94271 0.40438 lineto
0.94395 0.40467 lineto
0.94519 0.40491 lineto
0.94643 0.40511 lineto
0.94767 0.40525 lineto
0.94891 0.40536 lineto
Mistroke
0.95015 0.40542 lineto
0.95139 0.40543 lineto
0.95263 0.40542 lineto
0.95387 0.40536 lineto
0.95511 0.40527 lineto
0.95635 0.40514 lineto
0.96131 0.40433 lineto
0.96379 0.40376 lineto
0.96627 0.4031 lineto
0.97619 0.39973 lineto
Mfstroke
grestore
grestore
grestore
MathSubEnd
grestore
% End of sub-graphic
% Start of sub-graphic
gsave
0.54329 0.01605 0.97619 0.2836 MathSubStart
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.97619 0.0680272 0.309017 0.103006 [
[(-10)] 0.29592 0.30902 0 2 Msboxa
[(-5)] 0.63605 0.30902 0 2 Msboxa
[(-3)] 0.96369 0 1 0 Msboxa
[(-2)] 0.96369 0.10301 1 0 Msboxa
[(-1)] 0.96369 0.20601 1 0 Msboxa
[(1)] 0.96369 0.41202 1 0 Msboxa
[(2)] 0.96369 0.51503 1 0 Msboxa
[(3)] 0.96369 0.61803 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 0.61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
%%Object: Graphics
[ ] 0 setdash
gsave
gsave
0.002 setlinewidth
0.02381 0.30902 moveto
0.02381 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.09184 0.30902 moveto
0.09184 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.15986 0.30902 moveto
0.15986 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.22789 0.30902 moveto
0.22789 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.29592 0.30902 moveto
0.29592 0.31527 lineto
stroke
grestore
[(-10)] 0.29592 0.30902 0 2 Mshowa
gsave
0.002 setlinewidth
0.36395 0.30902 moveto
0.36395 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.43197 0.30902 moveto
0.43197 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.5 0.30902 moveto
0.5 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.56803 0.30902 moveto
0.56803 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.63605 0.30902 moveto
0.63605 0.31527 lineto
stroke
grestore
[(-5)] 0.63605 0.30902 0 2 Mshowa
gsave
0.002 setlinewidth
0.70408 0.30902 moveto
0.70408 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.77211 0.30902 moveto
0.77211 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.84014 0.30902 moveto
0.84014 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.90816 0.30902 moveto
0.90816 0.31527 lineto
stroke
grestore
gsave
0.002 setlinewidth
0 0.30902 moveto
1 0.30902 lineto
stroke
grestore
gsave
0.002 setlinewidth
0.97619 0 moveto
0.98244 0 lineto
stroke
grestore
[(-3)] 0.96369 0 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.10301 moveto
0.98244 0.10301 lineto
stroke
grestore
[(-2)] 0.96369 0.10301 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.20601 moveto
0.98244 0.20601 lineto
stroke
grestore
[(-1)] 0.96369 0.20601 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.41202 moveto
0.98244 0.41202 lineto
stroke
grestore
[(1)] 0.96369 0.41202 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.51503 moveto
0.98244 0.51503 lineto
stroke
grestore
[(2)] 0.96369 0.51503 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0.61803 moveto
0.98244 0.61803 lineto
stroke
grestore
[(3)] 0.96369 0.61803 1 0 Mshowa
gsave
0.002 setlinewidth
0.97619 0 moveto
0.97619 0.61803 lineto
stroke
grestore
grestore
0 0 moveto
1 0 lineto
1 0.61803 lineto
0 0.61803 lineto
closepath
clip
newpath
gsave
gsave
[ 0.03 0.01 ] 0 setdash
gsave
0.004 setlinewidth
0.02381 0.43653 moveto
0.04365 0.25217 lineto
0.05357 0.16921 lineto
0.05605 0.15331 lineto
0.05853 0.13991 lineto
0.06101 0.12923 lineto
0.06225 0.12496 lineto
0.06349 0.12142 lineto
0.06473 0.11865 lineto
0.06597 0.11663 lineto
0.06721 0.11539 lineto
0.06845 0.11493 lineto
0.06969 0.11524 lineto
0.07093 0.11633 lineto
0.07217 0.1182 lineto
0.07341 0.12083 lineto
0.07589 0.12835 lineto
0.07837 0.13877 lineto
0.08333 0.16759 lineto
0.08829 0.20544 lineto
0.09325 0.24992 lineto
0.10317 0.34712 lineto
0.10813 0.39363 lineto
0.1131 0.43474 lineto
0.11806 0.4678 lineto
0.12054 0.48064 lineto
0.12302 0.49073 lineto
0.12426 0.49468 lineto
0.1255 0.49789 lineto
0.12674 0.50035 lineto
0.12798 0.50203 lineto
0.12922 0.50294 lineto
0.13046 0.50308 lineto
0.1317 0.50243 lineto
0.13294 0.50101 lineto
0.13418 0.49881 lineto
0.13542 0.49586 lineto
0.1379 0.48771 lineto
0.14038 0.4767 lineto
0.14286 0.46299 lineto
0.15278 0.38625 lineto
0.1627 0.29009 lineto
0.16766 0.24225 lineto
0.17262 0.19868 lineto
0.17758 0.16216 lineto
0.18006 0.14729 lineto
0.18254 0.13502 lineto
0.18378 0.12992 lineto
0.18502 0.12554 lineto
0.18626 0.12189 lineto
Mistroke
0.1875 0.119 lineto
0.18874 0.11687 lineto
0.18998 0.11551 lineto
0.19122 0.11492 lineto
0.19246 0.11512 lineto
0.1937 0.11609 lineto
0.19494 0.11783 lineto
0.19742 0.12362 lineto
0.19866 0.12763 lineto
0.1999 0.13238 lineto
0.20238 0.14397 lineto
0.20734 0.17485 lineto
0.2123 0.21431 lineto
0.22222 0.30846 lineto
0.23214 0.40276 lineto
0.2371 0.44238 lineto
0.24206 0.47349 lineto
0.24454 0.48521 lineto
0.24578 0.49002 lineto
0.24702 0.4941 lineto
0.24826 0.49745 lineto
0.2495 0.50003 lineto
0.25074 0.50185 lineto
0.25198 0.5029 lineto
0.25322 0.50317 lineto
0.25446 0.50266 lineto
0.2557 0.50138 lineto
0.25694 0.49932 lineto
0.25818 0.4965 lineto
0.25942 0.49294 lineto
0.2619 0.4836 lineto
0.26438 0.47147 lineto
0.26687 0.45674 lineto
0.27183 0.42045 lineto
0.28175 0.32932 lineto
0.29167 0.23309 lineto
0.29663 0.19074 lineto
0.30159 0.15592 lineto
0.30407 0.14205 lineto
0.30655 0.13085 lineto
0.30779 0.12631 lineto
0.30903 0.12251 lineto
0.31027 0.11945 lineto
0.31151 0.11714 lineto
0.31275 0.11561 lineto
0.31399 0.11485 lineto
0.31523 0.11486 lineto
0.31647 0.11565 lineto
0.31771 0.11722 lineto
0.31895 0.11955 lineto
Mistroke
0.32143 0.12648 lineto
0.32267 0.13104 lineto
0.32391 0.13632 lineto
0.32639 0.14892 lineto
0.33135 0.18156 lineto
0.34127 0.26857 lineto
0.35119 0.36571 lineto
0.35615 0.41041 lineto
0.36111 0.44867 lineto
0.36359 0.4646 lineto
0.36607 0.47806 lineto
0.36855 0.48882 lineto
0.37103 0.49672 lineto
0.37227 0.49956 lineto
0.37351 0.50164 lineto
0.37475 0.50295 lineto
0.37599 0.50348 lineto
0.37723 0.50325 lineto
0.37847 0.50224 lineto
0.37971 0.50046 lineto
0.38095 0.49791 lineto
0.38343 0.49059 lineto
0.38591 0.48037 lineto
0.38839 0.46744 lineto
0.39087 0.45198 lineto
0.40079 0.37043 lineto
0.41071 0.27359 lineto
0.41567 0.22698 lineto
0.42063 0.18555 lineto
0.4256 0.15189 lineto
0.42808 0.13866 lineto
0.43056 0.12812 lineto
0.4318 0.12391 lineto
0.43304 0.12043 lineto
0.43428 0.1177 lineto
0.43552 0.11572 lineto
0.43676 0.1145 lineto
0.438 0.11405 lineto
0.43924 0.11437 lineto
0.44048 0.11545 lineto
0.46032 0.22475 lineto
0.47024 0.31962 lineto
0.48016 0.41191 lineto
0.48512 0.44987 lineto
0.4876 0.46571 lineto
0.49008 0.47912 lineto
0.49256 0.4899 lineto
0.4938 0.49424 lineto
0.49504 0.49788 lineto
0.49628 0.50078 lineto
Mistroke
0.49752 0.50294 lineto
0.49876 0.50436 lineto
0.5 0.50502 lineto
0.50124 0.50492 lineto
0.50248 0.50407 lineto
0.50372 0.50247 lineto
0.50496 0.50013 lineto
0.5062 0.49705 lineto
0.50744 0.49325 lineto
0.50992 0.48354 lineto
0.51488 0.4563 lineto
0.51984 0.42009 lineto
0.52976 0.33006 lineto
0.53968 0.23501 lineto
0.54464 0.19273 lineto
0.5496 0.15737 lineto
0.55208 0.14295 lineto
0.55456 0.131 lineto
0.55704 0.12168 lineto
0.55828 0.11805 lineto
0.55952 0.11513 lineto
0.56076 0.11291 lineto
0.562 0.11142 lineto
0.56324 0.11066 lineto
0.56448 0.11062 lineto
0.56572 0.11131 lineto
0.56696 0.11273 lineto
0.5682 0.11486 lineto
0.56944 0.1177 lineto
0.57192 0.12545 lineto
0.5744 0.13587 lineto
0.57937 0.16403 lineto
0.58929 0.24315 lineto
0.59921 0.33701 lineto
0.60913 0.42479 lineto
0.61409 0.46031 lineto
0.61657 0.47511 lineto
0.61905 0.48766 lineto
0.62153 0.49782 lineto
0.62277 0.50197 lineto
0.62401 0.50547 lineto
0.62525 0.50832 lineto
0.62649 0.51051 lineto
0.62773 0.51203 lineto
0.62897 0.51289 lineto
0.63021 0.51308 lineto
0.63145 0.51261 lineto
0.63269 0.51147 lineto
0.63393 0.50968 lineto
0.63641 0.50415 lineto
Mistroke
0.63765 0.50044 lineto
0.63889 0.49612 lineto
0.65873 0.36125 lineto
0.67857 0.18721 lineto
0.68353 0.15268 lineto
0.68849 0.12491 lineto
0.69097 0.11388 lineto
0.69345 0.10488 lineto
0.69469 0.10117 lineto
0.69593 0.09798 lineto
0.69717 0.09533 lineto
0.69841 0.09322 lineto
0.69965 0.09164 lineto
0.70089 0.0906 lineto
0.70213 0.0901 lineto
0.70337 0.09013 lineto
0.70461 0.09069 lineto
0.70585 0.09177 lineto
0.70709 0.09337 lineto
0.70833 0.09548 lineto
0.71329 0.10884 lineto
0.71577 0.1183 lineto
0.71825 0.12948 lineto
0.72817 0.18864 lineto
0.7381 0.2637 lineto
0.75794 0.42136 lineto
0.76786 0.4874 lineto
0.77282 0.51495 lineto
0.77778 0.53852 lineto
0.78274 0.55809 lineto
0.7877 0.57377 lineto
0.79266 0.58578 lineto
0.79514 0.59051 lineto
0.79762 0.59444 lineto
0.8001 0.59761 lineto
0.80258 0.60008 lineto
0.80382 0.60106 lineto
0.80506 0.60189 lineto
0.8063 0.60256 lineto
0.80754 0.60308 lineto
0.80878 0.60346 lineto
0.81002 0.6037 lineto
0.81126 0.60381 lineto
0.8125 0.6038 lineto
0.81374 0.60367 lineto
0.81498 0.60342 lineto
0.81622 0.60307 lineto
0.81746 0.6026 lineto
0.85714 0.55624 lineto
0.89683 0.49714 lineto
Mistroke
0.93651 0.44906 lineto
0.97619 0.41303 lineto
Mfstroke
grestore
grestore
grestore
MathSubEnd
grestore
% End of sub-graphic
grestore
% End of Graphics
MathPictureEnd
end

grestore


/TeXDict 200 dict def   % define a working dictionary
TeXDict begin           % start using it.

/bdf { bind def } def
                             % internal units are in "dots" (300/inch)
/Inch  {Resolution mul} bdf  % converts inches to "dots"
/Dots {72 div Resolution mul} bdf  % converts points to "dots"

/dopage
  { 72 Resolution div dup neg scale          % set scaling to 1.
    translate
  } bdf

/@letter
  { Resolution dup -10 mul dopage } bdf

/@note { @letter } bdf

/@a4
  { Resolution dup -10.6929133858 mul dopage } bdf

/@translate { translate } bdf

/@scale { scale } bdf

/@rotate { rotate } bdf

/@landscape
  { [ 0 1 -1 0 0 0 ] concat
    Resolution dup dopage } bdf

/@legal
  { Resolution dup -13 mul dopage } bdf

/@manualfeed
   { statusdict /manualfeed true put
   } bdf

        % n @copies -   set number of copies
/@copies
   { /#copies exch def
   } bdf

/@FontMatrix [1 0 0 -1 0 0] def
/@FontBBox [0 0 1 1] def

%%%%%%%%%%%%%%%%%%%% Procedure Definitions %%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   This procedure has changed a lot.  We make a new name for the
%   font, and define the string that we were given to select that
%   font.
%
/dmystr (ZZf@@) def       % define a place to put the new name
/newname {dmystr cvn} bdf % make it easy to get that name
/df       % id df -         -- initialize a new font dictionary
  { /fontname exch def
    dmystr 2 fontname cvx (@@@) cvs putinterval  % put name in template
    newname 7 dict def          % allocate new font dictionary
    newname load begin
        /FontType 3 def
	/FontMatrix @FontMatrix def
	/FontBBox @FontBBox def
        /BitMaps 256 array def
        /BuildChar {CharBuilder} def
        /Encoding TeXEncoding def
        end
    fontname { /foo setfont }       %  dummy macro to be filled in
       2 array copy cvx def         %  have to allocate a new one
    fontname load                   %  now we change it
       0 dmystr 5 string copy       %  get a copy of the font name
       cvn cvx put                  %  and stick it in
  } bdf

/dfe { newname newname load definefont setfont } bdf

% the following is the only character builder we need.  it looks up the
% char data in the BitMaps array, and paints the character if possible.
% char data  -- a bitmap descriptor -- is an array of length 6, of
%          which the various slots are:

/ch-image {ch-data 0 get} bdf   % the hex string image
/ch-width {ch-data 1 get} bdf   % the number of pixels across
/ch-height {ch-data 2 get} bdf  % the number of pixels tall
/ch-xoff  {ch-data 3 get} bdf   % number of pixels below origin
/ch-yoff  {ch-data 4 get} bdf   % number of pixels to left of origin
/ch-tfmw  {ch-data 5 get} bdf   % spacing to next character

/CharBuilder    % fontdict ch Charbuilder -     -- image one character
     {save 3 1 roll exch /BitMaps get exch get /ch-data exch def
      ch-data null ne
      {ch-tfmw 0 ch-xoff neg ch-yoff neg ch-width ch-xoff sub ch-height ch-yoff
sub
            setcachedevice
        ch-width ch-height true [1 0  0 1  ch-xoff ch-yoff]
            {ch-image} imagemask
     }if
     restore
  } bdf
                % in the following, the font-cacheing mechanism requires that
                % a name unique in the particular font be generated

/dc            % char-data ch dc -    -- define a new character bitmap in
%%current font
  { /ch-code exch def
% ++oystr 12-Feb-86++
    dup 0 get
    length 1 lt
      { pop [ <00> 1 1 0 0 8.00 ] } % replace <> with null
    if
% --oystr 12-Feb-86--
    /ch-data exch def
    newname load /BitMaps get ch-code ch-data put
  } bdf

/bop           % bop -              -- begin a brand new page
  {
    gsave /SaveImage save def
    0 0 moveto
  } bdf

/eop           % - eop -              -- end a page
  { % eop-aux  % -- to observe VM usage
    clear SaveImage restore
    showpage grestore
  } bdf

/@start         % - @start -            -- start everything
  {
    /Resolution exch def
    /TeXEncoding 256 array def
    0 1 255 {TeXEncoding exch 1 string dup 0 3 index put cvn put} for
  } bdf

%/p { show } bdf        %  the main character setting routine
% Changed 1/4/90 by Chuck Parons MITLNS to compensate for "light"
% printout on DEC ScriptPrinter everything is now printed twice! moved
% vertically by half a dot
/p {dup show dup stringwidth neg exch neg exch rmoveto
    0 .5 rmoveto show
    0 -.5 rmoveto} bdf


/RuleMatrix [ 1 0 0 -1 0 -1 ] def % things we need for rules
/BlackDots 8 string def
/v {                   % can't use ...fill; it makes rules too big
   gsave
      currentpoint translate
      false RuleMatrix { BlackDots } imagemask
   grestore
} bdf
/a { moveto } bdf    % absolute positioning
/delta 0 def         % we need a variable to hold space moves
%
%   The next ten macros allow us to make horizontal motions which
%   are within 4 of the previous horizontal motion with a single
%   character.  These are typically used for spaces.
%
/tail { dup /delta exch def 0 rmoveto } bdf
/b { exch p tail } bdf
/c { p delta 4 sub tail } bdf
/d { p delta 3 sub tail } bdf
/e { p delta 2 sub tail } bdf
/f { p delta 1 sub tail } bdf
/g { p delta 0 rmoveto } bdf
/h { p delta 1 add tail } bdf
/i { p delta 2 add tail } bdf
/j { p delta 3 add tail } bdf
/k { p delta 4 add tail } bdf
%
%   These next allow us to make small motions (-4..4) cheaply.
%   Typically used for kerns.
%
/l { p -4 0 rmoveto } bdf
/m { p -3 0 rmoveto } bdf
/n { p -2 0 rmoveto } bdf
/o { p -1 0 rmoveto } bdf
/q { p 1 0 rmoveto } bdf
/r { p 2 0 rmoveto } bdf
/s { p 3 0 rmoveto } bdf
/t { p 4 0 rmoveto } bdf
%
%   w is good for small vertical positioning.  x is good for small x
%   positioning.  And y is good for a print followed by a move.
%
/w { 0 rmoveto } bdf
/x { 0 exch rmoveto } bdf
/y { 3 -1 roll p moveto } bdf
%
%   These two commands bracket sections of downloaded characters.
%
/bos { /section save def } bdf
/eos { clear section restore } bdf

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%     the \special command junk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   The structure of the PostScript produced by dvi2ps for \special is:
%         @beginspecial
%           - any number of @hsize, @hoffset, @hscale, etc., commands
%         @setspecial
%           - the users file of PostScript commands
%         @endspecial

% The following are user settable options from the \special command.
/SDict 200 dict def
SDict begin

/@SpecialDefaults
  { /hs 8.5 Inch def
    /vs 11 Inch def
    /ho 0 def
    /vo 0 def
    /hsc 1 def
    /vsc 1 def
    /ang 0 def
    /CLIP false def
    /BBcalc false def
  } bdf

%       d @hsize   -    specify a horizontal clipping dimension
/@hsize {/hs exch def /CLIP true def} bdf
/@vsize {/vs exch def /CLIP true def} bdf
%       d @hoffset -    specify a shift for the drwgs
/@hoffset {/ho exch def} bdf
/@voffset {/vo exch def} bdf
%       s @hscale  -    set scale factor
/@hscale {@scaleunit div /hsc exch def} bdf
/@vscale {@scaleunit div /vsc exch def} bdf
%       a @angle   -    set rotation angle
/@angle {/ang exch def} bdf
%
%   This definition sets up the units that hscale/vscale are in.
%   For certain sites this might require change, but it is
%   recommended instead that any macro packages that require
%   hscale/vscale set the units appropriately via
%
%   \special{! /@scaleunit 1 def }
%
%   if global, or
%
%   \special{" /@scaleunit 1 def }
%
%   before each instance if multiple macro packages with
%   different requirements are being used.
%
/@scaleunit 100 def
%
%   Here we handle bounding box calculations, if necessary.
%
/@rwi { 10 div /rwi exch def } bdf
/@llx { /llx exch def } bdf
/@lly { /lly exch def } bdf
/@urx { /urx exch def } bdf
/@ury { /ury exch def /BBcalc true def } bdf
/@setclipper
  { BBcalc
      { rwi urx llx sub div dup scale
        llx neg lly neg translate }
      { hsc vsc scale }
    ifelse
    CLIP
      { newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
        closepath clip }
      { initclip }
    ifelse
  } bdf

end

/@MacSetUp
  { userdict /md known  % if md is defined
      { userdict /md get type /dicttype eq      % and if it is a dictionary
	{
	md begin                             % then redefine some stuff
	/letter {} def
	/note {} def
	/legal {} def
	/od{txpose
	    1 0 mtx defaultmatrix dtransform exch atan/pa exch def
	    newpath clippath mark
	    {transform{itransform moveto}}
	    {transform{itransform lineto}}
	    { 6 -2 roll transform
	      6 -2 roll transform
	      6 -2 roll transform
	      { itransform 6 2 roll
		itransform 6 2 roll
		itransform 6 2 roll
		curveto
	      }
	    }
	    {{closepath}}
	    pathforall newpath counttomark array astore /gc xdf
	    pop ct 39 0 put
	    10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if
	}def
	/txpose{
	    pxs pys scale ppr aload pop
	    por {
		noflips {
		    pop exch neg exch translate pop 1 -1 scale
		}if
		xflip yflip and {
		    pop exch neg exch translate 180 rotate 1 -1 scale
		    ppr 3 get ppr 1 get neg sub neg ppr 2 get
		    ppr 0 get neg sub neg translate
		}if
		xflip yflip not and {
		    pop exch neg exch translate pop 180 rotate
		    ppr 3 get ppr 1 get neg sub neg 0 translate
		}if
		yflip xflip not and {
		    ppr 1 get neg ppr 0 get neg translate
		}if
	    }
	    {
		noflips {
		    translate pop pop 270 rotate 1 -1 scale
		}if
		xflip yflip and {
		    translate pop pop 90 rotate 1 -1 scale
		    ppr 3 get ppr 1 get neg sub neg ppr 2 get
		    ppr 0 get neg sub neg translate
		}if
		xflip yflip not and {
		    translate pop pop 90 rotate ppr 3 get
		    ppr 1 get neg sub neg 0 translate
		}if
		yflip xflip not and {
		    translate pop pop 270 rotate ppr 2 get
		    ppr 0 get neg sub neg 0 exch translate
		}if
	    }ifelse
	    scaleby96 {
		ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy
		translate .96 dup scale neg exch neg exch translate
	    }if
	}def
	/cp {pop pop showpage pm restore}def
        end
      }if
    } if
  } def

%
%   We need the psfig macros.
%
% All software, documentation, and related files in this distribution of
% psfig/tex are Copyright (c) 1987 Trevor J. Darrell
%
% Permission is granted for use and non-profit distribution of psfig/tex
% providing that this notice be clearly maintained, but the right to
% distribute any portion of psfig/tex for profit or as part of any commercial
% product is specifically reserved for the author.
%
%
% psfigTeX PostScript Prolog
% $Header: tex.ps,v 1.15 88/01/21 23:44:40 van Exp $
%
/psf$TeXscale { 65536 div } def

%  x y bb-llx bb-lly bb-urx bb-ury startFig -
/startTexFig {
	/psf$SavedState save def
	userdict maxlength dict begin

        Resolution 72 div dup neg scale
        currentpoint translate    %set the current point as the user's origin

	/psf$ury exch psf$TeXscale def
	/psf$urx exch psf$TeXscale def
	/psf$lly exch psf$TeXscale def
	/psf$llx exch psf$TeXscale def
	/psf$y exch psf$TeXscale def
	/psf$x exch psf$TeXscale def

	currentpoint /psf$cy exch def /psf$cx exch def

	/psf$sx psf$x psf$urx psf$llx sub div def 	% scaling for x
	/psf$sy psf$y psf$ury psf$lly sub div def	% scaling for y

	psf$sx psf$sy scale			% scale by (sx,sy)

	psf$cx psf$sx div psf$llx sub
	psf$cy psf$sy div psf$ury sub translate

	/showpage {
	} def
	/erasepage {
	} def
	/copypage {
	} def
	@MacSetUp
} def

% llx lly urx ury doclip -	(args in figure coordinates)
/doclip {
        psf$llx psf$lly psf$urx psf$ury
	currentpoint 6 2 roll
	newpath 4 copy
	4 2 roll moveto
	6 -1 roll exch lineto
	exch lineto
	exch lineto
	closepath clip
	newpath
	moveto
} def
% - endTexFig -
/endTexFig { end psf$SavedState restore } def

% this will be invoked as the result of a \special command (for the
% inclusion of PostScript graphics).  The basic idea is to change all
% scaling and graphics back to defaults, but to shift the origin
% to the current position on the page.

/@beginspecial          % - @beginspecial -     -- enter special mode
  { SDict begin
    /SpecialSave save def
    gsave
    Resolution 72 div dup neg scale
    currentpoint translate    %set the current point as the user's origin
    @SpecialDefaults    % setup default offsets, scales, sizes, and angle
  } bdf

/@setspecial    % to setup user specified offsets, scales, sizes (for clipping)
  {
    ho vo translate @setclipper ang rotate
    /showpage {} def
    newpath
  } bdf

/@endspecial            % - @endspecial -       -- leave special mode
  { grestore clear SpecialSave restore
    end
  } bdf
/@defspecial
  {
    SDict begin
  } bdf
/@fedspecial
  {
    end
  } bdf

%%% macros for tpic
/li             % x y li -              -- draw line to
  { lineto
  } bdf

/rl             % dx dy rl -            -- draw relative line
  { rlineto
  } bdf

/rc             % x0 y0 x1 y1 y2 y2 rc  -- draw bezier curve
  { rcurveto
  } bdf

/np		% np -			-- start a new path and save currenpoint
  { /SaveX currentpoint /SaveY exch def def   % remember current point
    newpath
  } bdf

/st             % st -                  -- draw the last path and restore
%%currentpoint
  { stroke
    SaveX SaveY moveto                  % restore the current point
  } bdf

/fil             % fil                    -- fill the last path and restore
%%currentpoint
  { fill
    SaveX SaveY moveto                  % restore the current point
  } bdf

/ellipse        % xc yc xrad yrad startAngle endAngle ellipse
    {
        /endangle exch def
        /startangle exch def
        /yrad exch def
        /xrad exch def

        /savematrix matrix currentmatrix def

        translate
        xrad yrad scale
        0 0 1 startangle endangle arc
        savematrix setmatrix
    } bdf
%%% end of macros for tpic
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%     the lyman stuff for draft
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
/Mtrx 6 array def
/@DRAFTF { gsave initmatrix newpath
       /Helvetica-BoldOblique findfont setfont
       (Draft) dup stringwidth pop 8.875 exch div dup 72 mul dup scale
       52.3 rotate 2.5 exch div -.35 translate
       0.95 setgray
       0 0 moveto show
       grestore } bdf
/@DRAFTM { gsave initmatrix newpath
       /Helvetica-BoldOblique findfont 72 scalefont setfont
       0.95 setgray
       576 702 moveto
       (Draft) stringwidth exch neg exch neg rmoveto
       (Draft) show
       grestore } bdf

% n @bop0 -              -- begin the char def section of a new page
/@bop0 { pop } bdf

% n @bop1 -              -- begin a brand new page
/@bop1 { pop Mtrx setmatrix /SaveImage save def } def

/@draft {
       /bop           % bop -              -- begin a brand new page
         { @DRAFTF
          /@DRAFTF { @DRAFTM } def
         gsave /SaveImage save def
         0 0 moveto
         } bdf
  } bdf
%%% end of macros for lyman
end                     % revert to previous dictionary
%%EndDocument
TeXDict begin 300 @start /fa df[<7FFFE07FFFE001F80001F80001F80001F80001F80001
F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001
F80001F80001F80001F80001F80001F80001F800FFF800FFF80001F800007800001800>19 32
-4 0 28]49 dc[<FFFFE000FFFFE00007F0000007F0000007F0000007F0000007F0000007F000
0007F0000007F0000007F0180007F0180007F0180007F0180007F0380007F0780007FFF80007FF
F80007F0780007F0380007F0180007F0180007F0180C07F0180C07F0000C07F0000C07F0001C07
F0001C07F0001807F0003807F0007807F001F8FFFFFFF8FFFFFFF8>30 34 -2 0 35]70 dc[<00
FF0003FFC00FC0701F00303E00187E00007C00007C0000FC0000FC0000FC0000FFFFF8FFFFF8FC
00F8FC00F87C00F87C00F03E01F01E01E00F87C007FF8000FE00>21 22 -2 0 26]101 dc[<01
FF000FFFE03F01F87C007C78003CF0001EF0001EF0001EF0001E70003E3800FC1FFFFC0FFFF81F
FFF03FFF8038000030000030000013FC001FFF001F0F803E07C03C03C07C03E07C03E07C03E07C
03E07C03E03C03C03E07CF1F0F8F0FFF7F03FC1E>24 33 -2 11 28]103 dc[<FFE0FFE01F001F
001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F00FF00FF000000
0000000000000000000000001C003F007F007F007F003F001C00>11 36 -2 0 16]105 dc[<FF
F000FFF0001F00001F00001F00001F00001F00001F00001F00001F00001F00001F00001F00001F
00001F00001F83C01E87E01E87E01EC7E01E67E0FE3FC0FE1F00>19 22 -2 0 23]114 dc[<01
FC3FC007FF3FC00F81BE001F00FE001F007E001F003E001F003E001F003E001F003E001F003E00
1F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E
00FF01FE00FF01FE00>26 22 -2 0 31]117 dc dfe /fb df[<FFFFFFFCFFFFFFFC0000000000
0000000000000000000000000000000000000000000000000000000000000C0000003C000000F0
000003C000000F0000003C000000F0000007C000001F00000078000001E00000078000001E0000
0078000000E0000000780000001E0000000780000001E0000000780000001F00000007C0000000
F00000003C0000000F00000003C0000000F00000003C0000000C>30 39 -4 7 39]20 dc[<03F0
0003F8000FFE000FFC001807003F8F003001807E03802000C0F80080400061F00040400033E000
4080001FC0006080001F80002080000F80002080001F00002080003E00002080003F000020C000
7F0000204000F98000404001F0C000402003E0600080380FC03001801E3F801C030007FE000FFE
0003F80001F800>43 21 -3 0 50]49 dc dfe /fc df[<030000070000038000038000038000
01800001800001C00000C00000C00000C00000C000006000006000806000802000402000302000
3FFFF81FFFF807FFF8>21 21 -2 0 21]28 dc[<4020101008080404040474FCFCF870>6 15 -4
10 14]59 dc[<0F80000030E0000040380000401C0000F01E0000F80F0000F80F000078078000
3807800000078000000780000003C0000003C0000003C0000003C0000001E0000001E0000001E0
000001E0000000F0000000F0000000F0000000F000000078000000780000007800000078000000
3C0000003C0000003C0000003C0000001E0000001E0000001E000007FFE0>27 35 -3 1 27]74
dc[<FFFC07FF8007C000FC0003C000780003C000780003C000F80003C000F00001E000F00001E0
01E00001E001E00001E001E00000F003C00000F003C00000F007C00000F8078000007C07800000
7A0F000000788F000000785F0000003C3E0000003C1E0000003C040000003C020000001E010000
001E008000001E004000001E001000000F000800000F000400000F000200000F00018000078000
C000078000600007800078007FFC01FF>40 34 -2 0 41]75 dc[<1C0064006200E200E200E200
710070007000700038003800380038001C001C001C001C000E000E000E000E0007000700070007
00038003800380038001C001C001C01FC001E0>11 35 -3 0 15]108 dc[<3000F00070018800
3801840038038200380382003801C2001C01C1001C01C0001C00E0001C00E0000E0070000E0070
000E0070000E0070008700380087003800878038008740380047203000461860003C07C000>25
21 -2 0 29]110 dc[<3C1F00423080E17040F0E020F0E01060E0100070080070000070000070
00003800003800003800003800201C00201C18201C3C101A3C081A1C06310C01E0F0>22 21 -2
0 28]120 dc dfe /fd df[<7FE3FF800700780007007000070070000700700007007000070070
000700700007007000070070000700700007007000070070000700700007007000070070000700
7000070070000700700007007000FFFFFFC0070070000700700007007000070070000700700007
0070000700700007007000070070000380F0780180F87800C07C7800706E30001F83E0>29 35 0
0 28]11 dc[<7FE1FF800700380007003800070038000700380007003800070038000700380007
003800070038000700380007003800070038000700380007003800070038000700380007003800
0700380007007800FFFFF800070000000700000007000000070000000700000007000000070000
000700300007007800038078000180380000C0100000702000001FC000>25 35 0 0 27]12 dc[
<00200040008001000300060004000C000C00180018003000300030007000600060006000E000
E000E000E000E000E000E000E000E000E000E000E000E000E00060006000600070003000300030
00180018000C000C000400060003000100008000400020>11 50 -4 13 19]40 dc[<80004000
2000100018000C000400060006000300030001800180018001C000C000C000C000E000E000E000
E000E000E000E000E000E000E000E000E000E000E000C000C000C001C001800180018003000300
0600060004000C0018001000200040008000>11 50 -3 13 19]41 dc[<402010100808040404
0474FCFCF870>6 15 -4 10 14]44 dc[<70F8F8F870>5 5 -4 0 14]46 dc[<FFFE07C0038003
800380038003800380038003800380038003800380038003800380038003800380038003800380
038003800380038003800380F3800F8003800080>15 33 -4 0 24]49 dc[<FFFFC07FFFC03FFF
C030004018006008002004002002002001000001800000C000006000003000001800001C00000E
000007000007800003C00003C00003E02003E0F801E0F801E0F801E0F003E08003E04003C04003
C02007801007000C1C0003F000>19 33 -2 0 24]50 dc[<03F0000C1C00100F00200780400780
4003C0F003C0F803E0F803E07003E02003E00003E00003C00003C0000780000780000F00001C00
03F000003800000E00000F000007000007800007803807C07807C07803C07807C04007C0200780
1007000C1E0003F800>19 34 -2 1 24]51 dc[<03F0000C1C001006002007004003804003C080
01C0E001C0F001E0F001E07001E00001E00001E00001E00001E00001C00001C010038018038014
0700130E0010F80010000010000010000010000010000010000013E0001FF8001FFE001FFF001E
0700100080>19 34 -2 1 24]53 dc[<01F000070C000C06001C03001803803801C03801C07001
E07001E07001E0F001E0F001E0F001E0F001E0F001E0F001C0F801C0F80380F40300F40600F30C
00F0F8007000007000007800003800003800001801801C03C00E03C00601C003008001C100007E
00>19 34 -2 1 24]54 dc[<70F8F8F870000000000000000000000070F8F8F870>5 21 -4 0
14]58 dc[<FFFFFFFEFFFFFFFE0000000000000000000000000000000000000000000000000000
000000000000FFFFFFFEFFFFFFFE>31 12 -3 -6 38]61 dc[<FF800FFF3E0001F80C0000F00C
0000F0040001E0040001E0040003E0020003C0020003C0030007C0010007800100078000FFFF00
00800F0000800F0000401E0000401E0000401E0000203C0000203C0000203C0000107800001078
000010F8000008F0000008F000000DF0000005E0000005E0000003C0000003C0000003C0000001
80000001800000018000>32 35 -2 0 37]65 dc[<000FE00000783C0000E00E0003C007800780
03C00F0001E00F0001E01E0000F03E0000F83C0000787C00007C7C00007C7800003CF800003EF8
00003EF800003EF800003EF800003EF800003EF800003EF800003EF800003E7800003C7800003C
7C00007C7C00007C3C0000783C0000781E0000F00E0000E00F0001E0078003C003C0078000E00E
0000783C00000FE000>31 36 -3 1 38]79 dc[<03FFFF00000FC0000007800000078000000780
000007800000078000000780000007800000078000000780000007800000078000000780000007
800000078000000780000007800000078000000780000007800000078000000780008007800480
0780048007800480078004C007800C40078008400780084007800860078018780780787FFFFFF8
>30 34 -2 0 35]84 dc[<0FC1E03C2390781708F00F08F00708F00708F007087007007807003C
07001E070007C70000FF000007000007000007001807003C0E003C0C001838000FE000>21 21
-2 0 24]97 dc[<083F000C41C00C80600F00700E00380E003C0E001C0E001E0E001E0E001E0E
001E0E001E0E001E0E001E0E001C0E003C0E00380F00300E80600E61C00E1F000E00000E00000E
00000E00000E00000E00000E00000E00000E00000E00000E00001E0000FE00000E0000>23 35
-1 0 27]98 dc[<01F8000706000C01001C0080380040780040700000F00000F00000F00000F0
0000F00000F00000F000007000007800003803001C07800C078007030001FE00>18 21 -2 0 22
]99 dc[<01F0FE070CF00C02E01801E03800E07800E07000E0F000E0F000E0F000E0F000E0F000
E0F000E0F000E07000E07800E03800E01C01E00C02E00704E001F8E00000E00000E00000E00000
E00000E00000E00000E00000E00000E00000E00000E00001E0000FE00000E0>23 35 -2 0 27]
100 dc[<00FC000703000E00801C0040380020780020700000F00000F00000F00000F00000F000
00FFFFE0F000E07000E07801E03801C01C01C00C038007070001FC00>19 21 -1 0 22]101 dc[
<7FF8078007000700070007000700070007000700070007000700070007000700070007000700
0700FFF8070007000700070007000700070007000700030F038F018F00C6003C>16 35 0 0 15]
102 dc[<03FE000E03803800E0600030600030C00018C00018C000184000186000303800F00FFF
E00FFFC01FFE0018000018000010000010000019F0000F1C000E0E001C07001C07003C07803C07
803C07803C07801C07001C07000E0E18071E1801F198000070>21 33 -1 11 24]103 dc[<FFE7
FF0E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00
700E00700E00700F00700F00700E80E00E60C00E1F800E00000E00000E00000E00000E00000E00
000E00000E00000E00000E00000E00001E0000FE00000E0000>24 35 -1 0 27]104 dc[<FFC0
0E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E001E00FE000E
00000000000000000000000000000000001C001E003E001E001C00>10 34 -1 0 14]105 dc[<
FFE00E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E
000E000E000E000E000E000E000E000E000E000E000E000E001E00FE000E00>11 35 -1 0 14]
108 dc[<FFE3FF8FFE0E003800E00E003800E00E003800E00E003800E00E003800E00E003800E0
0E003800E00E003800E00E003800E00E003800E00E003800E00E003800E00E003800E00E003800
E00E003800E00F003C00E00F003C00E01E807201C0FE60E183800E1FC07F00>39 21 -1 0 42]
109 dc[<FFE7FF0E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00
700E00700E00700E00700E00700F00700F00701E80E0FE60C00E1F80>24 21 -1 0 27]110 dc[
<01FC000707000E03801C01C03800E07800F0700070F00078F00078F00078F00078F00078F000
78F000787000707000703800E01800C00C018007070001FC00>21 21 -1 0 24]111 dc[<FFE0
000E00000E00000E00000E00000E00000E00000E00000E00000E00000E3F000E41C00E80E00F00
700E00380E003C0E003C0E001E0E001E0E001E0E001E0E001E0E001E0E001E0E001C0E003C0E00
380F00700E8060FE61C00E1F00>23 31 -1 10 27]112 dc[<000FFE0000E00000E00000E00000
E00000E00000E00000E00000E00000E001F0E0070CE00C02E01C01E03801E07800E07000E0F000
E0F000E0F000E0F000E0F000E0F000E0F000E07800E07800E03801E01C01600E026007046001F8
20>23 31 -2 10 26]113 dc[<FFF00F000E000E000E000E000E000E000E000E000E000E000E00
0E000E000F000F060F0F1E8FFE460E3C>16 21 -1 0 19]114 dc[<8FC0D030E018C008C00C80
0C800C801C003C01F80FF03FE07F80F000E008C008C008C018601830780F88>14 21 -2 0 19]
115 dc[<01F0030807080E040E040E040E040E040E040E000E000E000E000E000E000E000E000E
000E000E00FFF83E001E000E000600060006000200020002000200>14 31 -1 0 19]116 dc[<
00FC7F0382780601700E00F00E00F00E00700E00700E00700E00700E00700E00700E00700E0070
0E00700E00700E00700E00700E00701E00F0FE07F00E0070>24 21 -1 0 27]117 dc[<001000
00380000380000380000740000740000E20000E20000E20001C10001C100038080038080038080
0700400700400E00200E00200E00301E0078FFC1FE>23 21 -1 0 26]118 dc[<006006000060
06000060060000F00F0000F00F0000F00D0001C81C8001C81C8001C81880038438400384384003
8430400702702007027020070260200E01E0100E01E0100E01C0181C01C0181E01E03CFF8FF8FF
>32 21 -1 0 35]119 dc[<3C0000430000F18000F08000F04000004000002000002000002000
00100000100000380000380000380000740000740000E20000E20000E20001C10001C100038080
0380800380800700400700400E00200E00200E00301E0078FFC1FE>23 31 -1 10 26]121 dc
dfe /fe df[<003000000038000000780000007C000000CC000000CE0000018600000187000003
03000003038000060180000601C0000C00C0000C00E00018006000D80070003000300000003800
00001C0000000C0000000E000000060000000700000003000000038000000180000001C0000000
C0000000E00000006000000070000000300000003800000018>29 34 -1 32 29]112 dc dfe
/ff df[<FF80FF001C003C000E0038000E0038000E0070000E0070000700E0000700E0000701C0
000781C00003A3800003938000038F00000383000001C1000001C0800001C0200001C0100000E0
080000E0020000E0010000E001C007FE03F8>29 23 -1 0 30]75 dc[<78F000C50800E70400C3
020003020003000001800001800001800041800020C30020C38011A1800F1F00>17 14 -1 0 20
]120 dc dfe /fg df[<0102040C1818303070606060E0E0E0E0E0E0E0E0E0E060606070303018
180C040201>8 34 -3 9 14]40 dc[<8040203018180C0C0E0606060707070707070707070706
06060E0C0C181830204080>8 34 -2 9 14]41 dc dfe end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300
 TeXDict begin @letter
%%EndSetup
%%Page: 17 1
 bop 0 342 a fc(J)5 b fd(\()p fc(x)p fd(\))956 b fc(K)t fd(\()p fc(x)p fd(\))
1831 730 y fc(x)780 861 y(x)725 1031 y ff(K)r fg(\()p ff(x)p fg(\))725 1043 y
80 2 v 741 1047 a fe(p)p 20 2 v 770 1071 a ff(x)1741 1031 y(K)r fg(\()p ff
(x)p fg(\))1741 1043 y 80 2 v 1756 1047 a fe(p)p 20 2 v 1785 1071 a ff(x)791
1366 y fc(l)q(n)8 b(x)938 b(l)q(n)8 b(x)0 1820 y fa(Figure)24 b(1)p fd(:)30 b
fc(J)5 b fd(\()p fc(x)p fd(\))p fc(;)28 b(K)t fd(\()p fc(x)p fd(\))21 b(as)h
(fun)o(tions)f(of)g fc(x)p fd(,)h(and)1012 1797 y ff(K)r fg(\()p ff(x)p fg
(\))1012 1809 y 80 2 v 1027 1813 a fe(p)q 20 2 v 1057 1838 a ff(x)1118 1820 y
fd(as)f(a)h(fun)o(tion)e(of)i fc(l)q(nx)e fd(for)i fc(x)f fb(\024)h fd(1.)36 b
(These)0 1920 y(graphs)20 b(are)f(obtained)g(b)o(y)f(in)o(tegrating)g(n)o(ume\
rically)e(eqs.)28 b(\(A2\),)19 b(starting)g(with)g(regular)f(b)q(oundary)0
2020 y(conditions)f(at)h(in\014nit)o(y)l(.)k(The)17 b(plain)g(and)g(dashed)h
(lines)e(represen)o(t)g(t)o(w)o(o)h(di\013eren)o(t)f(rates)i(of)f(approac)o
(h)0 2119 y(to)k(the)f(asymptotes)g(\(A3a\))h(at)f fc(x)h fd(=)g fb(1)f fd
(\(with)h fc(K)k fd(=)20 b(1\).)35 b(Our)20 b(n)o(umerical)e(results)i(sho)o
(w)h(that)g(the)0 2219 y(monop)q(ole)e(solution)g(\(A3b\))g(is)g(reac)o(hed)f
(at)i(the)f(origin.)29 b(The)20 b(\(analytical\))e(relation)h(\(A5\))g(b)q
(et)o(w)o(een)0 2318 y(the)e(p)q(erio)q(d)h fc(\034)23 b fd(of)18 b(the)f(osc\
illations)g(in)g fc(K)t fd(\()p fc(x)p fd(\))g(near)h(the)f(origin)g(and)h
(the)f(v)m(alue)h(of)f fc(J)5 b fd(\()p fc(x)p fd(\))17 b(at)h(the)f(origin)0
2418 y(is)f(satis\014ed.)947 2856 y( )g eop
%%Trailer
 end
%%EOF

