%Paper: hep-th/9212102
%From: Emmanuel Guitter <guitter@amoco.saclay.cea.fr>
%Date: 16 Dec 92 17:37:37+0100

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  WARNINGS!!!!!!
%
%    The file below contains:
%       1/ The tex file (from the delimiter `HEAD OF TEX FILE' to the
%          delimiter `END OF TEX FILE')
%       2/ The Postscript file of figure 1 (from %!PS-Adobe-2.0 to
%          the end), ready to be inserted in the text.
%
%    The tex file is plain tex and needs harvmac.tex
%
%    The figure file must be saved separately under the name `fig.ps'
%    and is automatically inserted by the \special command in the tex file
%
%
%    A Postscript file of this paper (with text + figure) is available by
%    anonymous ftp on amoco.saclay.cea.fr as 92-156.ps in the directory
%    pubs.spht
%
%
%%%%%%%%%%%%%%%%%%%%%%% HEAD OF TEX FILE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input harvmac
%\baselineskip =20pt plus 1 pt minus 1 pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\font\tay=eusb10
\font\tentit=cmmib10
\font\ninetit=cmmib9
\font\seventit=cmmib7
\font\fivetit=cmmib5
\newfam\titfam
\textfont\titfam=\tentit
\scriptfont\titfam=\seventit
\scriptscriptfont\titfam=\fivetit
\def\tit{\fam\titfam\tentit}
%
\def\CA{{\cal A}} \def\CB{{\cal B}} \def\CC{{\cal C}} \def\CD{{\cal D}}
\def\CE{{\cal E}} \def\CF{{\cal F}} \def\CG{{\cal G}} \def\CH{{\cal H}}
\def\CI{{\cal I}} \def\CJ{{\cal J}} \def\CK{{\cal K}} \def\CL{{\cal L}}
\def\CM{{\cal M}} \def\CN{{\cal N}} \def\CO{{\cal O}} \def\CP{{\cal P}}
\def\CQ{{\cal Q}} \def\CR{{\cal R}} \def\CS{{\cal S}} \def\CT{{\cal T}}
\def\CU{{\cal U}} \def\CV{{\cal V}} \def\CW{{\cal W}} \def\CX{{\cal X}}
\def\CY{{\cal Y}} \def\CZ{{\cal Z}}
%%%%%%%%%%%%%%%%%%%%
% personal macros  %
%%%%%%%%%%%%%%%%%%%%
\def\rvec{{\bf \vec r}}
\def\kvec{{\bf \vec k}}
%
\def\Tay{{\hbox{\tay T}}}
\def\RR{\relax{\rm I\kern-.18em R}}
\def\Rr{\relax{\ninerm I\kern-.22em \ninerm R}}
\def\setminusp{\hbox{$/ \kern -3pt {}_p$}}
\def\ssetminusp{\hbox{$\scriptstyle / \kern -2pt {}_p$}}
%
\def\figcap#1#2{\nfig#1{#2} \noindent {Fig.\ }\xfig#1{:\ }{\sl #2}}
%
\def\figinsert#1#2#3{\vbox to #1{\vfill \vbox {\hsize=#2
\baselineskip=10pt {#3} \par}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\draftmode
\Title{SPhT/92-156 hep-th/9212102}{
\vbox{
\vskip -2truecm
\centerline{Renormalization of Crumpled Manifolds}
}}
%
\centerline{Fran\c cois David\footnote{$^\dagger$}
{Member of CNRS},
Bertrand Duplantier{$^\dagger$}{} and Emmanuel Guitter}
\bigskip\centerline{Service de Physique Th\'eorique\footnote{$^\star$}{
Laboratoire de la Direction des Sciences de la Mati\`ere du Commissariat
\`a l'Energie Atomique}}
\centerline{C.E. Saclay}
\centerline{F-91191 Gif-sur-Yvette, France}
\vskip 1.truecm
%
\centerline{\bf Abstract}{
\ninerm
\textfont0=\ninerm
\font\ninemit=cmmi9
\font\sevenmit=cmmi7
\textfont1=\ninemit
\scriptfont0=\sevenrm
\scriptfont1=\sevenmit
\baselineskip=11pt
\bigskip
We consider a model of $D$-dimensional tethered manifold interacting
by excluded volume in \Rr${}^d$ with a single point. By use of
intrinsic distance geometry, we first provide a rigorous definition of the
analytic continuation of its perturbative expansion for arbitrary $D$,
\ $0\!<\!D\!<\!2$. We then construct explicitly a renormalization operation
{\ninebf R}, ensuring renormalizability to all orders.
This is the first example of mathematical construction and renormalization
for an interacting extended object with continuous internal dimension,
encompassing field theory. \par
}
%
\vskip .3in
\Date{PACS numbers: 05.20.-y, 11.10.Gh, 11.17.+y}
%for preliminary versions, specify \draftmode at some point
\hfuzz 1.pt
\vfill\eject
The Statistical Mechanics of random surfaces and membranes,
or more generally of extended objects, poses fundamental problems
\ref\Jerus{{\sl Statistical Mechanics of Membranes and
Surfaces}, Proceedings of the Fifth Jerusalem Winter School for Theoretical
Physics (1987), D. R. Nelson, T. Piran and S. Weinberg Eds., World Scientific,
Singapore (1989).}.
\nref\SirSam{S. F. Edwards, Proc. Phys. Soc. Lond. {\bf 85} (1965) 613.}
\nref\desClJan{J. des Cloizeaux and G. Jannink,
{\sl Polymers in Solution, their Modelling and Structure}, Clarendon
Press, Oxford (1990).}
\nref\Wenetal{X. Wen {\it et al.},
%C.W. Garland, T. Hwa, M. Kardar,
%E. Kokufuta, Y. Li, M. Orkisz and T. Tanaka,
Nature {\bf 355} (1992) 426.}
\nref\Spect{C.F. Schmidt {\it et al.},
%K. Svoboda, N. Lei, I.B. Petsche,
%L.E. Berman, C. Safinya and G. Grest,
unpublished.}
\hskip -20pt Among those, the study of {\it polymerized}
membranes, which are simple generalizations of linear polymers
[\xref\SirSam ,\xref\desClJan ] to two-dimensionally connected networks,
is prominent,
with a number of possible experimental
realizations [\xref\Wenetal ,\xref\Spect ]. From a theoretical point
of view, a clear challenge is to understand
self-avoidance (SA) effects in membranes.
\nref\KN{M. Kardar and D. R. Nelson, Phys. Rev. Lett. {\bf 58}
(1987) 1289, 2280(E); Phys. Rev. {\bf A 38}
(1988) 966.}
\nref\ArLub{J. A. Aronowitz and T. C. Lubensky, Europhys. Lett. {\bf 4}
(1987) 395.}
\kern -10pt Recently, a model was proposed [\xref\KN ,\xref\ArLub ]
which aimed to incorporate the advances made in polymer theory
by Renormalization Group methods into the field of polymerized, or
tethered membranes. These extended objects, a priori two-dimensional in nature,
are generalized for theoretical purposes to intrinsically
{\it $D$- dimensional manifolds} with internal points $x\in \RR^D$,
embedded in external $d$-dimensional space with position
vector $\rvec (x)\in \RR^d$.
The associated continuum Hamiltonian $\CH$ generalizes that of
Edwards for polymers \SirSam :
\eqn\Edwards{
\beta{\cal H}={1\over 2}\int d^Dx\,\Big(\nabla_x\rvec (x)\Big)^2+
{b\over 2}\int d^Dx\int d^Dx'\ \delta^{d}\big(\rvec (x)-\rvec (x')\big)
\ ,}
\nref\BDbis{B. Duplantier, Phys. Rev. Lett. {\bf 58} (1987) 2733;
and in \Jerus .}
\hskip -3pt with an elastic Gaussian term and a self-avoidance
two-body $\delta$-potential with interaction parameter $b>0$. For
$0<D<2$, the Gaussian manifold ($b=0$) is {\it crumpled} with a finite
Hausdorff dimension
$d_H=2D/(2-D)$; and the finiteness of the upper critical dimension
$d^\star = 2d_H$ for the SA-interaction
allows for an $\epsilon$-expansion about
$d^\star$ [\xref\KN --\nobreak\xref\BDbis ],
performed via a direct renormalization method
adapted from that of des Cloizeaux in polymer theory
\ref\desCloiz{J. des Cloizeaux, J. Phys. France {\bf 42} (1981) 635.}.

It should be stressed however that only the polymer case,
with an {\it integer} internal dimension $D=1$, can be
mapped,
following de Gennes \ref\DeGe{P.G. de Gennes, Phys. Lett.
{\bf A 38} (1972) 339.},
onto a standard field theory, namely a $({\bf \Phi}^2)^2$ theory
for a field ${\bf \Phi}$ with $n\to 0$ components. This
is instrumental to show that the direct renormalization
method for polymers is mathematically sound
\ref\BenMa{M. Benhamou and G. Mahoux, J. Phys. France {\bf 47} (1986) 559.},
and equivalent to rigorous renormalization schemes in standard
local field theory, such as the landmark
Bogoliubov Parasiuk Hepp Zimmermann (BPHZ) construction
\ref\BPHZ{N. N. Bogoliubov and O. S. Parasiuk, Acta Math. {\bf 97} (1957) 227;
\hfill\break\noindent
K. Hepp, Commun. Math. Phys. {\bf 2} (1966) 301;
\hfill\break\noindent
W. Zimmermann, Commun. Math. Phys. {\bf 15} (1969) 208.  }.
For manifold theory, we have to deal with {\it non-integer} internal
dimension $D$, $D\ne 1$, where no such mapping exists.
Therefore, two outstanding problems remain in the theory
of interacting manifolds: (a) the mathematical meaning of a
{\it continuous} internal dimension $D$;
(b) the actual {\it renormalizability} of the perturbative
expansion of a manifold model like \Edwards ,
implying scaling as expected on physical grounds.

A first answer was brought up in
\ref\BD{B. Duplantier, Phys. Rev. Lett. {\bf 62} (1989) 2337.},
where a simpler model
of a crumpled manifold interacting by excluded volume with
a fixed Euclidean subspace of $\RR^d$ was proposed. The direct
resummation of leading divergences of the perturbation series
indeed validates there {\it one-loop} renormalization,
a result later extended to the Edwards model \Edwards\
\ref\DHK{B. Duplantier, T. Hwa and M. Kardar, Phys. Rev. Lett. {\bf 64}
(1990) 2022.}.

In this Letter, we announce the results of an extensive study
of these questions
\ref\DDG{F. David, B. Duplantier and E. Guitter, Saclay preprint SPhT/92-124.}.
We first propose a mathematical
construction of the $D$-dimensional internal measure $d^Dx$
via distance geometry within the elastic manifold,
with expressions for manifold Feynman
integrals which generalize the $\alpha$-parameter representation
of field theory. In the case
of the manifold model of \BD\ , we then describe the essential
properties which make it indeed {\it renormalizable to all orders}
by a renormalization of the coupling constant,
and we directly construct a renormalization operation,
generalizing the BPHZ construction to manifolds.

The simplified model Hamiltonian introduced in \BD\ reads:
\eqn\HamM{
\beta{\cal H}={1\over 2}\int d^Dx\,\Big(\nabla_x \rvec (x)
\Big)^2+ {b}\int d^Dx\ \delta^{d}\big(\rvec (x)\big)
\ ,}
with now a pointwise interaction of the
Gaussian manifold with the origin.
Notice that this Hamiltonian also represents
%other geometrical situations, like
interactions of a fluctuating (possibly directed) manifold
with a nonfluctuating $D'$- Euclidean subspace of $\RR^{d+D'}$,
$\rvec $ then standing for the coordinates transverse to this subspace.
The excluded volume case ($b>0$) parallels
that of the Edwards model \Edwards\ for SA-manifolds,
while an attractive interaction ($b<0$) is also possible, describing
pinning phenomena.
The dimensions of $\rvec$ and $b$ are respectively
$[\rvec ]=[x^\nu]$ with a size exponent $\nu\equiv (2-D)/2$, and
$[b]=[x^{-\epsilon}]$ with $\epsilon\equiv D-\nu d$.
For fixed $D$ and $\nu$, the parameter $d$ (or equivalently $\epsilon$)
controls
the relevance of the interaction, with the exclusion of a point
only effective for $d\le d^\star=D/\nu$.

The model is described by its (connected) partition function
$\CZ=\CV^{-1}\int {\cal D}[\rvec]\exp(-\beta{\cal H})$ (here $\CV$
is the internal volume of the manifold)
and, for instance, by its one-point vertex function
$\CZ^{(0)}(\kvec )/\CZ=
\int d^Dx_0 \langle e^{i\kvec\cdot\rvec(x_0)} \rangle $, where the
(connected) average
$\langle \cdots \rangle$ is performed with \HamM .
Those functions are formally defined via their perturbative expansions
in the coupling constant $b$:
$\CZ=\sum_{N=1}^{\infty}{(-b)^N\over N! }\,\CZ_N$ and a similar equation for
$\CZ^{(0)}$ with coefficients $\CZ_N^{(0)}$.
The term of order $N$, $\CZ_N$, is a ($b=0$) Gaussian average
involving $N$ interaction points $x_i$.
This average is expressed
solely in terms of the Green function
$G(x,y)=-{1\over 2}A_D|x\nobreak -\nobreak y|^{2\nu}$,
solution of $-\Delta_x G(x,y)=\delta^D (x-y)$,
with $A_D$ a suitable normalization, hereafter omitted.
In the following, it is important to preserve the condition
$0<\nu <1$ ({\it i.e.} $0<D<2$), corresponding to
the actual case of a  crumpled manifold, and where $(-G)$ is
positive and ultraviolet (UV) finite.
A direct evaluation of $\CZ_N$ then leads to its integral
representation in terms of the normalized
$G_{ij}\equiv - {1\over 2} |x_i-x_j|^{2\nu}$ \BD :
\eqn\ZN{
\CZ_N\ =\,{1\over \CV}\, \int\prod_{i=1}^N d^Dx_i
\,\left(\det\left[ \Pi_{ij}\right]_{\scriptscriptstyle 1\le i,j\le  N-1 }
\right)^{-{d\over 2}}
}
where the matrix $[\Pi_{ij}]$ is simply defined as
$\Pi_{ij}\,\equiv\,G_{ij}-G_{Nj}-G_{iN}$ with a
reference point, $x_N$, the symmetry between the $N$ points being
restored in the determinant.
The integral representation of $\CZ_N^{(0)}$ is obtained from
that of $\CZ_N$ by multiplying the integrand in \ZN\ by
$\exp (-{1\over 2}\kvec^2 \Delta^{(0)} )$ with :
\eqn\forvertex{\Delta^{(0)}\equiv {\det[\Pi_{ij}]_{\scriptscriptstyle
0\le i,j \le N-1} \over
\det[\Pi_{ij}]_{\scriptscriptstyle 1\le i,j\le N-1} } \ ,}
and integrating over one more position, $x_0$.
The resulting expression is quite similar to that
of the manifold Edwards model \DHK .

\medskip
{\bf Analytic continuation in ${\tit D}$ of the Euclidean measure}.
Integrals like \ZN\ are {\it a priori} meaningful only for
integer $D$. Still, an analytic continuation in $D$ can be performed
by use of {\it distance geometry}. The key idea is to substitute
to the internal Euclidean coordinates $x_i$ the set of all mutual (squared)
distances $a_{ij}=(x_i-x_j)^2$. This is possible for integrands invariant
under the group of Euclidean motions (as in \ZN\ and \forvertex ). For
$N$ integration points, it also requires $D$ large enough, {\it i.e.}
$D\ge N-1$, such that $N-1$ relative vectors spanning these points
are linearly independent.
We define the graph $\CG$ as the set $\CG =\{1,\ldots ,N\}$
labelling the interaction points.
Vertices $i\in\CG$ will be remnants of the original Euclidean points
after analytic continuation, and index the distance matrix $[a_{ij}]$.
The change of variables $\{x_i\}_{i\in \CG}
\to a\equiv [a_{ij}]_{{i<j \hfill \atop i,j \in \CG}}$
reads explicitly \DDG :
\eqn\inta{
{1\over \CV}\int_{\RR^D} \prod_{i\in \CG} d^Dx_i\,\cdots \ =\
\int_{{\cal A}_\CG}\, d\mu_\CG^{(D)}(a)\,\cdots
\ ,}
with the measure
\eqn\measaij{
d\mu_\CG^{(D)}(a)\equiv
\prod_{{i<j  \atop i,j \in \CG}}da_{ij}\
\Omega_N^{(D)}\, \Big(P_\CG(a) \Big)^{D-N\over 2}
\ ,}
where $N=|\CG|$,
$\Omega_N^{(D)}\equiv \prod_{K=0}^{N-2} {S_{D-K}\over 2^{K+1}}$
(here $S_D={2\pi^{D/2}\over \Gamma(D/2)}$
is the volume of the unit sphere in $\RR^D$), and
\eqn\CayMen{
P_\CG(a)\equiv {(-1)^N \over 2^{N-1}}\,
\left| \matrix{0&1&1&\ldots&1\cr 1&0&a_{12}&\ldots&a_{1N}\cr
1&a_{12}&0&\ldots&a_{2N}\cr \vdots&\vdots&\vdots&\ddots&\vdots\cr 1&a_{1N}&
a_{2N}&\ldots&0\cr } \right|\ . }
The factor
$\Omega_N^{(D)}$ is the volume of the rotation group of the rigid
simplex spanning the points $x_i$. The ``Cayley-Menger determinant"
\ref\Blum{L. M. Blumenthal, {\sl Theory and Applications of Distance Geometry},
Clarendon Press, Oxford (1953).}
$P_\CG(a)$ is proportional to the squared Euclidean
volume of this simplex, a polynomial of degree $N-1$ in the $a_{ij}$.
The set $a$ of squared distances has to fulfill the triangular
inequalities and their generalizations:
$P_\CK(a)\ge 0$ for all subgraphs $\CK\subset \CG$,
which defines the domain of integration $\CA_\CG$ in \inta .
For real $D>|\CG|-2$, $d\mu_\CG^{(D)}(a)$ is a positive measure on $\CA_\CG$,
analytic in $D$.
It is remarkable that, as a distribution, it can be extended
to $0\le D\le |\CG|-2$ \DDG . For integer $D\le |\CG|-2$,
although the change of
variables from $x_i$ to $a_{ij}$ no longer exists, Eq.\measaij\
still reconstructs the correct measure, concentrated on
$D$-dimensional submanifolds of $\RR^{N-1}$,
{\it i.e.} $P_\CK=0$ if $D\le |\CK|-2$
\DDG .
For example, when $D\to 1$ for $N=3$ vertices, we have,
denoting the distances $|ij|=\sqrt{a_{ij}}$:
$${d\mu^{(D\to 1)}_{\{1,2,3\}}(a)\over
d{\scriptstyle |12|}
d{\scriptstyle |13|}
d{\scriptstyle |23|}}
=2\, \delta\big({\scriptstyle |12|+|23|-|13|}\big)
+\hbox{\ninerm perm}
$$
which indeed describes nested intervals in $\RR$.
\medskip
Another nice feature of this formalism is that
the interaction determinants in \ZN\ and \forvertex\ are
themselves Cayley-Menger determinants. We have indeed
$\det\left[\Pi_{ij}\right]_{1\le i,j \le N-1} = P_\CG(a^\nu)$
where $a^\nu\equiv[a_{ij}^\nu]_{{i<j\hfill \atop i,j \in \CG}}$ is obtained
by simply raising
each squared distance to the power $\nu$.
We arrive at the representation of ``Feynman diagrams" in
distance geometry:
\eqn\ZNa{\eqalign{
&\CZ_N=\int_{\CA_\CG}d\mu^{(D)}_\CG\, I_\CG\ ,\ \ \
I_{\CG} =\big(P_\CG(a^\nu)\big)^{ -{d\over 2}}  \cr
&\CZ_N^{(0)}=
\int_{\CA_{\CG \cup \{0\}}}d\mu^{(D)}_{\CG \cup \{0\}}
\, I_\CG^{(0)} \ ,\cr
&I_{\CG}^{(0)} =
I_\CG \ \exp \left(-{1\over 2}\kvec^2
{P_{\CG\cup\{0\}}(a^\nu)\over P_\CG(a^\nu)}
\right)\hfill ,\cr } }
which are $D$-dimensional extensions of the Schwinger $\alpha$-parameter
representation.
We now have to study the actual
convergence of these integrals and, possibly, their renormalization.
\medskip
{\bf Analysis of divergences}.
Large distance infrared (IR) divergences occur for
manifolds of infinite size. One can keep a
finite size, preserve symmetries and
avoid boundary effects by choosing as a manifold
the $D$-dimensional sphere $\CS_D$ of radius $R$
in $\RR^{D+1}$. This amounts \DDG\ in distance geometry
to substituting to $P_\CG(a)$ the ``spherical" polynomial
$P_\CG^{\CS}(a)\equiv  P_\CG(a)+{1\over R^2}\det(-{1\over 2}a)$,
the second term providing an IR cut-off, such that $a_{ij}\le 4R^2$.
In the following, this regularization
will be simply ignored when dealing with
short distance properties, where $P_\CG^\CS\sim P_\CG$.
\medskip
{\it Schoenberg's theorem}. This result of geometry \Blum\ states that
{\sl for $0<\nu<1$, the set
$a^\nu=[a^\nu_{ij}]_{{ i<j\hfill \atop i,j \in \CG}}$ can be realized as the
set of
squared distances
of a transformed simplex in $\RR^{N-1}$, whose volume
$P_\CG(a^\nu)$ is positive and vanishes if and only if at least one of
the mutual
original distances itself vanishes}, $a_{ij}=0$.
This ensures that, as in field theory, the only source
of divergences in $I_\CG$ and $I_\CG^{(0)}$ is at {\it short distances}.
Whether these UV singularities are integrable or not will depend
on whether the external space dimension $d< d^\star
=D/\nu$ or $d>d^\star$.

{\bf Factorizations}. The key to convergence and
renormalization is the following
short distance {\it factorization} property of $P_\CG(a^\nu)$.
Let us consider a subgraph $\CP\subset\CG$, with at least two vertices,
in which we
distinguish an element, the {\it root} $p$ of $\CP$, and let us denote by
$\CG\setminusp\CP \equiv (\CG\setminus\CP )\cup \{p\}$ the subgraph
obtained by replacing in $\CG$ the whole
subset $\CP$ by its root $p$.
In the original Euclidean formulation, the analysis of short distance
properties amounts to that of contractions of points $x_i$, labeled
by such a subset $\CP$, toward the point $x_p$, according to:
$x_i(\rho )=x_p+\rho (x_i-x_p)$ if $i\in \CP$,
where $\rho\to 0^{+}$ is the dilation factor, and
$x_i(\rho)=x_i$ if $i\notin \CP$.
This transformation has
an immediate correspondent
in terms of mutual distances:
$a_{ij}\ \to a_{ij}(\rho)$, depending on both $\CP$ and $p$.
%$$a_{ij}(\rho)\,=\,\left\{\matrix{
%\rho^2 a_{ij}\hfill\quad&i,j \in \CP\hfill\cr
%a_{pj}-\rho\,(a_{pi}+a_{pj}-a_{ij})+\rho^2\,a_{pi}\ &i
%\in \CP, j \notin \CP \cr
%a_{ij}\hfill\quad& i,j\notin \CP \hfill\ .\cr}\right. $$
Under this transformation, the interaction polynomial
$P_\CG(a^\nu)$ factorizes into \DDG :
\eqn\factdet{\eqalign{
P_\CG(a^\nu (\rho)) & =
P_\CP(a^\nu(\rho ))\,
P_{\CG\setminusp \CP}(a^\nu) \cr & \quad \quad  \times
\left\{1+{\cal O}(\rho^{2\delta})\right\}\
\ .\cr}}
with $ \delta=\min(\nu, 1-\nu)>0 $ and
where, by homogeneity,
$P_\CP(a^\nu(\rho ))=\rho^{2\nu (|\CP |-1)}\, P_\CP(a^\nu)$.
\midinsert
\figinsert{6.5truecm}{7.truecm}{%\hsize}{
\figcap\contf{Factorization property \factdet . }}
\special{psfile=fig.ps}
\endinsert
\noindent The geometrical interpretation of \factdet\ is quite
simple:
the contribution of the set $\CG$ splits into that of
the contracting subgraph $\CP$ multiplied by that of the whole set $\CG$ where
$\CP$ has been replaced by its root $p$ (\contf ),
all correlation distances between these subsets being suppressed.
This is just, in this model,
the rigorous expression of an {\it operator
product expansion} \DDG .
%which we may heuristically write:
%$$\prod_{i\in {\cal P}}\delta^d(\rvec(x_i))\ \buildrel {x_i\to x_p} \over
%\sim  \big({\hbox {d}}({\cal P})\big)^{-d\nu (|\CP |-1)}
%\,\delta^d(\rvec(x_p))
%\ ,$$
%where ${\hbox {d}}(\CP)$ is the diameter of $\CP$, {\it i.e.}
%its typical linear size.

The factorization property \factdet\ does not hold for $\nu = 1$,
preventing a factorization of the measure \measaij\
$d\mu^{(D)}_{\CG }(a)$ itself.
Still, the integral of the measure, when applied to a factorized
integrand, factorizes as:
\eqn\factint{\int_{\CA_\CG}d\mu^{(D)}_\CG\cdots=\int_{\CA_\CP}
d\mu^{(D)}_\CP\cdots\int_{\CA_{(\CG\ssetminusp \CP)}}
d\mu^{(D)}_{(\CG\setminusp\CP)}\cdots \ .}
This fact, explicit for integer $D$ with a
readily factorized measure $\prod_i d^Dx_i$,
is preserved \DDG\ by analytic
continuation only after integration over relative distances between the
two ``complementary" subsets $\CP$ and $\CG\setminusp \CP$.
\medskip
{\bf Renormalization}.
A first consequence of factorizations \factdet\ and \factint\ is
the absolute convergence of $\CZ_N$ and $\CZ_N^{(0)}$ for $\epsilon >0$.
Indeed, the superficial degree of divergence of $\CZ_N$ (in distance units)
is $(N-1)\epsilon$, as can be read from \ZNa , already ensuring the
superficial convergence when $\epsilon>0$.
The above factorizations ensure that the superficial degree of
divergence in $\CZ_N$ or $\CZ_N^{(0)}$ of any subgraph $\CP$ of $\CG$
is exactly that of $\CZ_{|\CP |} $ itself, {\it i.e.} $(|\CP |-1)\epsilon >0$.
By recursion, this ensures the absolute convergence of the manifold
Feynman integrals. A complete discussion has recourse to a generalized
notion of Hepp sectors and is given elsewhere \DDG . In the proof,
it is convenient to first consider $D$ large enough
where $d\mu_{\CG}^{(D)}$ is a non singular measure,
with a fixed $\nu$ considered as an independent variable $0<\nu<1$,
and to then continue to $D=2-2\nu$, $0<D<2$,
corresponding to the physical case.

When $\epsilon =0$, the integrals giving $\CZ_N$ and $\CZ_N^{(0)}$
are (logarithmically) divergent.
Another main consequence of Eqs. \factdet\ and \factint\ is then
the possibility
to devise a renormalization operation {\bf R},
as follows.
To each contracting rooted subgraph $(\CP,p)$ of $\CG$,
we associate a Taylor operator $\Tay_{(\CP,p)}$, performing
on interaction integrands the exact factorization corresponding to \factdet :
\eqn\TayI{\Tay_{(\CP,p)}I_\CG^{(0)}=I_{\CP}\,I_{\CG\setminusp \CP}^{(0)}
\ ,}
and similarly
$\Tay_{(\CP,p)}I_\CG=I_{\CP}\,I_{\CG\setminusp \CP}$.
As in standard field theory \BPHZ ,
the subtraction renormalization operator {\bf R}
is then organized in terms of forests \`a la Zimmermann.
In manifold theory, we define
a {\it rooted forest} as a set of rooted subgraphs $(\CP,p)$ such
that any two subgraphs are either disjoint or nested, {\it i.e.} never
partially overlap. Each of these subgraphs in the forest
will be contracted toward its root under the action \TayI\ of the
corresponding Taylor operator.
When two subgraphs $\CP\subset \CP'$ are nested, the smallest one
is contracted first toward its root $p$, the root
$p'$ of $\CP'$ being itself attracted toward $p$
if $p'$ happened to be in $\CP$.
This hierarchical structure is anticipated by choosing the roots of the
forest as {\it compatible}: in the case described above, if $p'\in \CP$,
then $p'\equiv p$.
Finally, the renormalization operator is written
as a sum over all such compatibly rooted forests of $\CG$, denoted by
$\CF_\oplus$:
\eqn\Roper{
{\bf R}\ =\
\sum_{\CF_{\oplus}}\,W(\CF_{\oplus})\,
\Bigg[ \prod_{(\CP,p)\in \CF_{\oplus}}
\!\big( -\Tay_{(\CP,p)} \big)\Bigg]
\ .}
Here $W$ is a necessary combinatorial weight associated with the degeneracy
of compatible rootings, $W(\CF_{\oplus})\,
=\, \prod_{ {p\ {\rm root}}\atop {{\rm of}\,\CF_{\oplus}} }
1 /|\CP(p)| $ with $\CP(p)$ being the largest subgraph of the forest
$\CF_\oplus$ whose root is $p$.
An important property is that, with compatible roots, the Taylor operators
of a given forest now commute \DDG .
The renormalized amplitudes are defined as
\eqn\ZMRen{
{\CZ^{\bf R}}_N^{(0)}\ \equiv \
\int_{\CA_{\CG \cup \{0\}}} d\mu^{(D)}_{\CG\cup\{0\}}\,{\bf R}\,[I_\CG^{(0)}]
\ .}
The same operation ${\bf R}$
acting on $I_\CG$ leads automatically by homogeneity to
$ {\bf R}\left[I_{\CG}\right]=0 $ for $|\CG|\ge 2$.
We state the essential result that now {\it the renormalized Feynman
integral} \ZMRen\ {\it is convergent}: ${\CZ^{{\bf R}}}_N^{(0)} < \infty$
for $\epsilon=0$.
A complete proof of this renormalizability property goes
well beyond the scope of this Letter and is given elsewhere \DDG .
the analysis being
inspired from the direct proof by Berg\`ere and Lam of the
renormalizability in field theory of Feynman amplitudes
in the $\alpha$-representation
\ref\BergLam{M. C. Berg\`ere and Y.-M. P. Lam,
J. Math. Phys. {\bf 17} (1976) 1546.}.
\medskip
The physical interpretation of the renormalized amplitude \ZMRen\ and
of \Roper\ is now fairly simple.
Eqs.\factint\ and \TayI\ show that
the substitution to the bare amplitudes \ZNa\ of the renormalized ones
\ZMRen\ amounts to a reorganization to all orders of the original
perturbation series in $b$,
leading to the remarkable identity:
\eqn\RenExp{
\CZ^{(0)}\ =\ \sum_{N=1}^\infty \,{(\CZ)^N\over N!}\,
{\CZ^{\bf R}}_N^{(0)}
\ .}
This actually extends
to any vertex function, showing that the theory is
made perturbatively finite (at $\epsilon = 0$)
by a simple renormalization of the coupling constant $b$
into $\CZ$ itself.  From this result,
one establishes the existence of a Wilson function
${\scriptstyle \CV} {\partial \CZ \over \partial \CV} \big|_b$,
describing the scaling properties of the interacting manifold
for $\epsilon$ close to zero \DDG .
For $\epsilon >0$, an IR
fixed point at $b>0$ yields universal excluded volume exponents;
for $\epsilon <0$, the associated UV fixed point at $b<0$ describes
a localization transition.

In summary, we have shown how to define an interacting manifold model with
continuous internal dimension, by use of distance geometry, a natural
extension of Schwinger representation of field theories. Furthermore,
in the case of a pointwise interaction, we have shown that the
manifold model is indeed renormalizable to all orders.
The main ingredients are the Schoenberg's theorem
of distance geometry, insuring that divergences occur only at short
distances for (finite) manifolds, and the short-distance factorization
of the generalized Feynman amplitudes. The renormalization operator
is a combination
of Taylor operators associated with rooted diagrams,
a specific feature of manifold models.
This is probably the
first example of a perturbative renormalization
established for extended geometrical objects.
This opens the way to a similar study of self-avoiding manifolds,
as well as to other generalizations of field theories.
\medskip
We thank M. Berg\`ere for helpful discussions.

\vfill\eject
\listrefs
%\listfigs
\bye
%%%%%%%%%%%%%%%%%%%%%% END OF TEX FILE %%%%%%%%%%%%%%%%%%%%%%%%
%!PS-Adobe-2.0
%%Title: /home/wasa2/guitter/publi/renorm/fig2bis.ps
%%Creator: IslandDraw for guitter
%%CreationDate: Tue Dec 15 12:14:23 1992
%%Pages: 1
%%BoundingBox: 0 0 612 792
%%EndComments
save
/fichier2 exch def

%% catch nocurrentpoint error for:
%% pathbbox
    /ncpoint errordict /nocurrentpoint get def
    errordict begin
    /nocurrentpoint {
	dup /pathbbox load eq
	{ pop 0 0 1 1 }
	{ ncpoint }
	ifelse
    } bind def
    end

    /image_raster { %% sw sh dw dh xs ys
	translate scale /sh exch def /sw exch def
	/imagebuf sw 7 add 8 idiv string def
	sw sh 1 [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop }
	image
    } bind def
    /m {moveto} bind def
    /l {lineto} bind def
    /c {curveto} bind def
    /n {newpath} bind def
    /cl {closepath} bind def
    /ar { %% sa ea sx sy rot tx ty
	matrix currentmatrix 8 1 roll translate rotate scale
	n 0 0 1 5 3 roll arc setmatrix
    } bind def
    /arn { %% sa ea sx sy rot tx ty
	matrix currentmatrix 8 1 roll translate rotate scale
	n 0 0 1 5 3 roll arcn setmatrix
    } bind def
    /el { %% sx sy rot tx ty
	matrix currentmatrix 6 1 roll translate rotate scale
	n 0 0 1 0 360 arc setmatrix cl
    } bind def
    /bp {setlinejoin setlinewidth setgray} bind def
    /lw {setlinewidth} bind def
    /lj {setlinejoin} bind def
    /gr {setgray} bind def
%% pattern stuff
    /BPSIDE 32 def	%% pixels per pattern side
    /PATFREQ 3.0 def	%% pattern pixels per mm
    /dp_mat [PATFREQ 0 0 PATFREQ 0 0] def
    /dp_pw BPSIDE def	%% pattern pixel width
    /dp_ph BPSIDE def	%% pattern pixel height
    /dp_w dp_pw PATFREQ div def	%% pattern mm width
    /dp_h dp_ph PATFREQ div def	%% pattern mm height
    /dp_bs 1 def		%% pattern bits per pixel

    /savemat matrix def
    /topmat matrix def
    /patmat matrix def

    /patpath {
	topmat setmatrix
	pathbbox	%% get lo - hi indecies
	/hy exch dp_h div floor cvi def
	/hx exch dp_w div floor cvi def
	/ly exch dp_h div floor cvi def
	/lx exch dp_w div floor cvi def
	lx 1 hx {
	    dp_w mul
	    ly 1 hy {
		dp_h mul
		exch dup 3 1 roll exch
		patmat currentmatrix pop
		translate
		dp_pw dp_ph dp_bs
		dp_mat dp_proc image
		patmat setmatrix
	    } for
	    pop
	} for
    } bind def
% setpattern  brush of patterns instead of gray
   /setpattern {
         /freq    exch def
         /bwidth  exch def
         /bpside  exch def
         /bstring exch def
         /onbits 0 def  /offbits 0 def
         freq 0 {/y exch def
                 /x exch def
                 /xindex x 1 add 2 div bpside mul cvi def
                 /yindex y 1 add 2 div bpside mul cvi def
                 bstring yindex bwidth mul xindex 8 idiv add get not
                 1 7 xindex 8 mod sub bitshift and 0 ne
                 {/onbits  onbits  1 add def 1}
                 {/offbits offbits 1 add def 0}
                 ifelse
                }
                setscreen
         {} settransfer
         offbits offbits onbits add div setgray
        } bind def
    /dmatrix matrix def
    /dpi    72 0 dmatrix defaultmatrix dtransform
        dup mul exch   dup mul add   sqrt
    def

    /B {gsave bp stroke grestore} bind def %% brush: gr lw lj
    /F {gsave setgray eofill grestore} bind def %% fill: gr
    /PB {gsave setlinejoin setlinewidth setpattern stroke grestore} bind def
    /PF {gsave eoclip patpath grestore} bind def
    /BB {gsave strokepath clip patpath grestore} bind def
%! IslandDraw text prolog Version 1.0
%%
/BLACK { 0.0 } bind def
/CP {closepath} bind def
/FI {eofill} bind def
/E {exch} bind def
/FF {findfont} bind def
/GR {grestore} bind def
/GS {gsave} bind def
/MF {makefont} bind def
/NP {newpath} bind def
/RO {rotate} bind def
/ST {stroke} bind def
/SC {scale} bind def
/SF {setfont} bind def
/SG {setgray} bind def
/SLC {setlinecap} bind def
/SLJ {setlinejoin} bind def
/SLW {setlinewidth} bind def
/TR {translate} bind def
/WHITE { 1.0 } bind def
/m {moveto} bind def
/r {rmoveto} bind def
/l {lineto} bind def
/sp {x 0 rmoveto} bind def
/rl {rlineto} bind def
/s {show} bind def
/box { NP m l l l CP } bind def
/pageboundary { NP m l l l CP } bind def
/BS {   % black stroke
GS SLJ SLW BLACK SG ST GR
} bind def
/WS {   % white stroke
GS SLJ SLW WHITE SG ST GR
} bind def
/reencode_small_dict 12 dict def
/ReencodeSmall {
reencode_small_dict begin
/new_codes_and_names E def
/new_font_name E def
/base_font_name E def
/base_font_dict base_font_name FF def
/newfont base_font_dict maxlength dict def
base_font_dict {
E dup /FID ne
{ dup /Encoding eq
{ E dup length array copy newfont 3 1 roll put }
{ E newfont 3 1 roll put }
ifelse
}
{ pop pop }
ifelse
} forall
newfont /FontName new_font_name put
new_codes_and_names aload pop
new_codes_and_names length 2 idiv
{ newfont /Encoding get 3 1 roll put }
repeat
new_font_name newfont definefont pop
end     %reencode_small_dict
} def
/extended_Zapf [
8#223 /a89
8#224 /a90
8#225 /a93
8#226 /a94
8#227 /a91
8#230 /a92
8#231 /a205
8#232 /a85
8#233 /a206
8#234 /a86
8#235 /a87
8#236 /a88
8#237 /a95
8#240 /a96
] def
/extended_Standard [
128 /Acircumflex
129 /Adieresis
130 /Agrave
131 /Aring
132 /Atilde
133 /Ccedilla
134 /Eacute
135 /Ecircumflex
136 /Edieresis
137 /Egrave
138 /Iacute
139 /Icircumflex
140 /Idieresis
141 /Igrave
142 /Ntilde
143 /Oacute
144 /Ocircumflex
145 /Odieresis
146 /Ograve
147 /Otilde
148 /Scaron
149 /Uacute
150 /Ucircumflex
151 /Udieresis
152 /Ugrave
153 /Ydieresis
154 /Zcaron
155 /aacute
156 /acircumflex
157 /adieresis
158 /agrave
159 /aring
160 /atilde
209 /ccedilla
210 /copyright
211 /eacute
212 /ecircumflex
213 /edieresis
214 /egrave
215 /iacute
216 /icircumflex
217 /idieresis
218 /igrave
219 /logicalnot
220 /minus
221 /ntilde
222 /oacute
223 /ocircumflex
224 /odieresis
228 /ograve
229 /otilde
230 /registered
231 /scaron
236 /trademark
237 /uacute
238 /ucircumflex
239 /udieresis
240 /ugrave
242 /ydieresis
243 /zcaron
244 /Aacute
] def
/extended_Symbol [
] def
/extend_font {  % stack: fontname newfontname
E dup (ZapfDingbats) eq
{ cvn E cvn extended_Zapf ReencodeSmall }
{ dup (Symbol) eq
{ cvn E cvn extended_Symbol ReencodeSmall }
{ cvn E cvn extended_Standard ReencodeSmall }
ifelse
}
ifelse
} bind def
/getfont {
/f E def f cvn where
{ begin f cvx cvn exec SF end }
% { f 0 f length 8 sub getinterval (LocalFont) extend_font
%/LocalFont FF
{ f 0 f length 8 sub getinterval dup
dup length 1 add string /localfont exch def
localfont exch 0 exch putinterval
localfont dup length 1 sub (X) putinterval
localfont extend_font
localfont FF
/xsz f f length 4 sub 4 getinterval cvi def
/ysz f f length 8 sub 4 getinterval cvi def
[ xsz 0 0 ysz neg 0 0 ] MF dup f cvn E def
SF
}
ifelse
} bind def
/ul { % space drop thickness
GS currentpoint currentlinewidth
currentpoint NP m 6 -3 roll
SLW 0 E r
0 rl ST SLW m
GR
} bind def
/ss { currentpoint pop E m } bind def
/image_raster { % sw sh dw dh xs ys
TR SC /sh E def /sw E def
/imagebuf sw 7 add 8 idiv string def
sw sh 1 [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop }
image
} bind def
/nx { /x E def } bind def
0. nx
%%EndFixedProlog
%%EndProlog

%%Page: 1 1
gsave
1.5 -1.5 scale 30 -120 translate
topmat currentmatrix pop
% Ellipse
n 105.88 37.4039 m 105.892 40.1875 104.69 42.952 102.613 44.912 c
100.555 46.8886 97.6511 48.0337 94.7273 48.0219 c
91.8035 48.0337 88.8998 46.8886 86.8411 44.912 c
84.7649 42.952 83.5622 40.1875 83.5745 37.4039 c
83.5622 34.6203 84.7649 31.8558 86.8411 29.8958 c
88.8998 27.9191 91.8035 26.7741 94.7273 26.7858 c
97.6511 26.7741 100.555 27.9191 102.613 29.8958 c
104.69 31.8558 105.892 34.6203 105.88 37.4039 c
cl /pat0 {<DDDDDDDDFFFFFFFF77777777FFFFFFFFDDDDDDDDFFFFFFFF77777777FFFFFFF
FDDDDDDDDFFFFFFFF77777777FFFFFFFFDDDDDDDDFFFFFFFF77777777FFFFFFFFDDDDDDD
DFFFFFFFF77777777FFFFFFFFDDDDDDDDFFFFFFFF77777777FFFFFFFFDDDDDDDDFFFFFFF
F77777777FFFFFFFFDDDDDDDDFFFFFFFF77777777FFFFFFFF>} def
/dp_proc pat0 def
PF
/pat1 {<000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000>} def
pat1 32 4 dpi 64 div .352778 0 PB
% Ellipse
n 90.7933 33.7372 m 90.794 33.9024 90.724 34.0665 90.6031 34.1828 c
90.4832 34.3002 90.3142 34.3681 90.144 34.3674 c
89.9737 34.3681 89.8047 34.3002 89.6848 34.1828 c
89.564 34.0665 89.4939 33.9024 89.4947 33.7372 c
89.4939 33.572 89.564 33.4079 89.6848 33.2916 c
89.8047 33.1743 89.9737 33.1063 90.144 33.107 c
90.3142 33.1063 90.4832 33.1743 90.6031 33.2916 c
90.724 33.4079 90.794 33.572 90.7933 33.7372 c
cl 0 F
0 .352778 0 B
% Ellipse
n 98.5433 30.9456 m 98.544 31.1108 98.474 31.2748 98.3531 31.3912 c
98.2332 31.5085 98.0642 31.5765 97.894 31.5758 c
97.7237 31.5765 97.5547 31.5085 97.4348 31.3912 c
97.314 31.2748 97.2439 31.1108 97.2447 30.9456 c
97.2439 30.7803 97.314 30.6163 97.4348 30.4999 c
97.5547 30.3826 97.7237 30.3146 97.894 30.3153 c
98.0642 30.3146 98.2332 30.3826 98.3531 30.4999 c
98.474 30.6163 98.544 30.7803 98.5433 30.9456 c
cl 0 F
0 .352778 0 B
% Ellipse
n 101.557 37.7789 m 101.558 37.9441 101.488 38.1082 101.367 38.2245 c
101.247 38.3418 101.078 38.4098 100.908 38.4091 c
100.738 38.4098 100.569 38.3418 100.449 38.2245 c
100.328 38.1082 100.258 37.9441 100.259 37.7789 c
100.258 37.6137 100.328 37.4496 100.449 37.3333 c
100.569 37.2159 100.738 37.148 100.908 37.1487 c
101.078 37.148 101.247 37.2159 101.367 37.3333 c
101.488 37.4496 101.558 37.6137 101.557 37.7789 c
cl 0 F
0 .352778 0 B
% Ellipse
n 87.4599 39.7233 m 87.4607 39.8885 87.3906 40.0526 87.2698 40.169 c
87.1499 40.2863 86.9808 40.3542 86.8106 40.3535 c
86.6404 40.3542 86.4714 40.2863 86.3515 40.169 c
86.2306 40.0526 86.1606 39.8885 86.1613 39.7233 c
86.1606 39.5581 86.2306 39.394 86.3515 39.2777 c
86.4714 39.1604 86.6404 39.0924 86.8106 39.0931 c
86.9808 39.0924 87.1499 39.1604 87.2698 39.2777 c
87.3906 39.394 87.4607 39.5581 87.4599 39.7233 c
cl 0 F
0 .352778 0 B
% Ellipse
n 98.4183 44.1122 m 98.419 44.2774 98.349 44.4415 98.2281 44.5578 c
98.1082 44.6752 97.9392 44.7431 97.769 44.7424 c
97.5987 44.7431 97.4297 44.6752 97.3098 44.5578 c
97.189 44.4415 97.1189 44.2774 97.1197 44.1122 c
97.1189 43.947 97.189 43.7829 97.3098 43.6666 c
97.4297 43.5493 97.5987 43.4813 97.769 43.482 c
97.9392 43.4813 98.1082 43.5493 98.2281 43.6666 c
98.349 43.7829 98.419 43.947 98.4183 44.1122 c
cl 0 F
0 .352778 0 B
% Ellipse
n 81.4063 27.4178 m 81.4075 27.6966 81.2894 27.9735 81.0854 28.1698 c
80.8832 28.3677 80.5979 28.4824 80.3106 28.4812 c
80.0234 28.4824 79.7381 28.3677 79.5358 28.1698 c
79.3319 27.9735 79.2137 27.6966 79.2149 27.4178 c
79.2137 27.139 79.3319 26.8621 79.5358 26.6658 c
79.7381 26.4678 80.0234 26.3531 80.3106 26.3543 c
80.5979 26.3531 80.8832 26.4678 81.0854 26.6658 c
81.2894 26.8621 81.4075 27.139 81.4063 27.4178 c
cl 0 F
0 .352778 0 B
% Ellipse
n 81.3647 45.4039 m 81.3659 45.6827 81.2477 45.9596 81.0437 46.1559 c
80.8415 46.3539 80.5562 46.4685 80.269 46.4674 c
79.9817 46.4685 79.6964 46.3539 79.4942 46.1559 c
79.2902 45.9596 79.172 45.6827 79.1733 45.4039 c
79.172 45.1251 79.2902 44.8482 79.4942 44.6519 c
79.6964 44.4539 79.9817 44.3392 80.269 44.3404 c
80.5562 44.3392 80.8415 44.4539 81.0437 44.6519 c
81.2477 44.8482 81.3659 45.1251 81.3647 45.4039 c
cl 0 F
0 .352778 0 B
% Ellipse
n 111.573 36.3483 m 111.574 36.6271 111.456 36.904 111.252 37.1003 c
111.05 37.2983 110.765 37.413 110.477 37.4118 c
110.19 37.413 109.905 37.2983 109.703 37.1003 c
109.499 36.904 109.38 36.6271 109.382 36.3483 c
109.38 36.0695 109.499 35.7926 109.703 35.5963 c
109.905 35.3984 110.19 35.2837 110.477 35.2849 c
110.765 35.2837 111.05 35.3984 111.252 35.5963 c
111.456 35.7926 111.574 36.0695 111.573 36.3483 c
cl 0 F
0 .352778 0 B
% Ellipse
n 108.17 46.39 m 108.171 46.6688 108.053 46.9457 107.849 47.142 c
107.647 47.34 107.362 47.4546 107.075 47.4535 c
106.787 47.4546 106.502 47.34 106.3 47.142 c
106.096 46.9457 105.978 46.6688 105.979 46.39 c
105.978 46.1112 106.096 45.8343 106.3 45.638 c
106.502 45.44 106.787 45.3253 107.075 45.3265 c
107.362 45.3253 107.647 45.44 107.849 45.638 c
108.053 45.8343 108.171 46.1112 108.17 46.39 c
cl 0 F
0 .352778 0 B
% Ellipse
n 95.6008 37.64 m 95.602 37.9188 95.4838 38.1957 95.2799 38.392 c
95.0776 38.59 94.7923 38.7046 94.5051 38.7035 c
94.2178 38.7046 93.9326 38.59 93.7303 38.392 c
93.5263 38.1957 93.4082 37.9188 93.4094 37.64 c
93.4082 37.3612 93.5263 37.0843 93.7303 36.888 c
93.9326 36.69 94.2178 36.5753 94.5051 36.5765 c
94.7923 36.5753 95.0776 36.69 95.2799 36.888 c
95.4838 37.0843 95.602 37.3612 95.6008 37.64 c
cl 0 F
0 .352778 0 B
% Ellipse
n 117.491 36.7164 m 117.517 42.1234 115.045 47.4933 110.779 51.3005 c
106.549 55.14 100.582 57.3642 94.5745 57.3414 c
88.5667 57.3642 82.6002 55.14 78.37 51.3005 c
74.1039 47.4933 71.6325 42.1234 71.6579 36.7164 c
71.6325 31.3093 74.1039 25.9395 78.37 22.1323 c
82.6002 18.2928 88.5667 16.0685 94.5745 16.0914 c
100.582 16.0685 106.549 18.2928 110.779 22.1323 c
115.045 25.9395 117.517 31.3093 117.491 36.7164 c
cl 0 .705556 0 B
% Ellipse
n 87.6579 76.7789 m 87.6702 79.5625 86.4675 82.327 84.3913 84.287 c
82.3326 86.2636 79.4289 87.4087 76.5051 87.3969 c
73.5813 87.4087 70.6776 86.2636 68.6189 84.287 c
66.5427 82.327 65.3399 79.5625 65.3523 76.7789 c
65.3399 73.9953 66.5427 71.2308 68.6189 69.2708 c
70.6776 67.2941 73.5813 66.1491 76.5051 66.1608 c
79.4289 66.1491 82.3326 67.2941 84.3913 69.2708 c
86.4675 71.2308 87.6702 73.9953 87.6579 76.7789 c
cl 0 .352778 0 B
% Ellipse
n 72.8957 73.265 m 72.8968 73.5128 72.7917 73.7589 72.6104 73.9334 c
72.4306 74.1094 72.1771 74.2114 71.9217 74.2103 c
71.6664 74.2114 71.4128 74.1094 71.233 73.9334 c
71.0517 73.7589 70.9467 73.5128 70.9478 73.265 c
70.9467 73.0172 71.0517 72.7711 71.233 72.5966 c
71.4128 72.4206 71.6664 72.3186 71.9217 72.3197 c
72.1771 72.3186 72.4306 72.4206 72.6104 72.5966 c
72.7917 72.7711 72.8968 73.0172 72.8957 73.265 c
cl 0 F
0 .352778 0 B
% Ellipse
n 80.6457 70.4733 m 80.6468 70.7212 80.5417 70.9673 80.3604 71.1418 c
80.1806 71.3177 79.9271 71.4197 79.6717 71.4186 c
79.4164 71.4197 79.1628 71.3177 78.983 71.1418 c
78.8017 70.9673 78.6967 70.7212 78.6978 70.4733 c
78.6967 70.2255 78.8017 69.9794 78.983 69.8049 c
79.1628 69.6289 79.4164 69.527 79.6717 69.528 c
79.9271 69.527 80.1806 69.6289 80.3604 69.8049 c
80.5417 69.9794 80.6468 70.2255 80.6457 70.4733 c
cl 0 F
0 .352778 0 B
% Ellipse
n 83.6596 77.3067 m 83.6607 77.5545 83.5556 77.8006 83.3743 77.9751 c
83.1945 78.1511 82.941 78.253 82.6856 78.252 c
82.4303 78.253 82.1767 78.1511 81.9969 77.9751 c
81.8156 77.8006 81.7106 77.5545 81.7117 77.3067 c
81.7106 77.0588 81.8156 76.8127 81.9969 76.6382 c
82.1767 76.4622 82.4303 76.3603 82.6856 76.3614 c
82.941 76.3603 83.1945 76.4622 83.3743 76.6382 c
83.5556 76.8127 83.6607 77.0588 83.6596 77.3067 c
cl 0 F
0 .352778 0 B
% Ellipse
n 69.2568 79.0983 m 69.2579 79.3462 69.1529 79.5923 68.9715 79.7668 c
68.7918 79.9427 68.5382 80.0447 68.2829 80.0436 c
68.0275 80.0447 67.7739 79.9427 67.5942 79.7668 c
67.4128 79.5923 67.3078 79.3462 67.3089 79.0983 c
67.3078 78.8505 67.4128 78.6044 67.5942 78.4299 c
67.7739 78.2539 68.0275 78.152 68.2829 78.153 c
68.5382 78.152 68.7918 78.2539 68.9715 78.4299 c
69.1529 78.6044 69.2579 78.8505 69.2568 79.0983 c
cl 0 F
0 .352778 0 B
% Ellipse
n 80.5207 83.64 m 80.5218 83.8878 80.4167 84.1339 80.2354 84.3084 c
80.0556 84.4844 79.8021 84.5864 79.5467 84.5853 c
79.2914 84.5864 79.0378 84.4844 78.858 84.3084 c
78.6767 84.1339 78.5717 83.8878 78.5728 83.64 c
78.5717 83.3922 78.6767 83.1461 78.858 82.9716 c
79.0378 82.7956 79.2914 82.6936 79.5467 82.6947 c
79.8021 82.6936 80.0556 82.7956 80.2354 82.9716 c
80.4167 83.1461 80.5218 83.3922 80.5207 83.64 c
cl 0 F
0 .352778 0 B
% Ellipse
n 77.0485 77.1122 m 77.0496 77.36 76.9445 77.6062 76.7632 77.7807 c
76.5834 77.9566 76.3298 78.0586 76.0745 78.0575 c
75.8192 78.0586 75.5656 77.9566 75.3858 77.7807 c
75.2045 77.6062 75.0995 77.36 75.1006 77.1122 c
75.0995 76.8644 75.2045 76.6183 75.3858 76.4438 c
75.5656 76.2678 75.8192 76.1659 76.0745 76.1669 c
76.3298 76.1659 76.5834 76.2678 76.7632 76.4438 c
76.9445 76.6183 77.0496 76.8644 77.0485 77.1122 c
cl 0 F
0 .352778 0 B
% Text
n savemat currentmatrix pop [.5 0 0 .5 72.2273 82.8136] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic12001200) getfont () s
0.00 SG  (p) s
savemat setmatrix
% Ellipse
n 135.019 78.5899 m
135.017 78.7203 l
135.013 78.8507 l
135.007 78.9809 l
135 79.1111 l
134.991 79.2412 l
134.98 79.3711 l
134.967 79.5009 l
134.958 79.5876 l
0 .352778 0 B
n 134.796 80.5741 m
134.778 80.6589 l
134.749 80.7861 l
134.718 80.913 l
134.685 81.0395 l
134.651 81.1656 l
134.615 81.2914 l
134.577 81.4167 l
134.539 81.5399 l
0 .352778 0 B
n 134.189 82.4762 m
134.116 82.6424 l
134.061 82.7619 l
134.005 82.8807 l
133.947 82.9989 l
133.888 83.1163 l
133.827 83.2331 l
133.765 83.3492 l
0 .352778 0 B
n 133.23 84.2282 m
133.14 84.359 l
133.064 84.4669 l
132.986 84.574 l
132.907 84.6802 l
132.826 84.7854 l
132.744 84.8896 l
132.66 84.9929 l
0 .352778 0 B
n 131.958 85.7675 m
131.752 85.9675 l
131.557 86.1504 l
131.457 86.24 l
131.357 86.3283 l
131.255 86.4154 l
0 .352778 0 B
n 130.426 87.0485 m
130.289 87.1419 l
130.176 87.2161 l
130.063 87.289 l
129.948 87.3606 l
129.833 87.4308 l
129.717 87.4996 l
129.6 87.5671 l
0 .352778 0 B
n 128.688 88.0327 m
128.634 88.0571 l
128.51 88.1121 l
128.385 88.1655 l
128.26 88.2176 l
128.134 88.2682 l
128.007 88.3173 l
127.88 88.365 l
127.761 88.4077 l
0 .352778 0 B
n 126.806 88.7018 m
126.709 88.7269 l
126.576 88.7595 l
126.443 88.7906 l
126.31 88.8201 l
126.176 88.8481 l
126.042 88.8745 l
125.908 88.8993 l
125.829 88.913 l
0 .352778 0 B
n 124.837 89.0391 m
124.687 89.0506 l
124.551 89.0592 l
124.414 89.0662 l
124.277 89.0716 l
124.14 89.0752 l
124.003 89.0772 l
123.866 89.0775 l
0 .352778 0 B
n 122.839 89.0342 m
122.772 89.0283 l
122.636 89.0147 l
122.5 88.9995 l
122.365 88.9827 l
122.229 88.9642 l
122.094 88.9442 l
121.959 88.9225 l
121.848 88.9034 l
0 .352778 0 B
n 120.872 88.6874 m
120.76 88.6571 l
120.629 88.6198 l
120.498 88.5811 l
120.368 88.5409 l
120.238 88.4991 l
120.109 88.4559 l
119.981 88.4112 l
119.918 88.3885 l
0 .352778 0 B
n 118.993 88.0092 m
118.852 87.943 l
118.73 87.8839 l
118.609 87.8233 l
118.489 87.7613 l
118.369 87.698 l
118.25 87.6332 l
118.132 87.5671 l
0 .352778 0 B
n 117.26 87.0162 m
117.112 86.9114 l
117.004 86.8319 l
116.896 86.7512 l
116.79 86.6692 l
116.685 86.5858 l
116.581 86.5013 l
116.478 86.4154 l
0 .352778 0 B
n 115.735 85.7271 m
115.601 85.5908 l
115.419 85.3957 l
115.243 85.1964 l
115.157 85.0951 l
115.072 84.9929 l
0 .352778 0 B
n 114.471 84.1812 m
114.371 84.0298 l
114.3 83.9183 l
114.23 83.8061 l
114.162 83.693 l
114.096 83.5792 l
114.031 83.4646 l
113.967 83.3492 l
0 .352778 0 B
n 113.522 82.4242 m
113.462 82.2804 l
113.413 82.1586 l
113.366 82.0362 l
113.321 81.9133 l
113.277 81.7898 l
113.235 81.6659 l
113.194 81.5415 l
113.177 81.4859 l
0 .352778 0 B
n 112.924 80.5188 m
112.901 80.4034 l
112.876 80.2752 l
112.853 80.1468 l
112.832 80.018 l
112.813 79.8891 l
112.795 79.7599 l
112.779 79.6305 l
112.768 79.5314 l
0 .352778 0 B
n 112.714 78.5334 m
112.713 78.4594 l
112.714 78.329 l
112.716 78.1986 l
112.72 78.0682 l
112.725 77.9379 l
112.733 77.8077 l
112.742 77.6777 l
112.753 77.5478 l
0 .352778 0 B
n 112.895 76.545 m
112.927 76.3876 l
112.954 76.26 l
112.984 76.1328 l
113.015 76.0059 l
113.047 75.8794 l
113.082 75.7532 l
113.118 75.6275 l
113.133 75.5743 l
0 .352778 0 B
n 113.465 74.6312 m
113.512 74.5172 l
113.563 74.3965 l
113.617 74.2765 l
113.671 74.157 l
113.727 74.0382 l
113.785 73.92 l
113.844 73.8025 l
113.885 73.7242 l
0 .352778 0 B
n 114.389 72.861 m
114.443 72.7785 l
114.517 72.6688 l
114.592 72.5599 l
114.668 72.4519 l
114.746 72.3449 l
114.826 72.2387 l
114.906 72.1335 l
114.973 72.0494 l
0 .352778 0 B
n 115.631 71.2973 m
115.788 71.1374 l
115.98 70.9513 l
116.175 70.7685 l
116.275 70.6789 l
116.355 70.6083 l
0 .352778 0 B
n 117.139 69.9881 m
117.222 69.9293 l
117.332 69.8525 l
117.444 69.7769 l
117.556 69.7027 l
117.67 69.6299 l
117.784 69.5583 l
117.899 69.4881 l
117.977 69.4421 l
0 .352778 0 B
n 118.859 68.9724 m
118.975 68.9181 l
119.098 68.8617 l
119.222 68.8068 l
119.347 68.7533 l
119.473 68.7013 l
119.599 68.6507 l
119.725 68.6016 l
119.779 68.5814 l
0 .352778 0 B
n 120.729 68.2705 m
120.892 68.2261 l
121.024 68.192 l
121.156 68.1593 l
121.289 68.1283 l
121.422 68.0987 l
121.556 68.0708 l
121.69 68.0444 l
0 .352778 0 B
n 122.692 67.8986 m
122.772 67.8906 l
122.909 67.8786 l
123.045 67.8683 l
123.182 67.8596 l
123.318 67.8527 l
123.455 67.8473 l
123.592 67.8437 l
123.69 67.8422 l
0 .352778 0 B
n 124.689 67.8685 m
124.824 67.8786 l
124.96 67.8906 l
125.096 67.9041 l
125.232 67.9194 l
125.368 67.9362 l
125.503 67.9546 l
125.638 67.9747 l
0 .352778 0 B
n 126.662 68.1806 m
126.841 68.2261 l
126.972 68.2618 l
127.103 68.299 l
127.234 68.3378 l
127.364 68.378 l
127.494 68.4197 l
127.622 68.4625 l
0 .352778 0 B
n 128.553 68.8257 m
128.634 68.8617 l
128.758 68.9181 l
128.88 68.9758 l
129.002 69.035 l
129.123 69.0956 l
129.244 69.1576 l
129.363 69.2209 l
129.449 69.2679 l
0 .352778 0 B
n 130.304 69.7871 m
130.4 69.8525 l
130.511 69.9293 l
130.62 70.0075 l
130.729 70.0869 l
130.836 70.1677 l
130.942 70.2497 l
131.048 70.333 l
131.107 70.3816 l
0 .352778 0 B
n 131.852 71.0481 m
131.944 71.1374 l
132.131 71.3281 l
132.313 71.5232 l
132.489 71.7225 l
132.536 71.7773 l
0 .352778 0 B
n 133.147 72.5688 m
133.216 72.6688 l
133.289 72.7785 l
133.362 72.8891 l
133.432 73.0006 l
133.502 73.1128 l
133.57 73.2259 l
133.636 73.3397 l
133.679 73.4148 l
0 .352778 0 B
n 134.129 74.3073 m
134.169 74.3965 l
134.22 74.5172 l
134.271 74.6385 l
134.319 74.7603 l
134.366 74.8827 l
134.411 75.0056 l
134.455 75.129 l
134.493 75.2385 l
0 .352778 0 B
n 134.764 76.2005 m
134.778 76.26 l
134.806 76.3876 l
134.832 76.5155 l
134.856 76.6437 l
134.879 76.7721 l
134.9 76.9008 l
134.919 77.0298 l
134.937 77.159 l
0 .352778 0 B
n 135.016 78.1811 m
135.019 78.329 l
135.019 78.4594 l
135.019 78.5899 l
0 .352778 0 B
% Ellipse
n 146.936 77.3136 m 146.961 82.7206 144.49 88.0905 140.223 91.8977 c
135.993 95.7372 130.027 97.9615 124.019 97.9386 c
118.011 97.9615 112.045 95.7372 107.814 91.8977 c
103.548 88.0905 101.077 82.7206 101.102 77.3136 c
101.077 71.9066 103.548 66.5367 107.814 62.7295 c
112.045 58.89 118.011 56.6657 124.019 56.6886 c
130.027 56.6657 135.993 58.89 140.223 62.7295 c
144.49 66.5367 146.961 71.9066 146.936 77.3136 c
cl 0 .705556 0 B
% Ellipse
n 111.156 68.3206 m 111.158 68.5994 111.039 68.8762 110.835 69.0725 c
110.633 69.2705 110.348 69.3852 110.061 69.384 c
109.773 69.3852 109.488 69.2705 109.286 69.0725 c
109.082 68.8762 108.964 68.5994 108.965 68.3206 c
108.964 68.0418 109.082 67.7649 109.286 67.5686 c
109.488 67.3706 109.773 67.2559 110.061 67.2571 c
110.348 67.2559 110.633 67.3706 110.835 67.5686 c
111.039 67.7649 111.158 68.0418 111.156 68.3206 c
cl 0 F
0 .352778 0 B
% Ellipse
n 111.115 86.3067 m 111.116 86.5855 110.998 86.8623 110.794 87.0587 c
110.591 87.2566 110.306 87.3713 110.019 87.3701 c
109.732 87.3713 109.446 87.2566 109.244 87.0587 c
109.04 86.8623 108.922 86.5855 108.923 86.3067 c
108.922 86.0279 109.04 85.751 109.244 85.5547 c
109.446 85.3567 109.732 85.242 110.019 85.2432 c
110.306 85.242 110.591 85.3567 110.794 85.5547 c
110.998 85.751 111.116 86.0279 111.115 86.3067 c
cl 0 F
0 .352778 0 B
% Ellipse
n 141.323 77.2511 m 141.324 77.5299 141.206 77.8068 141.002 78.0031 c
140.8 78.2011 140.515 78.3158 140.227 78.3146 c
139.94 78.3158 139.655 78.2011 139.453 78.0031 c
139.249 77.8068 139.13 77.5299 139.132 77.2511 c
139.13 76.9723 139.249 76.6954 139.453 76.4991 c
139.655 76.3011 139.94 76.1865 140.227 76.1876 c
140.515 76.1865 140.8 76.3011 141.002 76.4991 c
141.206 76.6954 141.324 76.9723 141.323 77.2511 c
cl 0 F
0 .352778 0 B
% Ellipse
n 137.92 87.2928 m 137.921 87.5716 137.803 87.8485 137.599 88.0448 c
137.397 88.2427 137.112 88.3574 136.825 88.3563 c
136.537 88.3574 136.252 88.2427 136.05 88.0448 c
135.846 87.8485 135.728 87.5716 135.729 87.2928 c
135.728 87.014 135.846 86.7371 136.05 86.5408 c
136.252 86.3428 136.537 86.2281 136.825 86.2293 c
137.112 86.2281 137.397 86.3428 137.599 86.5408 c
137.803 86.7371 137.921 87.014 137.92 87.2928 c
cl 0 F
0 .352778 0 B
% Ellipse
n 124.281 78.39 m 124.283 78.6688 124.164 78.9457 123.96 79.142 c
123.758 79.34 123.473 79.4547 123.186 79.4535 c
122.898 79.4547 122.613 79.34 122.411 79.142 c
122.207 78.9457 122.089 78.6688 122.09 78.39 c
122.089 78.1112 122.207 77.8343 122.411 77.638 c
122.613 77.44 122.898 77.3253 123.186 77.3265 c
123.473 77.3253 123.758 77.44 123.96 77.638 c
124.164 77.8343 124.283 78.1112 124.281 78.39 c
cl 0 F
0 .352778 0 B
% Polyline
n 93.38 70.8923 m
90.43 62.0423 l
96.33 62.0423 l
cl
0 F
n 93.38 58.7368 m
93.38 62.0423 l
0 2.11667 0 B
% Polyline
n 90.1822 33.6608 m
90.9177 34.3383 l
pat1 32 4 dpi 64 div .176388 0 PB
n 91.6532 35.0158 m
92.3888 35.6933 l
pat1 32 4 dpi 64 div .176388 0 PB
n 93.1243 36.3707 m
93.8599 37.0482 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 97.8592 31.0254 m
97.412 31.9198 l
pat1 32 4 dpi 64 div .176388 0 PB
n 96.9648 32.8143 m
96.5176 33.7087 l
pat1 32 4 dpi 64 div .176388 0 PB
n 96.0704 34.6031 m
95.6232 35.4975 l
pat1 32 4 dpi 64 div .176388 0 PB
n 95.176 36.392 m
94.7287 37.2864 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 100.838 37.7858 m
99.8387 37.7623 l
pat1 32 4 dpi 64 div .176388 0 PB
n 98.839 37.7388 m
97.8392 37.7153 l
pat1 32 4 dpi 64 div .176388 0 PB
n 96.8395 37.6917 m
95.8398 37.6682 l
pat1 32 4 dpi 64 div .176388 0 PB
n 94.8401 37.6447 m
94.3453 37.6331 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 97.821 44.0497 m
97.3717 43.1564 l
pat1 32 4 dpi 64 div .176388 0 PB
n 96.9223 42.263 m
96.473 41.3697 l
pat1 32 4 dpi 64 div .176388 0 PB
n 96.0236 40.4763 m
95.5743 39.5829 l
pat1 32 4 dpi 64 div .176388 0 PB
n 95.1249 38.6896 m
94.6756 37.7962 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 86.9738 39.7337 m
87.94 39.4758 l
pat1 32 4 dpi 64 div .176388 0 PB
n 88.9061 39.2178 m
89.8723 38.9598 l
pat1 32 4 dpi 64 div .176388 0 PB
n 90.8384 38.7018 m
91.8046 38.4439 l
pat1 32 4 dpi 64 div .176388 0 PB
n 92.7707 38.1859 m
93.7369 37.9279 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 97.0572 32.6296 m
96.5973 33.5176 l
pat1 32 4 dpi 64 div .176388 0 PB
n 96.1374 34.4056 m
95.9495 34.7685 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 91.7099 35.0358 m
92.4364 35.723 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 90.1822 38.8935 m
91.1437 38.6187 l
pat1 32 4 dpi 64 div .176388 0 PB
n 92.1052 38.344 m
92.321 38.2824 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 96.8662 42.2546 m
96.4249 41.3572 l
pat1 32 4 dpi 64 div .176388 0 PB
n 95.9835 40.4599 m
95.7203 39.9247 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 99.1578 37.7094 m
98.1578 37.7094 l
pat1 32 4 dpi 64 div .176388 0 PB
n 97.1578 37.7094 m
96.7898 37.7094 l
pat1 32 4 dpi 64 div .176388 0 PB
% Polyline
n 74.1023 87.1331 m
73.5625 90.3681 l
pat1 32 4 dpi 64 div .352778 0 PB
% Polyline
n 102.977 85.9872 m
98.3889 90.6736 l
pat1 32 4 dpi 64 div .352778 0 PB
% Polyline
n 116.575 30.7581 m
122.075 28.6192 l
pat1 32 4 dpi 64 div .352778 0 PB
% Text
n savemat currentmatrix pop [.478106 0 0 .465254 67.993 98.5267] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic14401440) getfont () s
0.00 SG  (P) s
savemat setmatrix
% Text
n savemat currentmatrix pop [.5 0 0 .5 123.144 29.9942] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic14401440) getfont () s
0.00 SG  (G) s
savemat setmatrix
% Group
n savemat currentmatrix pop [.5 0 0 .5 84.2551 98.3553] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic14401440) getfont () s
0.00 SG  (G) s
savemat setmatrix
n savemat currentmatrix pop [.478106 0 0 .465254 98.3888 98.7489] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic14401440) getfont () s
0.00 SG  (P) s
savemat setmatrix
n savemat currentmatrix pop [.5 0 0 .5 93.2273 101.98] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic12001200) getfont () s
0.00 SG  (p) s
savemat setmatrix
n savemat currentmatrix pop [.478106 0 0 .465254 91.8888 100.193] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (Helvetica14401440) getfont () s
0.00 SG  (/) s
savemat setmatrix
% Text
n savemat currentmatrix pop [.478106 0 0 .465254 81.9652 50.4712] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic14401440) getfont () s
0.00 SG  (P) s
savemat setmatrix
% Text
n savemat currentmatrix pop [.5 0 0 .5 90.1717 43.9247] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic12001200) getfont () s
0.00 SG  (p) s
savemat setmatrix
% Text
n savemat currentmatrix pop [.5 0 0 .5 118.505 84.7858] concat
25.4 1440 div 1.000000 mul dup scale 0 0 m

0 0 m 0 ss (ZapfChancery-MediumItalic12001200) getfont () s
0.00 SG  (p) s
savemat setmatrix
% Group
n 95.6347 75.2698 m 95.7095 75.3437 95.7399 75.461 95.7173 75.5882 c
95.6958 75.7153 95.6216 75.8507 95.5158 75.9559 c
91.5201 79.9516 l
91.4149 80.0574 91.2796 80.1316 91.1525 80.1531 c
91.0252 80.1757 90.908 80.1452 90.8341 80.0704 c
90.7593 79.9965 90.7289 79.8793 90.7514 79.752 c
90.7729 79.625 90.8471 79.4896 90.9529 79.3844 c
94.9486 75.3887 l
95.0538 75.2829 95.1892 75.2087 95.3163 75.1872 c
95.4435 75.1646 95.5608 75.1951 95.6347 75.2698 c
cl /dp_proc pat1 def
PF
pat1 32 4 dpi 64 div .352778 0 PB
n 90.8237 75.374 m 90.8976 75.2992 91.0148 75.2688 91.1421 75.2914 c
91.2691 75.3129 91.4045 75.3871 91.5097 75.4929 c
95.5054 79.4886 l
95.6112 79.5938 95.6854 79.7291 95.7069 79.8562 c
95.7295 79.9835 95.699 80.1007 95.6243 80.1746 c
95.5504 80.2494 95.4331 80.2798 95.3059 80.2572 c
95.1788 80.2358 95.0434 80.1616 94.9382 80.0557 c
90.9425 76.06 l
90.8367 75.9548 90.7625 75.8195 90.741 75.6924 c
90.7184 75.5651 90.7489 75.4479 90.8237 75.374 c
cl /dp_proc pat1 def
PF
pat1 32 4 dpi 64 div .352778 0 PB
userdict /#copies 1 put
grestore fichier2
%%Trailer
restore




