Recovery of small electromagnetic inhomogeneities from boundary measurements on part of the boundary
We consider for the Helmholtz equation the inverse problem of identifying locations and certain properties of the shapes of small dielectric inhomogeneities in a homogeneous background medium from boundary measurements on part of the boundary. Using as weights particular background solutions constructed by solving a minimization problem we develop an asymptotic (variational) method based on appropriate averaging of the partial boundary measurements.
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\begin{document}
\title{Recovery of small electromagnetic inhomogeneities from boundary
measurements on part of the boundary}


\author{H. Ammari \thanks{ Centre de Math\'ematiques Appliqu\'ees,
CNRS UMR 7641 \& Ecole Polytechnique, 91128 Palaiseau Cedex,
France (Email: ammari@cmapx.polytechnique.fr).}   \and A. G. Ramm
\thanks{Mathematics Department, Kansas State University,
Manhattan, KS 66506-2602, USA (Email: ramm@math.ksu.edu).}}
\maketitle \abstract{ We consider for the Helmholtz equation the
inverse problem of identifying locations
 and certain properties of the shapes of small dielectric inhomogeneities
in a homogeneous background medium from boundary measurements on
part of the boundary. Using as weights particular background
solutions constructed by solving a minimization problem we develop
an asymptotic (variational) method based on appropriate averaging
of the partial boundary measurements. }


\begin{center}
{\bf Identification de petites inhomog\'en\'eit\'es
di\'electriques \`a partir de mesures sur une partie du bord}
\end{center}

\begin{center}
{\bf R\'esum\'e}
\end{center}
Nous consid\'erons le probl\`eme d'identification de petites
inhomog\'en\'eit\'es di\'electriques \`a partir de mesures
(incompl\`etes) sur uniquement une partie du bord. Gr\^ace \`a un
r\'esultat de densit\'e, nous construisons une  fonction  dont
la transform\'ee de Fourier inverse permet de localiser les
petites inhomog\'en\'eit\'es.

\section*{Version fran\c caise abr\'eg\'ee} Soit $\Omega$ un ouvert
born\'e  de $\R^d$, $d\geq 2$, de classe $C^2$. Supposons que  $\Omega$
contient $m$ inhomog\'en\'eit\'es,$\{z_j + \alpha B_j\}_{j=1}^m$,
o\`u $\alpha$ est un petit param\`etre, $B_j \subset \R^d$ est un
ouvert born\'e et les points $\{z_j\}_{j=1}^m$ v\'erifient les
hypoth\`eses (\ref{f1}). Soient la perm\'eabilit\'e magn\'etique
$\mu_\alpha$ et la permittivit\'e \'electrique $\eps_\alpha$ de
forme (\ref{murhodef}).

Cette Note concerne le probl\`eme de reconstruction des points
$\{z_j\}_{j=1}^m$ et des tenseurs de polarization $M_j$ des
domaines $B_j$, d\'efinis par (\ref{Meq}), \`a partir de la mesure
de la d\'eriv\'ee normale du champ \'electrique $E_\alpha$,
solution de l'\'equation de Helmholtz (\ref{f2}), sur une partie
$\Gamma_1 \subset \subset \partial \Omega$. Gr\^ace \`a un
r\'esultat de densit\'e, \'etabli dans la Proposition \ref{lem2}, 
et \`a la formule asymptotique (\ref{eq1}), nous r\'eduisons ce
probl\`eme inverse au calcul de la transform\'ee de Fourier
inverse de la fonction $\Lambda_\alpha$, d\'efinie dans
(\ref{f3}). La fonction test $w_\alpha$,  utilis\'ee dans (\ref{f3}),
peut \^etre construite num\'eriquement en r\'esolvant le probl\`eme 
de minimisation
(\ref{f4}). Cette fonction v\'erifie les propri\'et\'es
d'approximation \'ennonc\'ees dans le Lemme \ref{lem1}. Cette Note
g\'en\'eralise la m\'ethode introduite dans \cite{AMV} aux
situations o\`u on ne disposerait pas de la mesure de la
d\'eriv\'ee normale du champ \'electrique $E_\alpha$ sur tout le
bord $\partial \Omega$.



\section{Introduction} Let $\Omega$ be a bounded $C^2$-domain
in $\R^d$, $d\geq 2$ and
 $\nu$ be the outward unit normal to
$\partial \Omega$. Assume  that $\Omega$ contains a finite
number of inhomogeneities, each of the form $z_j + \alpha B_j$,
where $B_j \subset \R^d$ is a bounded, smooth domain containing
the origin. The total collection of inhomogeneities is
 $ \ds {\cal B}_\alpha
 = \ds \cup_{j=1}^{m} (z_j  + \alpha B_j)$.
  The points $z_j \in \Omega, j=1, \ldots, m,$ which determine the
  location of the inhomogeneities, are assumed to satisfy the
  following inequalities:
\be
\label{f1}
 | z_j  - z_l | \geq c_0 > 0, \forall \; j \neq l  \quad
\mbox{ and } \mbox{ dist} (z_j, \partial \Omega) \geq c_0
> 0, \forall \; j.
\ee This assumption implies
 $m \leq \ds \frac{2 d |\Omega|}{\pi c_0^d}$. Assume
that $\alpha
>0$, the common order of magnitude of the diameters of the
inhomogeneities, is sufficiently small, that these
 inhomogeneities are disjoint,  and that
their distance to $\R^d \setminus \overline{\Omega}$ is larger than
$c_0/2$.  Let $\mu_0$ and $\eps_0$ denote the permeability and
the permittivity of the background medium, and assume that
{\it $\mu_0>0$ and $\eps_0>0$ are positive constants}.  Let $\mu_j>0$ and
$\eps_j>0$
denote the permeability and the permittivity of the j-th
inhomogeneity, $z_j+\alpha B_j$, these are also assumed to be
positive constants. Introduce the piecewise-constant magnetic permeability
\begin{equation}
\mu_\alpha(x)=\left \{ \begin{array}{*{2}{l}}
 \mu_0,\;\;& x \in \Omega \setminus \bar {\cal B}_\alpha,  \\
 \mu_j,\;\;& x \in z_j+\alpha B_j, \;j=1 \ldots m.
\end{array}
\right . \label{murhodef}
\end{equation}
If we allow the degenerate case $\alpha =0$, then the function
$\mu_0(x)$ equals the constant $\mu_0$. The piecewise constant
electric permittivity, $\eps_\alpha(x)$ is defined analogously.

Consider solutions to the time-harmonic Maxwell's equations
with $\exp (-i \omega t)$ time dependence.
 Let $E_\alpha$ be the electric field
in the presence of the inhomogeneities. It solves the Helmholtz
equation $$ \nabla \cdot (\frac{1}{\mu_\alpha} \nabla E_\alpha) +
\omega^2 \eps_\alpha E_\alpha = 0 \ \ \mbox{in } \Omega,$$ with
the boundary condition $E_\alpha = f$ on $\partial \Omega$,  where
$\omega > 0$ is a given frequency. The electric field, $E_0$, in
the absence of any inhomogeneities, satisfies the
following equation:
\be
\label{f2}
 \Delta E_0 + k^2 E_0 = 0  \ \ \mbox{in } \Omega,
 \ee
where $k^2 = \omega^2 \mu_0 \eps_0$, with $E_0 = f$ on $\partial
\Omega$. In order to insure well-posedness (also for the
$\alpha$-dependent case for $\alpha$ sufficiently small) we shall
assume that $k^2$ is not an eigenvalue for the operator $- \Delta$
in $L^2(\Omega)$
with the Dirichlet boundary conditions. It has been shown
in \cite{VV} that the following asymptotic formula  holds
uniformly on $\partial \Omega$ \be \label{eq1}
\begin{array}{l}
\ds \frac{\partial E_\alpha}{\partial \nu}(x) - \frac{\partial
E_0}{\partial \nu}(x) - 2 \int_{\partial \Omega}(\frac{\partial
E_\alpha}{\partial \nu}- \frac{\partial E_0}{\partial \nu})(y)
\frac{\partial G(x, y)}{\partial \nu} \ ds(y) \\ \nm \ \ \ \ \ds =
2 \alpha^d \sum_{j=1}^m (1 - \frac{\mu_0}{\mu_j}) \nabla_y
\frac{\partial G(x, z_j)}{\partial \nu(x)} \cdot
M_j(\frac{\mu_j}{\mu_0}) \nabla E_0(z_j) \\
 \nm \ \ \ \ \ds - 2 \alpha^d k^2 \sum_{j=1}^m (1 - \frac{\eps_j}{\eps_0})
\frac{\partial G(x, z_j)}{\partial \nu(x)} |B_j| E_0(z_j) +
o(\alpha^d),
\end{array}
\ee where the remainder $o(\alpha^d)$ is independent of the set of
points $\{z_j\}_{j=1}^m$ provided that (\ref{f1}) holds, $G(x, y)$
is a free space Green's function for $\Delta + k^2$, and each
$M_j$ is a $d \times d$, symmetric, positive definite matrix
associated with the j-th inhomogeneity, called the polarizability
tensor, which is given by
\begin{equation}
(M_j)_{ll^\prime}=|B_j|\delta_{ll^\prime}+
\left(\frac{\mu_j}{\mu_0} -1\right) \int_{\partial
B_j}y_l{\partial\phi_{l^\prime}^+\over{
\partial\nu_j}}d\sigma_y,\label{Meq}
\end{equation}
where, for $1 \leq l^\prime \leq d$, $\phi_{l^\prime}(y)$ is the
unique function which satisfies \[ \left\{ \begin{array}{lll}
 \Delta\phi_{l^\prime} &=& 0\ \ \ \ \mbox{in}\ \ \ B_j \ \ \mbox{and} \ \ \ \R^d \setminus
\overline{B_j},\\ \nm \ds \frac{1}{\mu_0}  {\partial
\phi_{l^\prime}^-\over{\partial\nu_j}}- \frac{1}{\mu_j}
{\partial\phi_{l^\prime}^+\over{\partial\nu_j}} &=& \ds -
\frac{1}{\mu_j} \nu_j \cdot e_{l^\prime} \ \ \ \ \mbox{on}\ \ \
\partial B_j,
\end{array}
\right.
\]
with $\phi_{l^\prime}$ continuous across $\partial B_j$ and
$\lim_{|y|\rightarrow\infty} \phi_{l^\prime}(y)=0.$ Here
$\{e_{l^\prime}\}_{l^\prime =1}^d$ is an orthonormal basis of
$\R^d$, $\nu_j$ denotes the outward unit normal to $\partial B_j$,
superscripts $-$ and $+$ indicate the limiting values as the point
approaches
$
\partial B_j$ from outside $B_j$, and from
inside $B_j$, respectively.

Our goal is to identify the locations
$\{z_j\}_{j=1}^m$ and the polarizability tensors $\{M_j\}_{j=1}^m$
of the small inhomogeneities ${\cal B}_\alpha$ from the boundary
measurements of $\frac{\partial E_\alpha}{\partial \nu}$ on a
given part $\Gamma_1 \subset \subset \partial \Omega$. This
inverse problem is more complicated from the mathematical point of
view and more interesting in applications
than the one solved in \cite{AMV}, because
in many applications
one cannot get measurements on the whole boundary.
As in \cite{AMV}, we
want to reduce this reconstruction problem to the calculation of
an inverse Fourier transform. Our work is
a generalization of \cite{AMV} to the case of the reconstruction
of
small inhomogeneities from the measurements on a part of the boundary.

In  \cite{395} a different method is proposed for finding small
inhomogeneities from the scattering data. In  \cite{396} it is proved
that the singular (delta-type) potentials are uniquely determined
by the scattering data. In \cite{399} a numerical realization of the
method proposed in \cite{395} is given. In \cite{364} a method for
finding small inhomogeneities from tomographic data is given. In
\cite{144} analytic formulas are derived for electric and magnetic
polarizability tensors for bodies of arbitrary shapes. In \cite{278}
a number of multi-dimensional inverse scattering problems are
studied and some of the methods developed there are related to our
approach.

\section{Identification procedure}
Before describing our identification procedure,  let us introduce
the sets $N(\Omega) = \{v: v \in H^1(\overline{\Omega}) \cap H^2(\Omega),
\Delta v + k^2 v = 0 \
\mbox{in} \ \Omega  \}$ and $\tilde{N}(\Omega) = \{v: v \in
 H^1(\overline{\Omega}) \cap H^2(\Omega),
\Delta v + k^2 v = 0 \ \mbox{in} \ \Omega, v= 0 \
\mbox{on} \ \ \Gamma_2 \}$, where $\Gamma_2
=
\partial \Omega \setminus \overline{\Gamma_1}$, where
$\Gamma_1$ is an open in $\partial \Omega$ subset.

The general
approach we use to recover the locations and the
polarizability  tensors of the small inhomogeneities is to integrate
the solution $E_\alpha$ against special test functions in the set
$\tilde{N}(\Omega)$. Let $v$ be any function  in
$\tilde{N}(\Omega)$.
  As in \cite{AMV}, the following estimate can
be derived from (\ref{eq1}): \be \label{eq3}
\begin{array}{lll} \ds
\int_{\Gamma_1} \frac{\partial E_\alpha}{\partial \nu} v \ \ ds -
\int_{\partial \Omega} \frac{\partial v}{\partial \nu} E_\alpha \
\ ds &=&\ds  \alpha^d \sum_{j=1}^m (1 - \frac{\mu_j}{\mu_0})
\nabla E_0(z_j) \cdot M_j(\frac{\mu_j}{\mu_0})  \nabla v(z_j) \\
 \nm
 &+& \ds  \alpha^d k^2 \sum_{j=1}^m (1 - \frac{\eps_j}{\eps_0})
|B_j| E_0(z_j) v(z_j) + o(\alpha^d),
\end{array}
\ee where $|B_j|$ stands for the volume of the set $B_j$. We want
to make suitable
choices for the test functions
$v$ in $\tilde{N}(\Omega)$ and the boundary condition $E_\alpha
|_{\partial \Omega}$ in order to get simple
equations for the unknown parameters, namely, for
the points
$\{z_j\}_{j=1}^m$ and matrices $\{M_j\}_{j=1}^m$. Similar
idea was used and the associated numerical experiments have been
successfully
conducted in the case of the conductivity problem \cite{AMV} with
boundary measurements on all of $\partial \Omega$.

Let us describe
our inversion method. Take $\eta$ to be a vector in $\R^d$,
$\eta^{\perp}$ a unit vector in $\R^d$ which is orthogonal to
$\eta$, and $\gamma$ a complex number. Then $e^{i(\eta+ \gamma
\eta^{\perp})\cdot x}$ is a solution to the Helmholtz equation in
$\R^d$ if and only if $\gamma^2 = k^2 - | \eta |^2$ and in this
case $e^{i(\eta - \gamma \eta^{\perp})\cdot x}$ is also a solution
to the Helmholtz equation in $\R^d$. For simplicity, let us
consider the case where all the $B_j$ are balls. In this case all
the matrices $M_j$ are  multiples of the identity matrix which
makes our analysis simpler.

If $\frac{\partial E_\alpha}{\partial \nu}$ is known on the whole
boundary $\partial \Omega$ then taking $E_\alpha = e^{i(\eta+
\gamma \eta^{\perp})\cdot x}$ on $\partial \Omega$ we know from
\cite{AMV} that
\[\begin{array}{l} \ds
\int_{\partial \Omega} \frac{\partial E_\alpha}{\partial \nu}(y)
e^{i(\eta -  \gamma \eta^{\perp})\cdot y}\ \ ds(y) -
\int_{\partial \Omega} \frac{\partial}{\partial \nu}(e^{i(\eta -
\gamma \eta^{\perp})\cdot y}) E_\alpha(y) \ \ ds(y)
\\ \nm \ \ \ \ = \ds \alpha^d \sum_{j=1}^m e^{2 i \eta \cdot z_j}
\Bigr[ (1 - \frac{\mu^j}{\mu^0}) M_j(\frac{\mu^j}{\mu^0}) (2 |
\eta|^2 - k^2) + \ds  k^2 (1 - \frac{\eps^j}{\eps^0}) |B_j| \Bigr]
+ o(\alpha^d).
\end{array}
\]
The main difficulty in generalizing this approach to the case when
$\frac{\partial E_\alpha}{\partial \nu}$ is known only on a part
$\Gamma_1 \subset \subset \partial \Omega$ is to construct a
function $w_\alpha(x)$ in $\tilde{N}(\Omega)$,  that is
asymptotically $e^{i(\eta -  \gamma \eta^{\perp})\cdot x}$ as
$\alpha$ approaches $0$. The following lemma holds.
\begin{lem} \label{lem1} Let $\Omega^\prime \subset \subset \Omega$ be
a $C^2$-domain. Let $\eta \in \R^d$ and $\eta^{\perp}$ be a
unit vector in $\R^d$ that is orthogonal to $\eta$. There exists
$w_\alpha \in \tilde{N}(\Omega)$ such that
\[ w_\alpha(x) = e^{i(\eta -  \gamma
\eta^{\perp})\cdot x} + o(\alpha^d) \ \mbox{and} \ \nabla
w_\alpha(x) = i(\eta -  \gamma \eta^{\perp}) e^{i(\eta -  \gamma
\eta^{\perp})\cdot x} + o(\alpha^d),
\]
uniformly in $\Omega^\prime$.
\end{lem}
This lemma is an immediate corollary of the following general
density result.
\begin{prop} \label{lem2}
The set $\tilde{N}(\Omega)$ is dense, in the $L^2(\Omega^\prime)$
norm, in the set  $N(\Omega)$.
\end{prop}
\proof Assume the contrary and let $v \in N(\Omega)$ be an element which
cannot be
approximated in $L^2(\Omega^\prime)$ by the functions from
$\tilde{N}(\Omega)$ with a prescribed accuracy. Then
there is an element in $N(\Omega)$, which we denote again $v$, such that
$\int_{\Omega^\prime}
v w dx = 0, \forall
\; w \in  \tilde{N}(\Omega)$. Let $G_0$ be the Dirichlet Green's
function in $\Omega$: $$\left\{
\begin{array}{l}
\Delta G_0 + k^2 G_0 = \delta_y(x) \ \ \mbox{in} \ \ \Omega,\\ G_0
= 0 \ \ \mbox{on} \ \ \partial \Omega.
\end{array}
\right.
 $$
Define $\tilde{H}^{3/2}(\Gamma_1) = \{ p \in H^{3/2}(\Gamma_1) \
\mbox{such that there exists} \ \tilde{p}  \in H^{3/2}(\partial \Omega),
\tilde{p}|_{\Gamma_2}  = 0,\tilde{p}|_{\Gamma_1}  = p \}$.  
 Since any $w \in \tilde{N}(\Omega)$ can be represented as
follows $$w(x) = \int_{\Gamma_1} \frac{\partial G_0(x,y)}{\partial
\nu(y)} \ p(y) \ ds(y), x \in \Omega,$$ where $p \in
\tilde{H}^{3/2}(\Gamma_1)$ is arbitrary, we have $$\int_{\Omega^\prime} v(y)
\frac{\partial G_0(x,y)}{\partial \nu(x)} \ dy =0, \ \forall \ x
\in \Gamma_1.$$ Define $u(x): = \int_{\Omega^\prime} v(y)G_0(x, y)
\ dy.$ Then $u|_{\partial \Omega} = 0, \frac{\partial u}{\partial
\nu} |_{\Gamma_1} = 0$, and $$\Delta u + k^2 u = \left\{
\begin{array}{l} v \ \  \mbox{in } \Omega^\prime,\\ 0  \ \
\mbox{in } \Omega \setminus \overline{\Omega^\prime}.
\end{array}
\right. $$ By the uniqueness of the solution to the Cauchy problem
for the Helmholtz equation, it follows that $u = 0$ in $\Omega \setminus
\overline{\Omega^\prime},$ and so $u =  \frac{\partial u}{\partial
\nu}  = 0$ on $\partial \Omega^\prime$. Since $\Delta u + k^2 u =
v $ in $\Omega^\prime$, it follows by multiplying this equation by
$v$ and  integrating by parts over $\Omega^\prime$ that $
\int_{\Omega^\prime} v^2 dx= 0.$ Thus, $v=0$ in $\Omega^\prime$,
and, by the unique continuation, $v=0$ in  $\Omega,$ which
proves Proposition 2.1. In the above argument we assumed that $u$
and $v$ are real valued. This can be done without loss of
generality since $k^2>0$.
 \square

From Proposition \ref{lem2} it follows that the function
$w_\alpha$ in Lemma \ref{lem1} can be constructed (numerically) by
solving the minimization problem: \be \label{f4} \min_{p \in
\tilde{H}^{3/2}(\Gamma_1)} \| \int_{\Gamma_1} \frac{\partial
G_0(x,y)}{\partial \nu(y)} \ p(y) \ ds(y) - e^{i(\eta -  \gamma
\eta^{\perp})\cdot x} \|_{L^2(\Omega^\prime)}. \ee
We can take $$
w_\alpha(x) = \int_{\Gamma_1} \frac{\partial G_0(x,y)}{\partial
\nu(y)} \ p_\alpha(y) \ ds(y),$$ where $p_\alpha$ is any function
in $\tilde{H}^{1/2}(\Gamma_1)$ such that \be \label{eq4}\| \int_{\Gamma_1}
\frac{\partial G_0(x, y)}{\partial \nu(y)} \ p_\alpha(y) \  ds(y)
- e^{i(\eta - \gamma \eta^{\perp})\cdot x} \|_{L^2(\Omega^\prime)}
=o(\alpha^d). \ee
 Since $(\Delta + k^2) (w_\alpha -e^{i(\eta -
\gamma \eta^{\perp})\cdot x}) = 0$ in $\Omega$, by the standard
elliptic interior estimates \cite{GT}, we obtain from (\ref{eq4})
that $\|w_\alpha -e^{i(\eta - \gamma \eta^{\perp})\cdot
x}\|_{H^s(\Omega^{\prime \prime})}$ is of order $o(\alpha^d)$ for any
$s$ and any $C^2$-domain $\Omega^{\prime \prime} \subset \subset
\Omega^\prime$. Lemma \ref{lem1} follows then immediately from the
Sobolev imbedding theorems.


 Now, if we choose $E_\alpha = e^{i(\eta+ \gamma \eta^{\perp})\cdot
x}$ on $\partial \Omega$ and $v = w_\alpha$ in $\Omega$ then,
since the points $\{z_j\}_{j=1}^m$ are away
from the boundary $\partial \Omega$, it follows from (\ref{eq3})
and Lemma \ref{lem1} that the following asymptotic expansion holds:
\be
\label{f3}
\begin{array}{lll} \ds \Lambda_\alpha(\eta) &=&\ds
\int_{\Gamma_1} \frac{\partial E_\alpha}{\partial \nu} w_\alpha  \
ds - \int_{\partial \Omega} \frac{\partial w_\alpha}{\partial \nu}
E_\alpha  \ ds \\ \nm &=&\ds \alpha^d \sum_{j=1}^m e^{2 i \eta
\cdot z_j} \Bigr[ (1 - \frac{\mu_j}{\mu_0})
M_j(\frac{\mu_j}{\mu_0}) (2 | \eta|^2 - k^2) + \ds  k^2 (1 -
\frac{\eps_j}{\eps_0}) |B_j| \Bigr] + o(\alpha^d).
\end{array}
\ee
 Recall that $e^{2 i \eta \cdot z_j}$ (up to a
multiplicative constant) is  the Fourier transform of the
Dirac function $\delta_{-2 z_j}$ (a point mass located at $-2
z_j$). Multiplication by powers of $\eta$ in the Fourier space
corresponds to differentiation of the Dirac function.  Therefore, the
function  $\Lambda_\alpha(\eta)$ is the inverse Fourier
transform of a distribution supported at the points
$z_j$. A numerical
Fourier inversion of a sample of $\Lambda_\alpha(\eta)$ will
yield the points $z_j$ with a small error as
$\alpha \to 0$. It is natural to use a
fast
Fourier transform for this inversion.
One can estimate the number of the
sampling points needed for an accurate discrete Fourier
inversion  using the Shannon's theorem \cite{D}, page 18.
This number is
 of order $\ds (\frac{h}{\delta})^3$. One needs this amount of sampled
values of  $\eta$ to reconstruct, with resolution $\delta$, a
collection of inhomogeneities that lie inside a square of side
$h$. Once the points $\{z_j\}_{j=1}^m$ are found, one may find
$\{M_j\}_{j=1}^m$ by solving appropriate linear system arising
from the asymptotic formula (\ref{eq3}). If $B_j$ are general
domains, our calculations become more complicated, and eventually we
have to deal with pseudo-differential operators (independent of
the space variable $x$) applied to the same Dirac functions.
 In view of the asymptotic results derived in \cite{AVV} a
 similar approach may be applied to the full Maxwell equations in the presence of
 small dielectric inhomogeneities.

\section*{Acknowledgements}
This work is partially supported by ACI Jeunes Chercheurs (0693)
from the Ministry of Education and Scientific Research, France.



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\end{document}

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