%Paper: hep-th/9312132
%From: eli@zeta.ecm.ub.es
%Date: Wed, 15 Dec 1993 14:55:05 +0100





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\begin{document}

%\hfill UB-ECM-PF 92/33

%\hfill December 1992

\vspace*{3mm}


\begin{center}

{\LARGE \bf
Dynamical Determination of the Metric Signature in Spacetime
of Nontrivial Topology}

\vskip4ex

\renewcommand\baselinestretch{0.8}
{\sc E. Elizalde}
\footnote{E-mail: eli @ ebubecm1.bitnet} \\
Department E.C.M. and I.F.A.E., Faculty of Physics,
University of  Barcelona, \\ Diagonal 647, 08028 Barcelona, \\
and Center for Advanced Studies, C.S.I.C., Cam\'{\i} de Santa B\`arbara, \\
17300 Blanes, Catalonia, Spain \\
{\sc S.D. Odintsov}\footnote{
On leave from Tomsk Pedagogical Institute, 634041 Tomsk, Russian
Federation. E-mail: odintsov @
ebubecm1.bitnet} and {\sc A. Romeo} \\
Department E.C.M., Faculty of Physics,
University of  Barcelona, \\  Diagonal 647, 08028 Barcelona, Catalonia,
Spain \\
%and \\
%{\rm August Romeo}, \\
%{\it Dept de Matem\`atica Aplicada i An\`alisi, \\
%Facultat de Matem\`atiques, Universitat de Barcelona, \\
%Gran Via de les Corts Catalanes 585, 08071 Barcelona, \\ Catalonia. \\

\ms


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\vspace{5mm}

{\bf Abstract}

\end{center}

%Supposing the metric signature of spacetime subject to quantum
%fluctuations, and the compactification of a space dimension,
%we discuss the dynamical determination of signature and
%dimension arising from a
%one-loop effective-potential analysis.
%This leads us to consider the resulting signature of spacetime as
%an effect which is both dynamical and topological.
The formalism of Greensite for treating the spacetime signature as
a dynamical degree of freedom induced by quantum fields is considered,
for spacetimes
with nontrivial topology of the kind ${\bf R}^{D-1} \times {\bf T}^1$,
for varying $D$. It is
shown that a dynamical origin for the Lorentzian signature is possible
in the five-dimensional space ${\bf R}^4 \times {\bf T}^1$ with small
torus radius (periodic boundary conditions), as well as in
four-dimensional
space with trivial topology. Hence, the possibility exists that the
early universe might have been of the Kaluza-Klein
type, \ie multidimensional and of Lorentzian signature.

\newpage

It is well-known that the field equations of general relativity do not
fix the spacetime signature. However, there exist attempts to understand
the signature dynamics or, more specifically, possible signature
transitions from an
Euclidean signature spacetime to a Lorentzian signature spacetime, and
vice-versa, both at a classical and quantum level
\cite{Sakh}-\cite{Der}.
Notwithstanding these efforts, the explanation of the origin of the
Lorentzian signature
of our physical spacetime is still missing.

 Recently, a very interesting
attempt to devise a quantum mechanism for the dynamical origin of the
Lorentzian signature has been made \cite{Gr1,Gr2}.
Let us briefly describe this formalism. Considering
the flat-space metric in the following form:
\beq
\eta_{ab}=\rm{diag}( e^{i\theta}, 1, 1, \dots, 1),
\label{etaab}\eeq
where $\theta\in[ -\pi, \pi]$, one can easily see that the Euclidean
path-integral theory is obtained for the Wick angle $\theta=0$, while
the Lorentzian signature corresponds to $\theta=\pi$.
Starting from this simple setting, it was suggested in ref. \cite{Gr1}
that the Wick angle $\theta$ in
\req{etaab} could be treated as a dynamical degree of freedom, which can
fluctuate in the
interval $\theta\in [-\pi, \pi]$. Then, in order to fix the value of
this degree
of freedom and to show that the Lorentzian signature is in fact the
preferred
choice, the effective potential $V( \theta )$ for $\theta$ has been
calculated in
\cite{Gr1,Gr2}  at one-loop level, under the following assumptions:
(i) For free fields of equal mass, the contributions to the whole
path integral from any propagating bosonic degree of freedom is equal
and inverse to the contribution of the corresponding fermionic
degree of freedom.
(ii) For scalars, the real-valued invariant volume (De Witt measure)
 of integration is used.

Under these conditions, the one-loop potential $V( \theta )$ induced by
a massless scalar in flat spacetime
(\ie $g_{\mu \nu}=e^a_{\mu} \eta_{ab} e^b_{\mu}=\eta_{\mu \nu}$) is
given by \cite{Gr1,Gr2}
\beq V(\theta )=
-{ \log \det^{-1/2} ( - \sqrt{\eta} \eta^{ab} \partial_a \partial_b )
\over
\int d^Dx }, \label{Vlogdet}\eeq
and use of heat-kernel regularization yields
\beq
V(\theta)=-{1 \over 2}\int_{\Lambda}^{\infty} {ds \over s}
\int {d^D p \over (2 \pi )^D}
\exp\left\{\ds -s
[ \alpha p_0^2 + \beta ( p_1^2+\dots +p_{D-1}^2 )] \right\} ,
%\hspace{1cm} \rep \alpha, \rep \beta >0,
\label{hkV}\eeq
where $\Lambda$ is a cutoff, $\alpha=e^{-i {\theta \over 2}}$,
$\beta=e^{i {\theta \over 2}}$. In \cite{Gr2} it is explained why
heat-kernel regularization is the best one to use in this context.
Taking into account assumption (i), the multiplier $(n_B-n_F)$
---where $n_F$($n_B$) is the fermionic (bosonic) number--- appears in
front of \req{hkV}. Finally, using the above formalism it was shown in
Refs.
\cite{Gr1,Gr2} that the Lorentzian signature is uniquely connected with
$D=4$ dimensions, as it is given by the stationary point of the
potential.

Now, our point is the following. Let us start discussing some
modifications of the
above formalism, for it is quite reasonable that the picture above
described
may be valid in the early universe, perhaps between the Quantum Gravity
and the GUTs epochs. However, at this stage of the evolution of the
early universe, its curvature and temperature were still
very strong, and an external electromagnetic field might have existed.
Moreover, most probably
the topology of the universe at this epoch was highly nontrivial.
Surely, all
these effects (and combinations thereof) may change the picture
described in
\cite{Gr1,Gr2}, even qualitatively. In this letter we will restrict
ourselves to consider
only the influence of nontrivial topology on the above formalism.

So, we will suppose that some massless fields, which have induced
the $V(\theta)$ of eq. \req{hkV}, live in a flat spacetime of topology
${\bf R}^{D-1}\times {\bf T}^1$ (for a general discussion of QFT
on topologically nontrivial backgrounds, see \cite{EORBZ}). Then,
expressions \req{Vlogdet} and \req{hkV} still conserve the same
form. However,
now one of the space coordinates, $x_1$, for example, is compactified,
so that the
 corresponding momentum component ---$p_1$--- is discretized
and one has to do replacements of the type
\beq \int {d p_1 \over 2 \pi} \to
{1 \over 2 \pi R} \sum_{n=-\infty}^{\infty}. \eeq
The specific discrete values of $p_1$ will depend on whether the
fields are subject to periodic or antiperiodic boundary conditions.

\ni{\bf (a) Periodic boundary conditions}:
$p_1^2={n^2 \over R^2}$, $n \in {\bf Z}$.

After integrating the non-compactified components of $p$, we find
\beq
V(\theta )=-{\pi^{D-1 \over 2} \over
2R (2 \pi)^D \alpha^{1 \over 2} \beta^{{D\over 2}-1} }
\int_{\Lambda}^{\infty} {ds \over s} \, s^{-{D-1 \over 2}}
\theta_3\left( 0 \left\vert {s \beta \over \pi R^2} \right. \right),
\label{Vper}
\eeq
where
\beq \theta_3(0 | z)\equiv\sum_{n=-\infty}^{\infty} e^{-\pi z n^2}
=1+2\sum_{n=1}^{\infty} e^{-\pi z n^2}.
\label{deftheta3}\eeq

Now, let us consider the different limits of expr. \req{Vper}.

\ni{\bf (a1)} {$R \to 0$}, more specifically,
$\left\vert {\beta \Lambda \over R^2} \right\vert \gg 1$.
That limit corresponds to very strong nontrivial topology.
Thus, we take the expression \req{deftheta3} itself
as a large-$z$ expansion. Integration proceeds term by term
with the help of
\beq \int_z^{\infty} dt \ t^{a-1} e^{-t}= \Gamma( a, z ) , \
\label{igamma} \eeq
where $\Gamma( a, z )$ is an incomplete gamma function \cite{Gra},
whose asymptotic behaviour for large arguments is given by
\beq
\Gamma( a, z ) \sim
z^{a-1} e^{-z}
\left[ 1 + O\left( 1 \over z \right) \right], |z| \gg 1.
\label{Gasymp} \eeq
As a result, the expansion
\beq
V(\theta)=
-{ 2 \over
(4 \pi)^{D+1 \over 2} R \alpha^{1 \over 2} \beta^{{D \over 2}-1}
\Lambda^{D-1 \over 2} }
\left[
{1 \over D-1}
+{1 \over {\beta\Lambda \over R^2}} e^{-{\beta\Lambda \over R^2}}
+O\left(
{1 \over \left( \beta\Lambda \over R^2 \right)^2 }
e^{-{\beta\Lambda \over R^2}}, \dots,
{1 \over { \beta\Lambda \over R^2}} e^{-{4 \beta\Lambda \over R^2}},
\dots
\right)
\right]
\label{persmallR}
\eeq
follows. Note that we have assumed $\rep\alpha >0$, $\rep\beta >0$.

\ni{\bf (a2)}  $R \to \infty$, more precisely
$\left\vert{ \pi^2 R^2 \over \beta \Lambda }\right\vert \gg 1$.
Such a limit corresponds to ``switching off" the compactification.
First, a
small-$z$ expansion of $\theta_3$ must be performed. It can be obtained
by the reciprocal transformation
\beq \theta_3(0|z)=
{1 \over \sqrt{z}}
\theta_3\left( 0 \left\vert {1 \over z} \right. \right) .
\label{recitheta3}\eeq
Then, we can proceed similarly to the previous case, with the difference
that now we are led to make variable changes of the type
$\ds u={\pi^2 R^2 n^2 \over \beta s}$ and split the integration domains
in the way
$\ds \int_0^{{\pi^2 R^2 n^2 \over \beta \Lambda}} du \ =
\int_0^{\infty}du \
- \int_{{\pi^2 R^2 n^2 \over \beta \Lambda}}^{\infty}du $.
Doing so, incomplete $\Gamma$ functions appear again. After using
the asymptotic expansion \req{Gasymp} for large values of their
arguments, one arrives at
\bea
V(\theta)\ds &=&-{1 \over
(4 \pi)^{D/2} \alpha^{1 \over 2} \beta^{D-1 \over 2} \Lambda^{D \over2}}
 \ds \left[ {1 \over D}
+{1 \over \left( \pi^2 R^2 \over \beta \Lambda \right)^{D \over 2}}
\Gamma\left( D \over 2 \right) \zeta(D) \right. \nn
&&\ds\left. -{1 \over { \pi^2 R^2 \over \beta \Lambda }}
e^{-{\pi^2 R^2 \over \beta \Lambda}}
+O\left(
{1 \over \left( \pi^2 R^2 \over \beta \Lambda \right)^2 }
e^{-{\pi^2 R^2 \over \beta \Lambda}}, \dots,
{1 \over { 4 \pi^2 R^2 \over \beta \Lambda } }
e^{-{4 \pi^2 R^2 \over \beta \Lambda}}, \dots
\right) \right] . \nn
\label{perlargeR}\eea
$\zeta$ being the Riemann zeta function. The term where it occurs
yields precisely the result which one would obtain after removing the
$n=0$ piece and performing zeta-function regularization of the rest.
As one can notice, its contribution is ---of course--- independent
of the cutoff $\Lambda$.

\ni{\bf (b) Antiperiodic boundary conditions}:
$p_1^2={\left( n+ {1 \over 2} \right)^2 \over R^2}$, $n \in {\bf Z}$.

Now, the integration of the non-compactified $p$-components yields
\beq
V(\theta)=-{\pi^{D-1 \over 2} \over
2R (2 \pi)^D \alpha^{1 \over 2} \beta^{{D\over 2}-1} }
\int_{\Lambda}^{\infty} {ds \over s} s^{-{D-1 \over 2}}
\theta_2\left( 0 \left\vert {s \beta \over \pi R^2} \right. \right),
\label{Vantiper}
\eeq
where
\beq \theta_2(0|z)\equiv
\sum_{n=-\infty}^{\infty} e^{-\pi z (n+{1 \over 2})^2}
=2\sum_{n=0}^{\infty} e^{-\pi z (n+{1 \over 2})^2} .
\label{deftheta2}\eeq
This last equality can be viewed as a large-$z$ expansion, and will
therefore
be used for calculating the small-$R$ expression in a way analogous to
the corresponding periodic case.

\ni{\bf (b1)}  For $R \to 0$, that is
for
$\left\vert{\beta \Lambda \over R^2}\right\vert \gg 1$, we obtain
\beq
V(\theta)=
-{ 2 \over
(4 \pi)^{D+1 \over 2} R \alpha^{1 \over 2} \beta^{{D \over 2}-1}
\Lambda^{D-1 \over 2} }
\left[
{1 \over {\beta\Lambda \over 4 R^2}} e^{-{\beta\Lambda \over 4 R^2}}
+O\left(
{1 \over \left( \beta\Lambda \over 4 R^2 \right)^2}
e^{-{\beta\Lambda \over 4 R^2}}, \dots,
{1 \over 9{ \beta\Lambda \over 4 R^2 }}
e^{-9 { \beta\Lambda \over 4 R^2}}, \dots \right)
\right].
\label{antipersmallR}
\eeq

\ni{\bf (b2)} For $R \to \infty$,
in order to obtain a small-$z$
expansion from \req{deftheta2} ---which is just the opposite--- this
time we take advantage of the transformation
\[ \theta_2\left( 0 \left\vert z \right. \right)=
{1 \over \sqrt{z} }
\theta_4\left( 0 \left\vert {1 \over z} \right. \right), \]
\beq \theta_4(0|z)\equiv
\sum_{n=-\infty}^{\infty}(-1)^n e^{-\pi z n^2}
=1+2\sum_{n=1}^{\infty}(-1)^n e^{-\pi z n^2} . \eeq
This is a useful relation, as well as its counterpart \req{recitheta3}
for the analogous periodic case, and both can
be regarded as  special forms of the general reciprocal transformation
for $\theta$ functions (see \eg \cite{epsma}).
%It is now easy
%to use the expression of $\theta_4$ as an expansion for small argument.
After integrating, one gets
\bea
V(\theta)&=&-{1 \over
(4 \pi)^{D/2} \alpha^{1\over 2} \beta^{D-1 \over 2} \Lambda^{D \over 2}}
 \ds \left[ {1 \over D}
-{1 \over \left( \pi^2 R^2 \over \beta \Lambda \right)^{D/2}}
\Gamma\left( D \over 2 \right) \eta(D) \right. \nn
&&\ds \left. +{1 \over { \pi^2 R^2 \over \beta \Lambda }}
e^{-{\pi^2 R^2 \over \beta \Lambda}}
+O\left(
{1 \over \left( \pi^2 R^2 \over \beta \Lambda \right)^2 }
e^{-{\pi^2 R^2 \over \beta \Lambda}}, \dots,
{1 \over { 4 \pi^2 R^2 \over \beta \Lambda }}
e^{-{4 \pi^2 R^2 \over \beta \Lambda}}, \dots
\right)
\right] , \nn
\label{antiperlargeR}
\eea
which is valid for $\left\vert{\pi^2 R^2 \over \beta \Lambda}\right\vert
\gg 1$. Here $\eta$ is the well-known Dirichlet series
\beq \eta(z)\equiv \sum_{n=0}^{\infty} (-1)^{n-1} n^{-z}
=(1-2^{1-z})\zeta(z). \eeq

Now we discuss the physical consequences of the results just
obtained. As we could see, the potential $V(\theta )$ is complex, so we
will
look for the value of $\theta$ determined by the following conditions
\cite{Gr1}:
\beq
\begin{array}{lll}
(i) & \imp V&\mbox{must be stationary, and} \\
(ii) & \rep V&\mbox{must be a minimum.}
\end{array}
\label{condi}
\eeq

Note also that, in accordance with the first assumption of \cite{Gr1},
our results \req{Vper}, \req{persmallR}, \req{perlargeR},
\req{Vantiper}, \req{antipersmallR} and \req{antiperlargeR}
should be multiplied by $(n_B-n_F)$.

We are ready to start the analysis of the effective potential.
For the
$R \to \infty$ limit (trivial topology), the analysis made in \cite{Gr1}
shows that there exists a unique solution
\beq n_F>n_B, \ \ \theta=\pm\pi, \ \ D=4. \label{solGr}\eeq
(Note that for $D=2$ or $n_F=n_B$ there is no preferable choice of
$\theta$, as $V$ ceases to depend on it. We shall call these cases
`special'). The solution \req{solGr}
leads the authors of \cite{Gr1,Gr2} to conclude that the Lorentzian
signature is chosen by quantum dynamics {\it only} in $D=4$.

Now we may consider the case when the radius of the compactified
dimension is small, and quantum fields satisfy periodic boundary
conditions. Then, using the leading term of \req{persmallR}
(with the multiplier $(n_B-n_F)$)
we  find the following unique solution satisfying the requirements
\req{condi} (apart from the `special' cases, which are here $D=3$ or
$n_F=n_B$)
\beq n_F>n_B, \ \ \theta=\pm\pi, \ \ D=5. \eeq
Thus, the Lorentzian signature is singled out {\it also} in $D=5$, but
only if
the fifth dimension is compactified with a very small radius. Therefore,
the formalism of the dynamical degree of freedom associated to the Wick
angle provides a window for Kaluza-Klein-type theories.
It is not difficult to show that, had we started from the topology
${\bf R}^{D-n} \times {\bf T}^n \ ( n<D-1)$, and did a similar
small-$R$ expansion (taking all the torus radii to be equal) we would
have found the following result:
\beq n_F>n_B, \ \ \theta=\pm\pi, \ \ D=4+n. \eeq
Hence, an early universe with a nontrivial topology could have been
multidimensional, and this is compatible with the Lorentzian signature.

As a last example, let us consider the potential \req{antipersmallR}
corresponding to small radius for the compactified dimension and
antiperiodic boundary conditions for the quantum fields.
Then, from \req{antipersmallR} we have
\beq
V(\theta) \sim
{ 8(n_F-n_B) R \ e^{-{\beta \Lambda \over 4 R^2}} \over
(4 \pi)^{D+1 \over 2} \alpha^{1 \over 2} \beta^{D \over 2}
\Lambda^{D+1 \over 2} } .
\label{LTantipersmallR}
\eeq
The conditions \req{condi} for the $V(\theta)$ in \req{LTantipersmallR}
mean that there should be some value of $\theta$, say $\bar\theta$,
simultaneously satisfying
\beq
\begin{array}{c}
\begin{array}{ll}
\ds (n_B-n_F) e^{ -{\Lambda \over 4R^2} \cos{\theta \over 2} }
&\ds\left[ {\Lambda \over 8R^2} \sin{\theta \over 2} \
\sin\left(
{D-1 \over 4}\theta + {\Lambda \over 4R^2} \sin{\theta \over 2}
\right) \right. \\
&\ds\left.\left. +\left( {D-1 \over 4}
+{\Lambda \over 8R^2}\cos{\theta \over 2} \right)
\cos\left(
{D-1 \over 4}\theta + {\Lambda \over 4R^2} \sin{\theta \over 2}
\right)
\right] \right\vert_{\theta=\bar\theta}=0,
\end{array} \\
%\ds {\rm min}_{\theta | -\pi \le \theta \le \pi}
\ds  (n_F-n_B)e^{-{\Lambda \over 4R^2}\cos{\theta \over 2} }
\cos\left(
{D-1 \over 4}\theta + {\Lambda \over 4R^2} \sin{\theta \over 2}
\right) \ \ \
\mbox{has a minimum at $\theta=\bar\theta$}.
\end{array}
\label{condii}
\eeq
In other words, one takes the potential \req{LTantipersmallR} and
requires the
coincidence of stationary points of its imaginary part with minima of
its real part, bearing in mind all the time that we are restricted to
$\theta\in [-\pi,\pi]$.
Now, this situation differs from the previous cases in the variable
nature of this $V(\theta)$ as ${\Lambda \over 4R^2}$ varies.
We have plotted the curves representing its rescaled real and imaginary
parts
for $D=4,5$ and for different values of $b\equiv {\Lambda \over 4R^2}$
(see Fig. 1).
The behaviour observed is drastically modified as this parameter
increases, going from a regime of noticeable oscillation to
one in which a plateau around the origin is formed by both $\rep V$
and $\imp V$.
After studying $V$ for other values of $b$ not shown in the figure,
we have detected that the width of this plateau increases as $b$ grows.
The flatness of this region
indicates an angular range where $V$ becomes practically independent
of $\theta$, and therefore,
within this range no particular spacetime signature is preferred.
(Notice that the situation where $b \gg 1$ is precisely
the one in which \req{LTantipersmallR} is an acceptable approximation
for $V(\theta)$).
Apart from this, and generally speaking, there is no genuine
coincidence of stationary points and minima, unless very specific
values of $b$ and $D$ are deliberately chosen for that purpose.

Summing up, we have discussed the possibility of a dynamical origin of
the
Lorentzian signature in a universe with nontrivial topology. Using very
precise mathematical techniques, we
have shown that the Lorentzian signature can actually be dynamically
induced
in a multidimensional universe with the topology ${\bf R}^4 \times {\bf
T}^n$, where the radii of the torus are small. It would certainly be of
interest to estimate the mass effects \cite{Gr2} on the above results,
since it was already shown in \cite{Gr2} that considering
massive
fields might increase the resulting $D$ until $D=6$. Hence, the
combination of both effects (topology and nonzero mass) may lead
to changes in the
above results. Finally, let us note that it would also be of great
interest
to understand the influence of quantum gravity on the dynamical origin
of the spacetime signature. This subject surely deserves further study.

\vskip3ex

\ni{\Large \bf Acknowledgements}

We would like to thank Robin Tucker for interesting correspondence. This
work has been supported by
DGICYT (Spain), project no. PB90-0022, and by
CIRIT (Generalitat de Catalunya).

\newpage

\begin{thebibliography}{[00]}

\bibitem{Sakh}
A.D. Sakharov, {\it Sov. Phys.  JETP} {\bf 60} (1984) 214.

\bibitem{GHr}
G.W. Gibbons and J.B. Hartle, {\it Phys. Rev. }{\bf D 42} (1990) 2458.

\bibitem{Ish}
C.J. Isham, {\it Class. Quant. Grav. }{\bf 6} (1989) 1509.

\bibitem{GHw}
G.W. Gibbons and S.W. Hawking, {\it Commun. Math. Phys. }{\bf 148}
(1992) 345.

\bibitem{Vil}
A. Vilenkin, {\it Phys. Rev. } {\bf D33} (1986) 3560.

\bibitem{Ell}
G.F.R. Ellis, A. Sumruk, D. Coule and C. Hellaby,
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\bibitem{Der} T. Dereli and R.W. Tucker, {\it Class. Quant. Grav. }
{\bf 10} (1993) 365;
T. Dray, C. Manogue and R.W. Tucker,
{\it Gen. Rel. Grav. } {\bf 23} (1991) 967.

\bibitem{Gr1}
J. Greensite, {\it Phys. Lett. } {\bf B 300} (1993) 34.

\bibitem{Gr2}
A. Carlini and J. Greensite, NBI preprint NBI-HE-93-37,
gr-qc/9308012, 1993.

\bibitem{epsma}
P. Epstein, {\it Mathematische Annalen }{\bf 56 } (1903) 615.
%A. Erd\'elyi, Editor, {\sl Bateman Manuscript Project,
%Higher Transcendental Functions}, Vol. 2, McGraw-Hill, New York, 1954.

\bibitem{Gra}
I.S. Gradshteyn and J.M. Ryzhik,
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London, 1980.

\bibitem{EORBZ} E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko
and S. Zerbini, {\it Zeta Regularization Techniques with Applications},
World Sci., Singapore, 1994.

\end{thebibliography}

\newpage

\ni{\Large \bf Figure captions}

\ni{\bf Figure 1}.
Curves representing $\rep v(\theta)$ ---solid line---
and $\imp v(\theta)$ ---dashed line--- with $v(\theta)$
denoting the {\it rescaled} potential
$e^{-b \cos{\theta \over 2}
+i\left( a \theta+ b\sin{\theta \over 2} \right) }$,
with $a\equiv{D-1 \over 4}$, $b\equiv{\Lambda \over 4 R^2}$
for $-\pi \le \theta \le \pi$.
The plots correspond to
$D=4, D=5$ and to two different values of $b$:
(a) $D=4, b=1$, (b) $D=4, b=10$, (c) $D=5, b=1$, (d) $D=5, b=10$.
In (b) and (d) the formation of a wide plateau around the origin for
large values of $b$ is already apparent.

\end{document}
\bye


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9 3443 D 5957 3355 D 5995 3269 D 6033 3184 D 6070 3101 D 6108 3020 D 6146 2940
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showpage
end
grestore


% PostScript Driver  Fig. 1b
% Copyright (C) 1988 Golden Software, Inc.
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3034 4843 M 3054 4826 D 3092 4801 D 3130 4778 D 3134 4777 D S
3245 4733 M 3281 4723 D 3318 4716 D 3356 4710 D 3362 4710 D S
3482 4700 M 3506 4699 D 3545 4698 D 3582 4698 D 3602 4698 D S
3722 4698 M 3733 4698 D 3770 4699 D 3808 4699 D 3842 4699 D S
3962 4700 M 3997 4700 D 4034 4700 D 4072 4700 D 4082 4700 D S
4202 4700 M 4223 4700 D 4261 4700 D 4298 4700 D 4322 4700 D S
4442 4700 M 4450 4700 D 4487 4700 D 4525 4700 D 4562 4700 D S
4682 4700 M 4713 4700 D 4750 4700 D 4789 4700 D 4802 4700 D S
4922 4700 M 4939 4700 D 4977 4700 D 5014 4700 D 5042 4700 D S
5162 4700 M 5166 4700 D 5203 4700 D 5241 4700 D 5278 4700 D 5282 4700 D S
5402 4701 M 5430 4701 D 5467 4702 D 5505 4702 D 5522 4702 D S
5642 4702 M 5655 4702 D 5694 4701 D 5731 4699 D 5762 4698 D S
5881 4685 M 5882 4684 D 5919 4677 D 5957 4667 D 5995 4655 D 5997 4654 D S
6105 4602 M 6108 4599 D 6146 4574 D 6183 4543 D 6200 4529 D S
6284 4443 M 6297 4430 D 6334 4385 D 6362 4352 D S
6437 4258 M 6447 4246 D 6485 4199 D S
4333 7476 M 4333 7356 D S
4338 7476 M 4338 7356 D S
4373 7442 M 4373 7396 D S
4315 7476 M 4407 7476 D 4407 7442 D 4401 7476 D S
4338 7419 M 4373 7419 D S
4315 7356 M 4355 7356 D S
4447 7476 M 4441 7470 D 4447 7465 D 4453 7470 D 4447 7476 D S
4447 7436 M 4447 7356 D S
4453 7436 M 4453 7356 D S
4430 7436 M 4453 7436 D S
4430 7356 M 4470 7356 D S
4527 7436 M 4515 7430 D 4510 7425 D 4504 7413 D 4504 7402 D 4510 7390 D 4515
7385 D 4527 7379 D 4538 7379 D 4550 7385 D 4555 7390 D 4561 7402 D 4561 7413 D
4555 7425 D 4550 7430 D 4538 7436 D 4527 7436 D S
4515 7430 M 4510 7419 D 4510 7396 D 4515 7385 D S
4550 7385 M 4555 7396 D 4555 7419 D 4550 7430 D S
4555 7425 M 4561 7430 D 4573 7436 D 4573 7430 D 4561 7430 D S
4510 7390 M 4504 7385 D 4498 7373 D 4498 7367 D 4504 7356 D 4521 7350 D 4550
7350 D 4567 7345 D 4573 7339 D S
4498 7367 M 4504 7362 D 4521 7356 D 4550 7356 D 4567 7350 D 4573 7339 D 4573
7333 D 4567 7322 D 4550 7316 D 4515 7316 D 4498 7322 D 4493 7333 D 4493 7339 D
4498 7350 D 4515 7356 D S
4744 7453 M 4755 7459 D 4773 7476 D 4773 7356 D S
4767 7470 M 4767 7356 D S
4744 7356 M 4795 7356 D S
4853 7476 M 4853 7356 D S
4858 7476 M 4858 7356 D S
4858 7419 M 4870 7430 D 4881 7436 D 4893 7436 D 4910 7430 D 4921 7419 D 4927
7402 D 4927 7390 D 4921 7373 D 4910 7362 D 4893 7356 D 4881 7356 D 4870 7362 D
4858 7373 D S
4893 7436 M 4904 7430 D 4915 7419 D 4921 7402 D 4921 7390 D 4915 7373 D 4904
7362 D 4893 7356 D S
4835 7476 M 4858 7476 D S
1021 4554 M 1089 4554 D S
1055 4554 M 1055 4698 D S
1021 4698 M 1089 4698 D S
1144 4554 M 1144 4630 D S
1144 4609 M 1165 4630 D 1178 4630 D 1199 4609 D 1199 4554 D S
1199 4609 M 1219 4630 D 1233 4630 D 1254 4609 D 1254 4554 D S
1439 4623 M 1446 4637 D 1459 4650 D 1473 4650 D 1480 4644 D 1487 4630 D 1487
4609 D 1473 4575 D S
1480 4644 M 1480 4616 D 1473 4589 D 1473 4561 D S
1480 4630 M 1466 4596 D 1466 4575 D 1473 4561 D 1487 4554 D 1501 4554 D 1514
4561 D 1528 4575 D 1542 4596 D 1549 4623 D 1549 4650 D 1542 4650 D 1542 4644 D
1549 4630 D S
1000 4770 M 1000 4914 D 1089 4914 D 1110 4894 D 1110 4860 D 1089 4846 D 1000
4846 D S
1034 4846 M 1110 4770 D S
1219 4791 M 1199 4770 D 1165 4770 D 1144 4791 D 1144 4825 D 1165 4846 D 1199
4846 D 1219 4825 D 1219 4805 D 1144 4805 D S
1439 4839 M 1446 4853 D 1459 4866 D 1473 4866 D 1480 4860 D 1487 4846 D 1487
4825 D 1473 4791 D S
1480 4860 M 1480 4832 D 1473 4805 D 1473 4777 D S
1480 4846 M 1466 4812 D 1466 4791 D 1473 4777 D 1487 4770 D 1501 4770 D 1514
4777 D 1528 4791 D 1542 4812 D 1549 4839 D 1549 4866 D 1542 4866 D 1542 4860 D
1549 4846 D S
1624 4770 M 1624 4791 D 1603 4791 D 1603 4770 D 1624 4770 D 1603 4750 D S
showpage
end
grestore

% PostScript Driver  Fig. 1c
% Copyright (C) 1988 Golden Software, Inc.
gsave
newpath
.072 .072 scale
/gsdict 4 dict def
gsdict begin
/M /moveto load def
/D /lineto load def
/S /stroke load def
/W /setlinewidth load def
S
3 W
S
2200 2300 M 7000 2300 D S
2200 2300 M 2200 2210 D S
1960 2104 M 2023 2104 D S
2154 2058 M 2154 2178 D 2080 2104 D 2171 2104 D S
2229 2058 M 2246 2058 D 2246 2076 D 2229 2076 D 2229 2058 D S
2320 2076 M 2411 2161 D S
2411 2150 M 2383 2178 D 2349 2178 D 2320 2150 D 2320 2087 D 2349 2058 D 2383
2058 D 2411 2087 D 2411 2150 D S
2320 2300 M 2320 2240 D S
2440 2300 M 2440 2240 D S
2560 2300 M 2560 2240 D S
2680 2300 M 2680 2240 D S
2800 2300 M 2800 2240 D S
2920 2300 M 2920 2240 D S
3040 2300 M 3040 2240 D S
3160 2300 M 3160 2240 D S
3280 2300 M 3280 2240 D S
3400 2300 M 3400 2210 D S
3160 2104 M 3223 2104 D S
3280 2150 M 3280 2161 D 3297 2178 D 3354 2178 D 3371 2161 D 3371 2133 D 3280
2076 D 3280 2058 D 3371 2058 D S
3429 2058 M 3446 2058 D 3446 2076 D 3429 2076 D 3429 2058 D S
3520 2076 M 3611 2161 D S
3611 2150 M 3583 2178 D 3549 2178 D 3520 2150 D 3520 2087 D 3549 2058 D 3583
2058 D 3611 2087 D 3611 2150 D S
3520 2300 M 3520 2240 D S
3640 2300 M 3640 2240 D S
3760 2300 M 3760 2240 D S
3880 2300 M 3880 2240 D S
4000 2300 M 4000 2240 D S
4120 2300 M 4120 2240 D S
4240 2300 M 4240 2240 D S
4360 2300 M 4360 2240 D S
4480 2300 M 4480 2240 D S
4600 2300 M 4600 2210 D S
4420 2076 M 4511 2161 D S
4511 2150 M 4483 2178 D 4449 2178 D 4420 2150 D 4420 2087 D 4449 2058 D 4483
2058 D 4511 2087 D 4511 2150 D S
4569 2058 M 4586 2058 D 4586 2076 D 4569 2076 D 4569 2058 D S
4660 2076 M 4751 2161 D S
4751 2150 M 4723 2178 D 4689 2178 D 4660 2150 D 4660 2087 D 4689 2058 D 4723
2058 D 4751 2087 D 4751 2150 D S
4720 2300 M 4720 2240 D S
4840 2300 M 4840 2240 D S
4960 2300 M 4960 2240 D S
5080 2300 M 5080 2240 D S
5200 2300 M 5200 2240 D S
5320 2300 M 5320 2240 D S
5440 2300 M 5440 2240 D S
5560 2300 M 5560 2240 D S
5680 2300 M 5680 2240 D S
5800 2300 M 5800 2210 D S
5620 2150 M 5620 2161 D 5637 2178 D 5694 2178 D 5711 2161 D 5711 2133 D 5620
2076 D 5620 2058 D 5711 2058 D S
5769 2058 M 5786 2058 D 5786 2076 D 5769 2076 D 5769 2058 D S
5860 2076 M 5951 2161 D S
5951 2150 M 5923 2178 D 5889 2178 D 5860 2150 D 5860 2087 D 5889 2058 D 5923
2058 D 5951 2087 D 5951 2150 D S
5920 2300 M 5920 2240 D S
6040 2300 M 6040 2240 D S
6160 2300 M 6160 2240 D S
6280 2300 M 6280 2240 D S
6400 2300 M 6400 2240 D S
6520 2300 M 6520 2240 D S
6640 2300 M 6640 2240 D S
6760 2300 M 6760 2240 D S
6880 2300 M 6880 2240 D S
7000 2300 M 7000 2210 D S
6894 2058 M 6894 2178 D 6820 2104 D 6911 2104 D S
6969 2058 M 6986 2058 D 6986 2076 D 6969 2076 D 6969 2058 D S
7060 2076 M 7151 2161 D S
7151 2150 M 7123 2178 D 7089 2178 D 7060 2150 D 7060 2087 D 7089 2058 D 7123
2058 D 7151 2087 D 7151 2150 D S
4610 1978 M 4591 1969 D 4572 1940 D 4563 1921 D 4553 1893 D 4544 1845 D 4544
1807 D 4553 1788 D 4563 1778 D 4582 1778 D 4601 1788 D 4620 1816 D 4629 1836 D
4639 1864 D 4649 1912 D 4649 1950 D 4639 1969 D 4629 1978 D 4610 1978 D S
4553 1883 M 4639 1883 D S
2200 2300 M 2200 7100 D S
2200 2300 M 2110 2300 D S
1598 2286 M 1661 2286 D S
1736 2331 M 1764 2360 D 1764 2240 D S
1867 2240 M 1884 2240 D 1884 2257 D 1867 2257 D 1867 2240 D S
1958 2257 M 2050 2343 D S
2050 2331 M 2021 2360 D 1987 2360 D 1958 2331 D 1958 2269 D 1987 2240 D 2021
2240 D 2050 2269 D 2050 2331 D S
2200 2420 M 2140 2420 D S
2200 2540 M 2140 2540 D S
2200 2660 M 2140 2660 D S
2200 2780 M 2140 2780 D S
2200 2900 M 2140 2900 D S
2200 3020 M 2140 3020 D S
2200 3140 M 2140 3140 D S
2200 3260 M 2140 3260 D S
2200 3380 M 2140 3380 D S
2200 3500 M 2110 3500 D S
1598 3486 M 1661 3486 D S
1718 3457 M 1810 3543 D S
1810 3531 M 1781 3560 D 1747 3560 D 1718 3531 D 1718 3469 D 1747 3440 D 1781
3440 D 1810 3469 D 1810 3531 D S
1867 3440 M 1884 3440 D 1884 3457 D 1867 3457 D 1867 3440 D S
1958 3457 M 1976 3440 D 2021 3440 D 2050 3469 D 2050 3486 D 2033 3503 D 1976
3503 D 1958 3486 D 1958 3560 D 2050 3560 D S
2200 3620 M 2140 3620 D S
2200 3740 M 2140 3740 D S
2200 3860 M 2140 3860 D S
2200 3980 M 2140 3980 D S
2200 4100 M 2140 4100 D S
2200 4220 M 2140 4220 D S
2200 4340 M 2140 4340 D S
2200 4460 M 2140 4460 D S
2200 4580 M 2140 4580 D S
2200 4700 M 2110 4700 D S
1718 4657 M 1810 4743 D S
1810 4731 M 1781 4760 D 1747 4760 D 1718 4731 D 1718 4669 D 1747 4640 D 1781
4640 D 1810 4669 D 1810 4731 D S
1867 4640 M 1884 4640 D 1884 4657 D 1867 4657 D 1867 4640 D S
1958 4657 M 2050 4743 D S
2050 4731 M 2021 4760 D 1987 4760 D 1958 4731 D 1958 4669 D 1987 4640 D 2021
4640 D 2050 4669 D 2050 4731 D S
2200 4820 M 2140 4820 D S
2200 4940 M 2140 4940 D S
2200 5060 M 2140 5060 D S
2200 5180 M 2140 5180 D S
2200 5300 M 2140 5300 D S
2200 5420 M 2140 5420 D S
2200 5540 M 2140 5540 D S
2200 5660 M 2140 5660 D S
2200 5780 M 2140 5780 D S
2200 5900 M 2110 5900 D S
1718 5857 M 1810 5943 D S
1810 5931 M 1781 5960 D 1747 5960 D 1718 5931 D 1718 5869 D 1747 5840 D 1781
5840 D 1810 5869 D 1810 5931 D S
1867 5840 M 1884 5840 D 1884 5857 D 1867 5857 D 1867 5840 D S
1958 5857 M 1976 5840 D 2021 5840 D 2050 5869 D 2050 5886 D 2033 5903 D 1976
5903 D 1958 5886 D 1958 5960 D 2050 5960 D S
2200 6020 M 2140 6020 D S
2200 6140 M 2140 6140 D S
2200 6260 M 2140 6260 D S
2200 6380 M 2140 6380 D S
2200 6500 M 2140 6500 D S
2200 6620 M 2140 6620 D S
2200 6740 M 2140 6740 D S
2200 6860 M 2140 6860 D S
2200 6980 M 2140 6980 D S
2200 7100 M 2110 7100 D S
1736 7131 M 1764 7160 D 1764 7040 D S
1867 7040 M 1884 7040 D 1884 7057 D 1867 7057 D 1867 7040 D S
1958 7057 M 2050 7143 D S
2050 7131 M 2021 7160 D 1987 7160 D 1958 7131 D 1958 7069 D 1987 7040 D 2021
7040 D 2050 7069 D 2050 7131 D S
2200 7100 M 7000 7100 D S
7000 2300 M 7000 7100 D S
2715 3403 M 2753 3322 D 2790 3250 D 2828 3189 D 2866 3138 D 2903 3096 D 2942
3064 D 2979 3041 D 3017 3028 D 3054 3024 D 3092 3029 D 3130 3042 D 3167 3065 D
3205 3095 D 3243 3133 D 3281 3178 D 3318 3230 D 3356 3288 D 3394 3353 D 3431
3423 D 3469 3498 D 3506 3578 D 3545 3662 D 3582 3749 D 3620 3838 D 3658 3931 D
3695 4026 D 3733 4122 D 3770 4218 D 3808 4314 D 3846 4411 D 3884 4507 D 3922
4601 D 3959 4694 D 3997 4783 D 4034 4870 D 4072 4954 D 4110 5035 D 4148 5111 D
4186 5182 D 4223 5249 D 4261 5310 D 4298 5366 D 4336 5416 D 4374 5459 D 4411
5497 D 4450 5527 D 4487 5552 D 4525 5569 D 4562 5579 D 4600 5583 D 4638 5579 D
4675 5569 D 4713 5552 D 4751 5527 D 4789 5497 D 4826 5459 D 4864 5416 D 4902
5366 D 4939 5310 D 4977 5249 D 5014 5182 D 5052 5111 D 5090 5035 D 5128 4954 D
5166 4870 D 5203 4783 D 5241 4694 D 5278 4601 D 5316 4507 D 5354 4411 D 5392
4314 D 5430 4218 D 5467 4122 D 5505 4026 D 5542 3931 D 5580 3838 D 5618 3749 D
5655 3662 D 5694 3578 D 5731 3498 D 5769 3423 D 5806 !
3353 D 5844 3288 D 5882 3230 D 591

9 3178 D 5957 3133 D 5995 3095 D 6033 3065 D 6070 3042 D 6108 3029 D 6146 3024
D 6183 3028 D 6221 3041 D 6258 3064 D 6297 3096 D 6334 3138 D 6372 3189 D 6410
3250 D 6447 3322 D 6485 3403 D S
2715 6719 M 2745 6603 D S
2775 6487 M 2790 6426 D 2805 6371 D S
2834 6254 M 2863 6138 D S
2893 6022 M 2903 5980 D 2922 5906 D S
2952 5790 M 2979 5686 D 2982 5673 D S
3013 5557 M 3017 5541 D 3043 5441 D S
3074 5325 M 3092 5259 D 3106 5210 D S
3138 5094 M 3167 4990 D 3171 4978 D S
3205 4863 M 3205 4862 D 3240 4749 D S
3277 4634 M 3281 4622 D 3314 4520 D S
3354 4407 M 3356 4403 D 3394 4303 D 3397 4295 D S
3442 4184 M 3469 4123 D 3492 4074 D S
3546 3968 M 3582 3906 D 3610 3866 D S
3684 3771 M 3695 3759 D 3733 3726 D 3770 3699 D 3777 3697 D S
3891 3670 M 3922 3673 D 3959 3685 D 3997 3704 D 4003 3709 D S
4094 3786 M 4110 3802 D 4148 3846 D 4171 3878 D S
4238 3978 M 4261 4013 D 4298 4078 D 4300 4081 D S
4357 4186 M 4374 4219 D 4411 4294 D S
4463 4402 M 4487 4453 D 4514 4510 D S
4563 4620 M 4600 4700 D 4613 4729 D S
4663 4838 M 4675 4866 D 4713 4947 D S
4765 5055 M 4789 5106 D 4818 5163 D S
4873 5270 M 4902 5322 D 4932 5374 D S
4996 5475 M 5014 5503 D 5052 5554 D 5068 5572 D S
5151 5658 M 5166 5670 D 5203 5696 D 5241 5715 D 5254 5719 D S
5371 5724 M 5392 5718 D 5430 5701 D 5467 5674 D 5475 5667 D S
5558 5580 M 5580 5550 D 5618 5494 D 5625 5481 D S
5683 5376 M 5694 5357 D 5731 5277 D 5735 5268 D S
5782 5158 M 5806 5097 D 5826 5046 D S
5867 4933 M 5882 4890 D 5906 4819 D S
5943 4706 M 5957 4661 D 5978 4590 D S
6013 4476 M 6033 4410 D 6046 4361 D S
6079 4245 M 6108 4141 D 6111 4130 D S
6142 4014 M 6146 4002 D 6174 3898 D S
6203 3782 M 6221 3714 D 6234 3666 D S
6263 3549 M 6293 3433 D S
6322 3317 M 6334 3271 D 6352 3200 D S
6382 3084 M 6410 2974 D 6411 2967 D S
6441 2851 M 6447 2826 D 6471 2735 D S
4333 7476 M 4333 7356 D S
4338 7476 M 4338 7356 D S
4373 7442 M 4373 7396 D S
4315 7476 M 4407 7476 D 4407 7442 D 4401 7476 D S
4338 7419 M 4373 7419 D S
4315 7356 M 4355 7356 D S
4447 7476 M 4441 7470 D 4447 7465 D 4453 7470 D 4447 7476 D S
4447 7436 M 4447 7356 D S
4453 7436 M 4453 7356 D S
4430 7436 M 4453 7436 D S
4430 7356 M 4470 7356 D S
4527 7436 M 4515 7430 D 4510 7425 D 4504 7413 D 4504 7402 D 4510 7390 D 4515
7385 D 4527 7379 D 4538 7379 D 4550 7385 D 4555 7390 D 4561 7402 D 4561 7413 D
4555 7425 D 4550 7430 D 4538 7436 D 4527 7436 D S
4515 7430 M 4510 7419 D 4510 7396 D 4515 7385 D S
4550 7385 M 4555 7396 D 4555 7419 D 4550 7430 D S
4555 7425 M 4561 7430 D 4573 7436 D 4573 7430 D 4561 7430 D S
4510 7390 M 4504 7385 D 4498 7373 D 4498 7367 D 4504 7356 D 4521 7350 D 4550
7350 D 4567 7345 D 4573 7339 D S
4498 7367 M 4504 7362 D 4521 7356 D 4550 7356 D 4567 7350 D 4573 7339 D 4573
7333 D 4567 7322 D 4550 7316 D 4515 7316 D 4498 7322 D 4493 7333 D 4493 7339 D
4498 7350 D 4515 7356 D S
4744 7453 M 4755 7459 D 4773 7476 D 4773 7356 D S
4767 7470 M 4767 7356 D S
4744 7356 M 4795 7356 D S
4910 7419 M 4904 7413 D 4910 7407 D 4915 7413 D 4915 7419 D 4904 7430 D 4893
7436 D 4875 7436 D 4858 7430 D 4847 7419 D 4841 7402 D 4841 7390 D 4847 7373 D
4858 7362 D 4875 7356 D 4887 7356 D 4904 7362 D 4915 7373 D S
4875 7436 M 4864 7430 D 4853 7419 D 4847 7402 D 4847 7390 D 4853 7373 D 4864
7362 D 4875 7356 D S
1021 4554 M 1089 4554 D S
1055 4554 M 1055 4698 D S
1021 4698 M 1089 4698 D S
1144 4554 M 1144 4630 D S
1144 4609 M 1165 4630 D 1178 4630 D 1199 4609 D 1199 4554 D S
1199 4609 M 1219 4630 D 1233 4630 D 1254 4609 D 1254 4554 D S
1439 4623 M 1446 4637 D 1459 4650 D 1473 4650 D 1480 4644 D 1487 4630 D 1487
4609 D 1473 4575 D S
1480 4644 M 1480 4616 D 1473 4589 D 1473 4561 D S
1480 4630 M 1466 4596 D 1466 4575 D 1473 4561 D 1487 4554 D 1501 4554 D 1514
4561 D 1528 4575 D 1542 4596 D 1549 4623 D 1549 4650 D 1542 4650 D 1542 4644 D
1549 4630 D S
1000 4770 M 1000 4914 D 1089 4914 D 1110 4894 D 1110 4860 D 1089 4846 D 1000
4846 D S
1034 4846 M 1110 4770 D S
1219 4791 M 1199 4770 D 1165 4770 D 1144 4791 D 1144 4825 D 1165 4846 D 1199
4846 D 1219 4825 D 1219 4805 D 1144 4805 D S
1439 4839 M 1446 4853 D 1459 4866 D 1473 4866 D 1480 4860 D 1487 4846 D 1487
4825 D 1473 4791 D S
1480 4860 M 1480 4832 D 1473 4805 D 1473 4777 D S
1480 4846 M 1466 4812 D 1466 4791 D 1473 4777 D 1487 4770 D 1501 4770 D 1514
4777 D 1528 4791 D 1542 4812 D 1549 4839 D 1549 4866 D 1542 4866 D 1542 4860 D
1549 4846 D S
1624 4770 M 1624 4791 D 1603 4791 D 1603 4770 D 1624 4770 D 1603 4750 D S
showpage
end
grestore

% PostScript Driver  Fig. 1d
% Copyright (C) 1988 Golden Software, Inc.
gsave
newpath
.072 .072 scale
/gsdict 4 dict def
gsdict begin
/M /moveto load def
/D /lineto load def
/S /stroke load def
/W /setlinewidth load def
S
3 W
S
2200 2300 M 7000 2300 D S
2200 2300 M 2200 2210 D S
1960 2104 M 2023 2104 D S
2154 2058 M 2154 2178 D 2080 2104 D 2171 2104 D S
2229 2058 M 2246 2058 D 2246 2076 D 2229 2076 D 2229 2058 D S
2320 2076 M 2411 2161 D S
2411 2150 M 2383 2178 D 2349 2178 D 2320 2150 D 2320 2087 D 2349 2058 D 2383
2058 D 2411 2087 D 2411 2150 D S
2320 2300 M 2320 2240 D S
2440 2300 M 2440 2240 D S
2560 2300 M 2560 2240 D S
2680 2300 M 2680 2240 D S
2800 2300 M 2800 2240 D S
2920 2300 M 2920 2240 D S
3040 2300 M 3040 2240 D S
3160 2300 M 3160 2240 D S
3280 2300 M 3280 2240 D S
3400 2300 M 3400 2210 D S
3160 2104 M 3223 2104 D S
3280 2150 M 3280 2161 D 3297 2178 D 3354 2178 D 3371 2161 D 3371 2133 D 3280
2076 D 3280 2058 D 3371 2058 D S
3429 2058 M 3446 2058 D 3446 2076 D 3429 2076 D 3429 2058 D S
3520 2076 M 3611 2161 D S
3611 2150 M 3583 2178 D 3549 2178 D 3520 2150 D 3520 2087 D 3549 2058 D 3583
2058 D 3611 2087 D 3611 2150 D S
3520 2300 M 3520 2240 D S
3640 2300 M 3640 2240 D S
3760 2300 M 3760 2240 D S
3880 2300 M 3880 2240 D S
4000 2300 M 4000 2240 D S
4120 2300 M 4120 2240 D S
4240 2300 M 4240 2240 D S
4360 2300 M 4360 2240 D S
4480 2300 M 4480 2240 D S
4600 2300 M 4600 2210 D S
4420 2076 M 4511 2161 D S
4511 2150 M 4483 2178 D 4449 2178 D 4420 2150 D 4420 2087 D 4449 2058 D 4483
2058 D 4511 2087 D 4511 2150 D S
4569 2058 M 4586 2058 D 4586 2076 D 4569 2076 D 4569 2058 D S
4660 2076 M 4751 2161 D S
4751 2150 M 4723 2178 D 4689 2178 D 4660 2150 D 4660 2087 D 4689 2058 D 4723
2058 D 4751 2087 D 4751 2150 D S
4720 2300 M 4720 2240 D S
4840 2300 M 4840 2240 D S
4960 2300 M 4960 2240 D S
5080 2300 M 5080 2240 D S
5200 2300 M 5200 2240 D S
5320 2300 M 5320 2240 D S
5440 2300 M 5440 2240 D S
5560 2300 M 5560 2240 D S
5680 2300 M 5680 2240 D S
5800 2300 M 5800 2210 D S
5620 2150 M 5620 2161 D 5637 2178 D 5694 2178 D 5711 2161 D 5711 2133 D 5620
2076 D 5620 2058 D 5711 2058 D S
5769 2058 M 5786 2058 D 5786 2076 D 5769 2076 D 5769 2058 D S
5860 2076 M 5951 2161 D S
5951 2150 M 5923 2178 D 5889 2178 D 5860 2150 D 5860 2087 D 5889 2058 D 5923
2058 D 5951 2087 D 5951 2150 D S
5920 2300 M 5920 2240 D S
6040 2300 M 6040 2240 D S
6160 2300 M 6160 2240 D S
6280 2300 M 6280 2240 D S
6400 2300 M 6400 2240 D S
6520 2300 M 6520 2240 D S
6640 2300 M 6640 2240 D S
6760 2300 M 6760 2240 D S
6880 2300 M 6880 2240 D S
7000 2300 M 7000 2210 D S
6894 2058 M 6894 2178 D 6820 2104 D 6911 2104 D S
6969 2058 M 6986 2058 D 6986 2076 D 6969 2076 D 6969 2058 D S
7060 2076 M 7151 2161 D S
7151 2150 M 7123 2178 D 7089 2178 D 7060 2150 D 7060 2087 D 7089 2058 D 7123
2058 D 7151 2087 D 7151 2150 D S
4610 1978 M 4591 1969 D 4572 1940 D 4563 1921 D 4553 1893 D 4544 1845 D 4544
1807 D 4553 1788 D 4563 1778 D 4582 1778 D 4601 1788 D 4620 1816 D 4629 1836 D
4639 1864 D 4649 1912 D 4649 1950 D 4639 1969 D 4629 1978 D 4610 1978 D S
4553 1883 M 4639 1883 D S
2200 2300 M 2200 7100 D S
2200 2300 M 2110 2300 D S
1598 2286 M 1661 2286 D S
1736 2331 M 1764 2360 D 1764 2240 D S
1867 2240 M 1884 2240 D 1884 2257 D 1867 2257 D 1867 2240 D S
1958 2257 M 2050 2343 D S
2050 2331 M 2021 2360 D 1987 2360 D 1958 2331 D 1958 2269 D 1987 2240 D 2021
2240 D 2050 2269 D 2050 2331 D S
2200 2420 M 2140 2420 D S
2200 2540 M 2140 2540 D S
2200 2660 M 2140 2660 D S
2200 2780 M 2140 2780 D S
2200 2900 M 2140 2900 D S
2200 3020 M 2140 3020 D S
2200 3140 M 2140 3140 D S
2200 3260 M 2140 3260 D S
2200 3380 M 2140 3380 D S
2200 3500 M 2110 3500 D S
1598 3486 M 1661 3486 D S
1718 3457 M 1810 3543 D S
1810 3531 M 1781 3560 D 1747 3560 D 1718 3531 D 1718 3469 D 1747 3440 D 1781
3440 D 1810 3469 D 1810 3531 D S
1867 3440 M 1884 3440 D 1884 3457 D 1867 3457 D 1867 3440 D S
1958 3457 M 1976 3440 D 2021 3440 D 2050 3469 D 2050 3486 D 2033 3503 D 1976
3503 D 1958 3486 D 1958 3560 D 2050 3560 D S
2200 3620 M 2140 3620 D S
2200 3740 M 2140 3740 D S
2200 3860 M 2140 3860 D S
2200 3980 M 2140 3980 D S
2200 4100 M 2140 4100 D S
2200 4220 M 2140 4220 D S
2200 4340 M 2140 4340 D S
2200 4460 M 2140 4460 D S
2200 4580 M 2140 4580 D S
2200 4700 M 2110 4700 D S
1718 4657 M 1810 4743 D S
1810 4731 M 1781 4760 D 1747 4760 D 1718 4731 D 1718 4669 D 1747 4640 D 1781
4640 D 1810 4669 D 1810 4731 D S
1867 4640 M 1884 4640 D 1884 4657 D 1867 4657 D 1867 4640 D S
1958 4657 M 2050 4743 D S
2050 4731 M 2021 4760 D 1987 4760 D 1958 4731 D 1958 4669 D 1987 4640 D 2021
4640 D 2050 4669 D 2050 4731 D S
2200 4820 M 2140 4820 D S
2200 4940 M 2140 4940 D S
2200 5060 M 2140 5060 D S
2200 5180 M 2140 5180 D S
2200 5300 M 2140 5300 D S
2200 5420 M 2140 5420 D S
2200 5540 M 2140 5540 D S
2200 5660 M 2140 5660 D S
2200 5780 M 2140 5780 D S
2200 5900 M 2110 5900 D S
1718 5857 M 1810 5943 D S
1810 5931 M 1781 5960 D 1747 5960 D 1718 5931 D 1718 5869 D 1747 5840 D 1781
5840 D 1810 5869 D 1810 5931 D S
1867 5840 M 1884 5840 D 1884 5857 D 1867 5857 D 1867 5840 D S
1958 5857 M 1976 5840 D 2021 5840 D 2050 5869 D 2050 5886 D 2033 5903 D 1976
5903 D 1958 5886 D 1958 5960 D 2050 5960 D S
2200 6020 M 2140 6020 D S
2200 6140 M 2140 6140 D S
2200 6260 M 2140 6260 D S
2200 6380 M 2140 6380 D S
2200 6500 M 2140 6500 D S
2200 6620 M 2140 6620 D S
2200 6740 M 2140 6740 D S
2200 6860 M 2140 6860 D S
2200 6980 M 2140 6980 D S
2200 7100 M 2110 7100 D S
1736 7131 M 1764 7160 D 1764 7040 D S
1867 7040 M 1884 7040 D 1884 7057 D 1867 7057 D 1867 7040 D S
1958 7057 M 2050 7143 D S
2050 7131 M 2021 7160 D 1987 7160 D 1958 7131 D 1958 7069 D 1987 7040 D 2021
7040 D 2050 7069 D 2050 7131 D S
2200 7100 M 7000 7100 D S
7000 2300 M 7000 7100 D S
2715 6714 M 2753 6232 D 2790 5864 D 2828 5582 D 2866 5365 D 2903 5198 D 2942
5069 D 2979 4970 D 3017 4894 D 3054 4837 D 3092 4794 D 3130 4762 D 3167 4738 D
3205 4722 D 3243 4710 D 3281 4703 D 3318 4698 D 3356 4695 D 3394 4694 D 3431
4694 D 3469 4694 D 3506 4695 D 3545 4696 D 3582 4697 D 3620 4698 D 3658 4698 D
3695 4699 D 3733 4700 D 3770 4700 D 3808 4700 D 3846 4701 D 3884 4701 D 3922
4701 D 3959 4701 D 3997 4700 D 4034 4700 D 4072 4700 D 4110 4700 D 4148 4700 D
4186 4700 D 4223 4700 D 4261 4700 D 4298 4700 D 4336 4700 D 4374 4700 D 4411
4700 D 4450 4700 D 4487 4700 D 4525 4700 D 4562 4700 D 4600 4700 D 4638 4700 D
4675 4700 D 4713 4700 D 4750 4700 D 4789 4700 D 4826 4700 D 4864 4700 D 4902
4700 D 4939 4700 D 4977 4700 D 5014 4700 D 5052 4700 D 5090 4700 D 5128 4700 D
5166 4700 D 5203 4700 D 5241 4701 D 5278 4701 D 5316 4701 D 5354 4701 D 5392
4700 D 5430 4700 D 5467 4700 D 5505 4699 D 5542 4698 D 5580 4698 D 5618 4697 D
5655 4696 D 5694 4695 D 5731 4694 D 5769 4694 D 5806 !
4694 D 5844 4695 D 5882 4698 D 591

9 4703 D 5957 4710 D 5995 4722 D 6033 4738 D 6070 4762 D 6108 4794 D 6146 4837
D 6183 4894 D 6221 4970 D 6258 5069 D 6297 5198 D 6334 5365 D 6372 5582 D 6410
5864 D 6447 6232 D 6485 6714 D S
2715 3394 M 2725 3514 D S
2735 3634 M 2745 3753 D S
2756 3873 M 2770 3992 D S
2784 4111 M 2790 4166 D 2802 4230 D S
2822 4348 M 2828 4386 D 2848 4465 D S
2883 4579 M 2903 4631 D 2938 4686 D S
3034 4751 M 3054 4756 D 3092 4756 D 3130 4753 D 3154 4749 D S
3271 4727 M 3281 4726 D 3318 4719 D 3356 4714 D 3390 4710 D S
3510 4702 M 3545 4700 D 3582 4699 D 3620 4698 D 3630 4698 D S
3750 4699 M 3770 4699 D 3808 4699 D 3846 4699 D 3870 4699 D S
3990 4700 M 3997 4700 D 4034 4700 D 4072 4700 D 4110 4700 D S
4230 4700 M 4261 4700 D 4298 4700 D 4336 4700 D 4350 4700 D S
4470 4700 M 4487 4700 D 4525 4700 D 4562 4700 D 4590 4700 D S
4710 4700 M 4713 4700 D 4750 4700 D 4789 4700 D 4826 4700 D 4830 4700 D S
4950 4700 M 4977 4700 D 5014 4700 D 5052 4700 D 5070 4700 D S
5190 4700 M 5203 4700 D 5241 4700 D 5278 4700 D 5310 4700 D S
5430 4701 M 5467 4702 D 5505 4702 D 5542 4702 D 5550 4702 D S
5670 4699 M 5694 4698 D 5731 4697 D 5769 4694 D 5790 4692 D S
5909 4676 M 5919 4674 D 5957 4668 D 5995 4661 D 6026 4654 D S
6146 4644 M 6146 4644 D 6183 4652 D 6221 4671 D 6250 4698 D S
6309 4802 M 6334 4866 D 6346 4915 D S
6375 5032 M 6395 5150 D S
6414 5269 M 6428 5388 D S
6442 5507 M 6447 5552 D 6454 5626 D S
6463 5746 M 6474 5866 D S
6483 5985 M 6485 6006 D S
4333 7476 M 4333 7356 D S
4338 7476 M 4338 7356 D S
4373 7442 M 4373 7396 D S
4315 7476 M 4407 7476 D 4407 7442 D 4401 7476 D S
4338 7419 M 4373 7419 D S
4315 7356 M 4355 7356 D S
4447 7476 M 4441 7470 D 4447 7465 D 4453 7470 D 4447 7476 D S
4447 7436 M 4447 7356 D S
4453 7436 M 4453 7356 D S
4430 7436 M 4453 7436 D S
4430 7356 M 4470 7356 D S
4527 7436 M 4515 7430 D 4510 7425 D 4504 7413 D 4504 7402 D 4510 7390 D 4515
7385 D 4527 7379 D 4538 7379 D 4550 7385 D 4555 7390 D 4561 7402 D 4561 7413 D
4555 7425 D 4550 7430 D 4538 7436 D 4527 7436 D S
4515 7430 M 4510 7419 D 4510 7396 D 4515 7385 D S
4550 7385 M 4555 7396 D 4555 7419 D 4550 7430 D S
4555 7425 M 4561 7430 D 4573 7436 D 4573 7430 D 4561 7430 D S
4510 7390 M 4504 7385 D 4498 7373 D 4498 7367 D 4504 7356 D 4521 7350 D 4550
7350 D 4567 7345 D 4573 7339 D S
4498 7367 M 4504 7362 D 4521 7356 D 4550 7356 D 4567 7350 D 4573 7339 D 4573
7333 D 4567 7322 D 4550 7316 D 4515 7316 D 4498 7322 D 4493 7333 D 4493 7339 D
4498 7350 D 4515 7356 D S
4744 7453 M 4755 7459 D 4773 7476 D 4773 7356 D S
4767 7470 M 4767 7356 D S
4744 7356 M 4795 7356 D S
4910 7476 M 4910 7356 D S
4915 7476 M 4915 7356 D S
4910 7419 M 4898 7430 D 4887 7436 D 4875 7436 D 4858 7430 D 4847 7419 D 4841
7402 D 4841 7390 D 4847 7373 D 4858 7362 D 4875 7356 D 4887 7356 D 4898 7362 D
4910 7373 D S
4875 7436 M 4864 7430 D 4853 7419 D 4847 7402 D 4847 7390 D 4853 7373 D 4864
7362 D 4875 7356 D S
4893 7476 M 4915 7476 D S
4910 7356 M 4933 7356 D S
1021 4554 M 1089 4554 D S
1055 4554 M 1055 4698 D S
1021 4698 M 1089 4698 D S
1144 4554 M 1144 4630 D S
1144 4609 M 1165 4630 D 1178 4630 D 1199 4609 D 1199 4554 D S
1199 4609 M 1219 4630 D 1233 4630 D 1254 4609 D 1254 4554 D S
1439 4623 M 1446 4637 D 1459 4650 D 1473 4650 D 1480 4644 D 1487 4630 D 1487
4609 D 1473 4575 D S
1480 4644 M 1480 4616 D 1473 4589 D 1473 4561 D S
1480 4630 M 1466 4596 D 1466 4575 D 1473 4561 D 1487 4554 D 1501 4554 D 1514
4561 D 1528 4575 D 1542 4596 D 1549 4623 D 1549 4650 D 1542 4650 D 1542 4644 D
1549 4630 D S
1000 4770 M 1000 4914 D 1089 4914 D 1110 4894 D 1110 4860 D 1089 4846 D 1000
4846 D S
1034 4846 M 1110 4770 D S
1219 4791 M 1199 4770 D 1165 4770 D 1144 4791 D 1144 4825 D 1165 4846 D 1199
4846 D 1219 4825 D 1219 4805 D 1144 4805 D S
1439 4839 M 1446 4853 D 1459 4866 D 1473 4866 D 1480 4860 D 1487 4846 D 1487
4825 D 1473 4791 D S
1480 4860 M 1480 4832 D 1473 4805 D 1473 4777 D S
1480 4846 M 1466 4812 D 1466 4791 D 1473 4777 D 1487 4770 D 1501 4770 D 1514
4777 D 1528 4791 D 1542 4812 D 1549 4839 D 1549 4866 D 1542 4866 D 1542 4860 D
1549 4846 D S
1624 4770 M 1624 4791 D 1603 4791 D 1603 4770 D 1624 4770 D 1603 4750 D S
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