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%From: hamberh@uciph0.ps.uci.edu (Herbert W. Hamber)
%Date: Fri, 20 Aug 93 11:09:25 -0700


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%Date: Wed, xx June 1993 xx:xx +0200

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\begin{document}
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\addtolength{\baselineskip}{0.20\baselineskip}
\hfill DAMTP-93-27

\hfill UCI-Th-93-16

\hfill June 1993

\begin{center}

\vspace{36pt}
{\Large \bf Simplicial Gravity Coupled to Scalar Matter }

\vspace{24pt}

{\sl Herbert W. Hamber
\footnote{Supported in part by the National Science
Foundation under grant PHY-9208386}  }

\vspace{12pt}

Department of Physics \\
University of California at Irvine \\
Irvine, California 92717 USA \\

\vspace{18pt}

{\sl Ruth M. Williams}

\vspace{12pt}

Department of Applied Mathematics and Theoretical Physics \\
Silver Street \\
Cambridge CB3 9EW, England \\

\end{center}

\vfill

\begin{center} {\bf ABSTRACT } \end{center}
\vspace{12pt}
\noi

A model for quantized gravity coupled to matter in the form of a single
scalar field is investigated in four dimensions.
For the metric degrees of freedom we
employ Regge's simplicial discretization, with the scalar fields
defined at the vertices of the four-simplices. We examine how the
continuous phase transition found earlier,
separating the smooth from the rough phase of quantized gravity,
is influenced by the presence of scalar matter.
A determination of the critical exponents seems to indicate that the
effects of matter are rather small, unless the number of scalar flavors is
large. Close to the critical point where the average curvature
approaches zero, the coupling of matter to gravity is found to be weak.
The nature of the phase diagram and the values for the critical
exponents suggest that gravitational interactions increase with
distance.

\vspace{24pt}
\vfill
\newpage

\vskip 10pt
\newsection{Introduction}


Any serious attempt at understanding the ground state properties
of quantized gravity has to include at some stage the consideration
of the effects of matter fields. While there are many choices
for the matter fields and for their interactions, the simplest
actions to deal with in the framework of a lattice model for gravity
are the ones that represent one (or more) scalar fields.
In this paper we will discuss a first attempt at determining those
effects.

Regge's model is the natural discretization for quantized gravity
\cite{regge}.
At the classical level, it is completely equivalent to general relativity,
and the correspondence is particularly transparent in the lattice
weak field expansion, with the invariant edge lengths playing
the role of infinitesimal geodesics in the continuum.
In the limit of smooth manifolds with small curvatures, the
continuous diffeomorphism invariance of the continuum theory
is recovered \cite{rw,hw3d}.
But in contrast to ordinary lattice gauge theories, the model is formulated
entirely in terms of coordinate invariant quantities, the edge
lengths, which form the elementary degrees of freedom in the
theory \cite{hartle,rw1}.

Recent work based on Regge's simplicial formulation of gravity has shown,
in pure gravity without matter,
the appearance in four dimensions of a phase transition in the
bare Newton's constant, separating a smooth phase with small negative
average curvature from a rough phase with large positive
curvature \cite{tala,ph1}.
While the fractal dimension is rather small in the rough phase,
indicating a tree-like geometry for the ground state, it is very close
to four in the smooth phase close to the critical point.
Furthermore, a calculation of the critical exponents in the smooth
phase close to the critical point indicates that the transition is apparently
second order with divergent curvature fluctuations, and that a
lattice continuum can be constructed.

Very similar results have recently been obtained in the dynamical
triangulation model for gravity, in the sense that a similar phase
transition was found separating what appear to be the same type
of phases \cite{dt4d}.
This development represents an alternative and complementary
approach to what is being discussed here.
However it has not been possible yet in these models to extract the
critical exponents, and it is therefore not clear
yet whether a continuum limit really exists. In particular it appears
that close to the transition, the dynamical triangulation model
does not give rise to the correct scaling properties for the curvature,
which are necessary to define a lattice continuum limit.
It is therefore unclear whether the transition is
first order as a consequence of the discreteness of the
curvatures, with no continuum limit (as one finds for
example in lattice gauge theories based on discrete subgroups of
$SU(N)$ \cite{br}).
While in two dimensions both lattice models lead to similar results
both in the absence and presence of scalar matter \cite{dt2d,hw2d,gh},
in three dimensions the
dynamical triangulation model has no continuum limit \cite{dt3d},
in apparent disagreement with the continuum expectations \cite{wei,kn},
and the simplicial Regge gravity results \cite{hw3d}, which suggest instead
that a well defined continuum limit exists (albeit trivial in
the absence of matter, with the scalar curvature playing the role of a
scalar field). These results are rather disappointing, since
it would be desirable to have two rather different, independent
discretizations for gravity, with the same lattice continuum limit.
It is not clear yet at this point whether these results indicate
a fundamental flaw in the model (lack of restoration of broken diffeomorphism
invariance), or simply a perhaps surmountable technical difficulty in
determining exponents. For a clear recent review of some of these
aspects in the dynamically triangulated models we refer the reader
to the last reference in \cite{dt4d}.

In this paper we will present some first result on the properties of
Regge's simplicial gravity coupled to a scalar field, as derived from
numerical studies on lattices of up to $24 \times 16^4 = 1,572,864$
simplices.
The paper is organized as follows. First we
discuss in Sec. 2 the simplicial action and measure for the combined
gravitational and scalar degrees of freedom. Then we digress in Sec. 3
on what is known about the effects of scalar matter fields in the continuum,
to the extent that the results will be relevant for our later
calculations. We then present in Sec. 4 the definition of physical
observables which can be measured when scalar fields are present,
besides the purely gravitational ones introduced previously, and how
these can be related to effective low energy couplings.
In Sec. 5 we present our results and their interpretation, and in
Sec. 6 we give a discussion on how other quantities such as the curvature
and volume distributions can be obtained close to the critical point.
Sec. 7 then contains our conclusions.


\vskip 10pt
\newsection{Action and Measure for the Scalar Field}

Following \cite{hw84}, the four-dimensional pure gravity action on the
lattice is written as
\beq
I_g [l] =  \sum_ {\rm hinges \, h } \Bigl [ \, \lambda \, V_h -
k \, A_h \d _h + a \, { A_ h^2  \delta _ h^2 \over  V _ h } \, \Bigr ] ,
\label{eq:acg}
\eeq
where $V_h$ is the volume per hinge (represented by a triangle in four
dimensions),
$A_h$ is the area of the hinge and $\delta_h$ the corresponding
deficit angle, proportional to the curvature at $h$.
The term proportional to $k$ is the original Regge action. In the lattice
weak field expansion, the last two terms both contain higher derivative
contributions \cite{rw,hw3d} (in the last term it is the leading
contribution).
This is a simple consequence of the fact that on the lattice finite
differences give rise, when Fourier transformed, to terms involving
trigonometric functions of the lattice momenta.
The higher order corrections are
in general expected to be irrelevant in the continuum limit, if one
can be found, and unless the coefficient $a$ is taken to be very large
in this limit.
Whenever systematic studies have been done, there are
indications that this is indeed the case \cite{gh,hw3d}, as one would
expect from the experience gained in other, simpler model field theories.
The results of ref. ~\cite{ph1} in four dimensions
also suggest that the corrections are
negligible in the lattice continuum limit ($k \rightarrow k_c $),
and that the `ghost mass' associated with the higher derivative
corrections remains of the order of the ultraviolet cutoff,
of the order of the inverse average lattice spacing,
$m_{ghost} \sim \pi / l_0 $  (for a general
discussion of some of these points in simpler field theory models, see
for example \cite{pbook}). In the context of the present work the
higher derivative terms will be considered as convenient invariant
regulators, in addition to the usual lattice cutoff.

In the classical continuum limit the above action is
equivalent \cite{rw,hw3d,lee,cms,hw84} to
\beq
I_g [g] = \int  d^4 x  \, \sqrt g \, \Bigl [ \, \lambda - \half k \, R
+ \, \quarter a \, R _ { \mu \nu \rho \sigma }  R ^ { \mu \nu \rho \sigma }
+ \cdots \, \Bigr ] ,
\label{eq:acgc}
\eeq
with a cosmological constant term (proportional to $\lambda$), the
Einstein-Hilbert term ($k = 1 / ( 8 \pi G ) $), and a higher derivative
term, and with the dots indicating higher order lattice corrections.
In the following we will follow the convention of choosing the fundamental
lattice spacing
to be equal to one; the correct power of the lattice spacing needed
to convert lattice to continuum quantities can always be restored
by invoking dimensional arguments (but we have to remember that
due to the dynamical nature of the lattice, the average distance between
sites, in units of the fundamental lattice spacing, will still depend on
the bare couplings and the measure).
For an appropriate choice of bare couplings,
the above lattice action is bounded below for a regular lattice, even
for $a=0$, due to the presence of the lattice momentum cutoff \cite{rw}.
For non-singular measures and in the presence of the $\lambda$-term
such a regular lattice can be shown to arise naturally.
The higher derivative terms can be set to zero ($a=0$),
but they nevertheless seem to be necessary for reaching the lattice continuum
limit, and are in any case generated by radiative corrections
already in weak coupling perturbation theory.
When scalar fields are introduced, higher derivative terms are
generated as well by the quantum fluctuations of the scalar field.
Renormalization group arguments then suggest that in general the continuum
limit should be explored in this enlarged multi-parameter space.
Some very interesting suggestions regarding properties of non-renormalizable
theories beyond perturbation theory have been put forward in \cite{parisi}.

Next a scalar field is introduced, as the simplest type of dynamical
matter that can be coupled to gravity.
Consider an $n_f$-component field $\phi_i^a$, $a=1,...,n_f$, and
define this field at the vertices of the simplices.
Introduce finite lattice differences defined in the usual way
\beq
( \Delta _ \mu  \phi^a )_i =
{ \phi^a _{ i + \mu } - \phi^a _ i \over l _ { i,i+ \mu } } .
\eeq
The index $\mu$ labels the possible directions in which one can move from
a point in a given triangle, and
$l _ { i , i + \mu }$ is the length of the edge connecting the two points.
For simplicity let us consider for now the case $n_f=1$.
Then add to the above discrete pure gravitational action the contribution
\beq
I_\phi [l, \phi ] = \half \sum_{<ij>} V_{ij} \,
\Bigl ( { \phi_i - \phi_j \over l_{ij} } \Bigr )^2 \, +
\half \sum_{i} V_{i} \, (m^2 + \xi R_i ) \phi_i^2  +
\sum_{i} V_{i} \, U( \phi_i )  + \cdots ,
\label{eq:acp}
\eeq
where $U(\phi)$ is a potential for the scalar field, and the term containing
the discrete analog of the scalar curvature involves
\beq
V_{i} R_i \equiv \sum_{ h \supset i } \delta_h A_h  \sim \sqrt{g} R
\eeq
In the expression for the scalar action,
$V_{ij}$ is the volume associated with the edge $l_{ij}$,
while $V_i$ is associated with the site $i$.
There is more than one way to define such a volume \cite{hw84,cfl,bi}, but
under reasonable assumptions, such as positivity,
one should get equivalent results in the continuum.
The agreement between different lattice actions in the smooth limit
can be shown explicitly in the lattice weak field expansion, but the
calculations can be rather tedious and we will present the results elsewhere.
Here we will restrict ourselves to the baricentric volume
subdivision \cite{hw84} which is the simplest to deal with.
The above lattice action then corresponds to the continuum expression
\beq
I_\phi [ g, \phi ] = \half \int \, \sqrt g \; [  \,
g ^ { \mu \nu } \, \partial _ \mu  \phi \, \partial _ \nu  \phi
+ ( m^2 + \xi  R ) \phi^2   ] + \int \, \sqrt g \,  U( \phi ) + \cdots ,
\eeq
with the induced metric related in the usual way to the edge lengths
\cite{rw,hw3d}.
As is already the case for the purely gravitational action,
the correspondence
between lattice and continuum operators is true classically only up to higher
derivative corrections. But such higher derivative corrections in the scalar
field action are expected to be irrelevant and we will not consider them
here any further.
The scalar field potential $U(\phi)$ could contain quartic contributions,
whose effects are of interest in the context of cosmological
models where spontaneously broken symmetries play an important role.
For the moment we will be considering a scalar field
without direct self-interactions, and will set $U$=0.

The lattice scalar action contains a mass parameter $m$, which has
to be tuned to zero in lattice units to achieve the lattice continuum
limit for scalar correlations.
The dimensionless coupling $\xi$ is arbitrary;
two special cases are the minimal ($\xi = 0$) and the conformal
($\xi = \sixth $) coupling case.
As an extreme case one could consider a situation in which
the matter action by itself is the only action contribution, without
any kinetic term for the gravitational field, but still with a
non-trivial
gravitational measure; integration over the scalar field would
then give rise to an effective non-local gravitational action.

Having discussed the action, let us turn now to the measure.
The discretized partition function can be written as
\beq
Z = \int d \mu [ l ] \, d \mu [ \phi ] \,
\exp \, \left \{ -I_g [ l ] -I_\phi [ l, \phi ] \right \} .
\eeq
It is well known that the continuum gravitational measure
is not unique, and different regularizations will lead to different
forms for the measure.
DeWitt has argued that the gravitational measure should have the
form \cite{dw,dwm}
\beq
\int d \mu [g] \, = \, \int \prod_x  g^{ (d-4)(d+1)/8 }
\prod_{\mu \ge \nu} d g_{\mu \nu} .
\eeq
The main difference between various Euclidean
measures seems to be in the power of
$\sqrt{g}$ in the prefactor, which on the lattice corresponds to some product
of volume factors.  On the lattice these volume factors do not give rise
to coupling terms, and are therefore strictly {\it local}.
It should also be clear that since diffeomorphism invariance is lost in
{\it all} lattice models of gravity, at least away from
smooth manifolds (the very definition of a lattice breaks
local Poincar\'e invariance), there is no clear criterion at this point
to help one decide which measure should be singled out.
We have argued before that the power appearing in the measure should
be considered as an additional, hopefully irrelevant, bare
parameter \cite{hw84}.

On the simplicial lattice the invariant edge lengths represent the
elementary degrees of
freedom, which uniquely specify the geometry for a given incidence matrix.
Since the induced metric at a simplex is linearly related to the edge
lengths squared within that simplex, one would expect the lattice
analog of the DeWitt metric to simply correspond to $dl^2$ \cite{hartle}.
We will therefore write the lattice measure as \cite{lesh,hw84,tala}
\beq
\int d \mu _ \epsilon [ l ] =
\prod _ {\rm edges \, ij}  \int_ 0 ^ \infty \,
V_{ij}^{ 2 \sigma} \, { d  l _ {ij} ^ 2 } \, F _ \epsilon [ l ] ,
\label{eq:meas}
\eeq
where $V_{ij}$ is the 'volume per edge',
$ F _ \epsilon [l] $  is
a function of the edge lengths which enforces the higher-dimensional
analogs of the triangle inequalities,
and $\sigma = 0$ for the lattice analog of the DeWitt measure.
The parameter $ \epsilon $ is introduced as an ultraviolet
cutoff at small edge lengths: the function $F _ \epsilon [l]$ is
zero if any of the edges are equal to or less than $\epsilon$.
In general it is needed for sufficiently singular measures; for the $\sigma=0$
measure such a parameter is not necessary since
the triangle inequalities already strongly suppress small edge
lengths \cite{ph1}, and so we will set it to zero.
Note therefore that {\it no} cutoff is imposed on small or large edge
lengths, if a non-singular measure such as $dl^2$ is used.
This fact is essential for the recovery of diffeomorphism invariance
close to the critical point, where on large lattices
a few rather long edges, as well as some rather short ones,
start to appear \cite{tala,ph1}.

Eventually it is of interest to
systematically explore the sensitivity of the results
to the type of gravitational measure employed.
This has been done to a certain extent
in two \cite{gh} and three \cite{hw3d} dimensions.
The conclusion seems to be that for non-singular measures
the results relevant for the lattice continuum limit
(i.e. the long distance properties of the theory, as characterized
for example by the critical exponents)
appear to be independent of $\sigma$.
{}From a general point of view it is difficult to see how local volume
factors, which involve no gradient terms,
can possibly affect the nature of the continuum limit, which
is expected to be dominated by shear-wave-like distortions of the geometry of
space-time. The experience gained so far
seems to indicate that the volume factors
coming from the measure will only affect the overall lattice scale and the
shape of the distribution for the edge lengths, and will lead therefore
to different renormalizations of the cosmological constant, but will leave
the long-wavelength excitation spectrum, which is determined by the
relatively small fluctuations in the edge lengths about the lattice
equilibrium position, unaffected. But of course these arguments
cannot be taken as a substitute for a systematic investigation of this
issue in four dimensions.

In the presence of matter, similar considerations apply.
If an $n_f$-component scalar field is coupled to gravity
the power $\sigma$ appearing in the measure has to be changed, due
to an additional factor of $\prod_{x} ( \sqrt{g} )^{n_f/2}$
in the continuum gravitational measure.
On the lattice one then has $ \sigma  = n_f /30 $,
since with our discretization of spacetime based on hypercubes
there are $2^d -1=15$ edges emanating from each lattice vertex.
The additional measure factor insures that
\beq
\int \prod_x \left \{ d \phi  ( \sqrt{g} )^{n_f / 2} \right \} \,
\exp \left ( - \half m^2 \int \sqrt{g} \, \phi^2 \right ) \, = \,
\left [ \left ( \frac{ 2 \pi}{m^2} \right )^{n_f/2} \right ]^V = const
\eeq
or that for large mass, the scalar field completely decouples, leaving
only the dynamics of the pure gravitational field.


\vskip 10pt
\newsection{Effects of Matter Fields}

As long as the scalar action is quadratic, one can formally integrate
out the matter fields and obtain an effective Lagrangean contribution
written entirely in terms of the metric field,
\bea
\int d \mu [ \phi ] \, e^{ - \half \int \, \sqrt g \, \phi M[g] \phi }
& \equiv & \int \prod_x \left \{ d \phi \, ( \sqrt{g} )^{n_f / 2} \right \}
\, \exp \left \{ - \half \int \, \sqrt g \, \phi M[g] \phi \right \}
\nonumber \\
& \sim & \left \{ \det M[g] \right \}^{-n_f / 2}
\sim e^{- I_{eff} [g] } .
\eea
Here we have from the scalar field action
\beq
< x \vert \, M[g] \, \vert y > \, \equiv \,
( - \partial^2 + \xi R + m^2 ) \, \delta (x-y) ,
\eeq
where $\partial^2$ is the usual covariant Laplacian,
\beq
\partial^2 \phi \equiv  \frac{1}{\sqrt{g}} \, \partial_\mu \sqrt{g} \,
g ^ { \mu \nu } \partial_\nu  \phi .
\eeq
The full effective action, with terms from
Eq. ~(\ref{eq:acgc}) included, can be obtained from the results
of Ref. \cite{gi} (after introducing a proper time short distance
cutoff of the order of $s_0 \sim 1 / \Lambda^2 $). One finds then
\beq
I_{eff} [g] = \int \sqrt{g} \, \left [ \, \lambda' - \half k' \, R + \quarter
a' \, R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} + \cdots \, \right ] ,
\eeq
with effective couplings (for one flavor, $n_f=1$)
\bea
\lambda' & = & \lambda + { 1 \over 64 \pi^2 } \, \Lambda^4
- { 1 \over 32 \pi^2 } \, m^2 \Lambda^2
+ { 1 \over 64 \pi^2 } \, m^4 \ln \Lambda^2 + \cdots
\nonumber \\
k' & = & k + { 1 \over 16 \pi^2 } (\xi - \sixth ) \, \Lambda^2
+ { 1 \over 16 \pi^2 } ( \xi - \sixth ) \, m^2 \ln \Lambda^2 + \cdots
\nonumber \\
a' & = & a + { 1 \over 1920 \pi^2 } \, \ln \Lambda^2 + \cdots .
\label{eq:effcoupl}
\eea
For a fixed cutoff these corrections
are quite small in magnitude compared to the corresponding gravitational
radiative corrections computed in the $2+\epsilon$ expansion \cite{wei,kn}
or in higher derivative theories \cite{hdqg}.
We will see later that this is also clearly the case for the lattice results.
As in ordinary gauge theories, matter vacuum polarization effects
are small unless one has a large number of matter fields
(in which case even a new phase might appear).
To the extent that the lattice scalar action is equivalent in the
lattice continuum limit to the corresponding continuum scalar action, the above
perturbative results, valid for small curvatures,
should be relevant for the lattice model as well.

The effects of matter fields are small also from the point of view
of the $2 + \epsilon$ perturbative expansion for gravity \cite{wei,kn}.
One analytically continues in the spacetime dimension by using
dimensional regularization, and applies perturbation theory about
$d=2$, where Newton's constant is dimensionless
(it is not clear whether this approach makes any sense beyond
perturbation theory).
In this expansion the dimensionful bare coupling is written as
$G_0 = \Lambda^{2-d} G $, where $\Lambda$
is an ultraviolet cutoff (corresponding on the lattice to a momentum
cutoff of the order of the inverse average
lattice spacing, $\Lambda \sim \, $$ \pi / <~l^2~>^{1/2}$)
and $G$ a dimensionless bare coupling constant.
A double expansion in $G$ and $\epsilon$
then leads in lowest order to
a nontrivial fixed point in $G$ above two dimensions,
where some local averages and their fluctuations
are expected to develop an algebraic singularity in $G$
(the problem of the unboundedness of the Euclidean gravitational
action does not appear in perturbation theory).
Close to two dimensions the gravitational
beta function is given to one loop by
\beq
\beta (G) \, \equiv \, { \partial G \over \partial log \Lambda } \, = \,
\epsilon \, G \, - \, \twoth (25- n_f ) \, G^2 \, + \cdots ,
\label{eq:beta}
\eeq
where $n_f$ is the number of massless scalar fields.
To lowest order the ultraviolet fixed point is at
\beq
G^* \, = \,
{ 3 \epsilon \over 2 (25 - n_f ) } \, + \, O( \epsilon^2 ) .
\eeq
Integrating Eq.~(\ref{eq:beta}) close to the non-trivial fixed point
in $2 + \epsilon $ dimensions we obtain
\beq
\mu_0 \, = \, \Lambda \, \exp \left ( { - \int^G \, {d G' \over \beta (G') } }
\right )
\, \mathrel{\mathop\sim_{G \rightarrow G^* }} \,
\Lambda \, | \, G - G^* |^{ - 1 / \beta ' (G^*) } \, \sim \,
\Lambda \, | \, G - G^* |^{ 1 / \epsilon } ,
\eeq
where $\mu_0$ is an arbitrary integration constant, with dimension
of a mass, and which should be identified with some physical scale.
The derivative of the beta function at the fixed point defines
the critical exponent $\nu$, which to this order is independent of $n_f$,
\beq
\beta ' (G^*) \, = \, - \epsilon \, = \, - 1/ \nu .
\eeq
The possibility of algebraic singularities
in the neighborhood of the fixed point, appearing in vacuum
expectation values such as the average curvature and its derivatives,
is then a natural one, at least from the point
of view of the $2+\epsilon$ expansion.

The previous results also illustrate
how in principle the lattice continuum limit should be taken \cite{pbook}.
It corresponds to $\Lambda \rightarrow \infty$,
$G \rightarrow G^*$ with $\mu_0$ held constant; for fixed lattice
cutoff the continuum limit is approached by tuning $G$ to $G^*$.
Alternatively, we can choose to compute dimensionless ratios
directly, and determine their limiting value as we approach the critical
point (we will show examples of this later).
Away from $G^*$ one will in general expect to encounter some
lattice artifacts, which reflect
the non-uniqueness of the lattice transcription of the continuum action
and measure, as well as its reduced symmetry properties.

Let us conclude this section by mentioning that the
Nielsen-Hughes formula \cite{nh} for the one-loop beta function associated
with a spin-$s$ particle in four dimensions provides for a physical
interpretation of the fact that the matter contribution is so small
compared to the gravitational one.
It appears that this result is related to the fact that the
spin of the graviton is not a small number.
Considering only spin 0 and 2, the formula gives the lowest order
result for the beta function coefficient as
\beq
16 \pi^2 \beta_0 = - \sum_s
(-1)^{2s} \, \left [ \, (2s)^2 - \third \, \right ]
= - \frac{1}{3 } \, \bigl ( 47 - n_f \bigr ) ,
\label{eq:nh}
\eeq
making the matter contribution quite negligible unless the number
of flavors is large.
In higher derivative theories one finds similar large coefficients.
It is encouraging that similar results are found
from the lattice calculations to be described below.
Furthermore, for a sufficiently
large number of flavors one would expect eventually a phase transition
(if these lowest order results are taken seriously), due
to the change of sign in the beta function.


\vskip 10pt
\newsection{ Observables }

When we consider gravity coupled to a scalar field, we can distinguish
two types of observables, those involving the metric field (the edge
lengths) only, and those involving also the scalar field.
Quantities such as the expectation value of the scalar curvature, the
fluctuations in the curvatures or
the curvature correlations belong to the first class, while scalar
field averages and scalar correlations belong to the second class.

Following \cite{tala}, we define the following gravitational physical
observables, such as the average curvature
\beq
\cR (\lambda, k, a) \, \sim \,
\, { < \int \sqrt{g} \, R > \over < \int \sqrt{g} > } ,
\eeq
and the fluctuation in the local curvatures
\beq
\chi_\cR  (\lambda, k, a) \, \sim \,
{  < ( \int \sqrt{g} \, R )^2 > - < \int \sqrt{g} \, R >^2
\over < \int \sqrt{g} > }
\sim  \frac{1}{V} \frac{\partial^2}{\partial k^2} \ln Z .
\eeq
The lattice analogs of these expressions are readily written down
by making use of the correspondences \cite{hw84,lesh}
\bea
\int d^4 x \, \sqrt{g} & \to & \sum_{\rm hinges \, h} V_h  \\
\int d^4 x \, \sqrt{g} \, R & \to & 2 \sum_{\rm hinges \, h} \delta_h A_h \\
\int d^4 x \, \sqrt{g} \, R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}
& \to & 4 \sum_{\rm hinges \, h} V_h \, ( \delta_h^2 A_h^2 / V_h^2 ) .
\eea
On the lattice we prefer to define quantities in such a way that
variations in the average lattice spacing $\sqrt{<l^2>}$ are compensated by
the appropriate factor as determined from dimensional considerations.
In the case of the average curvature we define therefore the lattice quantity
$\cR$ as
\beq
{\cal R} \; = \; <l^2> { < 2 \sum_h \delta_h A_h > \over < \sum_h V_h > } ,
\label{eq:avr}
\eeq
and similarly for the curvature fluctuation.
The curvature fluctuation is related to the (connected) scalar curvature
correlator at zero momentum
\beq
\chi_\cR \sim { \int d^4 x \int d^4 y < \sqrt{g} R (x) \sqrt{g} R (y) >_c
\over < \int d^4 x \sqrt{g} > } .
\eeq
A divergence in the fluctuation is then indicative of
long range correlations (a massless particle).
Close to the critical point one expects for large separations a power
law decay in the geodesic distance,
\beq
< \sqrt{g} R (x) \sqrt{g} R (y) >
\mathrel{\mathop\sim_{ \vert x - y \vert \rightarrow \infty}}
\frac{1}{ \vert x-y \vert^{2n} } ,
\eeq
which in turn leads to the expectation $ \chi_\cR \sim L^{d-2n}$, where
$L \sim V^{ 1 / d} $ is the linear size of the system. In \cite{tala,ph1}
it was found that $\chi_\cR $ diverges close to the critical point as
\beq
\chi_\cR
\mathrel{\mathop\sim_{ k=k_c, \, L \rightarrow \infty}}
L^{ d ( 1 - \delta ) / ( 1 + \delta ) } ,
\eeq
where $\delta $ is the curvature critical exponent introduced in \cite{tala},
and therefore $ n = \delta d /( 1 + \delta ) = d - 1 / \nu  $, with
the exponent $\nu $ defined as $\nu = (1 + \delta)/d $.
Note that for a {\it scalar} field in four dimensions
one would expect $\nu = 1/2$, whereas we find
$\delta \approx 0.63$ and therefore $\nu \approx 0.41 $.

It is of interest to contrast the behavior of the preceding quantities,
associated with the curvature, with the analogous quantities involving
the local volumes (or the square root of the determinant of the metric in
the continuum) only.
We can consider therefore the average volume $<V>$, and its
fluctuation defined as
\beq
\chi_{V}  (\lambda, k, a) \, \sim \,
{  < ( \int \sqrt{g} )^2 > - < \int \sqrt{g} >^2
\over < \int \sqrt{g} > }
\sim  \frac{1}{V} \frac{\partial^2}{\partial \lambda^2} \ln Z .
\eeq
The latter is then related to the connected volume correlator at zero momentum
\beq
\chi_V
\sim { \int d^4 x \int d^4 y < \sqrt{g(x)} \sqrt{g(y)} >_c
\over < \int d^4 x \sqrt{g} > } .
\eeq
We have argued before \cite{tala}
that fluctuations in the curvature are sensitive
to the presence of a spin two massless particle,
while fluctuations in the volume probe only the correlations in the
scalar channel.
In the case of gravity a dramatic difference is therefore expected in
the two type of correlations.
Indeed the numerical simulations show clearly
a divergence in the curvature fluctuations, but at the same
time no divergence in the volume fluctuations.
Other, more complex invariant correlation functions at fixed geodesic
distance can be written down and measured \cite{ph1}.

Let us turn now to the observables involving the scalar field.
Due to the form of the action, the average of the scalar field
is always zero, but one can compute the discrete analog of the
following coordinate invariant fluctuation
\bea
\chi_\phi = &&
{ < \int d^4 x \int d^4 y \sqrt{g(x)} \, \phi (x) \sqrt{g(y)} \, \phi (y) >
\over < \int d^4 x \sqrt{g(x)} > }
\nonumber \\ &&
- \, { < \int d^4 x \sqrt{g(x)} \, \phi (x) >
< \int d^4 y \sqrt{g(y)} \, \phi (y) >
\over < \int d^4 x \sqrt{g(x)} > }
\eea
(again, for the Gaussian scalar action we will be considering, the
second term on the r.h.s. will be zero).
On the lattice such an expression can be written as
\beq
\chi_\phi \sim
{ < \sum_{i j} V_i \phi_i V_j \phi_j >
\over < \sum_i V_i > } -
{ < \sum_i V_i \phi_i > < \sum_j V_j \phi_j >
\over < \sum_i V_i > } .
\label{eq:phichi}
\eeq
Since $\chi_\phi $ is the zero-momentum component of the scalar
particle propagator, it is expected to diverge like $m^{-2}$ for small mass,
up to anomalous dimensions.
Also of interest are the local coordinate invariant averages
\bea
< \phi^2 > & \equiv &
{ < \int d^4 x \sqrt{g} \, \phi^2 > \over < \int d^4 x \sqrt{g} > }
\nonumber \\
< \phi^4 > & \equiv &
{ < \int d^4 x \sqrt{g} \, \phi^4 > \over < \int d^4 x \sqrt{g} > } .
\label{eq:phi2}
\eea
For {\it free} fields one expects the following dependence on the scalar
field mass,
\beq
< \phi^2 > \, = \int {d^4k \over ( 2 \pi )^4 }
\frac{1}{ k^2 + m^2 } =
\frac{1}{16 \pi^2} \left [ \Lambda^2 - m^2 \ln \frac{\Lambda^2 + m^2}{m^2}
\right ] ,
\label{eq:freephi2}
\eeq
\beq
< \phi^4 > \, = 2 \int {d^4k \over ( 2 \pi )^4 }
\frac{1}{ (k^2 + m^2)^2 } =
\frac{1}{8 \pi^2} \left [ \ln \frac{\Lambda^2 + m^2}{m^2} +
\frac{m^2}{\Lambda^2 + m^2} - 1 \right ] ,
\label{eq:freephi4}
\eeq
where $\Lambda$ is the ultraviolet momentum cutoff. In the interacting case
one anticipates, among other effects, a multiplicative renormalization
of the mass parameter $m$. In the presence of gravity, the behavior
of these quantities will be discussed below.

We can write schematically the propagator for the scalar field in a
{\it fixed} background geometry specified by some distribution
of edge lengths as
\beq
G(d) \, = \, < y | { 1 \over  - \partial^2 + \xi R + m^2 } | x > ,
\eeq
where $d$ is the geodesic distance between the two spacetime points being
considered.
Now fix one point at the origin $0$, and use the discretized
form of the scalar field action of Eq.~(\ref{eq:acp}). Then the discrete
equation of motion for the field $\phi_i$ in the presence of a
$\delta$-function source of unit strength localized at the origin
gives us the sought-after Green's function.
For $\xi=0$ we write the equation as
\beq
\phi_i = \frac{1}{W_i} \, \Bigl ( \sum_{ j \not= i }
W_{ij}  \; \phi_j + \delta_{i0} \Bigr ) ,
\eeq
with the weights $W$ given by
\beq
W_i = \sum_{j \not= i } \, \Bigl ( \frac{m^2}{2} + \frac{1}{l_{ij}^2}
\Bigr ) \, V_{ij} \;\;\;\;\;\;\; W_{ij} = \frac{V_{ij}}{l_{ij}^2} .
\eeq
Here the sums extend over nearest-neighbor points only,
$V_{ij}$ is the volume associated via a baricentric subdivision
with the edge $ij$, and
$\delta_{i0} $ is a delta-function source localized at the origin
on site $0$.
The above equation for $\phi_i$ can then be solved by an iterative procedure,
taking $\phi_i = 0$ as an initial guess.
After the solution $\phi_i$ has been
determined by relaxation, at large distances from the origin one has
\beq
\phi_i \sim G(d_{i0}) \sim A \, \sqrt{m / d_{i0}^{\, 3} }  \, \,
exp \, ( - m d_{i0} ) ,
\eeq
which determines the geodesic distance $d_{i0}$ from lattice point $0$ to
lattice point $i$. This method is more efficient
and accurate than trying to determine the geodesic distance by sampling paths
connecting the two points as was done in \cite{ph1},
but is of course equivalent to it \cite{fi}.

In quantum gravity it is of great interest to try to determine the
value of the low energy, renormalized coupling constants, and in
particular the effective cosmological constant $\lambda_{eff}$ and the
effective Newton's constant $G_{eff} = (8 \pi k_{eff})^{-1}$.
Equivalently, one would like
to be able to determine the large distance limiting value of a
dimensionless ratio such as $\lambda_{eff} G_{eff}^2 $, and its
dependence on the linear size of the system $L = V^{1/4} $.
(In the real world one knows that
$G_{eff} = (1.6160 \times 10^{-33} cm )^2 $, while
$\lambda_{eff} G_{eff}^2 \sim 10^{-120} $ is very small).
The vacuum expectation value of the scalar curvature can be used as
a definition of the effective, long distance cosmological constant
\beq
{\cal R} \, \sim \, { < \int \sqrt{g} \, R > \over < \int \sqrt{g} > }
\sim \left ( { 4 \lambda \over k } \right )_{eff} .
\label{eq:effr}
\eeq
In the pure gravity case one finds that there is a critical point in
$k$ at which the curvature vanishes, and for $k < k_c$ one has
\beq
\cR
\mathrel{\mathop\sim_{ k \rightarrow k_c}}
- A_\cR \, ( k_c - k )^\delta
\eeq
and thus $ (\lambda / k )_{eff} \rightarrow 0 $ in lattice units.
The location of the critical point $k_c$ and the amplitude in general depend on
the higher derivative coupling $a$ and other non-universal parameters,
but the exponent is expected
to be universal, and was estimated previously to be about 0.63; more
details can be found in refs. ~\cite{tala,ph1}.

One immediate consequence of this result is that in the smooth phase
with $k < k_c$ (or $G > G_c \equiv G^*$) the gravitational coupling
constant $G$ must increase with distance (anti-screening), at least
for rather short distances.
Introducing an arbitrary momentum scale $\mu $, one has close to the
ultraviolet fixed point the following short-distance behavior for
Newton's constant
\beq
G ( \mu ) - G^* \, = \, \left [  G ( \Lambda ) - G^* \right ]
\left ( \Lambda \over \mu \right ) ^{ 1 / \nu }
\eeq
with $\Lambda $ the ultraviolet cutoff; the exponents $\delta$ and $\nu$
are calculable and are related to each other via the scaling relation
$\nu = (1 + \delta ) / 4 \approx 0.41$.
The opposite behavior (screening) would be true in the phase with
$k > k_c$, but such a phase is known not to be stable and leads to no lattice
continuum limit \cite{ph1}.

If the system is of finite extent, with linear dimensions $L=V^{1/4}$,
then the scaling laws for ${\cal R} $ should also give the volume
dependence of the effective cosmological constant at the fixed point.
For pure gravity one finds at the critical point
\beq
{\cal R} \sim \, \mathrel{\mathop\sim_{ L \gg l_0 }} \,
\left ( { 1 \over L } \right )^{ \delta / \nu} ,
\label{eq:rsize}
\eeq
with $l_0$ of the order of the average lattice spacing,
$l_0 = \sqrt{ < l^2 > } $, and $\delta / \nu \approx 1.54 $.
The critical point here is defined, as usual, as the point in bare
coupling constant space where the curvature fluctuations diverge in the
infinite volume limit. Similarly for the dimensionless coupling $G$ in a finite
volume, one expects the scaling behavior
\beq
G ( \mu )  \, \mathrel{\mathop\sim_{ L, \; 1 / \mu \gg l_0 }} \,
G_c + \left ( { 1 \over \mu L } \right )^{ 1 / \nu} ,
\label{eq:gsize}
\eeq
These results are all direct consequences of the scaling laws
and the values of the critical exponents \cite{ph1}.
An important issue is how these results are affected by the presence
of dynamical matter. This will be addressed later in the paper.

The gravitational exponent $\delta$ determines the universal scaling
behavior of a variety of observables. Among the simplest ones which
are relevant for simple cosmological models one can mention the FRW
scale factor $a(t)$, as it appears in the line element
\beq
d s^2 \, = \, - dt^2 + a^2 (t) \left \{ { dr^2 \over 1 - k r^2 } +
r^2 ( d \theta^2 + \sin^2 \theta d \phi^2 ) \right \} ,
\eeq
and which we would expect to scale at short distances according to the equation
\beq
{ a^2 (t) \over a^2 (t_0) } \; \mathrel{\mathop\sim_{ t \gg t_0 }} \;
\left ( { t \over t_0 } \right )^{ \delta / \nu }
\eeq
with $c t_0 = l_0 $.
It is amusing to note that in this model the scale factor cannot exhibit a
singularity for short times, $t \sim t_0$. For such short distances the
strong fluctuations in the metric field and the curvature
prevent this from happening. We should add though that
the scale factor itself is essentially a semiclassical quantity,
linked to a specific ansatz for the (classical) metric at large
distances. In the presence of strong metric fluctuations it is no
longer clear that it remains a well-defined concept.

The bare Newton's constant also describes the coupling of gravity to
matter at scales comparable to the ultraviolet cutoff.
Consider the classical equations of motion
for pure Einstein gravity with a cosmological constant term
\beq
R_{\mu\nu} - \half g_{\mu\nu} R + \Lambda g_{\mu\nu} = 8 \pi G T_{\mu\nu} .
\eeq
Here we have followed the usual conventions by defining
$ \Lambda = 8 \pi G \lambda $ (not to be confused with the ultraviolet
momentum cutoff introduced earlier).
In the presence of higher derivative terms and higher order lattice
corrections this is of course not the right equation (the equations of
motion for higher derivative gravity are substantially more complex),
but at sufficiently large distances it should be the appropriate
equation if the average curvature is small and a sensible continuum
limit can be found in the lattice theory.
If we have only one real scalar field, the energy-momentum tensor is given by
\beq
T_{\mu\nu} =  \partial_\mu \phi \, \partial_\nu \phi -
\half g_{\mu\nu} ( \partial_\lambda \phi \, \partial^\lambda \phi
+ m^2 \phi^2 )
\eeq
(we will consider from now on only the case $\xi=0$).
Taking the trace we obtain
\beq
R = 4 \Lambda - 8 \pi G \, T_\mu^\mu = 4 \Lambda + 8 \pi G \,
\left [ (\partial \phi )^2 + 2 m^2  \phi^2 \right ] .
\eeq
Now consider the effects of quantum fluctuations, and
separate the pure gravity and matter contributions to the scalar curvature,
by writing for the average curvature
$ < R > \, = \, < R_{gravity} > + < R_{matter} > $,
where  $ < R > $ is the average of the
total scalar curvature in the presence of matter,
and $ < R_{gravity} > $ is the same quantity in the absence of matter.
More specifically, by the expectation value $ < R_{gravity} > $
we will simply mean the averages obtained in the absence of
any matter fields, as computed in ref. \cite{ph1}.
We will see below that $ < R_{matter} > $ represents a rather small
contribution, unless there are many scalar fields contributing
to the vacuum polarization.
In the presence of quantum fluctuations, we can write therefore for the
matter correction
\beq
< R_{matter} > \, = 8 \pi G
< (\partial \phi )^2 + 2 m^2  \phi^2 >
\, = \, 8 \pi G \left [ 2 < I_{\phi} > + m^2  < \phi^2 > \right ] .
\label{eq:geff}
\eeq
In other words, the change in the average value of the
scalar curvature that arises when matter fields are included is
proportional to Newton's constant $G$, and it is expected to
be positive.
This is indeed what will be found in the numerical simulations
discussed
below, even though the magnitude of the correction is quite small
(in agreement with the perturbative arguments presented in the
previous section).
To the extent that the feedback of the scalar degrees of freedom
on the gravitational degrees of freedom appears to be rather small (almost
to the point of being difficult to measure), we shall
argue below that gravity is indeed 'weak', at least for the
type of scalar action we have investigated here.


\vskip 10pt
\newsection{Numerical Procedure}

In order to explore the ground state of four-dimensional simplicial
gravity coupled to matter beyond perturbation theory one has to resort
to numerical methods.
As in our previous work, the edge lengths and scalars are updated by a
standard Metropolis algorithm,
generating eventually an ensemble of configurations distributed
according to the action of Eqs.~(\ref{eq:acg}) and ~(\ref{eq:acp}),
with the inclusion
of the appropriate generalized triangle inequality constraints
arising from the nontrivial gravitational measure.
Further details of the method as applied to pure gravity
are discussed in \cite{ijsa}, and will not be repeated here, since
the scalar action contribution can be dealt with in essentially the same way.

We have not included here a term coupling the scalar field directly to the
curvature ($\xi =0$), since the continuum perturbative results discussed
previously appear rather similar for different values of $\xi \ne \sixth $,
and the scalar action becomes significantly simpler for $\xi =0$. Also we note
that, in the absence of matter,
$<R>$ itself vanishes at the critical point \cite{tala,ph1}.
In mean field theory, we can replace the term $R \phi^2 $ by $R < \phi^2 > $.
Since $< \phi^2 > $ is finite at the critical point (see discussion
below), we expect the inclusion of this term to mostly affect a
renormalization of the critical coupling $k_c$ (related to the
critical value of Newton's constant by $k_c = 1 / (8 \pi G_c)$),
which should not change the universal critical behavior.

Let us point out here only the fact that, while the scalar field
action of Eq.~(\ref{eq:acp}) looks rather innocuous, due to the
simplicial nature
of the lattice a large number of interaction terms are involved at
each site: at each vertex there are 15 edges emanating in the positive
lattice 'directions', and 15 in the negative lattice 'directions' \cite{rw}.
In the update of the scalar field each of the 30 edge volumes
$V_{ij}$ has to be re-computed, by adding together the contributions
from all the four-simplices that meet on that edge. For the edge
volume one has
\beq
V_{ij} \, = \, \tenth \sum_{\rm simplices \,\, s \supset ij } V_s
\eeq
since there are ten edges per simplex in four dimensions.
Here the volume of a n-simplex with edge lengths $l_{ij}$
is given as usual by the determinant
\beq
V_n = { (-1)^{n+1 \over 2} \over n!  2^{n/2} }
\left| \begin{array}{llll}
0      &    1     &    1     & \ldots \\
1      &    0     & l_{12}^2 & \ldots \\
1      & l_{21}^2 &    0     & \ldots \\
1      & l_{31}^2 & l_{32}^2 & \ldots \\
\ldots &  \ldots  &  \ldots  & \ldots \\
1      & l_{n1}^2 & l_{n2}^2 & \ldots \\
1      & l_{n+1,1}^2 & l_{n+1,2}^2 & \ldots \\
\end{array} \right| ^{1 \over 2} , \\
\eeq
and corresponds to the determinant of a $6 \times 6$ matrix in the case
of a four-simplex; when expanded out it contains 130 distinct terms.
Furthermore the number of four-simplices meeting on a given edge depends
on the type of edge. With our simplicial subdivision of the four-dimensional
hypercubes that make up the lattice, we have four body principals,
six face diagonals, four body
diagonals and one hyperbody diagonal per hypercube \cite{rw}.
For a body principal or hyperbody diagonal
there are 24 four-simplices meeting on it, while for a face or body
diagonal there are 12 four-simplices meeting on it.
When updating one scalar field by the multi-hit Monte Carlo or heat bath
method, the 30 neighboring
link contributions need to be computed once, with their associated link
volumes, and special care has to be taken of the order of the edge lengths
appearing in the simplex formulae.
When updating a given edge length, all the scalar field action
contributions involving that particular edge have to evaluated, in addition
to the purely gravitational part. For a body principal and hyperbody
diagonal there are 65 such contributions that have to be added up,
while for a face or body diagonal 35 such contributions have to be added up.
By assigning then special fixed values to the edge lengths, one can perform
a number of checks against the expected analytical result to verify that
the volumes are computed and added up correctly.
Even though the program is quite computing intensive, it is well suited
for a massively parallel machine. In the two parallel versions of the
program we have written, a large number (64-256) of independent edge
and scalar variables are all updated simultaneously in parallel.

We considered lattices of size between $4 \times 4 \times 4 \times 4$
(256 vertices, 3840 edges, 6144 simplices)
and $16 \times 16 \times 16 \times 16 $
(65536 vertices, 983040 edges, 1572864 simplices).
Even though these lattices are not very large, one should keep in mind
that due to the simplicial nature of the lattice there are many edges
per hypercube with many interaction terms, and as a consequence the
statistical fluctuations are comparatively small, unless one is very
close to a critical point.
In all cases the measure over the edge lengths was of the form
$ dl^2 V_l^{n_f/30} $ times the triangle inequality constraints
(see Eq.~(\ref{eq:meas})). We shall restrict here our attention to
the case $n_f=1$; results for larger values of $n_f$ will
be presented elsewhere.

The topology was restricted to a four-torus (periodic boundary conditions),
and it is expected that for this choice boundary effects on physical
observables should be minimized.
One could perform similar calculations with lattices of different
topology, but the universal infrared scaling properties of
the theory should be determined only by short-distance renormalization
effects, independently of the specific choice of boundary conditions.
This is a consequence of the fact that the renormalization group
equations are {\it independent} of the boundary conditions, which enter only
in their solution as it affects the correlation functions through
the presence of a new dimensionful parameter $L$.
Thus the four-torus should be as good as any other choice of topology,
as long as we consider the universal long distance properties.

Let us give here a few details about the runs performed to compute
the averages.
In the presence of matter fields, the
lengths of the runs are much shorter than in the pure gravity
case \cite{ph1}, since the scalar field update is rather time-consuming.
The couplings $\lambda$ and $a$ in the gravitational action
of Eq. ~(\ref{eq:acg}) were fixed, as in the pure gravity case, to
$1$ and $0.005$ respectively. For pure gravity this choice leads
to a well defined ground state for $k \le k_c \approx 0.244 $ (the system
then resides in the smooth phase, with a fractal dimension
very close to four).
In the presence of matter, we also restricted most of our runs to this
physically more interesting phase, where the curvature is small and negative.
We investigated five values of $k$ ($0.0,0.05,0.1,0.15,0.20$), and for
each value we looked at a scalar mass of $1.0$, $0.5$ and $0.2$ in
lattice units.
In addition, we have accurate results for infinite mass from the previous
{\it pure} gravity calculations. Besides the results on lattices with
$L=4$ for all the above values of $k$ and $m$,
we also have accurate results on lattices of size $L=8$ and 16 for
$m=0.5$, and of size $L=8$ for $m=0.2$.
For these values of the scalar mass, the scalar correlations only extend over
a few lattice spacings, and finite size effects should therefore be contained
(we have checked that this is indeed the case for the quantities we
have measured).
In general we are interested in a regime in which the scalar mass
is much larger than the infrared cutoff, but much smaller than
the lattice ultraviolet cutoff, or
\beq
\sqrt{< l^2 >} \; \ll \;  m^{-1} \; \ll \; V^{1/4} ,
\eeq
in order to avoid finite lattice spacing and finite volume effects.
Similarly, one should also impose the constraint that
the scale of the curvature in magnitude should be much smaller than
the average lattice spacing, but much larger than the size of the
system, or
\beq
< l^2 >  \; \ll \;\; < l^2 > | {\cal R} |^{-1} \; \ll \; V^{1/2} .
\eeq
It is equivalent to the statement that in momentum space the physical scales
should be much smaller that the ultraviolet cutoff, but much larger
than the infrared one.

The lengths of the runs typically varied between $2-6k$ Monte Carlo
iterations on the $4^4$ lattice, $1-2k$ on the $8^4$ lattice, and
$0.6-0.9k$ on the $16^4$ lattice.
The runs are comparatively longer on the larger lattices, since it
was possible in that case to use a fully parallel version of the program.
As input configurations, we used the thoroughly thermalized configurations
generated previously for pure gravity. These configurations
are rather 'close' to the ones that include the effects of matter, since
the feedback of matter turns out to be rather small.
On the larger lattices duplicated copies of the
smaller lattices are used as starting configurations
for each $k$, allowing for additional
equilibration sweeps after duplicating the lattice in all four directions.
This allows for a substantial savings in time, since the initial
edge length configuration on the larger lattice is already quite close to
a representative configuration. We have found that in the well behaved
phase ($k < k_c$) the auto-correlation times are contained, of the
order of at most about one hundred sweeps. When we duplicate the
smaller lattice to a larger lattice, almost no drift in the averages is
observed during later re-thermalization, which indicates that for
our parameters the finite size corrections are small. On the larger
lattices, because there are so many variables to average over, the
statistical fluctuations from configuration to configuration are of
course much smaller.

\vskip 10pt
\newsection{Results}

In the pure gravity case, one finds that for fixed positive $a$ and $\lambda$
(the latter can be set equal to one without loss of generality, since
it determines the overall scale)
and sufficiently small $k$, the curvature is small and negative (smooth
phase), and goes to zero at the critical point $k_c (a) $, where
the curvature fluctuation diverges.
In the pure gravity case we write therefore, for $k$ less than $k_c$
\beq
\cR (k, a)
\mathrel{\mathop\sim_{k \rightarrow k_c(a)}}
- A_\cR (a) \, \left ( k_c(a) - k \right )^\delta
\label{eq:rsing}
\eeq
\beq
\chi_\cR (k, a)
\mathrel{\mathop\sim_{k \rightarrow k_c(a)}}
A_\chi (a) \, \left ( k_c(a) - k \right )^{\delta-1}
\eeq
where $\delta$ is a universal curvature critical
exponent, characteristic of the gravitational
transition \cite{tala}.
Here we will only consider the case $a=0.005$, for which the phase transition
is second order, leading therefore to a well-defined continuum limit
at least in the pure gravity case \cite{ph1}.
For $k \ge k_c$ the curvature is very large (rough phase),
and the lattice tends to collapse into degenerate configurations
with very long, elongated simplices ( with $ <V_h> / <l^2>^2 \sim 0$).
(In ref. \cite{ph1} several values for $a$ were studied, and it was found
that the model actually exhibits multicritical behavior.
While for $a=0.005$ one finds a second order phase transition,
for $a=0$ the singularity
appears to be in fact logarithmic ($\delta=0$), suggesting
a first order transition with no continuum limit for sufficiently
small $a$, with a multicritical point separating the two transition
lines).

When including the effects of the scalar field, one finds that the largest
changes are in the average volumes (which decrease by about three
percent for a scalar mass $m=0.5$)
and the average edge lengths. But such changes
are somewhat uninteresting, since they correspond effectively to a shift
(here actually an increase) in the bare cosmological constant (also by
about the same percentage, since
$\delta V / V \sim - \delta \lambda / \lambda$).
We note here incidentally that such a small effect is consistent with
the perturbative result of
Eq. (\ref{eq:effcoupl}), which predicts an increase in the effective
cosmological constant $\lambda$ by about one percent, for a cutoff
$\Lambda \sim \pi / l_0 \sim 1 $.
Indeed before we have chosen to define observables in such
a way that these effects are largely compensated, by rescaling by an
appropriate power of the average lattice spacing, as in Eq. (\ref{eq:avr}).
Physically more interesting are the results for the average curvature
in the presence of the scalar field.
As can be seen from Fig. 1, the effects of the feedback of one scalar
field on the curvature are quite small. It is useful to display the results
as a function of $ z = 1 / (1 + m^2 )$, since this allows us to put the
results for infinite mass (no scalar feedback, from ref. \cite{ph1})
on the same graph. The most accurate results in the presence of the
scalar field are for $m=0.5$, where we have relatively accurate results
for three different lattice sizes ($L=4,8,16$) and the highest statistics.
The points for $m=1.0$ are for reference only, since they
are from an $L=4$ lattice only.
For $m=0.5$ and $m=0.2$ the results show a small but clear systematic
decrease in the magnitude of the average curvature in the smooth phase for all
values of $k$, at the level of a few percent; to see such a small
effect long runs were needed.
The results are in qualitative agreement with the expectation that
the presence of the scalar field should give a positive
contribution to the average curvature.
In any case, for all values of the mass we have considered, the effects are
rather small.

As should be clear from the discussion in the previous section, we are
interested in how the critical behavior of the theory is affected in the
neighborhood of the critical point by the presence of the scalar field.
We will write therefore again for the average curvature, now in the
presence of the scalar field,
\beq
\cR  \mathrel{\mathop\sim_{ k \rightarrow k_c }}
- A_{\cR} \, ( k_c - k )^\delta ,
\eeq
where now we expect $A_{\cR} , k_c , \delta $ to depend also on the number of
scalar flavors, $n_f$.
In the presence of the scalars we have to look at the scaling limit
$m \rightarrow 0$, which in practical terms corresponds to a mass much
smaller than the inverse average lattice spacing.
It is not clear if $m=0.5$ (where we have our most accurate results)
in our case corresponds already to such a
scaling region, but our results should not be too far off, if the experience
in other lattice models can be used here as a guide.
If we adopt the same procedure as for pure gravity, and fit the average
curvature for $m=0.5$ to an algebraic singularity, we find
$A_{\cR}=3.68(5) $, $k_c=0.243(2)$ and $\delta=0.61(6)$.
This should be compared to the estimates for pure gravity (and for the same
value of $a=0.005$),
$A_{\cR}=3.79(4) $, $k_c=0.244(1)$ and $\delta=0.63(3)$ \cite{ph1}.
In Fig. 2 we compare the results for the average curvature $\cR(k)$ with
and without the presence of the scalar fields.
In Fig. 3 the same data is used to display $ [ - \cR (k) ]^{1 / \delta} $
instead, which as can be seen from the graph
deviates very little from a straight line behavior in $k$, if
one uses $\delta = 0.63 $.

We conclude therefore that, within our errors, switching on the
scalar fields leaves the exponents almost unchanged, and the critical
point moves very little; our results suggests that
$k_c$ decreases when we include the effects of the scalar field.
Again we notice that such a small shift is not unexpected on the basis
of the perturbative result of Eq. (\ref{eq:effcoupl}), which also
suggests a small decrease in the effective $k$, for a cutoff
$\Lambda \sim \pi / l_0 \sim 1 $.
For small non-integer $n_f$ we can expand the amplitude, critical
value of $k$ and the exponent in powers of the number of flavors $n_f$,
\bea
& A_{\cR} & =  A_0 + n_f A_1 + O(n_f^2) \nonumber \\
& k_c & =  k_0 + n_f k_1 + O(n_f^2) \nonumber \\
& \delta & =  \delta_0 + n_f \delta_1 + O(n_f^2) ,
\eea
and for the average curvature itself we get
\beq
\cR \mathrel{\mathop\sim_{ n_f \rightarrow 0 }}
- A_0 ( k_0 - k )^{\delta_0} \left \{
1 + n_f \bigl [ \frac{A_1}{A_0} + \frac{\delta_0 k_1}{k_0-k}
+ \delta_1 \ln (k_0 -k) \bigr ] + O(n_f^2) \right \}  ,
\eeq
which shows that the $k_1$ renormalization is dominant for very small
$n_f$.
Since the results for $n_f=1$ indicate that the corrections due to the scalar
field are quite small, we would tend to conclude that coefficients
of the $n_f$ terms must be rather small, and that the pure gravity theory
is already a good approximation to the full theory including scalars,
provided $n_f$ is not too large.

Let us assume for the moment that $k_1$ and $\delta_1$ are so small that
they can be neglected to a first approximation when we consider a single
scalar matter field (in the $2+\epsilon$ expansion the matter corrections
are certainly very small, and the exponent is independent of the number
of matter fields to leading order in $\epsilon$).
Then the difference between the average curvature in the presence of the
scalar field and in pure gravity determines the ratio of curvature
amplitudes $A_1/A_0$,
\beq
{ \cR_{matter} \over \cR_{gravity} } \, = \,
{ \cR_{gravity+matter} - \cR_{gravity} \over \cR_{gravity} }
\mathrel{\mathop\sim_{ k \rightarrow k_c }}
{ A_1 \over A_0 }
\eeq
The difference in the numerator is of course quite small, and requires
a very accurate measurement of the average curvature in both cases.
At the same time it
provides a direct determination of the physical effects of dynamical
matter fields, on a quantity that represents a direct physical observable,
since the average curvature can in principle be measured by performing
parallel transports of vectors around large closed loops.
The calculated difference
$ \cR_{gravity+matter} - \cR_{gravity} $ is shown in Fig. 4, together
with a fit to a behavior $ \sim ( k_c - k )^{\delta} $, treating only the
amplitude as a free parameter.
To reduce any systematic effects coming from finite
volume corrections, it is advisable to subtract the
average curvatures on the {\it same} lattice size.
In addition, such a subtraction can be done without any assumption
about the (singular) behavior of the curvature at $k_c$.
One then estimates approximately for the ratio
$ A_1 / A_0 \approx 0.053 / 3.79 = 0.014 $;
we will leave a more accurate quantitative determination of this ratio
for future work. We note though that the sign of the matter correction
to the curvature is consistent with the fact that the effective
Newton's constant gives rise to an attractive interaction ($G_{eff} >
0)$, thereby adding a positive contribution to the pure gravity
average curvature.

For an explanation for the smallness of such a ratio, we can look again
at the formula of Eq. ~(\ref{eq:nh}).
There the relative smallness of the matter
contribution is simply a consequence of the particle's relative spin.
For spin zero and spin two, as we have here, the ratio of the matter
over gravity contributions is
$ \sim \third / ( 4 s^2 - \third ) = 0.021 $, indeed of the same order as the
ratio we computed. One can go perhaps as far as turning
this argument around, and argue that
the smallness of the vacuum polarization effects compared to the
purely gravitational contribution is an indirect indication of the spin-two
nature of the graviton (if we were to treat the value of the graviton
spin as an unknown parameter, we would obtain a value
very close to two, $s \sim 2.5 $).

Let us turn now to a discussion of the renormalization properties of
the couplings $ G$ and $\lambda$.
It is clear from the preceding discussion that the effects of scalar
matter are quite small. In the following we shall therefore not
distinguish between the cases with and without matter fields, assuming
that if there are only a few matter fields, the exponents will not
change drastically.

As we indicated previously,
using the methods of finite size scaling \cite{fss},
one can translate the dependence of the curvature on $k-k_c$
into a statement about the {\it volume}
dependence of the curvature at the critical point $k_c$.
In a finite volume, of linear size $L$, finite size scaling
(from Eqs. ~(\ref{eq:effr}) and ~(\ref{eq:rsize})) gives
\beq
( G \lambda )_{eff} ( L ) \,
\mathrel{\mathop\sim_{ L  \gg l_0 }} \,
l_0^{-2} \left ( { l_0 \over L } \right )^{ 4 - 1 / \nu} ,
\eeq
since essentially the correlation length $\xi$ saturates at the system size,
$ \xi \sim (k_c - k)^{-\nu} \sim L $.
Combining this result with Eq. ~(\ref{eq:gsize}), one obtains for
the dimensionful Newton's constant the following scale dependence,
valid for short distances $1 / \mu \ll L $,
\beq
G_{eff} ( \mu ) \, \mathrel{\mathop\sim_{ L , \; 1 / \mu \gg l_0 }} \,
l_0^{2} G_c + l_0^{2} \left ( { 1 \over \mu L } \right )^{ 1 / \nu} ,
\eeq
(with $ 1 / \nu \approx 2.46 $), and for the dimensionful cosmological constant
\beq
\lambda_{eff} ( \mu ) \, \mathrel{\mathop\sim_{ L , \; 1 / \mu \gg l_0 }} \,
l_0^{-4} \left ( { \mu l_0 } \right )^{ 4 - 1 / \nu}
\left [ G_c + \left ( { 1 \over \mu L } \right )^{ 1 / \nu} \right ]^{-1}
\eeq
(with $ 4 - 1 / \nu \approx 1.54 $),
Here again $l_0$ is of the order of the average lattice spacing, and we have
restored the correct dimensions for $G_{eff}$ (length squared) and
$ \lambda_{eff} $ (inverse length to the fourth power).
For the dimensionless ratio $ G^2 \lambda$ we then obtain the result
\beq
( G^ 2\lambda )_{eff} ( \mu ) \,
\mathrel{\mathop\sim_{ L , \; 1 / \mu \gg l_0 }} \,
\left ( { \mu l_0 } \right )^{ 4 - 1 / \nu}
\left [ G_c + \left ( { 1 \over \mu L } \right )^{ 1 / \nu} \right ]
\eeq
As a check, it is immediate to see that the exponent associated with
$G_{eff}$ is indeed what one would expect from the form of the
Einstein part of the gravitational action in Eq. ~(\ref{eq:acgc}) and
the value of the curvature critical exponent $\delta$,
irrespective of whether matter fields are present or not (the specific
values of $\delta$ and $\nu $ will depend of course on how many matter
fields are present).

In conclusion, it seems that the dimensionless ratio $G^2 \lambda$
can be made very
small, provided the momentum scale $\mu$ is small enough, or, in other
words, at sufficiently large distances.
We should add also that the fixed point value for the dimensionless
gravitational constant, $G_c$, is in general non-universal and
cutoff-dependent, and depends on the specific way in which
an ultraviolet cutoff is introduced in the theory (here via an average
lattice spacing).
In our model it is of order one for very small $a$, but for larger $a$
it decreases in magnitude.

One notices that the smaller $G_c$, the smaller the distance
dependence of $G(r)$, since one has for the distance variation the result
\beq
{ \delta G ( r ) \over G ( r ) } \, = \,
{ \nu^{-1} \over G_c ( L / r )^{ 1 / \nu} + 1 } \, { \delta r \over r } ,
\eeq
(we have set $r= 1 / \mu$), so in practice $G_c$ cannot be too large.
For small $G_c$, $l_0^2$ becomes substantially larger than the Planck length.
It should be pointed out here that there is apparently no reason why
in this model the effective coupling $G_{eff}$ should turn out to be
of the same order as the ultraviolet cutoff $l_0^{-1}$, and indeed it
does not; the previous results seem to indicate that the situation
is more subtle.
Let us also add that one does do not expect the results to depend
significantly on the form of the lattice scalar action we have used.
In particular the presence of additional higher derivative terms
involving the scalar fields should not affect the results close to the
continuum limit, since the corrections should be suppressed by inverse
powers of the ultraviolet cutoff.

Another simple way of interpreting the results related to the
scalar field is as follows.
Close to the critical point, the average curvature approaches zero, and
at large distances it is therefore legitimate to write
$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, where $\eta_{\mu\nu}$ is the
flat metric, and $h_{\mu\nu} $ is a small correction. Then the scalar field
action of Eq.~(\ref{eq:acp}) is, again at large distances, close to the
action describing a free scalar, and its coupling to gravity is
correspondingly weak. At short distances the geometry fluctuates
wildly, and the coupling between gravity and matter is strong, while
at large distances the fluctuations eventually average out to zero,
effectively reducing the coupling.

Turning to the behavior of the scalar field itself, we show in Fig. 5
the results for $< \phi^2 >$,
in Fig. 6 those for $< \phi^4 >$ (see Eq.~(\ref{eq:phi2})), and
in Fig. 7 for $ \chi_{\phi} $ (defined in Eq.~(\ref{eq:phichi})).
The behavior of these three quantities is qualitatively rather
similar to their free field behavior (Eqs.~(\ref{eq:freephi2})
and ~(\ref{eq:freephi4})),
and is not too sensitive, at the
level of our accuracy, to the value of $k$. We note in particular that
$< \phi^2 > $ approaches a constant at $m=0$, while both $< \phi^4 > $ and
$\chi_{\phi}$ diverge at $m=0$, in agreement with a multiplicative mass
renormalization (no shift in the critical point for the field $\phi$, which
remains at $m=0$).

%%% Fig. 1 R/m   -- Fig. 2 R(k)  -- Fig. 3 R^1/delta -- Fig. 4 Delta R
%%% Fig. 5 phi^2 -- Fig. 6 phi^4 -- Fig. 7 chi-phi   -- Fig. 8 phases

Let us conclude this section with a brief, qualitative discussion of
the phase diagram, reconsidered in light of the results obtained in
the presence of scalar matter.
In the case of pure gravity, the phase diagram shows a line of
critical points in the ($a,k$) plane separating the smooth from the
rough (or collapsed) phase of gravity.
The curvature vanishes along this line when it is approached from the
smooth phase, and for some sufficiently
negative $a < a_0 < 0 $ a stable ground state
ceases to exist entirely.
For $a=0$ or very small positive $a$, the transition from one phase
to the other is first order, with no continuum limit, while for larger
$a$ is becomes second order, with a well defined lattice continuum
limit, as we indicated previously. These findings in
particular would seem to indicate the presence of a multicritical point,
where the two transition lines intersect \cite{ph1}.

In the presence of scalar matter fields, and for sufficiently large $a$,
our new results presented here
seem to suggest that a continuum limit still exists.
In addition, we have found that in the smooth phase the average curvature
decreases in magnitude by a small but calculable relative amount.
A quantitative estimate for the amount of this decrease gives
$\Delta \cR / \cR \sim A_1 / A_0 \approx 0.014 $. As the number of
(degenerate) scalar fields increases, we expect this trend to continue, until
$\Delta \cR / \cR \sim n_f A_1 / A_0 \sim 1 $, at which point a
new phase transition might take place, in the sense that the
smooth phase disappears altogether (we expect that
the critical value $k_c$ will continue
to decrease, and might even become negative at some point).
The appearance of a new phase in the presence of matter,
with the geometry resembling
branched polymers, is a well known fact in two dimensions \cite{2dtrees}.
In Fig. 8 we have sketched what a possible phase diagram in the ($k,n_f$)
plane might look like.
Presumably this new phase is nothing but the rough phase found for
$n_f=0$ and sufficiently large $k$.
It is characterized by very long
elongated simplices, with very small volumes, and a fractal dimension
much smaller than four, reminiscent of a tree-like structure of space-time.
Given our rather limited results, a crude estimate
for the critical number of flavors at which this is expected to happen
would be $n_f \sim 71$, a rather large number.
But such an estimate is not inconsistent with the perturbative estimates
of Eqs.~(\ref{eq:beta}) and ~(\ref{eq:nh}), which also give such large
numbers (24 and 47, respectively).
And of course for such large values, we expect deviations
from linearity in $n_f$, and we will have to leave a direct investigation
of this issue for future work. Finally let us remark that since
the effects of fermions can be mimicked by having scalars with negative $n_f$,
the above conclusions would be rather different in that case, and their
presence should rather impede the appearance of this new phase transitions.
While scalars tend to make the geometry rougher, fermions should make
it smoother.


\vskip 10pt
\newsection{ Volume and Curvature Distributions }

In this section we will discuss the properties of volume and
curvature distributions, and how their behavior close to the critical
point, which defines the lattice continuum limit, can be related largely
to the critical exponents discussed previously.
Let us assume that close to the critical point $\lambda_c$ one has
for the average volume a singularity of the type
\beq
< V > \, \equiv \, < \int \sqrt{g} > \,
\sim - {\partial \over \partial \lambda } \ln Z
\mathrel{\mathop\sim_{ \lambda \rightarrow \lambda_c }}
{ V_0 \over ( \lambda - \lambda_c )^\omega } + {\rm reg.} ,
\eeq
with $\omega \neq 1 $, and ``${\rm reg.}$'' denotes the regular part.
For the volume fluctuation one then expects close to $\lambda_c$
\beq
< V^2 > - <V>^2 \, \sim \, {\partial^2 \over \partial \lambda^2 } \ln Z
\mathrel{\mathop\sim_{ \lambda \rightarrow \lambda_c }}
{ \omega V_0 \over ( \lambda - \lambda_c )^{\omega +1} } + {\rm reg.} ,
\eeq
and it follows that the partition function close to the singularity is
given by
\beq
Z_{sing.} ( \lambda ) \sim \exp \left \{ - \int^\lambda d \lambda'
{ V_0 \over ( \lambda' - \lambda_c )^\omega } + {\rm reg.} \right \} .
\eeq
Now let us introduce the quantity $N(V)$ defined by
\beq
N ( V ) = \int d \mu [g] \, \delta ( \int \sqrt{g} - V ) \, e^{-I[g]} .
\eeq
It can be evaluated from
\beq
N (V) = { 1 \over 2 \pi i } \int_{- i \infty }^{ + i \infty }
d \lambda \, Z ( \lambda ) \, e^{ \lambda V } ,
\eeq
to give, in the saddle point approximation, the following expression
for the density of states
\beq
N ( V ) \sim V^{ \gamma - 3 } \exp \left \{
\lambda_c V ( 1 + b / V^{1 / \omega } ) \right \} .
\eeq
where $b$ is a constant involving $\omega$, $V_0$ and
$\lambda_c$, and the exponent $\gamma$ parameterizes a possible power
law correction.
Let us denote by $<...>_V$ the averages obtained in the fixed volume
ensemble. Then it is easy to see, from the transformation properties of
the fixed-volume partition function under a change of scale, that one has
\beq
{ \partial \ln N(V) \over \partial V } =
- \frac{1}{V} + \frac{\sigma}{4}  + \frac{k}{2}
{ < \int \sqrt{g} R >_V \over V } ,
\eeq
which can be combined with the previous equation
to give the result, valid for large
volumes and in the fixed volume ensemble \cite{tala},
\beq
{ < \int \sqrt{g} R >_V \over V }
\mathrel{\mathop\sim_{ V \rightarrow \infty }}
c_0 - {  2 - \gamma \over V } + { c_1 \over V^{1 / \omega } } + \cdots .
\eeq
We have not calculated the above average in the fixed volume ensemble,
but in the {\it canonical} ensemble, where the volume is allowed to fluctuate,
one finds the following result close to the critical point \cite{ph1}
\beq
{ < \int \sqrt{g} R > \over < \int \sqrt{g} > }
\mathrel{\mathop\sim_{ V \rightarrow \infty }}
{ 1 \over V^{\delta / ( 1 + \delta ) } } ,
\eeq
with $\delta \approx 0.63$.
It is reasonable to assume that the exponent $\omega$ is the same in the
two ensembles, in which case one gets $\omega \approx 2.60 $.
But this result then implies that the volume fluctuations cannot drive
a continuous phase transitions. If this were the case, then the specific heat
exponent $\alpha \equiv 2 - 4 \nu = 1 + \omega $ would have to be
$\alpha < 1 $ or $\nu > 1/d = 1/4$, otherwise the transition is
expected to be first order \cite{nn},
in which case one would not be able to define a lattice continuum limit.
Indeed a direct determination of the volume fluctuations shows that
they are always finite, and in particular do not diverge at the
critical point at $k_c$, indicating that the mass associated with
the volume fluctuations (the conformal mode) is of the order of the
ultraviolet cutoff \cite{tala,ph1}.

Let us look for completeness at the analogous result for the curvature
distribution. Again the exponents appearing in this case can be
related to the curvature critical exponent $\delta$.
Let us assume, as seems to be the case, that close to the critical
point $k_c$ one has
\beq
\cR (k) \equiv { < \int \sqrt{g} R > \over < \int \sqrt{g} > }
\sim + { 1 \over V } {\partial \over \partial k } \ln Z
\mathrel{\mathop\sim_{ k \rightarrow k_c }} - A_{\cR} ( k_c - k )^\delta  .
\eeq
(see Eq. (\ref{eq:rsing})).
Then for the curvature fluctuation one expects close to $k_c$
\beq
\chi_{\cR} \sim { 1 \over V } {\partial^2 \over \partial k^2 } \ln Z
\mathrel{\mathop\sim_{ k \rightarrow k_c }}
{ \delta A_{\cR} \over ( k_c - k )^{1 - \delta} } .
\eeq
Here we are interested in the singular part of the free energy.
Close to the singularity the partition function is then given by
\beq
Z_{sing} ( k ) \sim \exp \left \{ - V \int^k d k' \,
A_{\cR} ( k_c - k' )^\delta + {\rm reg.}  \right \} .
\eeq
Now let us introduce the quantity $N(R)$ defined by
\beq
N (R) = { 1 \over 2 \pi i } \int_{- i \infty }^{ + i \infty }
d k \, Z ( k ) \, e^{ k R } ,
\eeq
with $ R = - V \cR $ ($R$ is therefore a positive quantity, related to
the magnitude of the curvature,  in the smooth phase where $\cR < 0 $).
In the saddle point approximation the density
of states is given by
\beq
N ( R ) \sim \exp \left \{ k_c R -
{\textstyle {\delta \over 1 + \delta} \displaystyle } \, R \,
[ R / ( V A_\cR ) ]^{ 1 / \delta }  \right \} .
\eeq
We find therefore that the full probability distribution for $R$
has an algebraic singularity close to $R=0$ of the type
\beq
\ln P(R) \equiv - k R + \ln N ( R ) \sim
( k_c - k) R - {\textstyle {\delta \over 1 + \delta} \displaystyle }
\, R \, [ R / ( V A_\cR ) \, ]^{ 1 / \delta } .
\eeq
Again there will also be a regular part, which we have omitted here.
One can verify that the stationary point of the distribution $P(R)$
gives indeed the singular behavior of Eq. (\ref{eq:rsing}).


\vskip 10pt
\newsection{Conclusions}

In the previous sections we have presented some first results
regarding the effects of scalar matter on quantized gravity, in the
context of a quantum gravity model based on Regge's simplicial
formulation. It was found that the feedback of the scalar fields on
the geometry is quite
small on purely gravitational quantities such as the average
curvature, in agreement with some of the perturbative predictions
in the continuum, which also seem to suggest that the scalar vacuum
polarization effects should be rather small.
The qualitative features of the phase diagram for gravity, and in
particular the appearance of a smooth and a rough phase, seem
unchanged, at least as long as one does not have too many matter fields.
It appears therefore that the approximation in which matter internal
loops are neglected (quenched approximation) could be considered a
reasonable one, and that quantities such as the critical exponents
should not be too far off in this case.
To the extent that the coupling between the scalar and metric
degrees of freedom is weak close to the critical point, we have argued
that gravity is indeed weak, and have presented a procedure by
which the effective low energy Newton's constant can be estimated
independently of the renormalized cosmological constant, which is
determined from the scaling behavior of the average curvature close to the
critical point.
Our results suggest that in this model the effective
gravitational coupling close to the ultraviolet fixed point grows
with distance, and is expected to depend in a
non-trivial way on the overall linear size of the system.
For the gravitational coupling we have found an infrared growth
away from the fixed point of the
type $ G ( \mu ) \sim \mu^{- 1 / \nu }$, while for the cosmological
constant we have found a decrease in the infrared,
$ \Lambda (\mu ) \sim \mu^{ 4 - 1 / \nu }$, with an exponent
$\nu$ given approximately by $\nu \approx 0.41$, and only weakly
dependent on the matter content.

Finally let us add that our results bear some similarity with the
results obtained recently from the dynamical
triangulation model in four dimensions \cite{dts}, where the scalar
field also seems to give a rather small contribution. On the other
hand the matter contribution does not seem to improve on the fact
that in these models, which only allow discrete local curvatures,
the average curvature does not show the correct scaling behavior
close to the critical point,
which is a necessary condition for defining a lattice continuum limit
(in these models at the critical point the curvature diverges in
physical units).
Clearly more work is needed in both models to further clarify these
issues.


\vspace{12pt}

{\bf Acknowledgements}

The numerical computations were performed at the NSF-sponsored SDSC, NCSA
and PSC Supercomputer Centers under a {\sl Grand Challenge} allocation grant.
The parallel MIMD version of the quantum gravity program was written
and optimized for the CM5-512 with Yasunari Tosa of TMC, and his
invaluable help is here gratefully acknowledged.

\vspace{24pt}

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\newpage

\noindent {\large \bf Figure Captions}

\vspace{12pt}

\begin{itemize}

\item[Fig.\ 1]
Average curvature $\cR $ as a function of the mass of the scalar field
$m$, for different values of $k = 1 / 8 \pi G $. From top to bottom
$k=$ 0.0, 0.05, 0.1, 0.15, 0.2. The values for pure gravity ($z=0$) are
included for comparison, and drawn also as lines of constant $\cR$.
The values for $m=1.0$ ($z=0.5$) and $m=0.2$ ($z=0.962$) are from
a relatively small lattice with $L=4$ and are therefore for reference only,
while the values for $m=0.5$ ($z=0.80$) are
averages from the $L=8$ and $L=16$ lattices, with much smaller uncertainties.
The slight but clear decrease in the magnitude of the curvature in the
presence of the scalar field should be noted.

\item[Fig.\ 2]
Comparison of the average curvature $\cR$ as a function of $k$ in
the presence ($\diamond$) and absence ($\Box$) of the scalar field, with
mass $m=0.5$. The results for pure gravity are from ref. ~\cite{ph1} on
an $L=16$ lattice. The line corresponds to a fit of the pure gravity
results to an algebraic singularity, as discussed in the text.

\item[Fig.\ 3]
Minus the average curvature $\cR$ raised to the power
$1 / \delta = 1/0.63 $.
Parameters and data are the same as in Fig. 2. The straight line is
a fit to the pure gravity results. The linearity is now quite striking.

\item[Fig.\ 4]
Difference $\Delta \cR (k) $ between the average curvature in the presence
and absence of one scalar field, again for $m=0.5$ and $L=8,16$.
The difference is small and positive. The curve represents a
behavior close to the critical point of the type
$ \Delta \cR (k) \sim A \, ( k_c - k )^\delta $, with
$\delta \approx 0.63 $ and $k_c \approx 0.244 $ (the values for
pure gravity).

\item[Fig.\ 5]
The scalar field average $<\phi^2>$ as a function of $m$, and
for different values of the bare gravitational coupling $k$
($k=0.0,0.05,0.10,0.15,0.20$).
The data for $m=1.0$ and $m=0.2$ is from a lattice with $L=4$,
while data for $m=0.5$ from lattices with $L=8$ and 16.
The line is a fit assuming the free-field dependence on the mass $m$.

\item[Fig.\ 6]
Same as in Fig. 5, but for the scalar field average $<\phi^4>$.

\item[Fig.\ 7]
Same as in Fig. 5, but for the scalar field fluctuation $\chi_\phi $.

\item[Fig.\ 8]
A possible schematic phase diagram for gravity coupled to $n_f$ scalar
fields. The presence of the scalar fields shifts the critical point
$k_c = 1 / 8 \pi G_c $ towards smaller values as the number of scalar
flavors is increased, until the smooth phase disappears entirely for some
large number of flavors.


\end{itemize}

\end{document}

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808400018400010400010C00030C00030C00020C00060C00060C000E0C000C0E001C0E001C0E00
380F0018060016147E931A>25 D<00004000004000008000008000008000008000010000010000
0100000100000200000200001FC000E27003841806040C0C040E1C040638080730080770080770
0807E0100EE0100EE0100CE0101C6020387020303020601821C00E470003F80000400000400000
800000800000800000800001000001000001000018297E9F1B>30 D<0F00008013C0010021C001
0001E0020000E0040000F008000070100000702000007840000038400000388000003D0000001E
0000001C0000001C0000000E0000001E0000002E0000004F000000870000010700000207800002
038000040380000803C0001001C0002001E2004000E20080003C00191D7F931C>I<70F8F8F870
05057C840D>58 D<000001C00000078000001E00000078000001E00000078000000E0000003800
0000F0000003C000000F0000003C000000F0000000F00000003C0000000F00000003C0000000F0
000000380000000E0000000780000001E0000000780000001E0000000780000001C01A1A7C9723
>60 D<000100030003000600060006000C000C000C001800180018003000300030006000600060
00C000C000C00180018001800300030003000600060006000C000C000C00180018001800300030
003000600060006000C000C000C000102D7DA117>I<E0000000780000001E0000000780000001
E0000000780000001C0000000700000003C0000000F00000003C0000000F00000003C0000003C0
00000F0000003C000000F0000003C00000070000001C00000078000001E00000078000001E0000
0078000000E00000001A1A7C9723>I<00007E0100038183000E00460038002E0070001E00E000
0E01C0000C0380000C0700000C0F00000C1E0000081E0000083C0000003C000000780000007800
00007800000078000000F0000000F0007FFCF00001E0F00001E0F00003C0700003C0700003C070
0003C038000780380007801C000F800C000B80060033000380C100007F000020217E9F24>71
D<01E0000FE00001C00001C00001C00001C0000380000380000380000380000700000700000701
E00706100E08700E10F00E20F00E40601C80001D00001E00001FC000387000383800383800381C
20703840703840703840701880E01880600F0014207E9F18>107 D<1E07C07C00231861860023
A032030043C0340300438038038043803803808700700700070070070007007007000700700700
0E00E00E000E00E00E000E00E00E000E00E01C101C01C01C201C01C038201C01C038401C01C018
4038038018801801800F0024147E9328>109 D<1E07802318C023A06043C07043807043807087
00E00700E00700E00700E00E01C00E01C00E01C00E03821C03841C07041C07081C030838031018
01E017147E931B>I<01E02003F04007F8C00C1F80080100000200000400000800001000006000
00C0000100000200000400800801001003003F060061FC0040F80080700013147E9315>122
D E /Fg 16 94 df<00008000000001C000000001C000000003E000000003E000000005F00000
0004F000000008F80000000878000000107C000000103C000000203E000000201E000000401F00
0000400F000000800F80000080078000010007C000010003C000020003E000020001E000040001
F000040000F000080000F80008000078001000007C001000003C002000003E002000001E007FFF
FFFF007FFFFFFF00FFFFFFFF8021207E9F26>1 D<0020004000800100020006000C000C001800
18003000300030007000600060006000E000E000E000E000E000E000E000E000E000E000E000E0
006000600060007000300030003000180018000C000C000600020001000080004000200B2E7DA1
12>40 D<800040002000100008000C00060006000300030001800180018001C000C000C000C000
E000E000E000E000E000E000E000E000E000E000E000E000C000C000C001C00180018001800300
0300060006000C00080010002000400080000B2E7DA112>I<0006000000060000000600000006
000000060000000600000006000000060000000600000006000000060000000600000006000000
06000000060000FFFFFFF0FFFFFFF0000600000006000000060000000600000006000000060000
0006000000060000000600000006000000060000000600000006000000060000000600001C207D
9A23>43 D<03F0000E1C001C0E00180600380700700380700380700380700380F003C0F003C0F0
03C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C070038070038070
03807807803807001806001C0E000E1C0003F000121F7E9D17>48 D<018003800F80F380038003
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C0000180000300000600400C00401800401000803FFF807FFF80FFFF80121E7E9D17>I<03F000
0C1C00100E00200F00780F80780780780780380F80000F80000F00000F00000E00001C00003800
03F000003C00000E00000F000007800007800007C02007C0F807C0F807C0F807C0F00780400780
400F00200E001C3C0003F000121F7E9D17>I<000600000600000E00000E00001E00002E00002E
00004E00008E00008E00010E00020E00020E00040E00080E00080E00100E00200E00200E00400E
00C00E00FFFFF0000E00000E00000E00000E00000E00000E00000E0000FFE0141E7F9D17>I<18
03001FFE001FFC001FF8001FE00010000010000010000010000010000010000011F000161C0018
0E001007001007800003800003800003C00003C00003C07003C0F003C0F003C0E0038040038040
0700200600100E000C380003E000121F7E9D17>I<007C000182000701000E03800C07801C0780
380300380000780000700000700000F1F000F21C00F40600F80700F80380F80380F003C0F003C0
F003C0F003C0F003C07003C07003C07003803803803807001807000C0E00061C0001F000121F7E
9D17>I<4000007FFFC07FFF807FFF804001008002008002008004000008000008000010000020
0000200000400000400000C00000C00001C0000180000380000380000380000380000780000780
00078000078000078000078000078000030000121F7D9D17>I<03F0000C0C0010060030030020
01806001806001806001807001807803003E03003F06001FC8000FF00003F80007FC000C7E0010
3F00300F806003804001C0C001C0C000C0C000C0C000C0C000806001802001001002000C0C0003
F000121F7E9D17>I<7FFFFFE0FFFFFFF000000000000000000000000000000000000000000000
00000000000000000000FFFFFFF07FFFFFE01C0C7D9023>61 D<FEFEC0C0C0C0C0C0C0C0C0C0C0
C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0FEFE072D7CA10D>91
D<FEFE060606060606060606060606060606060606060606060606060606060606060606060606
0606060606FEFE072D7FA10D>93 D E end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300
TeXDict begin @letter
%%EndSetup
%%Page: 1 1
bop 264 1973 20 2 v 220 1987 a Fg(0)p 1866 1973 V 264 1844
V 184 1858 a(0)p Ff(:)p Fg(2)p 1866 1844 V 264 1715 V 184 1729
a(0)p Ff(:)p Fg(4)p 1866 1715 V 264 1586 V 184 1600 a(0)p Ff(:)p
Fg(6)p 1866 1586 V 264 1457 V 184 1471 a(0)p Ff(:)p Fg(8)p
1866 1457 V 264 1328 V 220 1342 a(1)p 1866 1328 V 264 1200
V 184 1214 a(1)p Ff(:)p Fg(2)p 1866 1200 V 264 1071 V 184 1085
a(1)p Ff(:)p Fg(4)p 1866 1071 V 264 942 V 184 956 a(1)p Ff(:)p
Fg(6)p 1866 942 V 264 813 V 184 827 a(1)p Ff(:)p Fg(8)p 1866
813 V 264 684 V 220 698 a(2)p 1866 684 V 399 1973 2 20 v 388
2032 a(0)p 399 704 V 669 1973 V 229 w(0)p Ff(:)p Fg(2)p 669
704 V 939 1973 V 211 w(0)p Ff(:)p Fg(4)p 939 704 V 1211 1973
V 212 w(0)p Ff(:)p Fg(6)p 1211 704 V 1481 1973 V 211 w(0)p
Ff(:)p Fg(8)p 1481 704 V 1751 1973 V 229 w(1)p 1751 704 V 264
1973 1622 2 v 1886 1973 2 1289 v 264 684 1622 2 v 45 1339 a
Fe(\000R)p Fg(\()p Ff(k)q Fg(\))923 2075 y Ff(z)15 b Fg(=)e(1)p
Ff(=)p Fg(\(1)c(+)h Ff(m)1188 2059 y Fd(2)1208 2075 y Fg(\))1019
648 y Fc(Fig.)18 b(1)p 264 1973 2 1289 v 381 971 a Fb(3)1057
949 y(3)1463 980 y(3)182 b(3)381 1107 y(3)1057 1136 y(3)1463
1125 y(3)1682 1123 y(3)381 1254 y(3)1057 1220 y(3)1463 1260
y(3)1682 1265 y(3)381 1426 y(3)1057 1433 y(3)1463 1442 y(3)1682
1446 y(3)381 1637 y(3)1057 1645 y(3)1463 1638 y(3)1682 1645
y(3)p 399 962 2 12 v 389 962 20 2 v 389 951 V 1075 955 2 39
v 1065 955 20 2 v 1065 916 V 1481 972 2 13 v 1471 972 20 2
v 1471 960 V 1700 972 2 14 v 1690 972 20 2 v 1690 959 V 399
1098 2 10 v 389 1098 20 2 v 389 1088 V 1075 1142 2 39 v 1065
1142 20 2 v 1065 1103 V 1481 1116 2 12 v 1471 1116 20 2 v 1471
1105 V 1700 1116 2 14 v 1690 1116 20 2 v 1690 1103 V 399 1247
2 14 v 389 1247 20 2 v 389 1234 V 1075 1232 2 53 v 1065 1232
20 2 v 1065 1180 V 1481 1253 2 15 v 1471 1253 20 2 v 1471 1239
V 1700 1258 2 14 v 1690 1258 20 2 v 1690 1245 V 399 1419 2
15 v 389 1419 20 2 v 389 1405 V 1075 1432 2 26 v 1065 1432
20 2 v 1065 1406 V 1481 1437 2 19 v 1471 1437 20 2 v 1471 1418
V 1700 1438 2 14 v 1690 1438 20 2 v 1690 1425 V 399 1635 2
25 v 389 1635 20 2 v 389 1611 V 1075 1651 2 39 v 1065 1651
20 2 v 1065 1612 V 1481 1634 2 19 v 1471 1634 20 2 v 1471 1615
V 1700 1641 2 19 v 1690 1641 20 2 v 1690 1622 V 264 957 2 2
v 264 957 1622 2 v 264 1093 2 2 v 264 1093 1622 2 v 264 1240
2 2 v 264 1240 1622 2 v 264 1412 2 2 v 264 1412 1622 2 v 264
1623 2 2 v 264 1623 1622 2 v 964 2825 a Fg(1)p eop
%%Page: 2 2
bop 264 1973 20 2 v 220 1987 a Fg(0)p 1866 1973 V 264 1715
V 184 1729 a(0)p Ff(:)p Fg(5)p 1866 1715 V 264 1457 V 220 1471
a(1)p 1866 1457 V 264 1200 V 184 1214 a(1)p Ff(:)p Fg(5)p 1866
1200 V 264 942 V 220 956 a(2)p 1866 942 V 264 684 V 184 698
a(2)p Ff(:)p Fg(5)p 1866 684 V 264 1973 2 20 v 218 2030 a Fe(\000)p
Fg(0)p Ff(:)p Fg(2)p 264 704 V 426 1973 V 56 w Fe(\000)p Fg(0)p
Ff(:)p Fg(15)p 426 704 V 588 1973 V 57 w Fe(\000)p Fg(0)p Ff(:)p
Fg(1)p 588 704 V 750 1973 V 56 w Fe(\000)p Fg(0)p Ff(:)p Fg(05)p
750 704 V 912 1973 V 901 2032 a(0)p 912 704 V 1076 1973 V 111
w(0)p Ff(:)p Fg(05)p 1076 704 V 1238 1973 V 91 w(0)p Ff(:)p
Fg(1)p 1238 704 V 1400 1973 V 92 w(0)p Ff(:)p Fg(15)p 1400
704 V 1562 1973 V 91 w(0)p Ff(:)p Fg(2)p 1562 704 V 1724 1973
V 92 w(0)p Ff(:)p Fg(25)p 1724 704 V 1886 1973 V 91 w(0)p Ff(:)p
Fg(3)p 1886 704 V 264 1973 1622 2 v 1886 1973 2 1289 v 264
684 1622 2 v 45 1339 a Fe(\000R)p Fg(\()p Ff(k)q Fg(\))966
2073 y Ff(k)14 b Fg(=)f(1)p Ff(=)p Fg(8)p Ff(\031)r(G)1019
648 y Fc(Fig.)18 b(2)p 264 1973 2 1289 v 512 1595 a(Pure)d(Gra)o(vit)o(y)778
1600 y Fb(2)572 984 y(2)896 1168 y(2)977 1232 y(2)1059 1289
y(2)1140 1336 y(2)1221 1395 y(2)1302 1472 y(2)1383 1535 y(2)1464
1628 y(2)1545 1709 y(2)1626 1814 y(2)1687 1974 y(2)p 772 1586
66 2 v 772 1596 2 20 v 838 1596 V 588 979 2 18 v 578 979 20
2 v 578 961 V 912 1161 2 15 v 902 1161 20 2 v 902 1146 V 994
1228 2 22 v 984 1228 20 2 v 984 1207 V 1076 1287 2 25 v 1066
1287 20 2 v 1066 1263 V 1157 1332 2 22 v 1147 1332 20 2 v 1147
1311 V 1238 1396 2 31 v 1228 1396 20 2 v 1228 1366 V 1319 1473
2 32 v 1309 1473 20 2 v 1309 1442 V 1400 1537 2 33 v 1390 1537
20 2 v 1390 1505 V 1481 1629 2 32 v 1471 1629 20 2 v 1471 1598
V 1562 1711 2 34 v 1552 1711 20 2 v 1552 1678 V 1643 1811 2
22 v 1633 1811 20 2 v 1633 1790 V 1704 1965 2 12 v 1694 1965
20 2 v 1694 1954 V 500 1644 a Fc(Scalar)f(Matter)777 1645 y
Fb(3)895 1182 y(3)1058 1297 y(3)1220 1405 y(3)1382 1551 y(3)1544
1708 y(3)p 772 1631 66 2 v 772 1641 2 20 v 838 1641 V 912 1174
2 14 v 902 1174 20 2 v 902 1161 V 1076 1290 2 15 v 1066 1290
20 2 v 1066 1276 V 1238 1397 2 12 v 1228 1397 20 2 v 1228 1386
V 1400 1543 2 13 v 1390 1543 20 2 v 1390 1531 V 1562 1700 2
14 v 1552 1700 20 2 v 1552 1687 V 772 1676 66 2 v 264 795 2
2 v 264 795 3 2 v 266 796 V 268 797 V 270 798 V 272 799 V 274
800 V 276 801 V 278 802 V 280 803 2 2 v 281 804 V 283 805 V
285 806 V 287 807 V 289 808 V 291 809 V 293 810 V 295 811 V
296 812 3 2 v 299 813 V 301 814 V 303 815 V 305 816 V 307 817
V 309 818 V 311 819 V 313 820 V 315 821 V 317 822 V 319 823
V 321 824 V 323 825 V 325 826 V 327 827 V 329 828 2 2 v 330
829 V 332 830 V 334 831 V 336 832 V 338 833 V 340 834 V 342
835 V 344 836 V 345 837 V 347 838 V 349 839 V 351 840 V 353
841 V 354 842 V 356 843 V 358 844 V 360 845 V 361 846 3 2 v
364 847 V 366 848 V 368 849 V 370 850 V 372 851 V 374 852 V
376 853 V 379 854 2 2 v 380 855 V 382 856 V 384 857 V 386 858
V 387 859 V 389 860 V 391 861 V 393 862 V 394 863 3 2 v 397
864 V 399 865 V 401 866 V 403 867 V 405 868 V 407 869 V 409
870 V 411 871 2 2 v 412 872 V 414 873 V 416 874 V 418 875 V
420 876 V 422 877 V 424 878 V 426 879 V 427 880 V 429 881 V
431 882 V 433 883 V 435 884 V 436 885 V 438 886 V 440 887 V
442 888 V 443 889 V 445 890 V 447 891 V 449 892 V 451 893 V
452 894 V 454 895 V 456 896 V 458 897 V 459 898 V 461 899 V
463 900 V 465 901 V 467 902 V 469 903 V 471 904 V 473 905 V
475 906 V 476 907 V 478 908 V 480 909 V 482 910 V 484 911 V
485 912 V 487 913 V 489 914 V 491 915 V 492 916 3 2 v 495 917
V 497 918 V 499 919 V 501 920 V 503 921 V 505 922 V 507 923
V 510 924 2 2 v 511 925 V 513 926 V 515 927 V 517 928 V 518
929 V 520 930 V 522 931 V 524 932 V 525 933 V 527 934 V 529
935 V 531 936 V 533 937 V 534 938 V 536 939 V 538 940 V 540
941 V 541 942 V 543 943 V 545 944 V 547 945 V 548 946 V 550
947 V 552 948 V 553 949 V 555 950 V 557 951 V 559 952 V 560
953 V 562 954 V 564 955 V 566 956 V 567 957 V 569 958 V 571
959 V 573 960 V 574 961 V 576 962 V 578 963 V 580 964 V 582
965 V 583 966 V 585 967 V 587 968 V 589 969 V 590 970 V 592
971 V 594 972 V 596 973 V 598 974 V 600 975 V 602 976 V 604
977 V 606 978 V 608 979 V 609 980 V 611 981 V 613 982 V 615
983 V 616 984 V 618 985 V 620 986 V 622 987 V 623 988 V 625
989 V 627 990 V 629 991 V 630 992 V 632 993 V 634 994 V 635
995 V 637 996 V 639 997 V 641 998 V 642 999 V 644 1000 V 646
1001 V 648 1002 V 649 1003 V 651 1004 V 653 1005 V 655 1006
V 656 1007 V 658 1008 V 660 1009 V 661 1010 V 663 1011 V 664
1012 V 666 1013 V 668 1014 V 669 1015 V 671 1016 V 672 1017
V 674 1018 V 676 1019 V 678 1020 V 680 1021 V 682 1022 V 684
1023 V 686 1024 V 688 1025 V 690 1026 V 691 1027 V 693 1028
V 694 1029 V 696 1030 V 697 1031 V 699 1032 V 701 1033 V 702
1034 V 704 1035 V 705 1036 V 707 1037 V 709 1038 V 711 1039
V 713 1040 V 714 1041 V 716 1042 V 718 1043 V 720 1044 V 721
1045 V 723 1046 V 725 1047 V 727 1048 V 728 1049 V 730 1050
V 732 1051 V 733 1052 V 735 1053 V 737 1054 V 739 1055 V 740
1056 V 742 1057 V 743 1058 V 745 1059 V 746 1060 V 748 1061
V 750 1062 V 751 1063 V 753 1064 V 754 1065 V 756 1066 V 758
1067 V 760 1068 V 762 1069 V 764 1070 V 766 1071 V 768 1072
V 770 1073 V 772 1074 V 773 1075 V 775 1076 V 776 1077 V 778
1078 V 779 1079 V 781 1080 V 783 1081 V 784 1082 V 786 1083
V 787 1084 V 789 1085 V 791 1086 V 792 1087 V 794 1088 V 795
1089 V 797 1090 V 799 1091 V 800 1092 V 802 1093 V 803 1094
V 805 1095 V 807 1096 V 809 1097 V 810 1098 V 812 1099 V 814
1100 V 815 1101 V 817 1102 V 819 1103 V 821 1104 V 822 1105
V 824 1106 V 825 1107 V 827 1108 V 828 1109 V 830 1110 V 832
1111 V 833 1112 V 835 1113 V 836 1114 V 838 1115 V 840 1116
V 841 1117 V 843 1118 V 844 1119 V 846 1120 V 848 1121 V 849
1122 V 851 1123 V 852 1124 V 854 1125 V 856 1126 V 857 1127
V 859 1128 V 860 1129 V 862 1130 V 863 1131 V 865 1132 V 866
1133 V 868 1134 V 870 1135 V 871 1136 V 873 1137 V 874 1138
V 876 1139 V 877 1140 V 879 1141 V 881 1142 V 882 1143 V 884
1144 V 885 1145 V 887 1146 V 889 1147 V 891 1148 V 892 1149
V 894 1150 V 896 1151 V 897 1152 V 899 1153 V 901 1154 V 903
1155 V 904 1156 V 905 1157 V 907 1158 V 908 1159 V 910 1160
V 911 1161 V 913 1162 V 914 1163 V 916 1164 V 917 1165 V 918
1166 V 920 1167 V 922 1168 V 923 1169 V 925 1170 V 926 1171
V 928 1172 V 930 1173 V 931 1174 V 933 1175 V 934 1176 V 936
1177 V 938 1178 V 939 1179 V 941 1180 V 942 1181 V 944 1182
V 945 1183 V 947 1184 V 948 1185 V 950 1186 V 952 1187 V 953
1188 V 954 1189 V 956 1190 V 957 1191 V 959 1192 V 960 1193
V 962 1194 V 963 1195 V 965 1196 V 966 1197 V 967 1198 V 969
1199 V 971 1200 V 972 1201 V 974 1202 V 976 1203 V 978 1204
V 980 1205 V 981 1206 V 983 1207 V 984 1208 V 986 1209 V 988
1210 V 989 1211 V 991 1212 V 992 1213 V 994 1214 V 995 1215
V 997 1216 V 998 1217 V 1000 1218 V 1002 1219 V 1003 1220 V
1004 1221 V 1006 1222 V 1007 1223 V 1009 1224 V 1010 1225 V
1012 1226 V 1013 1227 V 1015 1228 V 1016 1229 V 1017 1230 V
1019 1231 V 1021 1232 V 1022 1233 V 1024 1234 V 1025 1235 V
1027 1236 V 1028 1237 V 1030 1238 V 1031 1239 V 1033 1240 V
1034 1241 V 1036 1242 V 1037 1243 V 1039 1244 V 1040 1245 V
1041 1246 V 1043 1247 V 1044 1248 V 1045 1249 V 1047 1250 V
1048 1251 V 1049 1252 V 1051 1253 V 1052 1254 V 1053 1255 V
1055 1256 V 1056 1257 V 1058 1258 V 1059 1259 V 1061 1260 V
1062 1261 V 1064 1262 V 1065 1263 V 1067 1264 V 1068 1265 V
1070 1266 V 1071 1267 V 1073 1268 V 1074 1269 V 1076 1270 V
1077 1271 V 1079 1272 V 1080 1273 V 1082 1274 V 1083 1275 V
1085 1276 V 1086 1277 V 1088 1278 V 1089 1279 V 1090 1280 V
1092 1281 V 1093 1282 V 1094 1283 V 1096 1284 V 1097 1285 V
1098 1286 V 1100 1287 V 1101 1288 V 1102 1289 V 1104 1290 V
1105 1291 V 1107 1292 V 1108 1293 V 1110 1294 V 1111 1295 V
1113 1296 V 1114 1297 V 1116 1298 V 1117 1299 V 1118 1300 V
1120 1301 V 1121 1302 V 1123 1303 V 1124 1304 V 1125 1305 V
1127 1306 V 1128 1307 V 1130 1308 V 1131 1309 V 1132 1310 V
1134 1311 V 1135 1312 V 1137 1313 V 1138 1314 V 1139 1315 V
1141 1316 V 1142 1317 V 1143 1318 V 1145 1319 V 1146 1320 V
1147 1321 V 1149 1322 V 1150 1323 V 1151 1324 V 1153 1325 V
1154 1326 V 1156 1327 V 1157 1328 V 1158 1329 V 1160 1330 V
1161 1331 V 1163 1332 V 1164 1333 V 1165 1334 V 1167 1335 V
1168 1336 V 1170 1337 V 1171 1338 V 1172 1339 V 1174 1340 V
1175 1341 V 1176 1342 V 1178 1343 V 1179 1344 V 1180 1345 V
1182 1346 V 1183 1347 V 1184 1348 V 1186 1349 V 1187 1350 V
1188 1351 V 1190 1352 V 1191 1353 V 1192 1354 V 1194 1355 V
1195 1356 V 1196 1357 V 1198 1358 V 1199 1359 V 1200 1360 V
1201 1361 V 1203 1362 V 1204 1363 V 1205 1364 V 1207 1365 V
1208 1366 V 1209 1367 V 1211 1368 V 1212 1369 V 1213 1370 V
1215 1371 V 1216 1372 V 1217 1373 V 1218 1374 V 1219 1375 V
1221 1376 V 1222 1377 V 1223 1378 V 1224 1379 V 1226 1380 V
1227 1381 V 1228 1382 V 1229 1383 V 1230 1384 V 1232 1385 V
1233 1386 V 1235 1387 V 1236 1388 V 1237 1389 V 1239 1390 V
1240 1391 V 1241 1392 V 1243 1393 V 1244 1394 V 1245 1395 V
1247 1396 V 1248 1397 V 1249 1398 V 1250 1399 V 1252 1400 V
1253 1401 V 1254 1402 V 1256 1403 V 1257 1404 V 1258 1405 V
1260 1406 V 1261 1407 V 1262 1408 V 1264 1409 V 1265 1410 V
1266 1411 V 1267 1412 V 1268 1413 V 1270 1414 V 1271 1415 V
1272 1416 V 1273 1417 V 1275 1418 V 1276 1419 V 1277 1420 V
1278 1421 V 1279 1422 V 1281 1423 V 1282 1424 V 1283 1425 V
1284 1426 V 1286 1427 V 1287 1428 V 1288 1429 V 1289 1430 V
1290 1431 V 1292 1432 V 1293 1433 V 1294 1434 V 1295 1435 V
1296 1436 V 1298 1437 V 1299 1438 V 1300 1439 V 1301 1440 V
1303 1441 V 1304 1442 V 1305 1443 V 1306 1444 V 1308 1445 V
1309 1446 V 1310 1447 V 1311 1448 V 1312 1449 V 1314 1450 V
1315 1451 V 1316 1452 V 1317 1453 V 1318 1454 V 1319 1455 V
1320 1456 V 1322 1457 V 1323 1458 V 1324 1459 V 1325 1460 V
1326 1461 V 1327 1462 V 1328 1463 V 1330 1464 V 1331 1465 V
1332 1466 V 1333 1467 V 1335 1468 V 1336 1469 V 1337 1470 V
1338 1471 V 1339 1472 V 1341 1473 V 1342 1474 V 1343 1475 V
1344 1476 V 1345 1477 V 1347 1478 V 1348 1479 V 1349 1480 V
1350 1481 V 1351 1482 V 1352 1483 V 1353 1484 V 1355 1485 V
1356 1486 V 1357 1487 V 1358 1488 V 1359 1489 V 1360 1490 V
1361 1491 V 1363 1492 V 1364 1493 V 1365 1494 V 1366 1495 V
1367 1496 V 1368 1497 V 1369 1498 V 1371 1499 V 1372 1500 V
1373 1501 V 1374 1502 V 1375 1503 V 1376 1504 V 1377 1505 V
1379 1506 V 1380 1507 V 1381 1508 V 1382 1509 V 1383 1510 V
1384 1511 V 1385 1512 V 1387 1513 V 1388 1514 V 1389 1515 V
1390 1516 V 1391 1517 V 1392 1518 V 1393 1519 V 1394 1520 V
1396 1521 V 1397 1522 V 1398 1523 V 1399 1524 V 1400 1525 V
1401 1526 V 1402 1527 V 1403 1528 V 1404 1529 V 1405 1530 V
1406 1531 V 1407 1532 V 1408 1533 V 1409 1534 V 1410 1535 V
1412 1536 V 1413 1537 V 1414 1538 V 1415 1539 V 1416 1540 V
1417 1541 V 1418 1542 V 1419 1543 V 1420 1544 V 1421 1545 V
1422 1546 V 1423 1547 V 1424 1548 V 1425 1549 V 1426 1550 V
1428 1551 V 1429 1552 V 1430 1553 V 1431 1554 V 1432 1555 V
1433 1556 V 1434 1557 V 1435 1558 V 1436 1559 V 1437 1560 V
1438 1561 V 1439 1562 V 1440 1563 V 1441 1564 V 1442 1565 V
1443 1566 V 1445 1567 V 1446 1568 V 1447 1569 V 1448 1570 V
1449 1571 V 1450 1572 V 1451 1573 V 1452 1574 V 1453 1575 V
1454 1576 V 1455 1577 V 1456 1578 V 1457 1579 V 1458 1580 V
1459 1581 V 1460 1582 V 1461 1583 V 1462 1584 V 1463 1585 V
1464 1586 V 1465 1587 V 1466 1588 V 1467 1589 V 1468 1590 V
1469 1591 V 1470 1592 V 1471 1593 V 1472 1594 V 1473 1595 V
1474 1596 V 1475 1597 V 1476 1598 V 1477 1601 V 1478 1602 V
1479 1603 V 1480 1604 V 1481 1605 V 1482 1606 V 1483 1607 V
1484 1608 V 1485 1609 V 1486 1610 V 1487 1611 V 1488 1612 V
1489 1613 V 1490 1614 V 1491 1615 V 1492 1616 V 1493 1618 V
1494 1619 V 1495 1620 V 1496 1621 V 1497 1622 V 1498 1623 V
1499 1624 V 1500 1625 V 1501 1626 V 1502 1627 V 1503 1628 V
1504 1629 V 1505 1630 V 1506 1631 V 1507 1632 V 1508 1633 V
1509 1635 V 1510 1636 V 1511 1637 V 1512 1638 V 1513 1639 V
1514 1640 V 1515 1641 V 1516 1642 V 1517 1643 V 1518 1644 V
1519 1645 V 1520 1646 V 1521 1647 V 1522 1648 V 1523 1649 V
1524 1650 V 1525 1651 V 1526 1653 V 1527 1654 V 1528 1655 V
1529 1656 V 1530 1657 V 1531 1659 V 1532 1660 V 1533 1661 V
1534 1662 V 1535 1663 V 1536 1665 V 1537 1666 V 1538 1667 V
1539 1668 V 1540 1669 V 1541 1670 V 1542 1672 V 1543 1673 V
1544 1674 V 1545 1675 V 1546 1676 V 1547 1677 V 1548 1678 V
1549 1679 V 1550 1681 V 1551 1682 V 1552 1683 V 1553 1684 V
1554 1685 V 1555 1686 V 1556 1687 V 1557 1688 V 1558 1689 V
1559 1691 V 1560 1692 V 1561 1693 V 1562 1694 V 1563 1696 V
1564 1697 V 1565 1698 V 1566 1699 V 1567 1701 V 1568 1702 V
1569 1703 V 1570 1704 V 1571 1706 V 1572 1707 V 1573 1708 V
1574 1709 V 1575 1711 V 1576 1712 V 1577 1713 V 1578 1715 V
1579 1716 V 1580 1717 V 1581 1719 V 1582 1720 V 1583 1721 V
1584 1723 V 1585 1724 V 1586 1725 V 1587 1727 V 1588 1728 V
1589 1729 V 1590 1730 V 1591 1732 V 1592 1733 V 1593 1734 V
1594 1736 V 1595 1737 V 1596 1738 V 1597 1740 V 1598 1741 V
1599 1742 V 1600 1743 V 1601 1745 V 1602 1746 V 1603 1747 V
1604 1749 V 1605 1750 V 1606 1751 V 1607 1752 V 1608 1754 V
1609 1755 V 1610 1757 V 1611 1758 V 1612 1760 V 1613 1761 V
1614 1763 V 1615 1764 V 1616 1766 V 1617 1767 V 1618 1769 V
1619 1770 V 1620 1772 V 1621 1773 V 1622 1775 V 1623 1776 V
1624 1778 V 1625 1780 V 1626 1781 V 1627 1783 V 1628 1785 V
1629 1786 V 1630 1788 V 1631 1789 V 1632 1791 V 1633 1793 V
1634 1794 V 1635 1796 V 1636 1798 V 1637 1799 V 1638 1801 V
1639 1802 V 1640 1804 V 1641 1806 V 1642 1807 V 1643 1809 V
1644 1811 V 1645 1812 V 1646 1814 V 1647 1816 V 1648 1817 V
1649 1819 V 1650 1821 V 1651 1822 V 1652 1824 V 1653 1826 V
1654 1827 V 1655 1829 V 1656 1830 V 1657 1832 2 3 v 1658 1834
V 1659 1836 V 1660 1838 V 1661 1840 V 1662 1842 V 1663 1844
V 1664 1846 V 1665 1848 V 1666 1850 V 1667 1852 V 1668 1854
V 1669 1856 V 1670 1858 V 1671 1860 V 1672 1862 V 1673 1865
V 1674 1867 V 1675 1869 V 1676 1871 V 1677 1874 V 1678 1876
V 1679 1878 V 1680 1880 V 1681 1883 V 1682 1885 V 1683 1887
V 1684 1889 V 1685 1892 V 1686 1894 V 1687 1896 V 1688 1898
V 1689 1900 V 1690 1905 2 5 v 1691 1909 V 1692 1913 V 1693
1918 V 1694 1922 V 1695 1926 V 1696 1931 V 1697 1935 V 1698
1939 V 1699 1944 V 1700 1948 V 1701 1952 V 1702 1957 V 1703
1961 V 1704 1965 V 1705 1969 V 964 2825 a Fg(2)p eop
%%Page: 3 3
bop 264 1973 20 2 v 220 1987 a Fg(0)p 1866 1973 V 264 1789
V 184 1803 a(0)p Ff(:)p Fg(5)p 1866 1789 V 264 1605 V 220 1619
a(1)p 1866 1605 V 264 1421 V 184 1435 a(1)p Ff(:)p Fg(5)p 1866
1421 V 264 1236 V 220 1250 a(2)p 1866 1236 V 264 1052 V 184
1066 a(2)p Ff(:)p Fg(5)p 1866 1052 V 264 868 V 220 882 a(3)p
1866 868 V 264 684 V 184 698 a(3)p Ff(:)p Fg(5)p 1866 684 V
264 1973 2 20 v 218 2030 a Fe(\000)p Fg(0)p Ff(:)p Fg(2)p 264
704 V 426 1973 V 56 w Fe(\000)p Fg(0)p Ff(:)p Fg(15)p 426 704
V 588 1973 V 57 w Fe(\000)p Fg(0)p Ff(:)p Fg(1)p 588 704 V
750 1973 V 56 w Fe(\000)p Fg(0)p Ff(:)p Fg(05)p 750 704 V 912
1973 V 901 2032 a(0)p 912 704 V 1076 1973 V 111 w(0)p Ff(:)p
Fg(05)p 1076 704 V 1238 1973 V 91 w(0)p Ff(:)p Fg(1)p 1238
704 V 1400 1973 V 92 w(0)p Ff(:)p Fg(15)p 1400 704 V 1562 1973
V 91 w(0)p Ff(:)p Fg(2)p 1562 704 V 1724 1973 V 92 w(0)p Ff(:)p
Fg(25)p 1724 704 V 1886 1973 V 91 w(0)p Ff(:)p Fg(3)p 1886
704 V 264 1973 1622 2 v 1886 1973 2 1289 v 264 684 1622 2 v
45 1343 a([)p Fe(\000R)p Fg(\()p Ff(k)q Fg(\)])206 1327 y Fd(1)p
Fa(=\016)966 2073 y Ff(k)14 b Fg(=)f(1)p Ff(=)p Fg(8)p Ff(\031)r(G)1019
648 y Fc(Fig.)18 b(3)p 264 1973 2 1289 v 572 930 a Fb(2)896
1220 y(2)977 1312 y(2)1059 1391 y(2)1140 1452 y(2)1221 1528
y(2)1302 1619 y(2)1383 1688 y(2)1464 1779 y(2)1545 1848 y(2)1626
1922 y(2)1687 1986 y(2)p 588 923 2 15 v 578 923 20 2 v 578
909 V 912 1213 2 15 v 902 1213 20 2 v 902 1198 V 994 1309 2
23 v 984 1309 20 2 v 984 1287 V 1076 1389 2 24 v 1066 1389
20 2 v 1066 1366 V 1157 1449 2 23 v 1147 1449 20 2 v 1147 1427
V 1238 1526 2 24 v 1228 1526 20 2 v 1228 1503 V 1319 1616 2
23 v 1309 1616 20 2 v 1309 1594 V 1400 1689 2 29 v 1390 1689
20 2 v 1390 1660 V 1481 1780 2 29 v 1471 1780 20 2 v 1471 1751
V 1562 1842 2 15 v 1552 1842 20 2 v 1552 1827 V 1643 1919 2
23 v 1633 1919 20 2 v 1633 1897 V 1704 1973 2 3 v 1694 1973
20 2 v 1694 1971 V 895 1240 a(3)1058 1403 y(3)1220 1541 y(3)1382
1705 y(3)1544 1846 y(3)p 912 1233 2 15 v 902 1233 20 2 v 902
1218 V 1076 1396 2 15 v 1066 1396 20 2 v 1066 1381 V 1238 1534
2 15 v 1228 1534 20 2 v 1228 1520 V 1400 1698 2 15 v 1390 1698
20 2 v 1390 1683 V 1562 1840 2 15 v 1552 1840 20 2 v 1552 1825
V 352 684 2 2 v 353 685 V 354 686 V 355 687 V 356 688 V 357
689 V 358 690 V 359 691 V 360 692 V 361 693 V 362 694 V 363
695 V 364 696 V 365 697 V 366 698 V 367 699 V 368 700 V 369
701 V 371 702 V 372 703 V 373 704 V 374 705 V 375 706 V 376
707 V 377 708 V 378 709 V 380 710 V 381 711 V 382 712 V 383
713 V 384 714 V 385 715 V 386 716 V 387 717 V 388 718 V 389
719 V 390 720 V 391 721 V 392 722 V 393 723 V 394 724 V 395
725 V 396 726 V 397 727 V 398 728 V 399 729 V 400 730 V 401
731 V 402 732 V 403 733 V 404 734 V 405 735 V 406 736 V 407
737 V 408 738 V 409 739 V 411 740 V 412 741 V 413 742 V 414
743 V 415 744 V 416 745 V 417 746 V 418 747 V 419 748 V 420
749 V 421 750 V 422 751 V 423 752 V 424 753 V 425 754 V 426
755 V 428 756 V 429 757 V 430 758 V 431 759 V 432 760 V 433
761 V 434 762 V 435 763 V 436 764 V 437 765 V 438 766 V 439
767 V 440 768 V 441 769 V 442 770 V 444 771 V 445 772 V 446
773 V 447 774 V 448 775 V 449 776 V 450 777 V 451 778 V 452
779 V 453 780 V 454 781 V 455 782 V 456 783 V 457 784 V 458
785 V 459 786 V 460 787 V 461 788 V 462 789 V 463 790 V 464
791 V 465 792 V 466 793 V 467 794 V 468 795 V 469 796 V 470
797 V 471 798 V 472 799 V 473 800 V 474 801 V 475 802 V 477
803 V 478 804 V 479 805 V 480 806 V 481 807 V 482 808 V 483
809 V 484 810 V 485 811 V 486 812 V 487 813 V 488 814 V 489
815 V 490 816 V 491 817 V 493 818 V 494 819 V 495 820 V 496
821 V 497 822 V 498 823 V 499 824 V 500 825 V 501 826 V 502
827 V 503 828 V 504 829 V 505 830 V 506 831 V 507 832 V 508
833 V 510 834 V 511 835 V 512 836 V 513 837 V 514 838 V 515
839 V 516 840 V 517 841 V 518 842 V 519 843 V 520 844 V 521
845 V 522 846 V 523 847 V 524 848 V 525 849 V 527 850 V 528
851 V 529 852 V 530 853 V 531 854 V 532 855 V 533 856 V 534
857 V 535 858 V 536 859 V 537 860 V 538 861 V 539 862 V 540
863 V 541 864 V 542 865 V 543 866 V 544 867 V 545 868 V 546
869 V 547 870 V 548 871 V 549 872 V 551 873 V 552 874 V 553
875 V 554 876 V 555 877 V 556 878 V 557 879 V 559 880 V 560
881 V 561 882 V 562 883 V 563 884 V 564 885 V 565 886 V 566
887 V 567 888 V 568 889 V 569 890 V 570 891 V 571 892 V 572
893 V 573 894 V 574 895 V 575 896 V 576 897 V 577 898 V 578
899 V 579 900 V 580 901 V 581 902 V 582 903 V 583 904 V 584
905 V 585 906 V 586 907 V 587 908 V 588 909 V 589 910 V 590
911 V 591 912 V 592 913 V 593 914 V 594 915 V 595 916 V 596
917 V 597 918 V 598 919 V 600 920 V 601 921 V 602 922 V 603
923 V 604 924 V 605 925 V 606 926 V 608 927 V 609 928 V 610
929 V 611 930 V 612 931 V 613 932 V 614 933 V 615 934 V 616
935 V 617 936 V 618 937 V 619 938 V 620 939 V 621 940 V 622
941 V 623 942 V 624 943 V 625 944 V 626 945 V 627 946 V 628
947 V 629 948 V 630 949 V 631 950 V 633 951 V 634 952 V 635
953 V 636 954 V 637 955 V 638 956 V 639 957 V 641 958 V 642
959 V 643 960 V 644 961 V 645 962 V 646 963 V 647 964 V 648
965 V 649 966 V 650 967 V 651 968 V 652 969 V 653 970 V 654
971 V 655 972 V 656 973 V 657 974 V 658 975 V 659 976 V 660
977 V 661 978 V 662 979 V 663 980 V 664 981 V 665 982 V 666
983 V 667 984 V 668 985 V 669 986 V 670 987 V 671 988 V 672
989 V 674 990 V 675 991 V 676 992 V 677 993 V 678 994 V 679
995 V 680 996 V 681 997 V 682 998 V 683 999 V 684 1000 V 685
1001 V 686 1002 V 687 1003 V 688 1004 V 690 1005 V 691 1006
V 692 1007 V 693 1008 V 694 1009 V 695 1010 V 696 1011 V 697
1012 V 698 1013 V 699 1014 V 700 1015 V 701 1016 V 702 1017
V 703 1018 V 704 1019 V 705 1020 V 707 1021 V 708 1022 V 709
1023 V 710 1024 V 711 1025 V 712 1026 V 713 1027 V 714 1028
V 715 1029 V 716 1030 V 717 1031 V 718 1032 V 719 1033 V 720
1034 V 721 1035 V 722 1036 V 723 1037 V 724 1038 V 725 1039
V 726 1040 V 727 1041 V 728 1042 V 729 1043 V 730 1044 V 731
1045 V 732 1046 V 733 1047 V 734 1048 V 735 1049 V 736 1050
V 737 1051 V 739 1052 V 740 1053 V 741 1054 V 742 1055 V 743
1056 V 744 1057 V 745 1058 V 746 1059 V 747 1060 V 748 1061
V 749 1062 V 750 1063 V 751 1064 V 752 1065 V 753 1066 V 754
1067 V 756 1068 V 757 1069 V 758 1070 V 759 1071 V 760 1072
V 761 1073 V 762 1074 V 763 1075 V 764 1076 V 765 1077 V 766
1078 V 767 1079 V 768 1080 V 769 1081 V 770 1082 V 772 1083
V 773 1084 V 774 1085 V 775 1086 V 776 1087 V 777 1088 V 778
1089 V 779 1090 V 780 1091 V 781 1092 V 782 1093 V 783 1094
V 784 1095 V 785 1096 V 786 1097 V 787 1098 V 789 1099 V 790
1100 V 791 1101 V 792 1102 V 793 1103 V 794 1104 V 795 1105
V 796 1106 V 797 1107 V 798 1108 V 799 1109 V 800 1110 V 801
1111 V 802 1112 V 803 1113 V 804 1114 V 805 1115 V 806 1116
V 807 1117 V 808 1118 V 809 1119 V 810 1120 V 811 1121 V 813
1122 V 814 1123 V 815 1124 V 816 1125 V 817 1126 V 818 1127
V 819 1128 V 821 1129 V 822 1130 V 823 1131 V 824 1132 V 825
1133 V 826 1134 V 827 1135 V 828 1136 V 829 1137 V 830 1138
V 831 1139 V 832 1140 V 833 1141 V 834 1142 V 835 1143 V 836
1144 V 837 1145 V 838 1146 V 839 1147 V 840 1148 V 841 1149
V 842 1150 V 843 1151 V 844 1152 V 845 1153 V 846 1154 V 847
1155 V 848 1156 V 849 1157 V 850 1158 V 851 1159 V 852 1160
V 853 1161 V 854 1162 V 855 1163 V 856 1164 V 857 1165 V 858
1166 V 859 1167 V 860 1168 V 862 1169 V 863 1170 V 864 1171
V 865 1172 V 866 1173 V 867 1174 V 868 1175 V 870 1176 V 871
1177 V 872 1178 V 873 1179 V 874 1180 V 875 1181 V 876 1182
V 877 1183 V 878 1184 V 879 1185 V 880 1186 V 881 1187 V 882
1188 V 883 1189 V 884 1190 V 885 1191 V 886 1192 V 887 1193
V 888 1194 V 889 1195 V 890 1196 V 891 1197 V 892 1198 V 893
1199 V 895 1200 V 896 1201 V 897 1202 V 898 1203 V 899 1204
V 900 1205 V 901 1206 V 903 1207 V 904 1208 V 905 1209 V 906
1210 V 907 1211 V 908 1212 V 909 1213 V 910 1214 V 911 1215
V 912 1216 V 913 1217 V 914 1218 V 915 1219 V 916 1220 V 917
1221 V 918 1222 V 919 1223 V 920 1224 V 921 1225 V 922 1226
V 923 1227 V 924 1228 V 925 1229 V 926 1230 V 927 1231 V 928
1232 V 929 1233 V 930 1234 V 931 1235 V 932 1236 V 933 1237
V 934 1238 V 936 1239 V 937 1240 V 938 1241 V 939 1242 V 940
1243 V 941 1244 V 942 1245 V 943 1246 V 944 1247 V 945 1248
V 946 1249 V 947 1250 V 948 1251 V 949 1252 V 950 1253 V 952
1254 V 953 1255 V 954 1256 V 955 1257 V 956 1258 V 957 1259
V 958 1260 V 959 1261 V 960 1262 V 961 1263 V 962 1264 V 963
1265 V 964 1266 V 965 1267 V 966 1268 V 967 1269 V 968 1270
V 969 1271 V 970 1272 V 971 1273 V 972 1274 V 973 1275 V 974
1276 V 976 1277 V 977 1278 V 978 1279 V 979 1280 V 980 1281
V 981 1282 V 982 1283 V 983 1284 V 984 1285 V 986 1286 V 987
1287 V 988 1288 V 989 1289 V 990 1290 V 991 1291 V 992 1292
V 993 1293 V 994 1294 V 995 1295 V 996 1296 V 997 1297 V 998
1298 V 999 1299 V 1000 1300 V 1002 1301 V 1003 1302 V 1004
1303 V 1005 1304 V 1006 1305 V 1007 1306 V 1008 1307 V 1009
1308 V 1010 1309 V 1011 1310 V 1012 1311 V 1013 1312 V 1014
1313 V 1015 1314 V 1016 1315 V 1017 1316 V 1019 1317 V 1020
1318 V 1021 1319 V 1022 1320 V 1023 1321 V 1024 1322 V 1025
1323 V 1026 1324 V 1027 1325 V 1028 1326 V 1029 1327 V 1030
1328 V 1031 1329 V 1032 1330 V 1033 1331 V 1035 1332 V 1036
1333 V 1037 1334 V 1038 1335 V 1039 1336 V 1040 1337 V 1041
1338 V 1042 1339 V 1043 1340 V 1044 1341 V 1045 1342 V 1046
1343 V 1047 1344 V 1048 1345 V 1049 1346 V 1050 1347 V 1052
1348 V 1053 1349 V 1054 1350 V 1055 1351 V 1056 1352 V 1057
1353 V 1058 1354 V 1059 1355 V 1060 1356 V 1061 1357 V 1062
1358 V 1063 1359 V 1064 1360 V 1065 1361 V 1066 1362 V 1067
1363 V 1068 1364 V 1069 1365 V 1070 1366 V 1071 1367 V 1072
1368 V 1073 1369 V 1074 1370 V 1075 1371 V 1076 1372 V 1077
1373 V 1078 1374 V 1079 1375 V 1080 1376 V 1081 1377 V 1082
1378 V 1084 1379 V 1085 1380 V 1086 1381 V 1087 1382 V 1088
1383 V 1089 1384 V 1090 1385 V 1091 1386 V 1092 1387 V 1093
1388 V 1094 1389 V 1095 1390 V 1096 1391 V 1097 1392 V 1098
1393 V 1099 1394 V 1101 1395 V 1102 1396 V 1103 1397 V 1104
1398 V 1105 1399 V 1106 1400 V 1107 1401 V 1108 1402 V 1109
1403 V 1110 1404 V 1111 1405 V 1112 1406 V 1113 1407 V 1114
1408 V 1115 1409 V 1116 1410 V 1117 1411 V 1118 1412 V 1119
1413 V 1120 1414 V 1121 1415 V 1122 1416 V 1123 1417 V 1125
1418 V 1126 1419 V 1127 1420 V 1128 1421 V 1129 1422 V 1130
1423 V 1131 1424 V 1132 1425 V 1134 1426 V 1135 1427 V 1136
1428 V 1137 1429 V 1138 1430 V 1139 1431 V 1140 1432 V 1141
1433 V 1142 1434 V 1143 1435 V 1144 1436 V 1145 1437 V 1146
1438 V 1147 1439 V 1148 1440 V 1149 1441 V 1150 1442 V 1151
1443 V 1152 1444 V 1153 1445 V 1154 1446 V 1155 1447 V 1156
1448 V 1158 1449 V 1159 1450 V 1160 1451 V 1161 1452 V 1162
1453 V 1163 1454 V 1164 1455 V 1165 1456 V 1167 1457 V 1168
1458 V 1169 1459 V 1170 1460 V 1171 1461 V 1172 1462 V 1173
1463 V 1174 1464 V 1175 1465 V 1176 1466 V 1177 1467 V 1178
1468 V 1179 1469 V 1180 1470 V 1181 1471 V 1182 1472 V 1183
1473 V 1184 1474 V 1185 1475 V 1186 1476 V 1187 1477 V 1188
1478 V 1189 1479 V 1190 1480 V 1191 1481 V 1192 1482 V 1193
1483 V 1194 1484 V 1195 1485 V 1196 1486 V 1197 1487 V 1199
1488 V 1200 1489 V 1201 1490 V 1202 1491 V 1203 1492 V 1204
1493 V 1205 1494 V 1206 1495 V 1207 1496 V 1208 1497 V 1209
1498 V 1210 1499 V 1211 1500 V 1212 1501 V 1213 1502 V 1215
1503 V 1216 1504 V 1217 1505 V 1218 1506 V 1219 1507 V 1220
1508 V 1221 1509 V 1222 1510 V 1223 1511 V 1224 1512 V 1225
1513 V 1226 1514 V 1227 1515 V 1228 1516 V 1229 1517 V 1230
1518 V 1231 1519 V 1232 1520 V 1233 1521 V 1234 1522 V 1235
1523 V 1236 1524 V 1237 1525 V 1238 1526 V 1239 1527 V 1240
1528 V 1241 1529 V 1242 1530 V 1243 1531 V 1244 1532 V 1245
1533 V 1246 1534 V 1248 1535 V 1249 1536 V 1250 1537 V 1251
1538 V 1252 1539 V 1253 1540 V 1254 1541 V 1255 1542 V 1256
1543 V 1257 1544 V 1258 1545 V 1259 1546 V 1260 1547 V 1261
1548 V 1262 1549 V 1264 1550 V 1265 1551 V 1266 1552 V 1267
1553 V 1268 1554 V 1269 1555 V 1270 1556 V 1271 1557 V 1272
1558 V 1273 1559 V 1274 1560 V 1275 1561 V 1276 1562 V 1277
1563 V 1278 1564 V 1279 1565 V 1281 1566 V 1282 1567 V 1283
1568 V 1284 1569 V 1285 1570 V 1286 1571 V 1287 1572 V 1288
1573 V 1289 1574 V 1290 1575 V 1291 1576 V 1292 1577 V 1293
1578 V 1294 1579 V 1295 1580 V 1297 1581 V 1298 1582 V 1299
1583 V 1300 1584 V 1301 1585 V 1302 1586 V 1303 1587 V 1304
1588 V 1305 1589 V 1306 1590 V 1307 1591 V 1308 1592 V 1309
1593 V 1310 1594 V 1311 1595 V 1312 1596 V 1314 1597 V 1315
1598 V 1316 1599 V 1317 1600 V 1318 1601 V 1319 1602 V 1320
1603 V 1321 1604 V 1322 1605 V 1323 1606 V 1324 1607 V 1325
1608 V 1326 1609 V 1327 1610 V 1328 1611 V 1329 1612 V 1330
1613 V 1331 1614 V 1332 1615 V 1333 1616 V 1334 1617 V 1335
1618 V 1336 1619 V 1337 1620 V 1338 1621 V 1339 1622 V 1340
1623 V 1341 1624 V 1342 1625 V 1343 1626 V 1344 1627 V 1346
1628 V 1347 1629 V 1348 1630 V 1349 1631 V 1350 1632 V 1351
1633 V 1352 1634 V 1353 1635 V 1354 1636 V 1355 1637 V 1356
1638 V 1357 1639 V 1358 1640 V 1359 1641 V 1360 1642 V 1361
1643 V 1363 1644 V 1364 1645 V 1365 1646 V 1366 1647 V 1367
1648 V 1368 1649 V 1369 1650 V 1370 1651 V 1371 1652 V 1372
1653 V 1373 1654 V 1374 1655 V 1375 1656 V 1376 1657 V 1377
1658 V 1378 1659 V 1379 1660 V 1380 1661 V 1381 1662 V 1382
1663 V 1383 1664 V 1384 1665 V 1385 1666 V 1387 1667 V 1388
1668 V 1389 1669 V 1390 1670 V 1391 1671 V 1392 1672 V 1393
1673 V 1394 1674 V 1396 1675 V 1397 1676 V 1398 1677 V 1399
1678 V 1400 1679 V 1401 1680 V 1402 1681 V 1403 1682 V 1404
1683 V 1405 1684 V 1406 1685 V 1407 1686 V 1408 1687 V 1409
1688 V 1410 1689 V 1411 1690 V 1412 1691 V 1413 1692 V 1414
1693 V 1415 1694 V 1416 1695 V 1417 1696 V 1418 1697 V 1420
1698 V 1421 1699 V 1422 1700 V 1423 1701 V 1424 1702 V 1425
1703 V 1426 1704 V 1427 1705 V 1429 1706 V 1430 1707 V 1431
1708 V 1432 1709 V 1433 1710 V 1434 1711 V 1435 1712 V 1436
1713 V 1437 1714 V 1438 1715 V 1439 1716 V 1440 1717 V 1441
1718 V 1442 1719 V 1443 1720 V 1444 1721 V 1445 1722 V 1446
1723 V 1447 1724 V 1448 1725 V 1449 1726 V 1450 1727 V 1451
1728 V 1452 1729 V 1453 1730 V 1454 1731 V 1455 1732 V 1456
1733 V 1457 1734 V 1458 1735 V 1459 1736 V 1460 1737 V 1461
1738 V 1462 1739 V 1463 1740 V 1464 1741 V 1465 1742 V 1466
1743 V 1467 1744 V 1469 1745 V 1470 1746 V 1471 1747 V 1472
1748 V 1473 1749 V 1474 1750 V 1475 1751 V 1476 1752 V 1478
1753 V 1479 1754 V 1480 1755 V 1481 1756 V 1482 1757 V 1483
1758 V 1484 1759 V 1485 1760 V 1486 1761 V 1487 1762 V 1488
1763 V 1489 1764 V 1490 1765 V 1491 1766 V 1492 1767 V 1493
1768 V 1494 1769 V 1495 1770 V 1496 1771 V 1497 1772 V 1498
1773 V 1499 1774 V 1500 1775 V 1501 1776 V 1502 1777 V 1503
1778 V 1504 1779 V 1505 1780 V 1506 1781 V 1507 1782 V 1508
1783 V 1510 1784 V 1511 1785 V 1512 1786 V 1513 1787 V 1514
1788 V 1515 1789 V 1516 1790 V 1517 1791 V 1518 1792 V 1519
1793 V 1520 1794 V 1521 1795 V 1522 1796 V 1523 1797 V 1524
1798 V 1526 1799 V 1527 1800 V 1528 1801 V 1529 1802 V 1530
1803 V 1531 1804 V 1532 1805 V 1533 1806 V 1534 1807 V 1535
1808 V 1536 1809 V 1537 1810 V 1538 1811 V 1539 1812 V 1540
1813 V 1541 1814 V 1543 1815 V 1544 1816 V 1545 1817 V 1546
1818 V 1547 1819 V 1548 1820 V 1549 1821 V 1550 1822 V 1551
1823 V 1552 1824 V 1553 1825 V 1554 1826 V 1555 1827 V 1556
1828 V 1557 1829 V 1559 1830 V 1560 1831 V 1561 1832 V 1562
1833 V 1563 1834 V 1564 1835 V 1565 1836 V 1566 1837 V 1567
1838 V 1568 1839 V 1569 1840 V 1570 1841 V 1571 1842 V 1572
1843 V 1573 1844 V 1574 1845 V 1575 1846 V 1576 1847 V 1577
1848 V 1578 1849 V 1579 1850 V 1580 1851 V 1581 1852 V 1582
1853 V 1583 1854 V 1584 1855 V 1585 1856 V 1586 1857 V 1587
1858 V 1588 1859 V 1589 1860 V 1590 1861 V 1592 1862 V 1593
1863 V 1594 1864 V 1595 1865 V 1596 1866 V 1597 1867 V 1598
1868 V 1599 1869 V 1600 1870 V 1601 1871 V 1602 1872 V 1603
1873 V 1604 1874 V 1605 1875 V 1606 1876 V 1608 1877 V 1609
1878 V 1610 1879 V 1611 1880 V 1612 1881 V 1613 1882 V 1614
1883 V 1615 1884 V 1616 1885 V 1617 1886 V 1618 1887 V 1619
1888 V 1620 1889 V 1621 1890 V 1622 1891 V 1623 1892 V 1625
1893 V 1626 1894 V 1627 1895 V 1628 1896 V 1629 1897 V 1630
1898 V 1631 1899 V 1632 1900 V 1633 1901 V 1634 1902 V 1635
1903 V 1636 1904 V 1637 1905 V 1638 1906 V 1639 1907 V 1640
1908 V 1641 1909 V 1642 1910 V 1643 1911 V 1644 1912 V 1645
1913 V 1646 1914 V 1647 1915 V 1649 1916 V 1650 1917 V 1651
1918 V 1652 1919 V 1653 1920 V 1654 1921 V 1655 1922 V 1656
1923 V 1658 1924 V 1659 1925 V 1660 1926 V 1661 1927 V 1662
1928 V 1663 1929 V 1664 1930 V 1665 1931 V 1666 1932 V 1667
1933 V 1668 1934 V 1669 1935 V 1670 1936 V 1671 1937 V 1672
1938 V 1673 1939 V 1674 1940 V 1675 1941 V 1676 1942 V 1677
1943 V 1678 1944 V 1679 1945 V 1680 1946 V 1682 1947 V 1683
1948 V 1684 1949 V 1685 1950 V 1686 1951 V 1687 1952 V 1688
1953 V 1689 1954 V 1691 1955 V 1692 1956 V 1693 1957 V 1694
1958 V 1695 1959 V 1696 1960 V 1697 1961 V 1698 1962 V 1699
1963 V 1700 1964 V 1701 1965 V 1702 1966 V 1703 1967 V 1704
1968 V 1705 1969 V 1706 1970 V 1707 1971 V 1708 1972 V 964
2825 a Fg(3)p eop
%%Page: 4 4
bop 264 1973 20 2 v 220 1987 a Fg(0)p 1866 1973 V 264 1844
V 139 1858 a(0)p Ff(:)p Fg(005)p 1866 1844 V 264 1715 V 162
1729 a(0)p Ff(:)p Fg(01)p 1866 1715 V 264 1586 V 139 1600 a(0)p
Ff(:)p Fg(015)p 1866 1586 V 264 1457 V 162 1471 a(0)p Ff(:)p
Fg(02)p 1866 1457 V 264 1328 V 139 1342 a(0)p Ff(:)p Fg(025)p
1866 1328 V 264 1200 V 162 1214 a(0)p Ff(:)p Fg(03)p 1866 1200
V 264 1071 V 139 1085 a(0)p Ff(:)p Fg(035)p 1866 1071 V 264
942 V 162 956 a(0)p Ff(:)p Fg(04)p 1866 942 V 264 813 V 139
827 a(0)p Ff(:)p Fg(045)p 1866 813 V 264 684 V 162 698 a(0)p
Ff(:)p Fg(05)p 1866 684 V 264 1973 2 20 v 218 2030 a Fe(\000)p
Fg(0)p Ff(:)p Fg(2)p 264 704 V 426 1973 V 56 w Fe(\000)p Fg(0)p
Ff(:)p Fg(15)p 426 704 V 588 1973 V 57 w Fe(\000)p Fg(0)p Ff(:)p
Fg(1)p 588 704 V 750 1973 V 56 w Fe(\000)p Fg(0)p Ff(:)p Fg(05)p
750 704 V 912 1973 V 901 2032 a(0)p 912 704 V 1076 1973 V 111
w(0)p Ff(:)p Fg(05)p 1076 704 V 1238 1973 V 91 w(0)p Ff(:)p
Fg(1)p 1238 704 V 1400 1973 V 92 w(0)p Ff(:)p Fg(15)p 1400
704 V 1562 1973 V 91 w(0)p Ff(:)p Fg(2)p 1562 704 V 1724 1973
V 92 w(0)p Ff(:)p Fg(25)p 1724 704 V 1886 1973 V 91 w(0)p Ff(:)p
Fg(3)p 1886 704 V 264 1973 1622 2 v 1886 1973 2 1289 v 264
684 1622 2 v 45 1344 a(\001)p Fe(R)966 2073 y Ff(k)14 b Fg(=)f(1)p
Ff(=)p Fg(8)p Ff(\031)r(G)1019 648 y Fc(Fig.)18 b(4)p 264 1973
2 1289 v 896 1420 a Fb(2)1059 1549 y(2)1221 1678 y(2)1383 1549
y(2)1545 1935 y(2)p 912 1509 2 206 v 902 1509 20 2 v 902 1303
V 1076 1638 2 206 v 1066 1638 20 2 v 1066 1432 V 1238 1870
2 414 v 1228 1870 20 2 v 1228 1457 V 1400 1741 2 414 v 1390
1741 20 2 v 1390 1328 V 1562 1973 2 309 v 1552 1973 20 2 v
1552 1664 V 264 1150 2 2 v 264 1150 3 2 v 266 1151 V 269 1152
V 271 1153 V 274 1154 V 277 1155 V 279 1156 V 282 1157 V 285
1158 V 288 1159 V 291 1160 V 294 1161 V 297 1162 V 299 1163
V 302 1164 V 304 1165 V 307 1166 V 310 1167 V 312 1168 V 315
1169 V 318 1170 V 320 1171 V 323 1172 V 326 1173 V 328 1174
V 331 1175 V 334 1176 V 337 1177 V 340 1178 V 343 1179 V 346
1180 V 348 1181 V 351 1182 V 353 1183 V 356 1184 V 359 1185
V 361 1186 V 364 1187 V 367 1188 V 370 1189 V 373 1190 V 376
1191 V 379 1192 V 381 1193 V 384 1194 V 386 1195 V 389 1196
V 392 1197 V 394 1198 V 397 1199 V 400 1200 V 402 1201 V 405
1202 V 408 1203 V 410 1204 V 413 1205 V 416 1206 V 419 1207
V 422 1208 V 425 1209 V 428 1210 V 430 1211 V 433 1212 V 435
1213 V 438 1214 V 441 1215 V 443 1216 V 446 1217 V 448 1218
V 450 1219 V 453 1220 V 455 1221 V 457 1222 V 459 1223 V 462
1224 V 465 1225 V 468 1226 V 471 1227 V 474 1228 V 477 1229
V 479 1230 V 482 1231 V 484 1232 V 487 1233 V 490 1234 V 492
1235 V 495 1236 V 498 1237 V 501 1238 V 504 1239 V 507 1240
V 510 1241 V 512 1242 V 515 1243 V 518 1244 V 520 1245 V 523
1246 V 526 1247 V 528 1248 V 530 1249 V 532 1250 V 535 1251
V 537 1252 V 539 1253 V 541 1254 V 544 1255 V 547 1256 V 550
1257 V 553 1258 V 556 1259 V 558 1260 V 561 1261 V 563 1262
V 565 1263 V 568 1264 V 570 1265 V 572 1266 V 574 1267 V 577
1268 V 580 1269 V 583 1270 V 585 1271 V 588 1272 V 591 1273
V 593 1274 V 596 1275 V 599 1276 V 602 1277 V 605 1278 V 607
1279 V 610 1280 V 612 1281 V 614 1282 V 617 1283 V 619 1284
V 621 1285 V 623 1286 V 626 1287 V 629 1288 V 632 1289 V 635
1290 V 638 1291 V 640 1292 V 643 1293 V 645 1294 V 647 1295
V 650 1296 V 652 1297 V 654 1298 V 656 1299 V 659 1300 V 661
1301 V 663 1302 V 666 1303 V 668 1304 V 670 1305 V 672 1306
V 675 1307 V 678 1308 V 681 1309 V 684 1310 V 687 1311 V 689
1312 V 692 1313 V 694 1314 V 696 1315 V 699 1316 V 701 1317
V 703 1318 V 705 1319 V 708 1320 V 710 1321 V 712 1322 V 715
1323 V 717 1324 V 719 1325 V 721 1326 V 724 1327 V 727 1328
V 730 1329 V 733 1330 V 736 1331 V 738 1332 V 741 1333 V 743
1334 V 745 1335 V 748 1336 V 750 1337 V 752 1338 V 754 1339
V 757 1340 V 759 1341 V 762 1342 V 764 1343 V 767 1344 V 769
1345 V 772 1346 V 774 1347 V 776 1348 V 778 1349 V 781 1350
V 783 1351 V 785 1352 V 787 1353 V 790 1354 V 792 1355 V 794
1356 V 797 1357 V 799 1358 V 801 1359 V 803 1360 V 806 1361
V 808 1362 V 811 1363 V 813 1364 V 816 1365 V 818 1366 V 821
1367 V 823 1368 V 825 1369 V 827 1370 V 830 1371 V 832 1372
V 834 1373 V 836 1374 V 839 1375 V 841 1376 V 843 1377 V 846
1378 V 848 1379 V 850 1380 V 852 1381 V 855 1382 V 857 1383
V 860 1384 V 862 1385 V 865 1386 V 867 1387 V 870 1388 V 872
1389 V 874 1390 V 876 1391 V 878 1392 V 880 1393 V 882 1394
V 884 1395 V 886 1396 V 888 1397 V 890 1398 V 893 1399 V 895
1400 V 898 1401 V 900 1402 V 903 1403 V 905 1404 V 907 1405
V 909 1406 V 912 1407 V 914 1408 V 916 1409 V 918 1410 V 921
1411 V 923 1412 V 925 1413 V 928 1414 V 930 1415 V 932 1416
V 934 1417 V 937 1418 V 939 1419 V 941 1420 V 943 1421 V 945
1422 V 947 1423 V 949 1424 V 952 1425 V 954 1426 V 956 1427
V 958 1428 V 961 1429 V 963 1430 V 965 1431 V 967 1432 V 970
1433 V 972 1434 V 974 1435 V 977 1436 V 979 1437 V 981 1438
V 983 1439 V 985 1440 V 987 1441 V 989 1442 V 991 1443 V 993
1444 V 995 1445 V 997 1446 V 999 1447 V 1002 1448 V 1004 1449
V 1006 1450 V 1008 1451 V 1011 1452 V 1013 1453 V 1015 1454
V 1017 1455 V 1020 1456 V 1022 1457 V 1024 1458 V 1026 1459
V 1028 1460 V 1030 1461 V 1032 1462 V 1035 1463 V 1037 1464
V 1039 1465 V 1041 1466 V 1043 1467 V 1045 1468 V 1047 1469
V 1049 1470 V 1051 1471 V 1053 1472 V 1055 1473 V 1057 1474
V 1059 1475 V 1061 1476 V 1063 1477 V 1065 1478 V 1067 1479
V 1069 1480 V 1071 1481 V 1073 1482 V 1075 1483 V 1077 1484
V 1079 1485 V 1081 1486 V 1084 1487 V 1086 1488 V 1088 1489
V 1090 1490 V 1092 1491 V 1094 1492 V 1096 1493 V 1098 1494
V 1100 1495 V 1102 1496 V 1104 1497 V 1106 1498 V 1108 1499
V 1110 1500 V 1112 1501 V 1114 1502 V 1116 1503 V 1118 1504
V 1120 1505 V 1122 1506 V 1124 1507 V 1126 1508 V 1128 1509
V 1130 1510 V 1133 1511 V 1135 1512 V 1137 1513 V 1139 1514
V 1141 1515 V 1143 1516 V 1145 1517 V 1147 1518 V 1149 1519
2 2 v 1150 1520 V 1152 1521 V 1154 1522 V 1156 1523 V 1158
1524 V 1160 1525 V 1162 1526 V 1164 1527 V 1166 1528 3 2 v
1168 1529 V 1170 1530 V 1172 1531 V 1174 1532 V 1176 1533 V
1178 1534 V 1180 1535 V 1182 1536 2 2 v 1183 1537 V 1185 1538
V 1187 1539 V 1189 1540 V 1190 1541 V 1192 1542 V 1194 1543
V 1196 1544 V 1198 1545 V 1199 1546 V 1201 1547 V 1203 1548
V 1205 1549 V 1207 1550 V 1209 1551 V 1211 1552 V 1213 1553
V 1215 1554 3 2 v 1217 1555 V 1219 1556 V 1221 1557 V 1223
1558 V 1225 1559 V 1227 1560 V 1229 1561 V 1231 1562 2 2 v
1232 1563 V 1234 1564 V 1236 1565 V 1238 1566 V 1239 1567 V
1241 1568 V 1243 1569 V 1245 1570 V 1247 1571 V 1248 1572 V
1250 1573 V 1252 1574 V 1254 1575 V 1256 1576 V 1258 1577 V
1260 1578 V 1262 1579 V 1264 1580 V 1265 1581 V 1267 1582 V
1268 1583 V 1270 1584 V 1271 1585 V 1273 1586 V 1275 1587 V
1276 1588 V 1278 1589 V 1279 1590 V 1281 1591 V 1283 1592 V
1285 1593 V 1287 1594 V 1289 1595 V 1291 1596 V 1293 1597 V
1295 1598 V 1297 1599 V 1298 1600 V 1300 1601 V 1302 1602 V
1304 1603 V 1305 1604 V 1307 1605 V 1309 1606 V 1311 1607 V
1313 1608 V 1314 1609 V 1316 1610 V 1317 1611 V 1319 1612 V
1320 1613 V 1322 1614 V 1324 1615 V 1325 1616 V 1327 1617 V
1328 1618 V 1330 1619 V 1332 1620 V 1334 1621 V 1335 1622 V
1337 1623 V 1339 1624 V 1340 1625 V 1342 1626 V 1344 1627 V
1345 1628 V 1347 1629 V 1349 1630 V 1350 1631 V 1352 1632 V
1353 1633 V 1355 1634 V 1357 1635 V 1358 1636 V 1360 1637 V
1361 1638 V 1363 1639 V 1365 1640 V 1366 1641 V 1368 1642 V
1369 1643 V 1371 1644 V 1373 1645 V 1374 1646 V 1376 1647 V
1377 1648 V 1379 1649 V 1381 1650 V 1383 1651 V 1384 1652 V
1386 1653 V 1388 1654 V 1389 1655 V 1391 1656 V 1393 1657 V
1394 1658 V 1396 1659 V 1397 1660 V 1399 1661 V 1400 1662 V
1402 1663 V 1403 1664 V 1405 1665 V 1406 1666 V 1408 1667 V
1409 1668 V 1411 1669 V 1412 1670 V 1414 1671 V 1416 1672 V
1417 1673 V 1419 1674 V 1421 1675 V 1422 1676 V 1424 1677 V
1426 1678 V 1427 1679 V 1429 1680 V 1430 1681 V 1432 1682 V
1433 1683 V 1435 1684 V 1436 1685 V 1438 1686 V 1439 1687 V
1441 1688 V 1442 1689 V 1444 1690 V 1445 1691 V 1446 1692 V
1448 1693 V 1449 1694 V 1450 1695 V 1452 1696 V 1453 1697 V
1454 1698 V 1456 1699 V 1457 1700 V 1458 1701 V 1460 1702 V
1461 1703 V 1463 1704 V 1464 1705 V 1466 1706 V 1467 1707 V
1469 1708 V 1470 1709 V 1472 1710 V 1473 1711 V 1475 1712 V
1476 1713 V 1478 1714 V 1479 1715 V 1481 1716 V 1482 1717 V
1483 1718 V 1485 1719 V 1486 1720 V 1487 1721 V 1489 1722 V
1490 1723 V 1491 1724 V 1493 1725 V 1494 1726 V 1495 1727 V
1497 1728 V 1498 1729 V 1499 1730 V 1501 1731 V 1502 1732 V
1503 1733 V 1505 1734 V 1506 1735 V 1507 1736 V 1509 1737 V
1510 1738 V 1511 1739 V 1512 1740 V 1514 1741 V 1515 1742 V
1516 1743 V 1518 1744 V 1519 1745 V 1520 1746 V 1522 1747 V
1523 1748 V 1524 1749 V 1526 1750 V 1527 1751 V 1528 1752 V
1529 1753 V 1530 1754 V 1532 1755 V 1533 1756 V 1534 1757 V
1535 1758 V 1537 1759 V 1538 1760 V 1539 1761 V 1540 1762 V
1541 1763 V 1543 1764 V 1544 1765 V 1545 1766 V 1547 1767 V
1548 1768 V 1549 1769 V 1551 1770 V 1552 1771 V 1553 1772 V
1555 1773 V 1556 1774 V 1557 1775 V 1559 1776 V 1560 1777 V
1561 1778 V 1562 1779 V 1563 1780 V 1564 1781 V 1565 1782 V
1566 1783 V 1568 1784 V 1569 1785 V 1570 1786 V 1571 1787 V
1572 1788 V 1573 1789 V 1574 1790 V 1576 1791 V 1577 1792 V
1578 1793 V 1579 1794 V 1580 1795 V 1581 1796 V 1582 1797 V
1583 1798 V 1584 1799 V 1585 1800 V 1586 1801 V 1587 1802 V
1588 1803 V 1589 1804 V 1590 1805 V 1592 1806 V 1593 1807 V
1594 1808 V 1595 1809 V 1596 1810 V 1597 1811 V 1598 1812 V
1599 1813 V 1600 1814 V 1601 1815 V 1602 1816 V 1603 1817 V
1604 1818 V 1605 1819 V 1606 1820 V 1608 1821 V 1609 1822 V
1610 1823 V 1611 1824 V 1612 1825 V 1613 1826 V 1614 1827 V
1615 1828 V 1616 1829 V 1617 1830 V 1618 1831 V 1619 1832 V
1620 1833 V 1621 1834 V 1622 1835 V 1623 1836 V 1624 1839 V
1625 1840 V 1626 1841 V 1627 1842 V 1628 1843 V 1629 1844 V
1630 1845 V 1631 1846 V 1632 1848 V 1633 1849 V 1634 1850 V
1635 1851 V 1636 1852 V 1637 1853 V 1638 1854 V 1639 1855 V
1640 1857 V 1641 1858 V 1642 1859 V 1643 1860 V 1644 1861 V
1645 1863 V 1646 1864 V 1647 1865 V 1648 1866 V 1649 1867 V
1650 1868 V 1651 1870 V 1652 1871 V 1653 1872 V 1654 1873 V
1655 1874 V 1656 1875 V 1657 1877 V 1658 1878 V 1659 1880 V
1660 1881 V 1661 1883 V 1662 1884 V 1663 1886 V 1664 1887 V
1665 1888 V 1666 1890 V 1667 1891 V 1668 1893 V 1669 1894 V
1670 1896 V 1671 1897 V 1672 1898 V 1673 1900 V 1674 1902 V
1675 1903 V 1676 1905 V 1677 1907 V 1678 1908 V 1679 1910 V
1680 1912 V 1681 1913 V 1682 1915 V 1683 1917 V 1684 1918 V
1685 1920 V 1686 1922 V 1687 1923 V 1688 1925 V 1689 1926 V
964 2825 a Fg(4)p eop
%%Page: 5 5
bop 264 1973 20 2 v 220 1987 a Fg(0)p 1866 1973 V 264 1715
V 184 1729 a(0)p Ff(:)p Fg(5)p 1866 1715 V 264 1457 V 220 1471
a(1)p 1866 1457 V 264 1200 V 184 1214 a(1)p Ff(:)p Fg(5)p 1866
1200 V 264 942 V 220 956 a(2)p 1866 942 V 264 684 V 184 698
a(2)p Ff(:)p Fg(5)p 1866 684 V 264 1973 2 20 v 253 2032 a(0)p
264 704 V 559 1973 V 254 w(0)p Ff(:)p Fg(2)p 559 704 V 853
1973 V 235 w(0)p Ff(:)p Fg(4)p 853 704 V 1149 1973 V 236 w(0)p
Ff(:)p Fg(6)p 1149 704 V 1444 1973 V 236 w(0)p Ff(:)p Fg(8)p
1444 704 V 1739 1973 V 254 w(1)p 1739 704 V 264 1973 1622 2
v 1886 1973 2 1289 v 264 684 1622 2 v 45 1343 a Ff(<)13 b(\036)120
1326 y Fd(2)153 1343 y Ff(>)1045 2081 y(m)1085 2064 y Fd(2)1019
648 y Fc(Fig.)18 b(5)p 264 1973 2 1289 v 1721 1636 a Fb(3)614
1260 y(3)305 899 y(3)p 1739 1628 2 12 v 1729 1628 20 2 v 1729
1617 V 632 1251 2 10 v 622 1251 20 2 v 622 1241 V 323 895 2
20 v 313 895 20 2 v 313 875 V 1721 1632 a Fg(+)615 1266 y(+)306
905 y(+)p 1739 1628 2 12 v 1729 1628 20 2 v 1729 1617 V 632
1261 2 10 v 622 1261 20 2 v 622 1251 V 323 906 2 22 v 313 906
20 2 v 313 885 V 1722 1636 a Fb(2)616 1265 y(2)307 909 y(2)p
1739 1628 2 12 v 1729 1628 20 2 v 1729 1617 V 632 1256 2 10
v 622 1256 20 2 v 622 1246 V 323 906 2 22 v 313 906 20 2 v
313 885 V 1721 1638 a Fe(\002)615 1261 y(\002)306 926 y(\002)p
1739 1633 2 12 v 1729 1633 20 2 v 1729 1622 V 632 1256 2 10
v 622 1256 20 2 v 622 1246 V 323 926 2 20 v 313 926 20 2 v
313 906 V 1718 1632 a(4)612 1261 y(4)303 911 y(4)p 1739 1628
2 12 v 1729 1628 20 2 v 1729 1617 V 632 1256 2 10 v 622 1256
20 2 v 622 1246 V 323 911 2 22 v 313 911 20 2 v 313 890 V 280
815 2 2 v 280 818 2 3 v 281 820 V 282 822 V 283 824 V 284 826
V 285 829 V 286 831 V 287 833 V 288 835 V 289 837 V 290 839
V 291 842 V 292 844 V 293 846 V 294 848 V 295 850 V 296 852
V 297 854 V 298 856 V 299 858 V 300 860 V 301 862 V 302 864
V 303 866 V 304 868 V 305 870 V 306 872 V 307 874 V 308 876
V 309 878 V 310 880 V 311 882 V 312 884 V 313 886 2 2 v 314
888 V 315 890 V 316 892 V 317 894 V 318 895 V 319 897 V 320
899 V 321 901 V 322 903 V 323 904 V 324 906 V 325 908 V 326
910 V 327 912 V 328 913 V 329 915 V 330 917 V 331 918 V 332
920 V 333 921 V 334 923 V 335 925 V 336 926 V 337 928 V 338
929 V 339 931 V 340 933 V 341 934 V 342 936 V 343 937 V 344
939 V 345 940 V 346 942 V 347 944 V 348 945 V 349 947 V 350
948 V 351 950 V 352 951 V 353 953 V 354 955 V 355 956 V 356
958 V 357 959 V 358 961 V 359 962 V 360 964 V 361 965 V 362
967 V 363 968 V 364 970 V 365 971 V 366 973 V 367 974 V 368
975 V 369 977 V 370 978 V 371 980 V 372 981 V 373 982 V 374
984 V 375 985 V 376 987 V 377 988 V 378 989 V 379 991 V 380
992 V 381 994 V 382 995 V 383 996 V 384 998 V 385 999 V 386
1000 V 387 1002 V 388 1003 V 389 1005 V 390 1006 V 391 1007
V 392 1009 V 393 1010 V 394 1011 V 395 1013 V 396 1014 V 397
1016 V 398 1017 V 399 1018 V 400 1020 V 401 1021 V 402 1022
V 403 1024 V 404 1025 V 405 1027 V 406 1028 V 407 1029 V 408
1031 V 409 1032 V 410 1033 V 411 1035 V 412 1036 V 413 1037
V 414 1038 V 415 1039 V 416 1041 V 417 1042 V 418 1043 V 419
1044 V 420 1045 V 421 1046 V 422 1048 V 423 1049 V 424 1050
V 425 1051 V 426 1052 V 427 1053 V 428 1055 V 429 1056 V 430
1057 V 431 1058 V 432 1059 V 433 1061 V 434 1062 V 435 1063
V 436 1064 V 437 1065 V 438 1067 V 439 1068 V 440 1069 V 441
1070 V 442 1071 V 443 1072 V 444 1074 V 445 1075 V 446 1076
V 447 1077 V 448 1078 V 449 1080 V 450 1081 V 451 1082 V 452
1083 V 453 1084 V 454 1086 V 455 1087 V 456 1088 V 457 1089
V 458 1090 V 459 1091 V 460 1093 V 461 1094 V 462 1095 V 463
1096 V 464 1097 V 465 1098 V 466 1099 V 467 1100 V 468 1101
V 469 1102 V 470 1103 V 471 1104 V 472 1105 V 473 1106 V 474
1107 V 475 1108 V 476 1109 V 477 1111 V 478 1112 V 479 1113
V 480 1114 V 481 1115 V 482 1116 V 483 1117 V 484 1118 V 485
1119 V 486 1120 V 487 1121 V 488 1122 V 489 1123 V 490 1124
V 491 1125 V 492 1126 V 493 1126 V 494 1127 V 495 1128 V 496
1129 V 497 1130 V 498 1131 V 499 1132 V 500 1133 V 501 1134
V 502 1135 V 503 1136 V 504 1137 V 505 1138 V 506 1139 V 507
1140 V 508 1141 V 509 1142 V 510 1143 V 511 1144 V 512 1145
V 513 1146 V 514 1147 V 515 1148 V 516 1149 V 517 1150 V 518
1151 V 519 1152 V 520 1153 V 521 1154 V 522 1155 V 523 1156
V 524 1157 V 525 1158 V 526 1159 V 527 1160 V 528 1161 V 529
1162 V 530 1163 V 531 1164 V 532 1165 V 533 1166 V 534 1167
V 535 1168 V 536 1169 V 537 1170 V 538 1171 V 539 1172 V 540
1173 V 541 1174 V 542 1175 V 543 1176 V 544 1177 V 545 1178
V 546 1179 V 547 1180 V 548 1181 V 549 1182 V 551 1183 V 552
1184 V 553 1185 V 554 1186 V 555 1187 V 556 1188 V 557 1189
V 559 1190 V 560 1191 V 561 1192 V 562 1193 V 563 1194 V 564
1195 V 565 1196 V 566 1197 V 567 1198 V 568 1199 V 569 1200
V 570 1201 V 571 1202 V 572 1203 V 573 1204 V 574 1205 V 576
1206 V 577 1207 V 578 1208 V 579 1209 V 580 1210 V 581 1211
V 583 1212 V 584 1213 V 585 1214 V 586 1215 V 587 1216 V 588
1217 V 589 1218 V 591 1219 V 592 1220 V 593 1221 V 594 1222
V 596 1223 V 597 1224 V 598 1225 V 600 1226 V 601 1227 V 602
1228 V 604 1229 V 605 1230 V 606 1231 V 607 1232 V 609 1233
V 610 1234 V 611 1235 V 612 1236 V 613 1237 V 614 1238 V 616
1239 V 617 1240 V 618 1241 V 619 1242 V 620 1243 V 621 1244
V 622 1245 V 624 1246 V 625 1247 V 626 1248 V 628 1249 V 629
1250 V 631 1251 V 632 1252 V 633 1253 V 635 1254 V 636 1255
V 638 1256 V 639 1257 V 641 1258 V 642 1259 V 643 1260 V 644
1261 V 645 1262 V 647 1263 V 648 1264 V 649 1265 V 650 1266
V 652 1267 V 653 1268 V 654 1269 V 655 1270 V 657 1271 V 658
1272 V 659 1273 V 660 1274 V 662 1275 V 663 1276 V 664 1277
V 666 1278 V 667 1279 V 668 1280 V 670 1281 V 671 1282 V 672
1283 V 674 1284 V 675 1285 V 677 1286 V 678 1287 V 680 1288
V 681 1289 V 682 1290 V 684 1291 V 685 1292 V 687 1293 V 688
1294 V 690 1295 V 691 1296 V 692 1297 V 694 1298 V 695 1299
V 697 1300 V 698 1301 V 700 1302 V 701 1303 V 703 1304 V 704
1305 V 705 1306 V 707 1307 V 708 1308 V 710 1309 V 711 1310
V 713 1311 V 714 1312 V 716 1313 V 717 1314 V 719 1315 V 720
1316 V 721 1317 V 723 1318 V 725 1319 V 726 1320 V 728 1321
V 729 1322 V 731 1323 V 732 1324 V 734 1325 V 735 1326 V 737
1327 V 739 1328 V 740 1329 V 741 1330 V 743 1331 V 744 1332
V 746 1333 V 747 1334 V 749 1335 V 750 1336 V 752 1337 V 753
1338 V 754 1339 V 756 1340 V 758 1341 V 760 1342 V 761 1343
V 763 1344 V 765 1345 V 766 1346 V 768 1347 V 770 1348 V 772
1349 V 773 1350 V 775 1351 V 776 1352 V 778 1353 V 779 1354
V 781 1355 V 783 1356 V 784 1357 V 786 1358 V 787 1359 V 789
1360 V 791 1361 V 792 1362 V 794 1363 V 795 1364 V 797 1365
V 799 1366 V 800 1367 V 802 1368 V 803 1369 V 805 1370 V 807
1371 V 809 1372 V 811 1373 V 813 1374 V 815 1375 V 817 1376
V 819 1377 V 821 1378 V 822 1379 V 824 1380 V 826 1381 V 828
1382 V 829 1383 V 831 1384 V 833 1385 V 835 1386 V 836 1387
V 838 1388 V 840 1389 V 842 1390 V 844 1391 V 845 1392 V 847
1393 V 849 1394 V 851 1395 V 852 1396 V 854 1397 V 856 1398
V 858 1399 V 860 1400 V 862 1401 V 864 1402 V 866 1403 V 868
1404 V 870 1405 3 2 v 872 1406 V 874 1407 V 876 1408 V 878
1409 V 880 1410 V 882 1411 V 884 1412 V 886 1413 V 888 1414
V 890 1415 V 892 1416 V 894 1417 V 896 1418 V 898 1419 V 900
1420 V 903 1421 V 905 1422 V 907 1423 V 909 1424 V 911 1425
V 913 1426 V 915 1427 V 917 1428 V 919 1429 V 921 1430 V 923
1431 V 925 1432 V 927 1433 V 929 1434 V 931 1435 V 933 1436
V 935 1437 V 937 1438 V 939 1439 V 941 1440 V 943 1441 V 945
1442 V 947 1443 V 949 1444 V 952 1445 V 954 1446 V 956 1447
V 958 1448 V 961 1449 V 963 1450 V 965 1451 V 967 1452 V 970
1453 V 972 1454 V 974 1455 V 978 1456 V 980 1457 V 982 1458
V 984 1459 V 987 1460 V 989 1461 V 992 1462 V 994 1463 V 997
1464 V 999 1465 V 1002 1466 V 1004 1467 V 1006 1468 V 1008
1469 V 1011 1470 V 1013 1471 V 1015 1472 V 1017 1473 V 1020
1474 V 1023 1475 V 1026 1476 V 1029 1477 V 1032 1478 V 1035
1479 V 1037 1480 V 1039 1481 V 1041 1482 V 1044 1483 V 1046
1484 V 1048 1485 V 1051 1486 V 1053 1487 V 1056 1488 V 1058
1489 V 1061 1490 V 1064 1491 V 1066 1492 V 1069 1493 V 1072
1494 V 1075 1495 V 1078 1496 V 1081 1497 V 1084 1498 V 1086
1499 V 1089 1500 V 1091 1501 V 1094 1502 V 1097 1503 V 1099
1504 4 2 v 1103 1505 V 1106 1506 V 1109 1507 V 1112 1508 V
1115 1509 3 2 v 1118 1510 V 1121 1511 V 1124 1512 V 1127 1513
V 1130 1514 V 1133 1515 4 2 v 1136 1516 V 1139 1517 V 1142
1518 V 1145 1519 V 1148 1520 V 1152 1521 V 1155 1522 V 1159
1523 V 1162 1524 V 1166 1525 V 1169 1526 V 1172 1527 V 1175
1528 V 1178 1529 V 1181 1530 V 1185 1531 V 1188 1532 V 1191
1533 V 1194 1534 V 1197 1535 V 1201 1536 V 1204 1537 V 1208
1538 V 1211 1539 V 1215 1540 V 1218 1541 V 1221 1542 V 1224
1543 V 1227 1544 V 1230 1545 5 2 v 1235 1546 V 1239 1547 V
1243 1548 V 1247 1549 V 1251 1550 V 1255 1551 V 1259 1552 V
1264 1553 V 1268 1554 V 1272 1555 V 1276 1556 V 1280 1557 V
1284 1558 V 1288 1559 V 1292 1560 V 1297 1561 V 1301 1562 V
1305 1563 V 1309 1564 V 1313 1565 V 1317 1566 V 1321 1567 V
1325 1568 V 1329 1569 6 2 v 1334 1570 V 1340 1571 V 1345 1572
5 2 v 1350 1573 V 1354 1574 V 1358 1575 V 1362 1576 6 2 v 1367
1577 V 1372 1578 V 1378 1579 V 1383 1580 V 1389 1581 V 1394
1582 V 1400 1583 V 1405 1584 V 1411 1585 V 1416 1586 V 1422
1587 V 1427 1588 V 1433 1589 V 1438 1590 V 1444 1591 V 1449
1592 V 1454 1593 V 1460 1594 9 2 v 1468 1595 V 1477 1596 6
2 v 1482 1597 V 1487 1598 V 1493 1599 8 2 v 1501 1600 V 1509
1601 9 2 v 1517 1602 V 1526 1603 8 2 v 1534 1604 V 1542 1605
9 2 v 1550 1606 V 1559 1607 8 2 v 1567 1608 V 1575 1609 V 1583
1610 V 1591 1611 9 2 v 1599 1612 V 1608 1613 16 2 v 1624 1614
8 2 v 1632 1615 V 1640 1616 17 2 v 1657 1617 16 2 v 1673 1618
9 2 v 1681 1619 V 1690 1620 16 2 v 1706 1621 V 1722 1622 34
2 v 1755 1623 16 2 v 1771 1624 34 2 v 1804 1625 V 1837 1626
49 2 v 964 2825 a Fg(5)p eop
%%Page: 6 6
bop 264 1973 20 2 v 220 1987 a Fg(0)p 1866 1973 V 264 1715
V 184 1729 a(0)p Ff(:)p Fg(5)p 1866 1715 V 264 1457 V 220 1471
a(1)p 1866 1457 V 264 1200 V 184 1214 a(1)p Ff(:)p Fg(5)p 1866
1200 V 264 942 V 220 956 a(2)p 1866 942 V 264 684 V 184 698
a(2)p Ff(:)p Fg(5)p 1866 684 V 264 1973 2 20 v 253 2032 a(0)p
264 704 V 559 1973 V 254 w(0)p Ff(:)p Fg(2)p 559 704 V 853
1973 V 235 w(0)p Ff(:)p Fg(4)p 853 704 V 1149 1973 V 236 w(0)p
Ff(:)p Fg(6)p 1149 704 V 1444 1973 V 236 w(0)p Ff(:)p Fg(8)p
1444 704 V 1739 1973 V 254 w(1)p 1739 704 V 264 1973 1622 2
v 1886 1973 2 1289 v 264 684 1622 2 v 45 1343 a Ff(<)13 b(\036)120
1326 y Fd(4)153 1343 y Ff(>)1045 2081 y(m)1085 2064 y Fd(2)1019
648 y Fc(Fig.)18 b(6)p 264 1973 2 1289 v 1721 1889 a Fb(3)614
1569 y(3)305 1043 y(3)p 1739 1880 2 10 v 1729 1880 20 2 v 1729
1870 V 632 1566 2 22 v 622 1566 20 2 v 622 1545 V 323 1045
2 32 v 313 1045 20 2 v 313 1014 V 1721 1885 a Fg(+)615 1565
y(+)306 1039 y(+)p 1739 1880 2 10 v 1729 1880 20 2 v 1729 1870
V 632 1566 2 22 v 622 1566 20 2 v 622 1545 V 323 1045 2 32
v 313 1045 20 2 v 313 1014 V 1722 1889 a Fb(2)616 1569 y(2)307
1043 y(2)p 1739 1880 2 10 v 1729 1880 20 2 v 1729 1870 V 632
1566 2 22 v 622 1566 20 2 v 622 1545 V 323 1045 2 32 v 313
1045 20 2 v 313 1014 V 1721 1885 a Fe(\002)615 1560 y(\002)306
1060 y(\002)p 1739 1880 2 10 v 1729 1880 20 2 v 1729 1870 V
632 1561 2 22 v 622 1561 20 2 v 622 1540 V 323 1066 2 32 v
313 1066 20 2 v 313 1035 V 1718 1885 a(4)612 1565 y(4)303 1045
y(4)p 1739 1880 2 10 v 1729 1880 20 2 v 1729 1870 V 632 1566
2 22 v 622 1566 20 2 v 622 1545 V 323 1050 2 32 v 313 1050
20 2 v 313 1019 V 283 697 2 13 v 284 709 V 285 721 V 286 733
V 287 745 V 288 757 V 289 769 V 290 782 V 291 794 V 292 806
V 293 818 V 294 830 V 295 842 V 296 854 V 297 862 2 8 v 298
870 V 299 877 V 300 885 V 301 892 V 302 900 V 303 907 V 304
915 V 305 923 V 306 930 V 307 938 V 308 945 V 309 953 V 310
960 V 311 968 V 312 975 V 313 981 2 6 v 314 986 V 315 991 V
316 997 V 317 1002 V 318 1007 V 319 1013 V 320 1018 V 321 1023
V 322 1029 V 323 1034 V 324 1039 V 325 1045 V 326 1050 V 327
1055 V 328 1060 V 329 1064 2 4 v 330 1068 V 331 1072 V 332
1076 V 333 1080 V 334 1084 V 335 1088 V 336 1092 V 337 1095
V 338 1099 V 339 1103 V 340 1107 V 341 1111 V 342 1115 V 343
1119 V 344 1123 V 345 1126 V 346 1130 V 347 1133 V 348 1137
V 349 1140 V 350 1143 V 351 1147 V 352 1150 V 353 1153 V 354
1157 V 355 1160 V 356 1164 V 357 1167 V 358 1170 V 359 1174
V 360 1177 V 361 1180 V 362 1183 2 3 v 363 1186 V 364 1188
V 365 1191 V 366 1194 V 367 1196 V 368 1199 V 369 1202 V 370
1204 V 371 1207 V 372 1210 V 373 1212 V 374 1215 V 375 1218
V 376 1220 V 377 1223 V 378 1225 V 379 1228 V 380 1230 V 381
1233 V 382 1235 V 383 1238 V 384 1240 V 385 1243 V 386 1245
V 387 1247 V 388 1250 V 389 1252 V 390 1255 V 391 1257 V 392
1260 V 393 1262 V 394 1264 V 395 1267 V 396 1269 V 397 1271
V 398 1273 V 399 1275 V 400 1277 V 401 1279 V 402 1281 V 403
1284 V 404 1286 V 405 1288 V 406 1290 V 407 1292 V 408 1294
V 409 1296 V 410 1298 V 411 1300 2 2 v 412 1302 V 413 1304
V 414 1306 V 415 1307 V 416 1309 V 417 1311 V 418 1313 V 419
1314 V 420 1316 V 421 1318 V 422 1320 V 423 1321 V 424 1323
V 425 1325 V 426 1327 V 427 1328 V 428 1330 V 429 1332 V 430
1334 V 431 1335 V 432 1337 V 433 1339 V 434 1341 V 435 1342
V 436 1344 V 437 1346 V 438 1348 V 439 1349 V 440 1351 V 441
1353 V 442 1355 V 443 1356 V 444 1358 V 445 1360 V 446 1361
V 447 1363 V 448 1364 V 449 1366 V 450 1367 V 451 1369 V 452
1371 V 453 1372 V 454 1374 V 455 1375 V 456 1377 V 457 1378
V 458 1380 V 459 1381 V 460 1383 V 461 1384 V 462 1385 V 463
1387 V 464 1388 V 465 1389 V 466 1391 V 467 1392 V 468 1393
V 469 1394 V 470 1396 V 471 1397 V 472 1398 V 473 1400 V 474
1401 V 475 1402 V 476 1403 V 477 1405 V 478 1406 V 479 1407
V 480 1409 V 481 1410 V 482 1411 V 483 1413 V 484 1414 V 485
1415 V 486 1417 V 487 1418 V 488 1419 V 489 1421 V 490 1422
V 491 1423 V 492 1424 V 493 1426 V 494 1427 V 495 1428 V 496
1429 V 497 1430 V 498 1432 V 499 1433 V 500 1434 V 501 1435
V 502 1436 V 503 1437 V 504 1439 V 505 1440 V 506 1441 V 507
1442 V 508 1443 V 509 1444 V 510 1446 V 511 1447 V 512 1448
V 513 1449 V 514 1450 V 515 1451 V 516 1452 V 517 1453 V 518
1455 V 519 1456 V 520 1457 V 521 1458 V 522 1459 V 523 1460
V 524 1461 V 525 1462 V 526 1462 V 527 1463 V 528 1464 V 529
1465 V 530 1466 V 531 1467 V 532 1468 V 533 1469 V 534 1470
V 535 1471 V 536 1472 V 537 1473 V 538 1474 V 539 1475 V 540
1476 V 541 1477 V 542 1478 V 543 1479 V 544 1480 V 545 1481
V 546 1482 V 547 1483 V 548 1484 V 549 1485 V 550 1486 V 551
1487 V 552 1488 V 553 1489 V 554 1490 V 555 1491 V 556 1492
V 557 1493 V 559 1494 V 560 1495 V 561 1496 V 562 1497 V 563
1498 V 564 1499 V 565 1500 V 566 1501 V 567 1502 V 568 1503
V 569 1504 V 570 1505 V 571 1506 V 572 1507 V 573 1508 V 574
1509 V 576 1510 V 577 1511 V 578 1512 V 579 1513 V 580 1514
V 581 1515 V 583 1516 V 584 1517 V 585 1518 V 586 1519 V 587
1520 V 588 1521 V 589 1522 V 591 1523 V 592 1524 V 593 1525
V 594 1526 V 596 1527 V 597 1528 V 598 1529 V 600 1530 V 601
1531 V 602 1532 V 604 1533 V 605 1534 V 606 1535 V 607 1536
V 609 1537 V 610 1538 V 611 1539 V 612 1540 V 614 1541 V 615
1542 V 616 1543 V 617 1544 V 619 1545 V 620 1546 V 621 1547
V 622 1548 V 624 1549 V 625 1550 V 626 1551 V 628 1552 V 629
1553 V 631 1554 V 632 1555 V 633 1556 V 635 1557 V 636 1558
V 638 1559 V 639 1560 V 641 1561 V 642 1562 V 643 1563 V 645
1564 V 646 1565 V 648 1566 V 649 1567 V 651 1568 V 652 1569
V 654 1570 V 655 1571 V 656 1572 V 658 1573 V 659 1574 V 661
1575 V 662 1576 V 664 1577 V 665 1578 V 667 1579 V 668 1580
V 670 1581 V 671 1582 V 672 1583 V 674 1584 V 676 1585 V 677
1586 V 679 1587 V 680 1588 V 682 1589 V 683 1590 V 685 1591
V 686 1592 V 688 1593 V 690 1594 V 691 1595 V 693 1596 V 694
1597 V 696 1598 V 697 1599 V 699 1600 V 701 1601 V 702 1602
V 704 1603 V 705 1604 V 707 1605 V 709 1606 V 711 1607 V 713
1608 V 714 1609 V 716 1610 V 718 1611 V 720 1612 V 721 1613
V 723 1614 V 725 1615 V 727 1616 V 729 1617 V 731 1618 V 733
1619 V 735 1620 V 737 1621 V 739 1622 V 740 1623 V 742 1624
V 744 1625 V 746 1626 V 747 1627 V 749 1628 V 751 1629 V 753
1630 V 754 1631 3 2 v 757 1632 V 759 1633 V 761 1634 V 763
1635 V 765 1636 V 767 1637 V 769 1638 V 772 1639 V 774 1640
V 776 1641 V 778 1642 V 780 1643 V 782 1644 V 784 1645 V 786
1646 V 788 1647 V 790 1648 V 792 1649 V 794 1650 V 796 1651
V 798 1652 V 800 1653 V 802 1654 V 804 1655 V 806 1656 V 808
1657 V 810 1658 V 812 1659 V 814 1660 V 816 1661 V 818 1662
V 821 1663 V 823 1664 V 825 1665 V 827 1666 V 830 1667 V 832
1668 V 834 1669 V 836 1670 V 839 1671 V 841 1672 V 843 1673
V 846 1674 V 848 1675 V 850 1676 V 852 1677 V 855 1678 V 857
1679 V 860 1680 V 862 1681 V 865 1682 V 867 1683 V 870 1684
V 872 1685 V 874 1686 V 876 1687 V 879 1688 V 881 1689 V 883
1690 V 885 1691 V 888 1692 V 891 1693 V 894 1694 V 897 1695
V 900 1696 V 902 1697 V 905 1698 V 908 1699 V 911 1700 V 913
1701 V 916 1702 V 919 1703 V 921 1704 V 924 1705 V 927 1706
V 929 1707 V 932 1708 V 935 1709 V 937 1710 V 940 1711 V 943
1712 V 946 1713 V 949 1714 V 951 1715 V 954 1716 V 957 1717
V 960 1718 V 962 1719 V 965 1720 V 968 1721 4 2 v 971 1722
V 974 1723 V 978 1724 V 981 1725 V 985 1726 V 988 1727 V 991
1728 V 995 1729 V 998 1730 V 1002 1731 3 2 v 1004 1732 V 1007
1733 V 1010 1734 V 1012 1735 V 1015 1736 V 1018 1737 4 2 v
1021 1738 V 1024 1739 V 1028 1740 V 1031 1741 V 1035 1742 5
2 v 1039 1743 V 1043 1744 V 1047 1745 V 1051 1746 4 2 v 1054
1747 V 1057 1748 V 1060 1749 V 1063 1750 V 1066 1751 V 1070
1752 V 1073 1753 V 1077 1754 V 1080 1755 V 1084 1756 5 2 v
1088 1757 V 1092 1758 V 1096 1759 V 1100 1760 4 2 v 1103 1761
V 1106 1762 V 1109 1763 V 1112 1764 V 1115 1765 5 2 v 1120
1766 V 1124 1767 V 1128 1768 V 1133 1769 V 1137 1770 V 1141
1771 V 1145 1772 V 1149 1773 V 1153 1774 V 1157 1775 V 1161
1776 V 1166 1777 V 1170 1778 V 1174 1779 V 1178 1780 V 1182
1781 V 1186 1782 V 1190 1783 V 1194 1784 V 1198 1785 V 1202
1786 V 1206 1787 V 1210 1788 V 1215 1789 V 1219 1790 V 1223
1791 V 1227 1792 V 1231 1793 6 2 v 1236 1794 V 1241 1795 V
1247 1796 5 2 v 1251 1797 V 1255 1798 V 1259 1799 V 1264 1800
6 2 v 1269 1801 V 1274 1802 V 1280 1803 5 2 v 1284 1804 V 1288
1805 V 1292 1806 V 1297 1807 6 2 v 1302 1808 V 1307 1809 V
1313 1810 V 1318 1811 V 1323 1812 V 1329 1813 V 1334 1814 V
1340 1815 V 1345 1816 V 1351 1817 V 1356 1818 V 1362 1819 V
1367 1820 V 1372 1821 V 1378 1822 V 1383 1823 V 1389 1824 V
1394 1825 V 1400 1826 V 1405 1827 V 1411 1828 V 1416 1829 V
1422 1830 V 1427 1831 V 1433 1832 V 1438 1833 V 1444 1834 8
2 v 1452 1835 V 1460 1836 6 2 v 1465 1837 V 1471 1838 V 1476
1839 V 1482 1840 V 1487 1841 V 1493 1842 8 2 v 1501 1843 V
1509 1844 6 2 v 1514 1845 V 1520 1846 V 1525 1847 8 2 v 1534
1848 V 1542 1849 9 2 v 1550 1850 V 1559 1851 6 2 v 1564 1852
V 1569 1853 V 1575 1854 8 2 v 1583 1855 V 1591 1856 9 2 v 1599
1857 V 1608 1858 8 2 v 1616 1859 V 1624 1860 6 2 v 1629 1861
V 1634 1862 V 1640 1863 9 2 v 1648 1864 V 1657 1865 8 2 v 1665
1866 V 1673 1867 9 2 v 1681 1868 V 1690 1869 8 2 v 1698 1870
V 1706 1871 V 1714 1872 V 1722 1873 9 2 v 1730 1874 V 1739
1875 16 2 v 1755 1876 8 2 v 1763 1877 V 1771 1878 9 2 v 1779
1879 V 1788 1880 8 2 v 1796 1881 V 1804 1882 17 2 v 1821 1883
8 2 v 1829 1884 V 1837 1885 V 1845 1886 V 1853 1887 17 2 v
1870 1888 8 2 v 1878 1889 V 964 2825 a Fg(6)p eop
%%Page: 7 7
bop 264 1973 20 2 v 220 1987 a Fg(0)p 1866 1973 V 264 1758
V 220 1772 a(5)p 1866 1758 V 264 1543 V 197 1557 a(10)p 1866
1543 V 264 1328 V 197 1342 a(15)p 1866 1328 V 264 1114 V 197
1128 a(20)p 1866 1114 V 264 899 V 197 913 a(25)p 1866 899 V
264 684 V 197 698 a(30)p 1866 684 V 264 1973 2 20 v 253 2032
a(0)p 264 704 V 559 1973 V 254 w(0)p Ff(:)p Fg(2)p 559 704
V 853 1973 V 235 w(0)p Ff(:)p Fg(4)p 853 704 V 1149 1973 V
236 w(0)p Ff(:)p Fg(6)p 1149 704 V 1444 1973 V 236 w(0)p Ff(:)p
Fg(8)p 1444 704 V 1739 1973 V 254 w(1)p 1739 704 V 264 1973
1622 2 v 1886 1973 2 1289 v 264 684 1622 2 v 45 1331 a Ff(\037)73
1338 y Fa(\036)1045 2081 y Ff(m)1085 2064 y Fd(2)1019 648 y
Fc(Fig.)18 b(7)p 264 1973 2 1289 v 1721 1940 a Fb(3)614 1812
y(3)305 1003 y(3)p 1739 1928 2 5 v 1729 1928 20 2 v 1729 1924
V 632 1802 2 8 v 622 1802 20 2 v 622 1794 V 323 1096 2 215
v 313 1096 20 2 v 313 882 V 1721 1940 a Fg(+)615 1809 y(+)306
1141 y(+)p 1739 1933 2 5 v 1729 1933 20 2 v 1729 1928 V 632
1803 2 9 v 622 1803 20 2 v 622 1794 V 323 1238 2 215 v 313
1238 20 2 v 313 1023 V 1722 1936 a Fb(2)616 1825 y(2)307 1110
y(2)p 1739 1924 2 5 v 1729 1924 20 2 v 1729 1920 V 632 1815
2 8 v 622 1815 20 2 v 622 1807 V 323 1204 2 215 v 313 1204
20 2 v 313 989 V 1721 1938 a Fe(\002)615 1826 y(\002)306 1162
y(\002)p 1739 1930 2 5 v 1729 1930 20 2 v 1729 1926 V 632 1820
2 9 v 622 1820 20 2 v 622 1811 V 323 1260 2 215 v 313 1260
20 2 v 313 1045 V 1718 1940 a(4)612 1816 y(4)303 1257 y(4)p
1739 1932 2 5 v 1729 1932 20 2 v 1729 1927 V 632 1811 2 9 v
622 1811 20 2 v 622 1802 V 323 1354 2 215 v 313 1354 20 2 v
313 1139 V 300 706 2 22 v 301 728 V 302 750 V 303 772 V 304
794 V 305 816 V 306 838 V 307 860 V 308 882 V 309 904 V 310
926 V 311 948 V 312 969 V 313 982 2 13 v 314 995 V 315 1008
V 316 1020 V 317 1033 V 318 1046 V 319 1059 V 320 1071 V 321
1084 V 322 1097 V 323 1110 V 324 1122 V 325 1135 V 326 1148
V 327 1161 V 328 1173 V 329 1181 2 8 v 330 1189 V 331 1197
V 332 1205 V 333 1213 V 334 1221 V 335 1229 V 336 1237 V 337
1244 V 338 1252 V 339 1260 V 340 1268 V 341 1276 V 342 1284
V 343 1292 V 344 1300 V 345 1307 V 346 1313 2 6 v 347 1319
V 348 1325 V 349 1331 V 350 1337 V 351 1343 V 352 1349 V 353
1355 V 354 1361 V 355 1367 V 356 1373 V 357 1379 V 358 1385
V 359 1391 V 360 1397 V 361 1402 V 362 1407 2 5 v 363 1411
V 364 1415 V 365 1419 V 366 1423 V 367 1428 V 368 1432 V 369
1436 V 370 1440 V 371 1444 V 372 1448 V 373 1453 V 374 1457
V 375 1461 V 376 1465 V 377 1469 V 378 1473 V 379 1477 2 4
v 380 1480 V 381 1484 V 382 1487 V 383 1491 V 384 1494 V 385
1498 V 386 1501 V 387 1504 V 388 1508 V 389 1511 V 390 1515
V 391 1518 V 392 1522 V 393 1525 V 394 1528 V 395 1531 2 3
v 396 1534 V 397 1537 V 398 1539 V 399 1542 V 400 1545 V 401
1548 V 402 1550 V 403 1553 V 404 1556 V 405 1559 V 406 1561
V 407 1564 V 408 1567 V 409 1570 V 410 1572 V 411 1575 V 412
1577 V 413 1579 V 414 1581 V 415 1583 V 416 1585 V 417 1587
V 418 1589 V 419 1592 V 420 1594 V 421 1596 V 422 1598 V 423
1600 V 424 1602 V 425 1604 V 426 1606 V 427 1608 V 428 1610
2 2 v 429 1612 V 430 1614 V 431 1616 V 432 1618 V 433 1620
V 434 1622 V 435 1623 V 436 1625 V 437 1627 V 438 1629 V 439
1631 V 440 1633 V 441 1635 V 442 1637 V 443 1638 V 444 1640
V 445 1642 V 446 1643 V 447 1645 V 448 1647 V 449 1648 V 450
1650 V 451 1651 V 452 1653 V 453 1655 V 454 1656 V 455 1658
V 456 1660 V 457 1661 V 458 1663 V 459 1664 V 460 1666 V 461
1667 V 462 1668 V 463 1670 V 464 1671 V 465 1672 V 466 1674
V 467 1675 V 468 1676 V 469 1677 V 470 1679 V 471 1680 V 472
1681 V 473 1683 V 474 1684 V 475 1685 V 476 1686 V 477 1688
V 478 1689 V 479 1690 V 480 1691 V 481 1692 V 482 1693 V 483
1694 V 484 1695 V 485 1697 V 486 1698 V 487 1699 V 488 1700
V 489 1701 V 490 1702 V 491 1703 V 492 1704 V 493 1704 V 494
1705 V 495 1706 V 496 1707 V 497 1708 V 498 1709 V 499 1710
V 500 1711 V 501 1712 V 502 1713 V 503 1714 V 504 1715 V 505
1716 V 506 1717 V 507 1718 V 508 1719 V 509 1720 V 510 1721
V 511 1722 V 512 1723 V 513 1724 V 514 1725 V 515 1726 V 516
1727 V 517 1728 V 518 1729 V 519 1730 V 520 1731 V 521 1732
V 522 1733 V 523 1734 V 524 1735 V 525 1736 V 527 1737 V 528
1738 V 529 1739 V 531 1740 V 532 1741 V 533 1742 V 535 1743
V 536 1744 V 537 1745 V 539 1746 V 540 1747 V 541 1748 V 543
1749 V 544 1750 V 546 1751 V 547 1752 V 549 1753 V 550 1754
V 551 1755 V 553 1756 V 554 1757 V 556 1758 V 557 1759 V 559
1760 V 560 1761 V 561 1762 V 563 1763 V 564 1764 V 566 1765
V 567 1766 V 569 1767 V 570 1768 V 572 1769 V 573 1770 V 574
1771 V 576 1772 V 578 1773 V 580 1774 V 582 1775 V 583 1776
V 585 1777 V 587 1778 V 589 1779 V 590 1780 V 592 1781 V 594
1782 V 596 1783 V 598 1784 V 600 1785 V 602 1786 V 604 1787
V 606 1788 V 608 1789 3 2 v 610 1790 V 612 1791 V 614 1792
V 617 1793 V 619 1794 V 621 1795 V 623 1796 V 626 1797 V 628
1798 V 630 1799 V 632 1800 V 634 1801 V 636 1802 V 638 1803
V 641 1804 V 643 1805 V 646 1806 V 649 1807 V 651 1808 V 654
1809 V 657 1810 V 659 1811 V 662 1812 V 665 1813 V 667 1814
V 670 1815 V 673 1816 V 675 1817 V 678 1818 V 681 1819 V 684
1820 V 687 1821 V 689 1822 4 2 v 693 1823 V 696 1824 V 699
1825 V 702 1826 V 706 1827 V 709 1828 V 712 1829 V 715 1830
V 718 1831 V 722 1832 V 725 1833 V 728 1834 V 732 1835 V 735
1836 V 739 1837 5 2 v 743 1838 V 747 1839 V 751 1840 V 755
1841 V 759 1842 V 763 1843 V 767 1844 V 772 1845 V 776 1846
V 780 1847 V 784 1848 V 788 1849 6 2 v 793 1850 V 798 1851
V 803 1852 5 2 v 808 1853 V 812 1854 V 816 1855 V 821 1856
6 2 v 826 1857 V 831 1858 V 836 1859 V 842 1860 V 847 1861
V 852 1862 V 858 1863 V 864 1864 V 870 1865 V 875 1866 V 880
1867 V 885 1868 9 2 v 894 1869 V 903 1870 6 2 v 908 1871 V
913 1872 V 918 1873 8 2 v 927 1874 V 935 1875 9 2 v 943 1876
V 952 1877 8 2 v 960 1878 V 968 1879 V 977 1880 V 985 1881
9 2 v 993 1882 V 1002 1883 8 2 v 1010 1884 V 1018 1885 9 2
v 1026 1886 V 1035 1887 8 2 v 1043 1888 V 1051 1889 16 2 v
1067 1890 9 2 v 1075 1891 V 1084 1892 16 2 v 1100 1893 8 2
v 1108 1894 V 1116 1895 17 2 v 1133 1896 8 2 v 1141 1897 V
1149 1898 17 2 v 1166 1899 16 2 v 1182 1900 V 1198 1901 9 2
v 1206 1902 V 1215 1903 16 2 v 1231 1904 V 1247 1905 17 2 v
1264 1906 16 2 v 1280 1907 17 2 v 1297 1908 16 2 v 1313 1909
V 1329 1910 17 2 v 1346 1911 16 2 v 1362 1912 V 1378 1913 34
2 v 1411 1914 17 2 v 1428 1915 16 2 v 1444 1916 34 2 v 1477
1917 16 2 v 1493 1918 V 1509 1919 34 2 v 1542 1920 17 2 v 1559
1921 33 2 v 1591 1922 34 2 v 1624 1923 16 2 v 1640 1924 34
2 v 1673 1925 V 1706 1926 V 1739 1927 33 2 v 1771 1928 34 2
v 1804 1929 V 1837 1930 49 2 v 964 2825 a Fg(7)p eop
%%Page: 8 8
bop 264 1973 20 2 v 220 1987 a Fg(0)p 1866 1973 V 264 1844
V 162 1858 a(0)p Ff(:)p Fg(05)p 1866 1844 V 264 1715 V 184
1729 a(0)p Ff(:)p Fg(1)p 1866 1715 V 264 1586 V 162 1600 a(0)p
Ff(:)p Fg(15)p 1866 1586 V 264 1457 V 184 1471 a(0)p Ff(:)p
Fg(2)p 1866 1457 V 264 1328 V 162 1342 a(0)p Ff(:)p Fg(25)p
1866 1328 V 264 1200 V 184 1214 a(0)p Ff(:)p Fg(3)p 1866 1200
V 264 1071 V 162 1085 a(0)p Ff(:)p Fg(35)p 1866 1071 V 264
942 V 184 956 a(0)p Ff(:)p Fg(4)p 1866 942 V 264 813 V 162
827 a(0)p Ff(:)p Fg(45)p 1866 813 V 264 684 V 184 698 a(0)p
Ff(:)p Fg(5)p 1866 684 V 264 1973 2 20 v 253 2032 a(0)p 264
704 V 534 1973 V 236 w(10)p 534 704 V 804 1973 V 224 w(20)p
804 704 V 1076 1973 V 225 w(30)p 1076 704 V 1346 1973 V 224
w(40)p 1346 704 V 1616 1973 V 224 w(50)p 1616 704 V 1886 1973
V 224 w(60)p 1886 704 V 264 1973 1622 2 v 1886 1973 2 1289
v 264 684 1622 2 v 45 1344 a Ff(k)1050 2065 y(n)1077 2072 y
Fa(f)1019 648 y Fc(Fig.)18 b(8)544 1728 y(Smo)q(oth)13 b(Phase)1107
951 y(Rough)g(Phase)p 264 1973 2 1289 v 264 1335 2 2 v 264
1335 5 2 v 268 1336 V 272 1337 V 276 1338 V 280 1339 V 284
1340 V 288 1341 V 292 1342 V 297 1343 V 301 1344 V 305 1345
V 309 1346 V 313 1347 V 317 1348 V 321 1349 V 325 1350 V 329
1351 V 333 1352 V 337 1353 V 341 1354 V 346 1355 V 350 1356
V 354 1357 V 358 1358 V 362 1359 V 366 1360 V 370 1361 V 374
1362 V 379 1363 V 383 1364 V 387 1365 V 391 1366 V 395 1367
V 399 1368 V 403 1369 V 407 1370 V 411 1371 V 415 1372 V 419
1373 V 423 1374 V 428 1375 V 432 1376 V 436 1377 V 440 1378
V 444 1379 V 448 1380 V 452 1381 V 456 1382 V 460 1383 V 464
1384 V 468 1385 V 472 1386 V 477 1387 4 2 v 480 1388 V 483
1389 V 486 1390 V 489 1391 V 493 1392 5 2 v 497 1393 V 501
1394 V 505 1395 V 510 1396 V 514 1397 V 518 1398 V 522 1399
V 526 1400 4 2 v 529 1401 V 532 1402 V 535 1403 V 538 1404
V 542 1405 5 2 v 546 1406 V 550 1407 V 554 1408 V 559 1409
V 563 1410 V 567 1411 V 571 1412 V 575 1413 4 2 v 578 1414
V 581 1415 V 584 1416 V 587 1417 V 591 1418 5 2 v 595 1419
V 599 1420 V 603 1421 V 608 1422 4 2 v 611 1423 V 614 1424
V 617 1425 V 620 1426 V 624 1427 5 2 v 628 1428 V 632 1429
V 636 1430 V 641 1431 4 2 v 644 1432 V 647 1433 V 650 1434
V 653 1435 V 657 1436 5 2 v 661 1437 V 665 1438 V 669 1439
V 673 1440 4 2 v 676 1441 V 679 1442 V 683 1443 V 686 1444
V 690 1445 V 693 1446 V 696 1447 V 699 1448 V 702 1449 V 706
1450 5 2 v 710 1451 V 714 1452 V 718 1453 V 722 1454 4 2 v
725 1455 V 728 1456 V 732 1457 V 735 1458 V 739 1459 V 742
1460 V 745 1461 V 748 1462 V 751 1463 V 755 1464 V 758 1465
V 761 1466 V 765 1467 V 768 1468 V 772 1469 V 775 1470 V 778
1471 V 781 1472 V 784 1473 V 788 1474 V 791 1475 V 794 1476
V 797 1477 V 800 1478 V 804 1479 V 807 1480 V 810 1481 V 814
1482 V 817 1483 V 821 1484 V 824 1485 V 827 1486 V 830 1487
V 833 1488 V 837 1489 V 840 1490 V 843 1491 V 846 1492 V 849
1493 V 853 1494 V 856 1495 V 859 1496 V 863 1497 V 866 1498
V 870 1499 V 873 1500 V 876 1501 V 879 1502 V 882 1503 V 886
1504 3 2 v 888 1505 V 891 1506 V 894 1507 V 897 1508 V 900
1509 V 902 1510 4 2 v 906 1511 V 909 1512 V 912 1513 V 915
1514 V 919 1515 V 922 1516 V 925 1517 V 928 1518 V 931 1519
V 935 1520 3 2 v 937 1521 V 940 1522 V 943 1523 V 946 1524
V 949 1525 V 951 1526 V 954 1527 V 957 1528 V 960 1529 V 962
1530 V 965 1531 V 968 1532 4 2 v 971 1533 V 974 1534 V 978
1535 V 981 1536 V 985 1537 3 2 v 987 1538 V 990 1539 V 993
1540 V 996 1541 V 999 1542 V 1001 1543 V 1004 1544 V 1007 1545
V 1010 1546 V 1012 1547 V 1015 1548 V 1018 1549 4 2 v 1021
1550 V 1024 1551 V 1028 1552 V 1031 1553 V 1035 1554 3 2 v
1037 1555 V 1040 1556 V 1042 1557 V 1045 1558 V 1048 1559 V
1050 1560 V 1053 1561 V 1056 1562 V 1058 1563 V 1061 1564 V
1064 1565 V 1066 1566 V 1069 1567 V 1071 1568 V 1074 1569 V
1076 1570 V 1079 1571 V 1081 1572 V 1084 1573 V 1086 1574 V
1089 1575 V 1091 1576 V 1094 1577 V 1097 1578 V 1099 1579 V
1102 1580 V 1105 1581 V 1107 1582 V 1110 1583 V 1113 1584 V
1115 1585 V 1118 1586 V 1121 1587 V 1124 1588 V 1127 1589 V
1130 1590 V 1133 1591 V 1135 1592 V 1137 1593 V 1139 1594 V
1142 1595 V 1144 1596 V 1146 1597 V 1149 1598 V 1151 1599 V
1153 1600 V 1156 1601 V 1158 1602 V 1161 1603 V 1163 1604 V
1166 1605 V 1168 1606 V 1171 1607 V 1173 1608 V 1176 1609 V
1179 1610 V 1181 1611 V 1184 1612 V 1186 1613 V 1188 1614 V
1191 1615 V 1193 1616 V 1195 1617 V 1198 1618 V 1200 1619 V
1202 1620 V 1205 1621 V 1207 1622 V 1210 1623 V 1212 1624 V
1215 1625 V 1217 1626 V 1219 1627 V 1221 1628 V 1224 1629 V
1226 1630 V 1228 1631 V 1231 1632 V 1233 1633 V 1235 1634 V
1237 1635 V 1239 1636 V 1241 1637 V 1243 1638 V 1245 1639 V
1247 1640 V 1249 1641 V 1251 1642 V 1254 1643 V 1256 1644 V
1259 1645 V 1261 1646 V 1264 1647 V 1266 1648 V 1268 1649 V
1270 1650 V 1272 1651 V 1274 1652 V 1276 1653 V 1278 1654 V
1280 1655 V 1282 1656 V 1284 1657 V 1286 1658 V 1288 1659 V
1290 1660 V 1292 1661 V 1294 1662 V 1297 1663 V 1299 1664 V
1301 1665 V 1303 1666 V 1305 1667 V 1307 1668 V 1309 1669 V
1311 1670 V 1313 1671 V 1315 1672 V 1317 1673 V 1319 1674 V
1321 1675 V 1323 1676 V 1325 1677 V 1327 1678 V 1329 1679 2
2 v 1330 1680 V 1332 1681 V 1334 1682 V 1336 1683 V 1338 1684
V 1340 1685 V 1342 1686 V 1344 1687 V 1346 1688 3 2 v 1348
1689 V 1350 1690 V 1352 1691 V 1354 1692 V 1356 1693 V 1358
1694 V 1360 1695 V 1362 1696 2 2 v 1363 1697 V 1365 1698 V
1366 1699 V 1368 1700 V 1369 1701 V 1371 1702 V 1373 1703 V
1374 1704 V 1376 1705 V 1377 1706 V 1379 1707 V 1381 1708 V
1383 1709 V 1385 1710 V 1387 1711 V 1389 1712 V 1391 1713 V
1393 1714 V 1395 1715 V 1396 1716 V 1398 1717 V 1399 1718 V
1401 1719 V 1402 1720 V 1404 1721 V 1406 1722 V 1407 1723 V
1409 1724 V 1410 1725 V 1412 1726 V 1414 1727 V 1416 1728 V
1417 1729 V 1419 1730 V 1421 1731 V 1422 1732 V 1424 1733 V
1426 1734 V 1427 1735 V 1429 1736 V 1431 1737 V 1432 1738 V
1434 1739 V 1435 1740 V 1437 1741 V 1439 1742 V 1440 1743 V
1442 1744 V 1443 1745 V 1445 1746 V 1446 1747 V 1448 1748 V
1449 1749 V 1451 1750 V 1452 1751 V 1454 1752 V 1455 1753 V
1457 1754 V 1458 1755 V 1460 1756 V 1461 1757 V 1462 1758 V
1464 1759 V 1465 1760 V 1467 1761 V 1468 1762 V 1469 1763 V
1471 1764 V 1472 1765 V 1474 1766 V 1475 1767 V 1476 1768 V
1478 1769 V 1479 1770 V 1480 1771 V 1481 1772 V 1483 1773 V
1484 1774 V 1485 1775 V 1486 1776 V 1488 1777 V 1489 1778 V
1490 1779 V 1491 1780 V 1492 1781 V 1494 1782 V 1495 1783 V
1496 1784 V 1497 1785 V 1499 1786 V 1500 1787 V 1501 1788 V
1502 1789 V 1504 1790 V 1505 1791 V 1506 1792 V 1507 1793 V
1508 1794 V 1510 1795 V 1511 1796 V 1512 1797 V 1513 1798 V
1515 1799 V 1516 1800 V 1517 1801 V 1518 1802 V 1519 1803 V
1521 1804 V 1522 1805 V 1523 1806 V 1524 1807 V 1525 1808 V
1527 1809 V 1528 1810 V 1529 1811 V 1530 1812 V 1531 1813 V
1532 1814 V 1533 1815 V 1534 1816 V 1535 1817 V 1536 1818 V
1537 1819 V 1538 1820 V 1539 1821 V 1540 1822 V 1541 1823 V
1542 1826 V 1543 1827 V 1544 1828 V 1545 1829 V 1546 1830 V
1547 1831 V 1548 1832 V 1549 1833 V 1550 1834 V 1551 1835 V
1552 1836 V 1553 1837 V 1554 1838 V 1555 1839 V 1556 1840 V
1557 1841 V 1558 1842 V 1559 1844 V 1560 1845 V 1561 1846 V
1562 1847 V 1563 1849 V 1564 1850 V 1565 1851 V 1566 1852 V
1567 1854 V 1568 1855 V 1569 1856 V 1570 1857 V 1571 1859 V
1572 1860 V 1573 1861 V 1574 1862 V 1575 1864 V 1576 1866 V
1577 1867 V 1578 1869 V 1579 1870 V 1580 1872 V 1581 1873 V
1582 1875 V 1583 1877 V 1584 1878 V 1585 1880 V 1586 1881 V
1587 1883 V 1588 1884 V 1589 1886 V 1590 1887 V 1591 1890 2
3 v 1592 1892 V 1593 1894 V 1594 1896 V 1595 1898 V 1596 1900
V 1597 1902 V 1598 1904 V 1599 1907 V 1600 1909 V 1601 1911
V 1602 1913 V 1603 1915 V 1604 1917 V 1605 1919 V 1606 1921
V 1607 1923 V 1608 1923 2 2 v 964 2825 a Fg(8)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF


--- Cut here ------------------------------------------------------------------

