text:
\section{Introduction}
Channel capacity for MIMO  channels (multiple input multiple output) has been the subject of intense study in
recent years. Most of the papers have been concerned with a strictly information theoretic analysis
\cite{telatar95, foschini98}. On the other hand, channel capacity can be treated as an optimization problem
subject to the constraint imposed by the Maxwell equations \cite{loyka2005}. In this paper, a mathematical
framework for MIMO capacity is provided using Electromagnetic constraints of the channel. It is shown  that this
problem can be rewritten as an optimal control problem where the control is the source measure subject to moment
constraints equivalent to transmitter power constraints. \par Rest of the paper is organized as follows:  The
current section ends after a brief list of notations. In section 2, we present the dynamic models of
electro-magnetic channels. Both distributed and boundary sources are considered, and existence and regularity
properties of solutions of the dynamic  systems   are presented.  In section 3, communication problems are
formulated. Section 4 deals with the solution of the problems proving existence of control measures from the
admissible class at which channel capacity is attained. In section 5, necessary and sufficient conditions of
optimality are  presented whereby a numerical  algorithm can be developed for capacity computation. The paper is
concluded with some comments in section 6. \vskip6pt \noindent {\bf Some notations:} Let $\Xi$ denote an
arbitrary set and ${\cal F}$ the Borel algebra of susbsets of the set $\Xi.$ We call the pair $(\Xi,{\cal F})$ a
measurable space. Let $\{\mu,\nu\}$ be any two regular Borel measures on the measurable space $(\Xi,{\cal F})$.
We let $\mu\prec \nu$ to denote the absolute continuity of the measure $\mu$ with respect to the measure $\nu.$
The Radon-Nikodym derivative, if it exists, of $\mu$ with respect to $\nu$ is denoted by $
\frac{\mu(dx)}{\nu(dx)} \equiv g(x),$ where $g \in L_1(\Xi,\nu).$
\par For any pair of Banach spaces $X,Y$, we let ${\cal L}(X,Y)$ denote the space  of bounded linear operators
from $X$ to $Y.$  For any bounded open connected domain $\Omega \subset R^n$ with sufficiently smooth boundary
$\partial \Omega,$ $H^s(\Omega,R^m) \subset L_2(\Omega,R^m), s\geq 0,$ will denote the standard Sobolev spaces
of functions defined on $\Omega$ and taking values from $R^m$ whose generalized derivatives up to order $s$
belong to $L_2(\Omega,R^m.$  Similarly $H^{-s}(\Omega,R^m), s \geq 0,$ will denote the Sobolev spaces with
negative exponents. These are distributions and, under some assumptions, are the topological duals of
$H^s(\Omega,R^m).$ By Sobolev's embedding theorem, it is known that for $s\geq (n/2) + k,$ $H^s(\Omega,R^m)
\hookrightarrow C^k(\Omega,R^m). $ Thus the Dirac measure $\delta_{\omega}(dx)$ with mass concentrated at
$\omega \in \Omega,$ $ b \delta_{\omega} \in H^{-s}(\Omega,R^m)$ for any  $b \in C(\Omega,R^m)$ and $s
>(n/2).$ Note that for $s \geq 0,$ a continuous linear functional  $\ell$ on $H^s(\Omega,R^m)$ has the
representation $$ \ell(\varphi) = \int_{\Omega} (\varphi,\psi) dx $$ for some $\psi \in H^{-s}(\Omega,R^m).$

For example,  for any $f \in L_2(\Omega,R^m)$  with $a_{\alpha}$ being constants, the function  $\psi,$ given by
$\psi \equiv \sum_{|\alpha|\leq s} a_{\alpha} D^{\alpha} f,$ is an element of $H^{-s}(\Omega,R^m).$ Here $\alpha
\equiv  (\alpha_1,\alpha_2,\cdots,\alpha_n)$ stands for the multi index and $|\alpha| = \sum_{i=1}^n
\alpha_i,\alpha_i \geq 0 $ and $D^{\alpha}f$ denotes the distributional derivative of $f$ of order $|\alpha|.$
For fractional $s,$ the Sobolev spaces are defined by use of Fourier transform.
\section{Channel Dynamics}
In this section we present  several models that describe the channel dynamics.  The first model is assumed to
satisfy  homogeneous Neumann  boundary condition (no leakage) with input source being  a vector of current  and
charge density.  The second model consists of  homogeneous wave equation  describing  the electric field with
nonhomogeneous  Dirichlet boundary  data whereby the input or source is provided.

 \subsection{\bf Channel with Current and charge as input Sources}  First we consider channels with homogeneous Neumann  boundary
condition. In this case the system is governed by a system of  wave equations arising from Lorenz transformation
of Maxwell's equations.  The  electromagnetic waves are  generated by input sources such as current  and charge
densities  and are confined in a wave guide. The electrical signals in the wave guide are governed by Maxwell's
equations. Using the vector and scalar potentials denoted by  $(a,\varphi)$ and the Lorentz gauge, the Maxwell's
equations  are  given by a system of wave equations:  \begin{eqnarray} &~& \partial^2 a/ \partial t^2  -  (1/\mu
\epsilon) \triangle a =(1/\epsilon )i, t \geq  0,\xi \in \Omega \subset R^3 \label{eqEM1}  \\ &~&
\partial^2\varphi/\partial t^2 - (1/\mu \epsilon) \triangle \varphi = (1/\mu \epsilon)   \rho ,\label{eqEM2} \end{eqnarray}
where $i$ and $ \rho$ are the sources, the first denoting the current density (vector) and the second the charge
density.  These are the sources that can be controlled to produce desirable field distributions inside the wave
guide. The field variables $\{E,B\}$  are related to  the potentials by the following equations:
\begin{eqnarray*} E = -(\dot a + \nabla \varphi), ~~ B =  \nabla \times a. \end{eqnarray*} These models are useful
in various fields of  communication such as radar,  optical fibre etc  \cite{ahmed03} (see references therein.).
We let $\Omega \subset R^3 $ denote an  open bounded  connected domain (representing the waveguide) having
piecewise smooth boundary.
\par Define  $ H \equiv L_2(\Omega,R^ 3)  \times L_2(\Omega,R) = L_2(\Omega,R^4)$ , denote $ y \equiv  (a,
\varphi)$ , and define the formal differential operator  $ C$  by  $ Cy \equiv (1/\mu \epsilon)(\triangle a,
\triangle \varphi) $ and let $B$ denote the Neumann  boundary operator and set $B(a,\varphi) =0 .$   This
operator is simply the outward normal derivative of the  arguments  at every point on the boundary of the wave
guide.   Then introduce the operator A as follows:
\begin{eqnarray*} &~& D(A) \equiv  \{  y  \in  H :  B(y) = 0 ~ \hbox{and}~ Cy \in  H\}   \subset   H ^2(\Omega
,R^3) \times H^2(\Omega ,R)
\\  &~&  \hbox{and set }  Az = Cz ~ \hbox{for}~  z \in  D(A).  \end{eqnarray*} Under  the given
boundary condition, $ -A $ is an unbounded positive self-adjoint operator in $H.$ Define the state space as  $
{\cal H}  \equiv  D(\sqrt -A) \times  H$  and the state as  $ z =(y, \dot y).$  This is the energy space.
Furnished with the scalar product and the associated  norm as presented below,\begin{eqnarray*} &~& (x,z)_{\cal
H} = (\sqrt -A x_1, \sqrt-A z_1)_{H} + (x_2,z_2)_{H} \\ &~&  \parallel x\parallel_{\cal H} \equiv \biggl(
\parallel \sqrt - A x_1\parallel_H^2 + \parallel x_2\parallel_H^2 \biggr)^{1/2},  \end{eqnarray*} ${\cal H}$ is
a Hilbert space. Note that the first term represents the potential energy and the second the kinetic energy
(magnetic field energy). Then we define the system  operator  ${\cal A}$  and the control operator ${\cal B}$ as
follows
\begin{eqnarray*} {\cal A} \equiv \biggl(\begin{array}{clcr}  0 & I \\ A & 0 \end{array} \biggr),
\end{eqnarray*}
\begin{eqnarray*} ~ {\cal B} \equiv  \left( \begin{array} {clcr} 0&0&0&0 \\ 0&0&0&0\\  0&0&0&0 \\ 0&0&0&0\\
(1/ \epsilon) &0&0&0 \\ 0 & (1/\epsilon) & 0 & 0 \\ 0&0&(1/ \epsilon) \\ 0&0&0& (1/\mu \epsilon)
\end{array} \right). \end{eqnarray*} Define the input or the control vector as $$ u \equiv
\left(\begin{array}{clcr} i_1 \\ i_2 \\  i_3 \\  \rho\end{array}\right). $$ Using these  notations the system of
wave equations given by (1) and (2) can be written as an abstract differential equation on the Hilbert space
${\cal H} $  as follows   \begin{eqnarray} \dot z = {\cal A}z + {\cal B}u, t \geq 0, \label {eqEM3}
\end{eqnarray}   where ${\cal  A }$  is an unbounded operator with domain and range in ${\cal H}.$  In practice
the input is localized. Let  $\Omega_0 \subset \Omega$ be a part of the domain at the input end of the wave
guide and consider the Hilbert space $U \equiv L_2(\Omega_0, R^4)$ with the standard scalar product. We may
assume that the  controls are functions of time taking values from the Hilbert space $U.$  Thus  our admissible
source is a proper subset ${\cal U}_{ad} \subset L_2(I,U).$  It can be shown that on the Hilbert space ${\cal
H}$ the system operator ${\cal A}$ is skew adjoint  and hence $i{\cal A}$ is self adjoint. Thus it follows from
semigroup theory (\cite{ahmed91},Theorem 3.1.4, p71),
in particular Stones theorem, that ${\cal A}$ generates a unitary group of operators ${\cal S}(t), t \in R . $
Using this unitary group of operators we can write the solution (mild) of equation (\ref{eqEM3}) as follows
\begin{eqnarray} z(t)= {\cal S}(t)z_0 + \int_0^t {\cal S}(t-s) {\cal B} u(s) ds, t \geq 0.\label{eqEM4} \end{eqnarray}
Note that in the absence of external input $u,$  the system is conservative and $$ \parallel
z(t)\parallel_{{\cal H}} = \parallel {\cal S}(t)z_0 \parallel_{{\cal H}}=\parallel z_0\parallel_{{\cal H}}
~\forall ~ t \in R.$$ This can be proved by simply  scalar multiplying in ${\cal H}$ on either side of the
equation
$$ \dot z = {\cal A}z , z(0) = z_0$$  by $z$ and integrating  and noting that
 $({\cal A}\xi,\xi)_{\cal H}= 0, \forall ~\xi \in D({\cal A}).$  We summarize  the above results in   the
 following theorem. \vskip6pt \noindent {\bf Theorem 2.1} For every input $u \in L_2(I,U)$ and initial state
 $z_0 \in {\cal H}$, the system (\ref{eqEM3}) has a unique mild solution $z \in C(I,{\cal H}).$ Further the solution
 is given by the expression (\ref{eqEM4}). This in turn implies that the system of wave equations
 (\ref{eqEM1})-(\ref{eqEM2}) has a unique mild  solution for every given initial state in the energy space and every
 given finite energy input.

\subsection{Channel with Dirichlet  Data  as Input Source} In cgs units, the Maxwell equations for electric field
$E$ and  magnetic field $B$ are given by\begin{eqnarray} &~& \nabla \times B = (1/c)
\partial E/\partial t + (4\pi/c) i \label {eqEM5} \\ &~& \nabla \times E = -(1/c) \partial B/\partial t,
\label{eqEM6}  \\ &~& \nabla \cdot E = 4\pi \rho,~ \nabla \cdot B = 0,  \label{eqEM7} \end{eqnarray}  where $c$
denotes the velocity of light and the pair $\{i,\rho\}$ denotes the current density vector and charge density
respectively. Here we have used standard notations for $ curl \phi \equiv \nabla \times \phi $ and $ div \phi
\equiv \nabla \cdot \phi .$ Using the first identity of equation (\ref{eqEM7}), the reader can easily verify
that $$ \nabla \times \nabla \times E = - \triangle E + 4\pi (\nabla \rho).$$ Now applying the curl operator on
either side of equation (\ref {eqEM6}) and using equation (\ref{eqEM5}) one can easily verify that the electric
field $E$ satisfies the following wave equation
\begin{eqnarray}  \partial^2 E /
\partial t^2 - c^2 \triangle E = - 4\pi (
\partial i/\partial t  + c^2 \nabla \rho ). \label {eqEM8} \end{eqnarray}  Since here we are interested in
boundary data, we assume that both the current and charge densities are identically zero. In order to solve such
equations in any bounded domain one must specify the initial and boundary  conditions. Hence the complete system
equation is given by \begin{eqnarray}  &~&  \partial^2 E / \partial t^2 - c^2 \triangle E = 0, \xi \in \Omega, t
\geq 0, \label {eqEM9}  \\  &~& E(0,\xi) = E_0(\xi) , \dot E(0,\xi) = E_1(\xi) , \xi \in \Omega, \label {eqEM10}
\\  &~& E(t,\xi)|_{\partial \Omega} = u(t,\xi), \xi \in \partial \Omega, t \geq 0. \label {eqEM11}
\end{eqnarray}  This is a initial boundary value problem  with nonhomogeneous Dirichlet boundary condition.   In
general the source  $u$ carries the information  to be transmitted  over the wave guide channel $\Omega.$ \par
There are two possible ways of attacking this problem. One is the semigroup approach (\cite{ahmed88}) and the
other is based on  the principle of transposition~(\cite{lions70}).
\par\noindent  {\bf Method A}
(Semigroup Approach): The first method is based on a well known technique (\cite{ahmed88},p59-63
(and the references therein)  whereby one can transfer the boundary data to the righthand side of the original
differential equation. We write equation (\ref{eqEM9})-(\ref{eqEM11})  as a system
\begin{eqnarray} &~& \partial e/\partial t  = Le, \label{eqEM9A}
e(0) = e_0 \\ &~& Be = Tr  e_1 = u \label {eqEM9B}  \end{eqnarray} where $e \equiv (E,\dot E),$  $Tr \phi \equiv
\phi|_{\partial \Omega},$ and
\begin{eqnarray*} {L} \equiv \biggl(\begin{array}{clcr}  0 & I \\ c^2 \triangle    & 0 \end{array} \biggr). \end{eqnarray*}Note
that  this is a $6 \times 6$ matrix  with the elements of the first and the fourth diagonal blocks being all
zero and the second block being a $3\times3$ identity matrix and the third diagonal block is a $3\times 3$
diagonal matrix with the elements being  the Laplacian  $c^2 \triangle.$ To avoid introducing new notations, we
use the same symbols to define the operators $A, {\cal A}, B_0$ by
$$ A \equiv c^2  I \triangle|_{ker B}~~ \hbox{and}~~ {\cal A} \equiv L|_{ker B}~~\hbox{and}~~ B_0 \equiv B|_{ker
L}.$$  Note that the operator $A$  is a negative self adjoint unbounded operator  on  $ H \equiv
L_2(\Omega,R^3).$ The domain of the operator ${\cal A}$ is given by $$ D({\cal A}) = H^2(\Omega,R^3) \cap
H_0^1(\Omega,R^3) \times L_2(\Omega,R^3) \subset {\cal H} $$ where ${\cal H}$  is  the energy space, $$ {\cal H}
\equiv D(\sqrt -A)\times H,$$ considered here as the state space. Now returning to the system model, it is not
difficult to verify that the operator ${\cal A}$ is closed and
 densely defined and that    for any  $ R \ni \lambda \ne 0$,
 $$  \parallel R(\lambda,{\cal A}\parallel  \equiv  \parallel (\lambda I - {\cal
A})^{-1}\parallel_{{\cal H}}~ \leq ~(1/|\lambda|).$$ Thus by Hille-Yosida theorem (\cite{ahmed91},Theorem
2.2.8,p27)
, ${\cal A}$ is the infinitesimal generator of a $C_0$-group ${\cal S}(t), t \in R,$  of contractions in ${\cal
H}.$ Further, it is easy to verify that the operator ${\cal A}$ is skew adjoint and hence by Stones
theorem~(\cite{ahmed91},Theorem 3.1.4, p 71
, it is the infinitesimal generator of a unitary group ${\cal S}(t), t \in R,$ on ${\cal H}.$ Our objective is
to convert the initial boundary value problem (\ref{eqEM9A})-(\ref{eqEM9B}) into a Cauchy problem (initial value
problem). Define
$${\cal W} \equiv H^2(\Omega,R^3)\times L_2(\Omega,R^3)\subset {\cal H}$$ and set ${\cal W}_1 \equiv Ker L ,
{\cal W}_2 \equiv Ker B.$ For any $\lambda( \in R)\ne 0$, define $ P \equiv R(\lambda,{\cal A}) (\lambda I -
L),$ and notice that $ P|_{{\cal W}_2} = I,$ the identity and that $P^2 = P.$ Thus ${\cal W}$ admits the direct
sum decomposition as follows, $$ {\cal W} = {\cal W}_1 \oplus {\cal W}_2.$$ Clearly $R(\lambda,{\cal A}) \in
{\cal L}({\cal H}, {\cal W}_2).$ For the source space, let $U$ be a linear subspace of $H^{3/2}(\partial
\Omega,R^3)$ carrying the structure of a Banach space  such that $B_0: {\cal W}_1 \longrightarrow U$ is
surjective and  ${\Re } \equiv (B_0)^{-1} \in {\cal L}(U,{\cal W}_1).$  Now going back to our original problem
(\ref{eqEM9A})-(\ref{eqEM9B}), we can rewrite the first equation in the equivalent form
\begin{eqnarray} \partial e/\partial t =  {\cal A}e  + (\Pi - (\lambda I - L))e  \label{eqEM9C}
\end{eqnarray} with $ \Pi \equiv (\lambda I -{\cal A}).$ For $\lambda \in \rho({\cal A})$, the resolvent set of
${\cal A}$, the operator $\Pi$ has bounded inverse giving the resolvent $R(\lambda,{\cal A}).$ Using the direct
sum decomposition, we can express the solution as  the sum given by $e = e^1+ e^2, e^1 \in {\cal W}_1, e^2 \in
{\cal W}_2.$ Substituting this in
 equation (\ref{eqEM9C}), and following similar  steps as presented in (\cite{ahmed88}, p59-62
 , we arrive at the following abstract  Cauchy problem
\begin{eqnarray} &~&  \dot \zeta   = {\cal A} \zeta  + \Lambda {\Re} u, \zeta_0 \equiv  \zeta (0)
= R(\lambda,{\cal A})e_0,  \label{eqEM9D} \\
 &~& \Lambda  \equiv (I- R(\lambda,{\cal A})(\lambda I - L)), \label{eqEM9E} \\ &~&  e =
\Pi  \zeta . \label{eqEM9F} \end{eqnarray} Using the unitary group  introduced above, the  mild solution of the
system (\ref{eqEM9D})-(\ref{eqEM9E})- (\ref{eqEM9F})  is given by
\begin{eqnarray} &~& \zeta (t) =  {\cal S}(t) \zeta_0 +  \int_0^t {\cal S}(t-s) \Lambda {\Re}  u(s) ds, t \in I,
\label{eqEM9G}\\
&~& e(t) = \Pi \zeta (t), t \in I. \label{eqEM9H}\end{eqnarray}  Briefly  this  is the first method. It is clear
from the expression (\ref{eqEM9G}) that, for every $u \in L_2(I,U)$ and $ \zeta_0 \in D({\cal A}),$  $\zeta \in
C(I, D({\cal A}))$ and hence it follows from (\ref{eqEM9H}) that $e \in C(I,{\cal H}).$  We collect these facts
together in the following theorem.
\par \noindent {\bf Theorem 2.2} For every $e_0 \in {\cal H}$ and $u \in L_2(I,U)$,  the initial boundary value
problem (\ref{eqEM9A})-(\ref{eqEM9B}) has a unique mild solution  $e \in C(I,{\cal H}),$ and it is given by the
expressions (\ref{eqEM9G}) and (\ref{eqEM9H}).
\par\noindent  {\bf Method B} (Principle of Transposition):  The second method, which admits much more general boundary
data, is the method of transposition~(\cite{lions70}, p231, p283)
This method admits $L_2(\partial \Omega)$ data and, more generally, data from Sobolev spaces with negative norm
like $H^{-1/2}(\partial \Omega).$ The method consists of constructing a suitable isomorphism and then
transposing the isomorphism  for the solution of nonhomogeneous Dirichlet problems like
(\ref{eqEM9})-(\ref{eqEM11}). Consider the homogeneous Dirichlet problem
\begin{eqnarray}  &~& L\psi \equiv  \partial^2 \psi  / \partial t^2 - c^2 \triangle \psi = f,
\xi \in \Omega, t \geq 0, \label {eqEM15}  \\  &~& \psi(T,\xi) = 0 , \dot \psi(T,\xi) = 0, \xi \in \Omega,
\label {eqEM16}  \\  &~& \psi(t,\xi)|_{\partial \Omega} = 0, \xi \in \partial \Omega, t \geq 0, \label {eqEM17}
\end{eqnarray} for $f \in L_2(Q,R^3) \equiv L_2(I, L_2(\Omega,R^3)).$  Reversing the flow of time, it follows from the
results of the  previous subsection that for every $f \in L_2(Q, R^3)$ this problem has a unique solution $\psi
\in H^2(Q,R^3).$ Now  introduce the vector space $\Psi$ by \begin{eqnarray*} \Psi \equiv \biggl\{ \psi \in
L_2(Q,R^3): L\psi \in L_2(Q,R^3), \psi(T,\cdot) = 0, \dot \psi(T,\cdot) = 0, \psi|_{I\times \partial \Omega} = 0
\biggr\}\end{eqnarray*} and furnish it with the norm topology given by
\begin{eqnarray} \parallel \psi \parallel_{\Psi} \equiv \parallel L\psi \parallel_{L_2(Q,R^3)}\label{eqEM19} .\end{eqnarray}
The reader can easily verify that $\Psi$ is a  normed linear space. Since $L$ is a closed operator, it  follows
that $\Psi$ is a Banach space, in fact a Hilbert space. Thus it follows from the given norm topology that $L$ is
an iosmetric isomorphism of $\Psi$ onto $L_2(Q,R^3) \equiv L_2(I,L_2(\Omega,R^3)).$ For convenience of notation
we may express this fact by stating that  $$ L \in Iso(\Psi,L_2(Q,R^3)).$$ This  is known as the adjoint
isomorphism. Transposing this isomorphism, we can  settle the question of existence of solution of our original
nonhomogeneous boundary value problem (\ref{eqEM9})-(\ref{eqEM11}). This is stated in the following Theorem.
 \par\noindent {\bf Theorem 2.3} Consider the system (\ref{eqEM9})-(\ref{eqEM11}) and suppose $E_0 \in
H^{-1/2}(\Omega,R^3)$, $E_1 \in H^{-3/2}(\Omega,R^3)$ and $u \in L_2(I,H^{-1/2}(\partial \Omega,R^3)) \subset
H^{-1/2}(I\times \partial \Omega,R^3)$. Then the system (\ref{eqEM9})-(\ref{eqEM11}) has a unique solution $E
\in L_2(Q,R^3) = L_2(I,L_2(\Omega,R^3)).$ \vskip6pt\noindent {\bf Proof} Formally, scalar multiplying equation
(\ref{eqEM9}) by any $\psi \in \Psi $ and using Greens formula for integration by parts, one can easily derive
the following identity
\begin{eqnarray} &~& \int_{I\times \Omega} (E,L\psi) d\xi dt \nonumber  \\ &~& = \int_{\Omega}(E_1,\psi(0)) d\xi - \int_{\Omega}
(E_0,\dot \psi(0))d\xi - c^2 \int_{I\times \partial \Omega} (\partial \psi/\partial \nu,u(t,\xi) d\sigma(\xi)dt
,\label{eqEM19}\end{eqnarray} where $\partial \psi/\partial \nu$ denotes the partial derivative of $\psi$  in
the outward direction  of the unit normal vector $\nu$ at any position on the boundary $\partial \Omega$ and
$\sigma$ denotes  the surface (Lebesgue) measure on the boundary.  Define the functional
\begin{eqnarray} \ell(\psi) \equiv   \int_{\Omega}(E_1,\psi(0)) d\xi - \int_{\Omega}
(E_0,\dot \psi(0))d\xi - c^2 \int_{I\times \partial \Omega} (\partial \psi/\partial \nu,u(t,\xi)) d\sigma(\xi)dt
.\label{eqEM20}\end{eqnarray} Clearly this is a linear functional. Since $\psi \in H^2(Q,R^3)$, it follows from
standard trace theorems for Sobolev spaces  that  $\psi(0) \equiv \psi(0,\xi), \xi \in \Omega,$ is an element of
$H^{3/2}(\Omega,R^3),$  $\dot \psi(0) \in H^{1/2}(\Omega,R^3)$ and $\partial \psi/\partial \nu \in
H^{1/2}(I\times \partial \Omega,R^3).$ Thus for the given data $\{E_0,E_1,u\}$ with the regularities as
specified in the statement of the theorem, the scalar products on the righthand side of the identity
(\ref{eqEM20}) have the correct duality pairings. Hence we conclude that the given data determines a continuous
and hence a bounded linear functional  $\ell$ on the Banach space $\Psi.$ Since $ L \in Iso (\Psi, L_2(Q,R^3))$,
this means that  the composition map $(\ell o  L^{-1})$ is a continuous linear functional on $L_2(Q,R^3).$
Hence, by Riesz representation theorem, there exists an unique  $ E \in L_2(Q,R^3)$ such that  \begin{eqnarray}
(\ell o L^{-1})(f) = (E,f)_{L_2(Q,R^3)}~~\forall~~ f \in L_2(Q,R^3).\label{eqEM21} \end{eqnarray} Since $L$ is
an isomorphism, this is equivalent to saying that \begin{eqnarray} \ell(\psi) =
(E,L\psi)_{L_2(Q,R^3)}~~\forall~~ \psi \in \Psi .\label{eqEM21} \end{eqnarray} This also  verifies the validity
of the formal identity (\ref{eqEM19}) obtained by  integration by parts.  The uniqueness is a consequence of the
fact that $L \in Iso(\Psi,L_2(Q,R^3)).$  This completes the proof. $\bullet $
 \vskip6pt \noindent {\bf Remark 2.4} It
is clear from the above result that our nonhomogeneous Dirichlet initial boundary value problem
(\ref{eqEM9})-(\ref{eqEM11}) has a unique solution $E \in L_2(Q,R^3)$ for  a very general set of
 data form the class of generalized functions $$\{ E_0,E_1,u\} \in H^{-1/2}(\Omega,R^3) \times H^{-3/2}(\Omega,R^3) \times
 H^{-1/2}(I\times \partial \Omega,R^3).$$ For practical applications we  may limit our data from the Hilbert
 spaces $ L_2(\Omega,R^3)\times L_2(\Omega,R^3) \times L_2(I,L_2(\partial \Omega,R^3))$. In this case, off course, we expect
 our solutions to be much more regular or smooth.
\vskip6pt \noindent  Note that the data to solution map $\{E_0,E_1,u\} \longrightarrow E,$ which we denote by
$G,$ is a continuous linear map from  $H^{-1/2}(\Omega,R^3) \times H^{-3/2}(\Omega,R^3) \times H^{-1/2}(I\times
\partial \Omega,R^3)$ to $L_2(Q,R^3) \equiv L_2(I,L_2(\Omega,R^3))$ and hence there exists a
constant $K> 0$ such that \begin{eqnarray*} &~&  \hskip-30pt\parallel G(E_0,E_1,u)\parallel_{L_2(Q,R^3)} ~\leq \\
&~& K\biggl\{ \parallel E_0\parallel_{H^{-1/2}(\Omega,R^3)} + \parallel
E_1\parallel_{H^{-3/2}(\Omega,R^3)}\parallel + \parallel u\parallel_{H^{-1/2}( I\times \partial
\Omega,R^3)}\biggr\}. \end{eqnarray*} Since for $s>0$, the embeddings $ L_2(Q)\hookrightarrow H^{-s}(Q)$ are
continuous, it follows from the above result that for $\{E_0,E_1,u\} \in L_2(\Omega,R^3) \times
L_2(\Omega,R^3)\times L_2(I,L_2(\partial \Omega,R^3))$
\begin{eqnarray*} &~&  \hskip-30pt\parallel G(E_0,E_1,u)\parallel_{L_2(Q,R^3)} ~\leq \\ &~&
\tilde K \biggl\{ \parallel E_0\parallel_{L_2(\Omega,R^3)} + \parallel E_1\parallel_{L_2(\Omega,R^3)}\parallel +
\parallel u\parallel_{L_2( I, L_2(\partial \Omega,R^3)) }\biggr\}, \end{eqnarray*} where $\tilde K$ depends on
$K$ and the embedding constants $$ L_2(\Omega,R^3) \hookrightarrow
H^{-1/2}(\Omega,R^3),L_2(I,L_2(\partial\Omega,R^3)) \hookrightarrow  H^{-1/2}(I\times
\partial \Omega,R^3). $$

\vskip6pt\noindent In the study of communication problems we will set $E_0 = E_1 = 0$ and consider the boundary
data as the input source giving $E = G(u) \in L_2(I,H).$  A complete characterization of the input-output  map
is given in section 3. \vskip6pt \noindent {\bf Remark 2.5} It is interesting to note that Theorem 2.2 dealing
with the question of existence and regularity properties of solutions also provides a clue to numerical
technique for solving the basic problem (\ref{eqEM9})-(\ref{eqEM11}). Let $\{f_i\} \subset L_2(Q,R^3)$ be a
complete orthonormal set (orthonormality is not essential though linear independence is). Then note that
\begin{eqnarray*} (\ell o L^{-1})(f_i) = (E,f_i)_{L_2(Q,R^3)} \equiv c_i, i \in N. \end{eqnarray*}
These are precisely the
Fourier coefficients of $E$ with respect to the complete set $\{f_i\}$ as indicated by the righthand expression,
and hence $E$ is given by $ E = \sum_{i =1}^{\infty}  c_i f_i.$ Further it is clear that $c_i$,s are determined
 entirely by the data $\{E_0,E_1,u\} $ of the problem.
\vskip6pt\noindent  {\bf Remark 2.6} Comparing method A (Semigroup Approach)  with  method B (Principle of
Transposition), it is apparent that the later admits much more general data. At least for linear
initial-boundary value problems, semigroup theory seems to be  less powerful.

\section{ Formulation of Communication Problems}
\subsection{ Distributed Source:}{\bf   Transmit End  (T):} First we consider the system model described
by equation (\ref{eqEM3}) with the distributed source or control space $U = L_2(\Omega_0,R^4),$ that is, $$ U
\equiv \{ u \in L_2(\Omega,R^4): u(\xi) = 0, \forall ~ \xi \in \Omega \setminus \Omega_0\}.$$  For application
to communication problems we may simplify the source further by taking a finite number of disjoint closed
subsets $\{\sigma_i\}_{i=1}^n \subset \Omega_0$ and consider input source to be composed of the sum
\begin{eqnarray} u \equiv \sum_{i =1}^n x_i(t) \varphi_i(\xi), t \in I, \xi \in \Omega_0
\label{eqEM24}\end{eqnarray} where the functions $\varphi_i \in L_2(\Omega_0,R^4)$ vanishing outside $\sigma_i.$
 In other words, these functions have $\sigma_i$ as their supports and $x_i \in L_2(I)$ are scalar valued functions which are the
signals. These represent message signals radiated by the strategically located $n$-transmit antennas.
\vskip6pt\noindent {\bf Receiver End (R):} Let $S_0$ denote the receiving end of the wave guide. Sensors are
located on this set. Again let $\{\beta_i\}_{i=1}^m $ be a family of disjoint closed subsets of the set $S_0$
where the sensors are located. These sensors are assumed to be able to measure the electric field distribution
on these patches. These represent receiving antennas. In terms of the vector and scalar potential
$\{a,\varphi\}$ we have already seen that the electric field vector is given by
\begin{eqnarray*} E = -(\dot a + \nabla \varphi).
\end{eqnarray*} Hence in terms of the state variable we have
\begin{eqnarray} E \equiv -  \biggl(\begin{array}{clcr}  z_5+ \partial_1z_4  \\ z_6 + \partial_2z_4\\ z_7 + \partial_3 z_4
\end{array} \biggr) \equiv \Gamma z, \label{eqEM25} \end{eqnarray} where $\Gamma$ is the matrix of differential
operators easily determined by the above relation. Clearly, $\Gamma$ is a bounded linear operator from ${\cal
H}$ to $L_2(\Omega,R^3).$  The outputs are the integrals of weighted sensor response to the electric field
distribution on the patches. These are given by
\begin{eqnarray} y_i(t) = \int_0^t \biggl( \int_{\beta_i} <\alpha_i(\xi), E(s,\xi)>_{R^3} d\sigma(\xi)\biggr) ds
+ w_i(t),~ t \in I, i = 1,2\cdots,m ,\label{eqEM26}\end{eqnarray} where $\alpha_i$ is  the vector of  weight
given to the measured electric field distribution on  i-th site and  $w_i$ represents the measurement noise of
this site. The weight vector $\alpha_i$ may be assumed to be supported on the set $\beta_i.$   We assume that
$\{w_i\}_{i=1}^m$ are mutually independent standard Brownian motions with $W$ denoting the corresponding $m$
vector Brownian motion. Throughout the paper we use $(\Xi,{\cal F}, {\cal F}_t,P)$ to denote the filtered
probability space where ${\cal F}_t, t \geq 0,$ is an increasing family of right continuous subsigma algebras of
the sigma algebra ${\cal F}$. All random processes arising in this paper will be assumed to be based on this
complete filtered probability space.
\par  Now returning to our  problem and  using the source and the output models as described above, the state and the measurement
dynamics turn out to be  \begin{eqnarray} &~& \dot z = {\cal A} z  +  {\cal C}x, z(0) = z_0, \label{eqEM27}\\
&~& dy = {\cal G} \Gamma z dt + dW \label{eqEM28} \end{eqnarray} where the operators $\{ {\cal C}, {\cal G}\}$
are given by
$$ {\cal C} x \equiv  \sum_{i=1}^n x_i {\cal B}\varphi_i, ~\hbox{and}~ ({\cal G}_iE)(t) \equiv  \int_{\beta_i}
<\alpha_i(\xi),
 E(t,\xi)> d\sigma(\xi), i = 1,2,\cdots,m. $$ The reader can easily verify that the operators ${\cal C}: R^n \longrightarrow {\cal
 H}~\hbox{ and}~ ({\cal G} \Gamma): {\cal H} \longrightarrow R^m$  and they are bounded linear operators. Using the
 semigroup ${\cal S}(t),t\geq 0,$ corresponding to the operator ${\cal A},$ it follows from the expression
 (\ref{eqEM4}), that  \begin{eqnarray} z(t) = K_t(x) \equiv  {\cal S}(t)z_0 + \int_0^t {\cal S}(t-r){\cal C}x(r) dr
. \label{eqEMS1} \end{eqnarray} Define the composition map $F$ with values
  \begin{eqnarray} F_t(x) \equiv ({\cal G} \Gamma K_t)(x). \label{eqEMS2} \end{eqnarray}
 Clearly $F$ is a nonanticipative (causal) operator mapping $L_2(I,R^n)$ to $L_2(I,R^m)\cap C(I,R^m) $ and,
  being the composition of
 bounded linear operators, it is also a  bounded linear operator.   Thus the output
 equations
  (\ref{eqEM26}) can be written as as a linear  stochastic differential equation in $R^m$,
    \begin{eqnarray} dy = F_t(x) dt + dW, y(0) = 0,  t \in I. \label{eqEMS3}
  \end{eqnarray} In case of method B, the map $F$ is given by the composition map $F_t(x) = ({\cal G} G {\cal
  C})_ t(x),$ which is a bounded linear operator from $L_2(I,R^n)$ to $L_2(I,R^m).$

\subsection{Boundary Source:}  Next we consider the model described by the boundary value problem
(\ref{eqEM9})-(\ref{eqEM11}). Here we consider a part  $\partial \Omega_0 \subset \partial \Omega$ of the
boundary $\partial \Omega$ where the source is active.  Then for   the source space  we take  $L_2(I,U) $ where
$U$ is a closed linear  subspace of $ H^{3/2}(\partial \Omega_0,R^3) \equiv \{\varphi \in H^{3/2}(\partial
\Omega,R^3): \varphi (\xi) = 0 ~\hbox{for}~\xi \not \in \partial \Omega_0\}.$ Again we let $\{\sigma_i\} $
denote a family of disjoint subsets of the set $\partial \Omega_0$ and model the input as
\begin{eqnarray} u(t,\xi) \equiv \sum_{i=1}^n x_i(t) \psi_i(\xi), t \in I, \xi \in \partial
\Omega_0  \label{eqEM29}\end{eqnarray} where $\psi_i  \in  U $ and supported on the set $\sigma_i$ and $x_i \in
L_2(I).$ The complete system model is then given by
\begin{eqnarray}  &~&  \partial^2 E / \partial t^2 - c^2 \triangle E = 0,
\xi \in \Omega, t \geq 0, \label {eqEM30}  \\  &~& E(0,\xi) = 0 , \dot E(0,\xi) = 0 , \xi \in \Omega, \label
{eqEM31}  \\  &~& E(t,\xi)|_{\partial \Omega} = u(t,\xi) = \sum_{i=1}^n x_i(t) \psi_i(\xi), \xi \in \partial
\Omega, t \geq 0. \label {eqEM32} \end{eqnarray} For this model, we can use the representations
(\ref{eqEM9G})-(\ref{eqEM9H}) and (\ref{eqEM32}) to construct the output equation. Define the map
\begin{eqnarray} K_t(x) \equiv \Pi \biggl( {\cal S}(t)\zeta_0 + \int_0^t {\cal S}(t-r) \Lambda {\Re}{\cal C} x(r)
dr \biggr), t \in I. \label{eqEMS4} \end{eqnarray} Let $\Gamma$ denote the projection  map $\Gamma e \equiv e_1$
which projects $e \equiv  (e_1,e_2) =  (E,\dot E)$ to the first component $e_1 = E.$  Using these maps, again we
can write the output
equation  in the same general form  (\ref{eqEMS3}), \begin{eqnarray} dy  &=& ({\cal G}\Gamma K_t)(x) dt + dW \nonumber \\
&=& F_t(x) dt + dW, t \in I. \label{eqEMS5} \end{eqnarray} The existence of the map $F$ is  assured by the
expressions (\ref{eqEM9G})-(\ref{eqEM9H}) as presented in the semigroup approach (method A).
\subsection{Noisy Source}  So far we have assumed that the source is noise free. In order to admit noisy source one must
add some compatible  additional terms to the evolution equations (\ref{eqEM3}) and (\ref{eqEM9D}).  For the
distributed source, we replace the evolution equation  (\ref{eqEM3}) by the stochastic differential equation
\begin{eqnarray} dz = {\cal A}z dt + {\cal B}u dt + \sigma dW^o, t \geq 0, \label{eqES1} \end{eqnarray}
on the Hilbert space ${\cal H},$  where the operator $\sigma$ is given by a $8\times 4 $ matrix of operators
with the first four rows being all zero and the remaining $4\times 4$ matrix is a diagonal matrix of operators
$\{\sigma_i, i =1,2,3,4\}.$ The  Brownian motion $W^o$ is given by the vector $ W^o \equiv \hbox{col}
\{W_1^o,W_2^o,W_3^o,W_4^o\}$ of independent Brownian motions $$ W^o_i(t,\cdot) \equiv \{W^o_i(t,\xi),\xi \in
\Omega\} , i =1,2,3,4,$$ each  taking values possibly from $L_2(\Omega,R).$  In reference to the field equations
(\ref{eqEM1}) and (\ref{eqEM2}), this means adding distributed white noise on the righthand side of each of the
equations in the form $ \sigma_i \dot W^o_i(t)  \equiv \sigma_i(\cdot ) \dot W^o_i(t,\cdot), i = 1,2,3,4.$ This
model allows one to deal with  localized as well as distributed noise around the wave guide. Letting $V$ denote
any  separable Hilbert space, for example a closed linear subspace of  $L_2(\Omega,R^4),$  we may assume $W^o$
to be a
   $V$ valued Brownian motion,
independent of the Brownian motion $W$ (receiver noise), with covariance operator denoted by $Q^o$ and $\sigma
\in {\cal L}(V,{\cal H})$ so that $ Q^o_{\sigma} \equiv \sigma Q^o \sigma^*$ is a positive nuclear operator in
${\cal H}$.  A natural choice for the space $V$ is $L_2(\Omega_0,R^4) \equiv U,$ same as the source space,  and
$\sigma = {\cal B}.$ In any case this choice is determined primarily by physical requirements and mathematical
simplicities.  Since ${\cal S}(t),t \in R,$ is a unitary group, it is easy to verify that ${\cal
S}(t)Q^o_{\sigma}S^*(t)$ is a positive nuclear operator in ${\cal H}$ for all $t\in R$ given that $Q^o_{\sigma}$
is. Thus for each $u \in L_2(I,U),$ equation (\ref{eqES1}) has a unique mild solution $z,$ which belongs to  $
C(I,{\cal H})$  with probability one, possessing  bounded second moments. In this case the map $K_t(x), t \geq
0,$ is given by
\begin{eqnarray} K_t(x) \equiv  z(t) = {\cal S}(t)z_0 + \int_0^t {\cal S}(t-r){\cal C}x(r) dr + \int_0^t {\cal
S}(t-r) \sigma dW^o. \label{eqES2} \end{eqnarray}   \par Considering the boundary value problems,  for noisy
boundary source, equation (\ref{eqEM9D}) is replaced by \begin{eqnarray}  d\zeta  = ({\cal A}\zeta  + \Lambda
\Re u) dt + \Lambda \Re \sigma dW^o, t \geq 0. \label{eqES3} \end{eqnarray}   For  the boundary source we had
chosen $U \subset  H^{3/2}(\partial \Omega_0,R^3).$ Thus it is necessary that  $\sigma \in {\cal L}(V,U)$ where
$V$ is any separable Hilbert space supporting the Brownian motion $W^o.$ For example, $V \equiv L_2(\partial
\Omega_0,R^3)$ or any closed linear subspace thereof.  Again for the existence of mild solutions $e \equiv \Pi
\zeta \in C(I, {\cal H}),$ it suffices if $\Lambda \Re \sigma Q^o\sigma^* \Re^* \Lambda^*$ is a positive nuclear
operator in ${\cal H}.$ In this case, equation (\ref{eqEMS4}) is replaced; and the process $K_t(x), t \geq 0,$
is given by
\begin{eqnarray} K_t(x) \equiv \Pi \biggl( {\cal S}(t)\zeta_0 + \int_0^t {\cal S}(t-r) \Lambda {\Re}{\cal C}
x(r) dr  + \int_0^t {\cal S}(t-r) \Lambda {\Re} \sigma dW^o\biggr), t \in I. \label{eqES4} \end{eqnarray}
\vskip6pt \noindent {\bf Remark 3.1.} In case the sensors (receiving antennas) are nonlinear, the operator
${\cal G}$ is nonlinear and hence the composition map $F_t, t \in I,$ is also nonlinear. The results presented
in this paper remain valid provided this nonlinearity is  uniformly Lipschitz having at most linear growth.
\vskip6pt
\section{Channel Capacity}  In view of the preceding  discussions, we notice that the input and output spaces are
given by $X \equiv L_2(I,R^n)$ and  $Y \equiv C(I,R^m).$ Suppose these spaces are furnished with the
(topological) Borel algebra turning them into measurable spaces $(X,B_X)$ and $(Y,B_Y).$ Let $M(X)$  and $M(Y)$
denote the space of Borel probability measures on $(X,B_X)$ and $(Y,B_Y)$ respectively.  Considering the source
space, let $M_2(X) \subset M(X)$ denote the class of probability measures having  finite second moments, that
is, $$ \mu \in M_2(X) ~\hbox{if and only if}~ \int_{X} |x|_X^2~ \mu(dx) < \infty.$$ Since normally the source
power is limited, we consider a  bounded subset of $M_2(X)$ given by $$ S_r \equiv \{ \mu \in M_2(X): \int_X
|x|_X^2 ~ \mu(dx) \leq r T\}$$ where  $r>0$ is the power constraint and $T$ is   the length of the time interval
$I$ denoting the duration of the message source. For admissible source measures, we can choose  any set ${\cal
M}_{ad}$ which is a weakly compact and convex  subset of the set  $S_r$. For example, ${\cal M}_{ad} = wc\ell
(\Gamma_r)$ where $\Gamma_r$ is any  convex subset of the set  $S_r$ satisfying
$$ \lim_{n\rightarrow \infty} \sup_{\mu \in \Gamma_r} \int_X \sum_{i\geq n}(x,e_i)^2 \mu(dx) = 0$$ for any
orthonormal basis  $\{e_i\}$ of the Hilbert space $X.$ Under this  assumption, the set $\Gamma_r$ is uniformly
tight and hence conditionally weakly compact. Thus its weak closure is weakly compact. For more concrete
examples of compact sets ${\cal M}_{ad}$ see Remark 4.2 following theorem 4.1.
\par Considering the output space $(Y,B_Y),$ let $M(Y)$ denote the space of regular Borel probability measures
on it. Let $M(X\times Y)$ denote the space of joint Borel probability measures on the product sigma algebra $B_X
\times B_Y.$ We have seen in section 3, that the output signal $y$ is related to the input process $x$  through
the communication system (\ref{eqEM27})-(\ref{eqEM28}) leading to (\ref{eqEMS3}) for the distributed source; and
(\ref{eqEM30})-(\ref{eqEM32}) leading to (\ref{eqEMS5})  for the Dirichlet source.  In other words, for a given
probability measure $\mu \in S_r \subset M_2(X)$ on the input space, there is a unique measure $\nu \in M(Y)$ on
the output space $Y$ induced by the channel.  Let $\gamma \in M(X\times Y)$ denote the joint probability measure
and $\mu \times \nu$ the product measure with $\mu$ and $\nu$ being the marginals of $\gamma.$  The relative
entropy of $\gamma$ with respect to the product measure $\mu\times\nu,$ denoted by $I({\cal X},{\cal Y}),$  is
called the mutual information which is a measure of the amount of information carried by the observable  noisy
output ${\cal Y}$  about the
 input message (source) ${\cal X}$.  This is given by the following expression,
  \begin{eqnarray} I({\cal X},{\cal Y} ) \equiv \int_{X\times Y} log \biggl( \frac {\gamma(dx\times
dy)}{\mu(dx)\times \nu(dy)} \biggr)~ \gamma(dx\times dy), \label{eqEM34}\end{eqnarray} where  $ \Upsilon(x,y)
\equiv \frac {\gamma(dx\times dy)}{\mu(dx)\times \nu(dy)}$ denotes the Radon-Nikodym derivative of $\gamma$ with
respect to the product measure.  Clearly this requires that $\gamma$ be absolutely continuous with respect to
the product probability measure $\mu \times \nu.$ Note that the output measure $\nu$ is related to the input
measure $\mu$ through the channel operator and it is given by
\begin{eqnarray} \nu(D) = \gamma(X\times D) = \int_X q(x,D) \mu(dx), ~\forall ~ D\in B_Y \label{eqEM35}
\end{eqnarray} where $$ q(x,D) = Pr\{ y \in D| x\} $$ is the conditional probability of the output $y$ being in $D
\in B_Y$ given that the input  $x \in X .$  This is precisely the action of the channel on the input and, as we
have seen in the preceding sections, it is determined by the dynamic models of the channel.  A closed form
expression for this will follow shortly. Substituting the expression (\ref{eqEM35}) into the expression
(\ref{eqEM34}) we obtain
\begin{eqnarray} I({\cal X},{\cal Y}) \equiv J(\mu) \equiv \int_{X\times Y} log \biggl( \frac {q(x,dy)}
{\int_X q(\xi,dy)\mu(d\xi)}\biggr) q(x,dy) \mu(dx) \label{eqEM36}\end{eqnarray} which is a functional of the
 measures $q$ and $\mu.$   Since the channel dynamics is given, this is a functional of the
source measure only  as indicated above.  We have seen in the preceding section that, for both  the distributed
and the boundary sources, the output equation  has the general form given by  a linear  stochastic differential
equation in $R^m$,
\begin{eqnarray} dy = F_t(x) dt + dW, y(0) = 0, \label{eqEM37}\end{eqnarray} where $F$ is the causal (nonanticipative)  map  $F: L_2(I,R^n)
\longrightarrow L_2(I,R^m)$ as described earlier. This is a continuous linear map.  Now it  follows from
equation (\ref{eqEM37}) that for every given $x \in X$, $q(x,\cdot) $ is a Gaussian measure on $Y$ with mean
trajectory given by
\begin{eqnarray} \bar F(x) \equiv \{ \int_0^t F_s(x) ds, t \in I\} \in Y \label{eqEM38}
\end{eqnarray} while the covariance operator $Q_1$ is given by \begin{eqnarray*}  (Q_1\xi,\xi) \equiv
\int_{I^2}(K_1(t,s)\xi(s),\xi(t)) ds dt, \xi \in Y ,
\end{eqnarray*} with the kernel of the operator $Q_1$ being   $ K_1(t,s) \equiv (t\wedge s) I_m, (t,s) \in I\times I,$
and $I_m$ is the identity matrix of dimension $m.$    Thus the Channel Kernel is given by the conditional
Gaussian measure
\begin{eqnarray} q(x,D) \equiv N_G(\bar F(x),Q_1)(D), x \in X, D \in B_Y.  \label{eqEM39} \end{eqnarray}
Since $\bar F$ is a continuous linear map from $X$ to $Y$, and   $Range (\bar F(x)) \subset Range (Q_1^{1/2})$
for all $ x \in X,$ it is clear that,  for every $D \in B_Y$,  the map $x \longrightarrow q(x,D)$ is continuous
from $X$ to the interval $[0,1].$  Using the expression (\ref{eqEM39}) in the expression for the mutual
information given by (\ref{eqEM36}) we obtain the following equivalent expression,
\begin{eqnarray}  J(\mu) \equiv \int_{X\times Y} log \biggl( \frac {N_G(\bar F(x),Q_1)(dy)}
{\int_X N_G(\bar F(\xi),Q_1)(dy) \mu(d\xi)}\biggr) N_G(\bar F(x),Q_1)(dy) \mu(dx), \label{eqEM40}\end{eqnarray}
which is clearly dependent on the channel  operator $F.$ Denoting the convolution $$ \int_X N_G(\bar
F(x),Q_1)(D) \mu(dx) \equiv \nu_G(D), D \in B_Y$$ the reader can easily verify that $$ N_G(\bar F(x),Q_1)(\cdot)
\prec \nu_G(\cdot)$$ for $\mu$ almost all $x \in X.$ Thus the Radon-Nikodym derivative  of $N_G$ with respect to
the measure $\nu_G$ exists and hence $J(\mu)$ given by (\ref{eqEM40}) is well defined.  Now our objective is to
determine the capacity of the channel by maximizing the above functional  over a set of admissible measures on
the source space subject to  power constraints. That is we must find
\begin{eqnarray} C \equiv \sup \{ J(\mu), \mu \in  {\cal M}_{ad}\}  \label{eqEM41} \end{eqnarray}
 where the set ${\cal M}_{ad},$ as
defined before, is any weakly compact convex subset of the set
\begin{eqnarray}  S_r \equiv \{ \mu \in M_2(X): \int_X |x|_X^2 ~ \mu(dx) \leq r T\}. \label{eqEM42}
\end{eqnarray}   The first question that we must  address is: does the supremum exist
and, if  it does, is it attained on the set ${\cal M}_{ad}.$  Without much additional assumptions we can prove
the following result.
 \vskip6pt \noindent {\bf Theorem 4.1}  Suppose ${\cal M}_{ad}$  is a  weakly compact and convex subset of the set
  $S_r \subset M_2(X)$. Then, there  exists a unique  $\mu^o \in {\cal M}_{ad}$ at which $J$
attains its supremum. In other words capacity is attained. \vskip6pt \noindent {\bf Proof.} For the existence of
supremum on ${\cal M}_{ad}$ it suffices to prove that $J$ is weakly upper semicontinuos and bounded away from
$+\infty.$  For uniqueness we show that $J$ is strictly concave.  First we prove that the functional $\mu
\longrightarrow J(\mu)$ is concave and weakly upper semicontinuous on $M(X).$ For simplicity of notation we
revert back to our original notation and set $ N_G(\bar F(x),Q_1)(D) \equiv q(x.D), D \in B_Y.$ For the first
statement,  we show that the functional
\begin{eqnarray}  J(\mu) \equiv \int_{X\times Y} log \biggl( \frac {q(x,dy)}
{\int_X q(\xi,dy)\mu(d\xi)}\biggr)~ q(x,dy) \mu(dx) \label{eqEM43}\end{eqnarray} is concave. Define the measure
$\tilde \mu \in M(Y)$ by the convolution
$$\tilde \mu(D) \equiv \int_X q(x,D) \mu(dx), D\in B_Y. $$ Since, for each $D \in B_Y,$  $x \longrightarrow q(x,D)$ is
 continuous and bounded, with values from
$[0,1],$ this is well defined.   Choose any  $v \in M(Y)$ so that $q(x,\cdot) \prec v(\cdot)$ for all $x \in X.$
Such a choice is assured since $q(x,\cdot)$ is a regular Borel probability measure on $B_Y$ induced by  the
output process $\{y\}$ having  continuous sample paths.   Clearly $\tilde \mu \prec v$ also. Using this measure
we can express (\ref{eqEM43}) as the sum of two terms as follows,
\begin{eqnarray*}  J(\mu) &~& \equiv \int_{X\times Y} log \biggl( \frac {q(x,dy)} {\tilde
\mu(dy)}\biggr)~ q(x,dy) \mu(dx) \nonumber \\ &~& = \int_{X\times Y}\biggl\{  log \biggl(
\frac{q(x,dy)}{v(dy)}\biggr)  - log  \biggl( \frac {\tilde \mu(dy)}{v(dy)}\biggr)\biggr\}~  q(x,dy)
\mu(dx).\end{eqnarray*}  Then using Fubini's theorem  and interchanging the order of integration in the second
term we find that
\begin{eqnarray}  J(\mu) &=& \int_{X\times Y}\biggl\{ log \biggl( \frac{q(x,dy)}{v(dy)}\biggr)\biggr\}q(x,dy)
\mu(dx) - \int_Y log \biggl\{ \biggl( \frac {\tilde \mu(dy)}{v(dy)}\biggr)\biggr\}  \tilde \mu(dy)\nonumber \\
 &=& J_1(\mu) - J_2(\mu) \equiv J_1(\mu) - I_2(\tilde \mu).  \label{eqEM44}\end{eqnarray} Consider the function
 $$ \eta(x) \equiv \int_{Y} \log\biggl( \frac{q(x,dy)}{v(dy)}\biggr) q(x,dy).$$  We have already noted that for
 every $D \in B_Y$,
 $x \longrightarrow q(x,D)$ is continuous on  $X$ with values in $[0,1].$  This implies continuity of $\eta$.
 Indeed, $\eta$ is the uniform limit of the sequence of bounded continuous functions $\{\eta_m\}$ given by  $$ \eta_m(x) \equiv \sum_{i=1}^m \log\biggl(
 \frac{q(x,Y_i)}{v(Y_i)}\biggr) q(x,Y_i)$$ where  $\{Y_i\}$ is a partition of $Y$ by  pairwise disjoint
members $ Y_i \in B_Y.$
 Thus    $\mu
\longrightarrow J_1(\mu)$ is linear and  bounded. By definition, the second term  is  the relative entropy of
$\tilde \mu$ with respect to the measure $v.$  For an arbitrary but fixed $v\in M(Y)$, $\nu \longrightarrow
I_2(\nu) $ is  a strictly convex functional (\cite{dupuis97},Lemma 1.4.3, p36) on the set  $\{ \nu \in M(Y):
I_2(\nu) <\infty\}.$ This can be easily verified by use of Gibb's formula and strict convexity of the function
$\eta(\xi) \equiv \xi \log \xi, \xi \geq 0.$  Thus $I_2$ is strictly convex and hence $-I_2$ is strictly
concave.
  Combining these facts we conclude that
$\mu \longrightarrow J(\mu)$ is strictly concave proving the first part of the statement.  Now we consider the
question of  continuity. It is clear from the expression (\ref{eqEM44}) that $J_1$ is bounded linear and hence
continuous with respect to the weak topology. The functional $J_2,$ or equivalently $I_2,$ gives   the relative
entropy of $\tilde \mu$ with respect to the measure $v$. Again, it is well known that relative entropy is weakly
lower semicontinuous (\cite{dupuis97},Lemma 1.4.3, p36). Thus $J_2$ is weakly lower semicontinuous and hence
$-J_2$ is weakly upper semicontinuous. Hence the functional $J$ given by their sum  is weakly upper
semicontinuous proving the second part of the statement. Thus we conclude that $\mu \longrightarrow J(\mu) $ is
strictly concave and weakly upper semicontinuous. Next we verify that $\sup \{ J(\mu), \mu \in {\cal M}_{ad}\} <
\infty.$ Clearly it suffices to verify that $\sup \{ J(\mu), \mu \in S_r\} < \infty.$  An alternative expression
for the mutual information, well known in the literature,  ~(\cite{duncan}, Duncan,Theorem 2, p269),  and
(\cite{lipster78},Lipster-Shirayev, Theorem 16.3, p174) is given by $$ I({\cal X},{\cal Y}) = (1/2) E\biggl\{
\int_I \biggl( |F_t(x)|_{R^m}^2 - |\hat F_t(y)|_{R^m}^2\biggr) dt\biggr\} $$ where $\hat F_t(y) \equiv E\biggl\{
F_t(x)|{\cal F}_t^y\biggr\}$ with ${\cal F}_t^y$ denoting the smallest sigma algebra with respect to which the
process  $\{y(s), s \leq t\} $ is measurable.  This identity holds for all finite dimensional stochastic
differential equations like (\ref {eqEMS3}) and (\ref{eqEMS5}) with $F$ nonanticipative  and $F_t$ taking values
from $R^m.$ Under the assumption, $P \biggl\{ \int_I | F_t(x)|_{R^m}^2 dt < \infty \biggr\}=1, $ the proof is
identical. We give a brief outline. Let $\gamma(dx\times dy)$ with its marginals, $\mu(dx)$ and $\nu(dy),$  be
the measures as introduced  earlier and let $\beta(dy)$ denote the Wiener measure on $Y$. For the system
$$ dy = F_t(x) dt + dW, y(0) = 0,$$ it follows from absolute continuity  of $\gamma$ with respect to $\mu \times \beta,$
 and  $\nu$ with respect to $\beta,$ and  Girsanov measure substitution that  the Radon-Nikodym
derivative of $\gamma$ with respect to $\mu\times \beta$ is given by \begin{eqnarray*}  \bigl(\gamma(dx\times
dy)/\mu(dx)\times \beta(dy)\bigr) (x,y) &=& \exp\biggl\{ \int_I (F_t(x),dy) -(1/2) \int_I |F_t(x)|_{R^m}^2 dt\biggr\} \\
&=& \exp\biggl\{ (1/2) \int_I |F_t(x)|^2 dt + \int_I (F_t(x),dW) \biggr\}. \end{eqnarray*} Similarly, the RND of
$\nu$ with respect $\beta$ is given by \begin{eqnarray*}(\nu(dy)/\beta(dy))(y)=  \rho(y)  &\equiv& \exp\biggl\{
\int_I (\hat F_t(y),dy) -(1/2) \int_I |\hat F_t(y)|^2 dt \biggr\} \\ &=& \exp\biggl \{ \int_I (\hat
F_t(y),F_t(x))dt -(1/2) \int_I |\hat F_t(y)|^2 dt + \int_I <\hat F_t(y), dW)  \biggr\},
\end{eqnarray*} where $ \hat F_t(y) = E\bigl\{ F_t(x)|{\cal F}_t^y \bigr\}.$  Since by assumption $P \biggl \{
\int_I |F_t(x)|_{R^m}^2 dt < \infty \biggr\} =1$ it is clear that $\beta \{ \rho(y) = 0\} =0$ and hence $\beta\{
1/\rho(y) < \infty\} = 1.$ From these it follows that the RND of $\gamma$ with respect to the product measure
$\mu\times \nu$ is given by   \begin{eqnarray*} &~& (\gamma(dx\times dy)/\mu(dx)\times \nu(dy)\bigr)(x,y) =  \\
&~& ~~~~~~~~~~~~~~~= \exp\biggl\{ (1/2) \int_I |F_t(x)-\hat F_t(y)|_{R^m}^2 dt + \int(F_t(x)-\hat
F_t(y),dW)\biggr\}.\end{eqnarray*} Using this expression and the  definition of mutual information along with
standard properties of
 conditional expectations, we obtain
 \begin{eqnarray*} &~&  I({\cal X},{\cal Y}) \equiv E \biggl\{ \log \{ (\gamma(dx\times dy)/\mu(dx)\times \nu(dy)\bigr)(x,y)\}\biggr\}   =  \\
&~& ~~~~~~~~~~~~~~~= (1/2) E \biggl\{  \int_I |F_t(x)|^2 -\hat F_t(y)|^2 dt\biggr\} .\end{eqnarray*} This ends
the outline. Conditional expectation is a contraction map and so $I({\cal X},{\cal Y})\geq 0,$ as expected, and
we have $$ J(\mu) \equiv I({\cal X},{\cal Y}) \leq  (1/2) E \int_I |F_t(x)|_{R^m}^2 dt  \leq  (1/2) \int_X
\biggl\{ \int_I | F_t(x)|_{R^m}^2 dt\biggr\} \mu(dx).$$  Since $F$ is a bounded linear operator from $X$ to $Y$
there exists a finite positive number $K,$ independent of $t \in I,$  so that $ |F_t(x)|_{R^m}^2   \leq K ^2
|x|_{L_2([0,t],R^n)}^2 , t \in I.$  Hence it follows from the above inequality that for $\mu \in M_2(X)$,
$$ J(\mu) \leq (K^2T/2) \int_X |x|^2 \mu(dx) < \infty.$$ Thus on $S_r \subset M_2(X),$ we have
$$ \sup_{\mu \in S_r} J(\mu) \leq (K^2T^2 r/2) < \infty $$  proving that the functional $\mu
\longrightarrow J(\mu)$ is  bounded away from $+\infty$  on $S_r.$ Since $J$ is weakly upper semicontinuous and
bounded away from $+\infty$ and,  by our choice, the admissible set ${\cal M}_{ad} \subset S_r $ is weakly
compact, $J$ attains its supremum on ${\cal M}_{ad}.$  This proves the existence of a  $\mu^o \in {\cal M}_{ad}$
at which the capacity $C$ is attained, that is, $C = J(\mu^o).$ The uniqueness follows  from strict concavity
of  $ \mu\longrightarrow J(\mu)$ and convexity of ${\cal M}_{ad}.$  This completes the proof.  $\bullet$
\vskip6pt \noindent {\bf Remark 4.2 (Some Examples of Compact sets):} Here we present  some  simple  examples of
weakly compact subsets ${\cal M}_{ad}$ of the set $S_r.$ {\bf (E1):} Let $\{\mu_k\} \subset S_r $ be a family of
distinct measures, in the sense that $\mu_k \not \equiv \mu_m $ on $B_X$ for  $k \not = m,$ and $\Lambda $ a
subset of $\ell_1$ satisfying the following properties:
\begin{eqnarray*} &~& (1): \alpha_k \geq 0, \sum_{k=1}^{\infty} \alpha_k = 1  ~\hbox{for}~ \alpha \in \Lambda,
\\ &~& (2): \lim_{N \rightarrow \infty} \sum_{k \geq N} \alpha_k  = 0 , ~\hbox{uniformly in}~ \alpha \in
\Lambda.
\end{eqnarray*} Define the set $M_{\Lambda} \equiv \{ \mu \in M(X) : \mu = \sum_{k\geq 1} \alpha_k \mu_k, \alpha
\in \Lambda \}.$ The reader can easily verify that $M_{\Lambda} \subset S_r$ and it  is  uniformly tight and so
relatively weakly compact. Since $\Lambda$ is compact, $M_{\Lambda}$ is closed and hence it is    weakly
compact.
 {\bf (E2):} A variant of this example for which the  same conclusion
 holds is as follows. Let $X_0$ be  a   countable  dense subset of the closed ball $ B_{a}(X)$ of  $X$ of radius
$a \leq rT.$ Then the  set   $ M_{\Lambda}(X_0)  \equiv \{ \mu \in M(X) :  \mu = \sum_{k\geq 1} \alpha_k
 \delta_{x_k}, x_k \in X_0,
 \alpha \in \Lambda \}$ is a  weakly compact subset of $S_r.$ {\bf (E3):}  If $\Upsilon$ is any  uniformly tight subset
 of $S_r$, then the set ${\cal M}_{ad} \equiv  wc\ell(\Upsilon)$ given by the weak closure of $\Upsilon$   is  weakly
compact. {\bf (E4):} Let $D$ be a compact subset of the  Hilbert space $X$ and $M(D)= \{\mu \in  M(X):
\mu(X\setminus D) = 0\}.$ Clearly $M(D)$ is a weakly compact subset of $M(X).$  We may choose $D$ such  that $
S_r \cap M(D) \neq \emptyset $ and then take ${\cal M}_{ad} \equiv  S_r \cap  M(D).$  Since $X = L_2(I,R^n)$, it
follows from Sobolev embedding theorems that  for any finite interval $I$ and any $p \in [2,\infty)$ the
embedding $W^{1,p}(I,R^n) \hookrightarrow L_2(I,R^n)$ is continuous and compact. Thus a good example of a
compact set $D$ of $X$ is any closed bounded subset $D$ of $W^{1,p}(I,R^n).$
\section{Maximizing Source Measure}
In this section we wish to present necessary conditions that a (maximizing) measure, subject to energy
constraints and  determining the channel capacity, must satisfy. Such conditions are called necessary conditions
of optimality. In fact in the following result we have both necessary and sufficient conditions of optimality.
\vskip6pt \noindent {\bf Theorem 5.1} In order for $\mu^o \in {\cal M}_{ad} \subset M_2(X)$ to be the optimum
source measure, it is necessary and sufficient that the following inequality holds
\begin{eqnarray}&~&  \int_{X\times Y} \biggl\{ \log \biggl( \frac{N_G(\bar F(x),Q_1)(dy)}{\int_X
N_G(\bar F(\xi),Q_1)(dy) \mu^o(d\xi)} \biggr)\biggr\} N_G(\bar F(x),Q_1)(dy) \mu^o(dx)\nonumber \\ &~& ~~\geq
\int_{X\times Y} \biggl\{\log \biggl( \frac{N_G(\bar F(x),Q_1)(dy)}{\int_X N_G(\bar F(\xi),Q_1)(dy) \mu^o(d\xi)}
\biggr)\biggr\} N_G(\bar F(x),Q_1)(dy) \mu(dx)
 \label{eqEM45}\end{eqnarray} for all $\mu \in {\cal M}_{ad}.$\vskip6pt \noindent {\bf Proof}
  Let $\mu^o\in {\cal M}_{ad} $ denote the optimizer, that is,
$$  J(\mu^o) = C \equiv \sup \{ J(\mu), \mu \in  {\cal M}_{ad}\}.$$ Then for any $\mu \in  {\cal M}_{ad},$   define
$\mu^{\varepsilon} \equiv \mu^o + \varepsilon (\mu-\mu^o)$ for $\varepsilon \in (0,1).$  Since $ {\cal M}_{ad}$
is convex, it is clear that $\mu^{\varepsilon} \in {\cal M}_{ad}.$  Clearly  $\mu^o$ being  the maximal element,
we have $J(\mu^o) \geq J(\mu^{\varepsilon}).$ Let  $DJ(\mu)$ denote the Gateaux gradient of $J$ at $\mu$
whenever it exists. Then computing  the limit, $$ \lim_{\varepsilon \downarrow 0}\biggl\{ \frac{
J(\mu^{\varepsilon})-J(\mu^o)}{ \varepsilon} \biggr\} = <DJ(\mu^o),\mu-\mu^o>,$$ and noting that for optimality,
\begin{eqnarray}   <DJ(\mu^o),\mu-\mu^o> ~ \leq ~ 0, \label{eqEM46} \end{eqnarray}  it is easy to verify that
\begin{eqnarray} \int_{X\times Y} \biggl\{ 1 - \log\biggl( \frac{N_G(\bar F(x),Q_1)(dy)}{\int_X N_G(\bar
F(\xi),Q_1)(dy) \mu^o(d\xi)} \biggr) \biggr\} N_G(\bar F(x),Q_1)(dy) (\mu-\mu^o)(dx) \geq 0,\nonumber \\
\label{eqEM47}
\end{eqnarray} $ \forall~ \mu \in  {\cal M}_{ad}.$ The inequality (\ref{eqEM45}) now easily follows from (\ref{eqEM47}).
  For the sufficiency, recall that $\mu\rightarrow
J(\mu)$ is concave. Hence \begin{eqnarray}  <DJ(\mu),\nu-\mu> ~ \geq ~J(\nu)-J(\mu)~~ \forall \mu,\nu \in  {\cal
M}_{ad}. \label{eqEM48}\end{eqnarray} Taking $\mu = \mu^o$ it follows from this inequality  and (\ref{eqEM45}),
or equivalently  (\ref{eqEM46}),   that
\begin{eqnarray} 0 ~ \geq ~ <DJ(\mu^o),\nu-\mu^o> ~ \geq ~ J(\nu)-J(\mu^o)~~ \forall~~\nu \in
 {\cal M}_{ad} .\label{eqEM48}\end{eqnarray} and hence $J(\mu^o) \geq J(\nu) ~\forall~ \nu \in {\cal M}_{ad}.$
  This proves the
sufficiency of condition (\ref{eqEM45}) thereby  completing the proof. $\bullet $ \vskip6pt \noindent {\bf
Remark 5.2} ~  So far in sections 4 and 5,  we have assumed the source of electro-magnetic fields to be  noise
free. As seen in section 3.3, to include noisy source, we must  consider the  evolution equations (\ref{eqES1})
and (\ref{eqES3}) which are the  stochastic versions of Maxwell (field)  equations. With the  source noise
included, the results of Theorem 4.1 and Theorem 5.1 remain valid with the replacement of the Channel Kernel
$N_G(\bar F(x),Q_1)(\cdot)$ by  $N_G(\bar F(x),Q_1 + Q_2)(\cdot)$ where $Q_2$ is the covariance operator
associated with the source noise. For the distributed source model (\ref{eqES1}), the covariance operator $Q_2$
is given by
\begin{eqnarray} (Q_2\xi,\xi) = \int_{I\times I} <K_2(t,\tau) \xi(t),\xi(\tau)> dt d\tau
\end{eqnarray} where the kernel $K_2$ is given in terms of the system parameters as follows:
 \begin{eqnarray} K_2(t,\tau) =  \int_0^{t\wedge \tau} dr R(t,r)
R^*(\tau,r), \end{eqnarray} with $R(t,r), 0 \leq r \leq t \leq T,$ given by  \begin{eqnarray}  R(t,r) \equiv
{\cal G} \Gamma\int_r^t {\cal S}(\theta-r) \sigma d\theta = \int_r^t {\cal G} \Gamma {\cal S}(\theta-r) \sigma
d\theta.\end{eqnarray} The last identity follows from the fact that ${\cal G}\Gamma$ is a bounded linear
operator  from the state space ${\cal H}$ to $R^m$  and hence closed and so commutes with the integral
operation.
 Similar expressions can be derived for the boundary source.  These  conclusions  are  based on the properties of
 stochastic convolutions,
see (\cite{daprato92},Theorem 5.2, p119)
which use stochastic Fubini's theorem~(\cite{daprato92},Theorem 4.18, p109)

\vskip6pt \noindent {\bf  Remark  5.3 (Numerical Algorithm for Computation):} Based on the necessary (and
sufficient) conditions of optimality as presented above, we can develop a gradient based algorithm for numerical
computation. In particular, for simple sources like those of examples, (E1) and (E2), the functional $J(\mu)$ on
${\cal M}_{ad}$ can be redefined as being a functional on  $\Lambda$ and, with slight abuse of notation, we may
denote it by $J(\alpha).$ Our problem is to find an $\alpha^o \in \Lambda$ at which $J$ attains its supremum.
For the source (E1), it follows from the necessary conditions that in order for $\alpha^o \in \Lambda $ to be
the optimal distribution of weights assigned to the family of measures $\{\mu^k\}$, it is necessary and
sufficient that we have,
\begin{eqnarray} \sum_{k\geq 1} (\alpha_k -\alpha_k^o) \int_X   L_{\alpha^o}(x)~  \mu^k(dx) \geq 0 , ~~\forall~~
\alpha \in \Lambda, \end{eqnarray} where $$ L_{\alpha^o} (x) \equiv \int_Y \log\biggl ( q(x,dy)/ \sum_{k \geq 1}
\alpha_k^o ~ \nu^k(dy)\biggr) q(x,dy),~ \nu^k(dy) = \int_X q(x,dy) \mu^k(dx). $$  For the  source (E2), this
further reduces to $$ \sum_{k\geq 1} (\alpha_k-\alpha_k^o) L_{\alpha^o}(x_k) \geq 0, ~\forall~ \alpha \in
\Lambda,$$ where $$L_{\alpha^o}(x_k) \equiv \int_Y \log\biggl( q(x_k,dy)/\sum_{k\geq 1}\alpha_k^o~
q(x_k,dy)\biggr) q(x_k,dy).$$  For these simple sources, one can write a gradient based numerical algorithm for
finding the optimum source. Considering  the first source, the gradient of $J$ at the n-th iteration is given by
$$ \triangle J(\alpha^n) =\{ \triangle_kJ(\alpha^n) =  \int L_{\alpha^n}(x) \mu^k(dx), k = 1,2,\cdots \}.$$
Choose  $$  \alpha_k^{n+1} \equiv \alpha_k^n + \varepsilon_n \triangle_k J(\alpha^n), k \in N, $$ with
$\varepsilon_n
>0$ sufficiently small so that $\alpha^{n+1} \in \Lambda.$   Using this $\alpha^{n+1}$, we compute the objective functional
 giving  $$ J(\alpha^{n+1}) =  J(\alpha^n) + \varepsilon \sum_{k=1}^{\infty} (\triangle_k
J(\alpha^n))^2 + 0(\varepsilon).$$ For $\varepsilon_n >0$ sufficiently small the series converges guaranteeing
improvement of  the objective functional at each step. Similar conclusion holds for the source (E2) with the
gradient vector given by $$ \triangle J(\alpha^n) =\{ \triangle_kJ(\alpha^n) =  L_{\alpha^n}(x_k) , k =
1,2,\cdots \}.$$.

\section{Conclusion} We have presented a complete dynamic model for MIMO channels
(wave guides, cavities)  based on deterministic as well as stochastic Maxwell's equations. To the best  of our
knowledge, this  formulation  has not been considered in the literature. Both distributed and boundary sources
have been considered. Proof of existence of maximizing source measure subject to power constraints has been
presented. Optimality conditions have been developed which can be used for numerical computations as indicated
in remark 5.3 for simple source spaces. The model used for sensors  in the above formulation can be extended to
cover linear (sensor) dynamics without any difficulty. This requires inclusion of the  associated transition
operator for the construction of the output map $F_t(x).$
 It would be interesting to study similar problems with nonlinear sensor dynamics.
\par\noindent {\bf Acknowledgement} (1):  The first author would like to thank his colleague,  Professor D. A. McNamara, for many valuable
discussions on physical properties of waveguide channels. (2):  Also thanks are due to National Science and
Engineering Research Council of Canada for partially supporting this research under grant no A7109.

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James Gilbert
Department of Biological and Marine Science (1)
Faculty of Science and Engineering (8)
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Sporopollenin as a dilution agent in artificial diets for solitary bees (2020)
Tainsh, F., Woodmansey, S. R., Austin, A. J., Bagnall, T. E., & Gilbert, J. D. (in press). Sporopollenin as a dilution agent in artificial diets for solitary bees. Apidologie, https://doi.org/10.1007/s13592-020-00801-1
Nutritional studies often require precise control of nutrients via dilution of artificial diets with indigestible material, but such studies in bees are limited. Common diluents like cellulose typically result in total mortality of bee larvae, making... Read More about Sporopollenin as a dilution agent in artificial diets for solitary bees.
Predation drives recurrent convergence of an interspecies mutualism (2018)
Feeney, W. E., Brooker, R. M., Johnston, L. N., Gilbert, J. D., Besson, M., Lecchini, D., …Manica, A. (2019). Predation drives recurrent convergence of an interspecies mutualism. Ecology letters, 22(2), 256-264. https://doi.org/10.1111/ele.13184
Mutualisms are fundamental ecological interactions that underpin much of the world's biodiversity. Recent studies have demonstrated how external pressures, such as predation, can regulate the dynamics of interspecific interactions and cause the break... Read More about Predation drives recurrent convergence of an interspecies mutualism.
Skew in ovarian activation depends on domicile size in phyllode-glueing thrips (2018)
Gilbert, J. D., Wells, A., & Simpson, S. J. (2018). Skew in ovarian activation depends on domicile size in phyllode-glueing thrips. Scientific reports, 8(1), https://doi.org/10.1038/s41598-018-21635-z
Costs and benefits of group living are a fundamental topic in behavioural ecology. Resource availability affects individuals' breeding prospects alone and in groups, as well as how reproduction is distributed within groups ("reproductive skew"). Here... Read More about Skew in ovarian activation depends on domicile size in phyllode-glueing thrips.
Direct Speech Reconstruction from Articulatory Sensor Data by Machine Learning (2017)
Gonzalez, J. A., Cheah, L. A., Gomez, A. M., Green, P. D., Gilbert, J. M., Ell, S. R., …Holdsworth, E. (2017). Direct Speech Reconstruction from Articulatory Sensor Data by Machine Learning. IEEE/ACM transactions on audio, speech, and language processing, 25(12), 2362-2374. https://doi.org/10.1109/TASLP.2017.2757263
© 2014 IEEE. This paper describes a technique that generates speech acoustics from articulator movements. Our motivation is to help people who can no longer speak following laryngectomy, a procedure that is carried out tens of thousands of times per... Read More about Direct Speech Reconstruction from Articulatory Sensor Data by Machine Learning.
Everything is connected: network thinking in entomology (2017)
Evans, D. M., Gilbert, J. D., & Port, G. R. (2017). Everything is connected: network thinking in entomology. Ecological entomology, 42, 1-3. doi:10.1111/een.12449
Editorial.
Aposematism in the burying beetle? Dual function of anal fluid in parental care and chemical defense (2017)
Lindstedt, C., Boncoraglio, G., Cotter, S., Gilbert, J., & Kilner, R. M. (2017). Aposematism in the burying beetle? Dual function of anal fluid in parental care and chemical defense. Behavioral ecology, 28(6), 1414-1422. https://doi.org/10.1093/beheco/arx100
© 2017 The Author. Published by Oxford University Press on behalf of the International Society for Behavioral Ecology. All rights reserved. Burying beetles (Nicrophorus vespilloides) bear distinctive and variable orange-black patterning on their ely... Read More about Aposematism in the burying beetle? Dual function of anal fluid in parental care and chemical defense.
Nonlinear Modeling and Verification of a Heaving Point Absorber for Wave Energy Conversion (2017)
Guo, B., Patton, R., Jin, S., Gilbert, J., & Parsons, D. (2017). Nonlinear Modeling and Verification of a Heaving Point Absorber for Wave Energy Conversion. IEEE Transactions on Sustainable Energy, 9(1), 453-461. https://doi.org/10.1109/tste.2017.2741341
Although the heaving Point Absorber (PA) concept is well known in wave energy conversion research, few studies focus on appropriate modelling of non-linear fluid viscous and mechanical friction dynamics. Even though these concepts are known to have n... Read More about Nonlinear Modeling and Verification of a Heaving Point Absorber for Wave Energy Conversion.
Male genital titillators and the intensity of post-copulatory sexual selection across bushcrickets (2017)
Lehmann, G. U., Gilbert, J. D., Vahed, K., & Lehmann, A. W. (2017). Male genital titillators and the intensity of post-copulatory sexual selection across bushcrickets. Behavioral ecology, 28(5), 1198-1205. doi:10.1093/beheco/arx094
Animal genitalia are diverse and a growing body of evidence suggests that they evolve rapidly under post-copulatory sexual selection. This process is predicted to be more intense in polyandrous species, although there have been very few comparative s... Read More about Male genital titillators and the intensity of post-copulatory sexual selection across bushcrickets.
Larval exposure to field-realistic concentrations of clothianidin has no effect on development rate, over-winter survival or adult metabolic rate in a solitary bee, Osmia bicornis (2017)
Nicholls, E., Fowler, R., Niven, J. E., Gilbert, J. D., & Goulson, D. (2017). Larval exposure to field-realistic concentrations of clothianidin has no effect on development rate, over-winter survival or adult metabolic rate in a solitary bee, Osmia bicornis. PeerJ, 5(6), e3417. https://doi.org/10.7717/peerj.3417
© 2017 Nicholls et al. There is widespread concern regarding the effects of agro-chemical exposure on bee health, of which neonicotinoids, systemic insecticides detected in the pollen and nectar of both crops and wildflowers, have been the most stron... Read More about Larval exposure to field-realistic concentrations of clothianidin has no effect on development rate, over-winter survival or adult metabolic rate in a solitary bee, Osmia bicornis.
Restoring speech following total removal of the larynx by a learned transformation from sensor data to acoustics (2017)
Gilbert, J. M., Gonzalez, J. A., Cheah, L. A., Ell, S. R., Green, P., Moore, R. K., & Holdsworth, E. (2017). Restoring speech following total removal of the larynx by a learned transformation from sensor data to acoustics. The Journal of the Acoustical Society of America, 141(3), EL307-EL313. https://doi.org/10.1121/1.4978364
Total removal of the larynx may be required to treat laryngeal cancer: speech is lost. This article shows that it may be possible to restore speech by sensing movement of the remaining speech articulators and use machine learning algorithms to derive... Read More about Restoring speech following total removal of the larynx by a learned transformation from sensor data to acoustics.
Macroecology of parental care in arthropods: higher mortality risk leads to higher benefits of offspring protection in tropical climates (2016)
Santos, E. S., Bueno, P. P., Gilbert, J. D. J., & Machado, G. (2017). Macroecology of parental care in arthropods: higher mortality risk leads to higher benefits of offspring protection in tropical climates. Biological Reviews, 92(3), 1688-1701 . https://doi.org/10.1111/brv.12303
The intensity of biotic interactions varies around the world, in such a way that mortality risk imposed by natural enemies is usually higher in the tropics. A major role of offspring attendance is protection against natural enemies, so the benefits o... Read More about Macroecology of parental care in arthropods: higher mortality risk leads to higher benefits of offspring protection in tropical climates.
Analytical and finite-element study of optimal strain distribution in various beam shapes for energy harvesting applications (2016)
Ooi, B. L., Gilbert, J. M., & Aziz, A. R. A. (2016). Analytical and finite-element study of optimal strain distribution in various beam shapes for energy harvesting applications. Acta Mechanica Sinica, 32(4), 670-683. https://doi.org/10.1007/s10409-016-0557-3
Due to the increasing demand for harvesting energy from environmental vibration, for use in self-powered electronic applications, cantilever-based vibration energy harvesting has attracted great interest from various parties and become one of the mos... Read More about Analytical and finite-element study of optimal strain distribution in various beam shapes for energy harvesting applications.
A silent speech system based on permanent magnet articulography and direct synthesis (2016)
Gonzalez, J. A., Cheah, L. A., Gilbert, J. M., Bai, J., Ell, S. R., Green, P. D., & Moore, R. K. (2016). A silent speech system based on permanent magnet articulography and direct synthesis. Computer speech & language, 39, 67-87. https://doi.org/10.1016/j.csl.2016.02.002
In this paper we present a silent speech interface (SSI) system aimed at restoring speech communication for individuals who have lost their voice due to laryngectomy or diseases affecting the vocal folds. In the proposed system, articulatory data cap... Read More about A silent speech system based on permanent magnet articulography and direct synthesis.
Modelling and optimisation of a bimorph piezoelectric cantilever beam in an energy harvesting application (2016)
Gilbert, J., Liu, J., Ooi, B. L., & Thein, C. K. (2016). Modelling and optimisation of a bimorph piezoelectric cantilever beam in an energy harvesting application. Journal of Engineering Science and Technology, 11(2), 212-227
Piezoelectric materials are excellent transducers in converting vibrational energy into electrical energy, and vibration-based piezoelectric generators are seen as an enabling technology for wireless sensor networks, especially in self-powered device... Read More about Modelling and optimisation of a bimorph piezoelectric cantilever beam in an energy harvesting application.
Integrating user-centred design in the development of a silent speech interface based on permanent magnetic articulography (2015)
Cheah, L. A., Gilbert, J. M., Gonzalez, J. A., Bai, J., Ell, S. R., Fagan, M. J., …Rychenko, S. I. (2015). Integrating user-centred design in the development of a silent speech interface based on permanent magnetic articulography. In Biomedical Engineering Systems and Technologies; Communications in Computer and Information Science, 324-337. Springer Verlag. https://doi.org/10.1007/978-3-319-27707-3_20
Abstract: A new wearable silent speech interface (SSI) based on Permanent Magnetic Articulography (PMA) was developed with the involvement of end users in the design process. Hence, desirable features such as appearance, port-ability, ease of use and... Read More about Integrating user-centred design in the development of a silent speech interface based on permanent magnetic articulography.
Sex-biased parental care and sexual size dimorphism in a provisioning arthropod (2015)
Field, J., Shreeves, G., Kennedy, M., Brace, S., & Gilbert, J. D. J. (2015). Sex-biased parental care and sexual size dimorphism in a provisioning arthropod. Behavioral Ecology and Sociobiology, 69(12), 1897-1906 . https://doi.org/10.1007/s00265-015-2002-1
The diverse selection pressures driving the evolution of sexual size dimorphism (SSD) have long been debated. Whilst the balance between fecundity selection and sexual selection has received much attention, explanations based on sex-specific ecology... Read More about Sex-biased parental care and sexual size dimorphism in a provisioning arthropod.
Questioning evidence of group selection in spiders (2015)
Grinsted, L., Bilde, T., & Gilbert, J. D. (2015). Questioning evidence of group selection in spiders. Nature, 524(7566), E1-E3. https://doi.org/10.1038/nature14595
Brief correspondence arising from J. N. Pruitt & C. J. Goodnight Nature 514, 359–362 (2014); doi:10.1038/nature13811
The evolution of parental care in insects: a test of current hypotheses (2015)
Gilbert, J. D. J., & Manica, A. (2015). The evolution of parental care in insects: a test of current hypotheses. Evolution, 69(5), 1255-1270. https://doi.org/10.1111/evo.12656
Which sex should care for offspring is a fundamental question in evolution. Invertebrates, and insects in particular, show some of the most diverse kinds of parental care of all animals, but to date there has been no broad comparative study of the ev... Read More about The evolution of parental care in insects: a test of current hypotheses.
Functional equivalence of grasping cerci and nuptial food gifts in promoting ejaculate transfer in katydids (2014)
Vahed, K., Gilbert, J. D. J., Weissman, D. B., & Barrientos-Lozano, L. (2014). Functional equivalence of grasping cerci and nuptial food gifts in promoting ejaculate transfer in katydids. Evolution, 68(7), 2052-2065. https://doi.org/10.1111/evo.12421
The function of nuptial gifts has generated longstanding debate. Nuptial gifts consumed during ejaculate transfer may allow males to transfer more ejaculate than is optimal for females. However, gifts may simultaneously represent male investment in o... Read More about Functional equivalence of grasping cerci and nuptial food gifts in promoting ejaculate transfer in katydids.
Design of wideband vibration-based electromagnetic generator by means of dual-resonator (2014)
Ooi, B. L., & Gilbert, J. M. (2014). Design of wideband vibration-based electromagnetic generator by means of dual-resonator. Sensors and Actuators A: Physical, 213, 9-18. https://doi.org/10.1016/j.sna.2014.03.037
This paper describes the design of a wideband electromagnetic energy harvester that utilizes a novel dual-resonator method to improve the operational frequency range of the vibration-based generator. The device consists of two separate resonator syst... Read More about Design of wideband vibration-based electromagnetic generator by means of dual-resonator.
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redpajama_set_name: RedPajamaCommonCrawl


For this week, we offer 2 articles from our Featured Journal, Advances in School Mental Health Promotion that tackle the all-important question: do mental health interventions improve the outcomes that schools focus on the most i.e. grades, attendance, and behavior? These 3 studies offer a heavily qualified "yes" to this question.
"There was a significant association between improvement on educational and mental health outcomes (p.5)." Mental health interventions can target both academic and mental health outcomes, and they can also be delivered in both school and out-of-school settings and impact these outcomes.
Interventions targeted a wide range of presenting issues that school social workers deal with every day, with the top 3 areas in these studies being ones we all recognize: disruptive behavior, anxiety, and inattention/hyperactivity. Mental health interventions "work" with these problems, and can also make a difference on student educational outcomes for these students, too.
Finally, the trend in intervention research is a promising one in this area, with over 33% of the total studies the authors found having been conducted in the past 15 years, indicating that while more study is always needed, there appears to be some momentum towards studying the link between school mental health interventions and educational outcomes.
While the first study is encouraging, a lot of the studies focused on elementary-age children. The momentum of school mental health research needs to step up big-time for adolescents who are on the edge academically. In the second study, "School mental health early interventions and academic outcomes for at-risk high school students: a meta-analysis," Iachini and colleagues (2015) conducted a meta-analysis of published studies of school mental health interventions targeting high school students who were at-risk for school failure. Their findings are concerning: only 7 studies (!) met the criteria for review, and as the authors state in their abstract: "A meta-analysis found no statistically significant effect on the academic outcomes most commonly assessed in the studies (i.e., GPA, attendance, and discipline). Findings suggest the need for more rigorous research in this area (p. 156)." We have a lot more work to do to establish the link between school mental health interventions and improved academic outcomes for all youth, particularly adolescents.


text:
Q: tle error in algorithm design  given binary string of size n ,there are two types of queries

1.D for input string D display longest subarray length of consecutive 1's

2.C X for input of string C and integer x change string position x to 1 if zero

 my attempt
int main(){
    int n,k,i,j,y,count=0;
    string s,x;
    int maxcount=0;
    cin>>n>>k;
    cin>>s;
    for(j=0;j<k;j++)
    {
        cin>>x;
        if(x=='D')
        {
            maxcount=0;
            for(i=0;i<n;i++)
            {
                if(s[i]=='1')
                {
                    count++;
                } else{
                    maxcount = max(maxcount,count);
                    count=0;
                }
            }
            maxcount = max(maxcount,count);
            cout<<maxcount<<endl;

        } else{
            cin>>x;
            s[x-1] = '1';

        }
    }
return 0;
}

 i think it could be done using offline queries ,can anyone suggest approach or solution

A: You may use disjoint set data structure for your problem. Each substring of consecutive '1's will be one component. When you change some char '0' to '1', you may create a new set with this single element. There may be several cases:

*

*prior and next char is '1'. So you can merge these three sets.

*prior is '1'. You can merge this one with prior set.

*next is '1'. You can merge this with next set.

redpajama_set_name: RedPajamaStackExchange


Todays winner is : Commerzbank AG which gained in total of 3,80 procent since last trading day.
The Stocks which lost most today was : EOAN, RWE AG and finaly on third place Beiersdorf AG. The absolute loser was Commerzbank AG and lost 3,80 procent since yesterday.
Most traded stock today is Deutsche Bank AG with s total of 9219469 stocks.


text:
Sciencenter Names New Executive Director
| Friday, November 29, 2019 |
By Kelly Barclay
The Board of Trustees of the Sciencenter announced that after several months of careful deliberation, Dean Briere has decided not to renew his contract to serve as Executive Director of the Sciencenter. The Board selected Michelle Kortenaar, who has served the Sciencenter in various capacities since 2010, including, most recently, as Vice President of Strategic Development, to succeed Mr. Briere as Executive Director.
"We are very grateful to Dean for his many contributions to the growth and success of the Sciencenter during his tenure," said Melinda Oakes, the Sciencenter Board Chair. "Under his leadership, the Sciencenter strengthened partnerships with key members of the community, significantly increased outreach to and access for rural communities, and created interactive programming for adults. Dean also played a critical leadership role in, among other things, the design and development of a number of new exhibits, the revitalization of the Sciencenter's strategic plan, the renovation of the early-childhood space, and in overseeing the steady growth of the Sciencenter's endowment. We would like to thank him for his dedication and wish him the best in his future endeavors."
"I am honored to have served the Sciencenter and the Ithaca community for the past three years," said Briere. "And I am thrilled that the Board of Trustees has chosen Michelle to succeed me. Her background and experience in education, exhibits, guest services, and development uniquely qualify her to lead the Sciencenter forward as it cultivates a broad community of curious, confident, critical thinkers."
The transition from Briere to Kortenaar is effective as of November 22, though Briere has agreed to serve in a consulting capacity to help ensure a smooth transition for the Sciencenter's new leadership.
The Board stated: "We are very pleased to announce Michelle Kortenaar's appointment as the new Executive Director of the Sciencenter. Her years of service as a member of the Sciencenter's leadership team give her a breadth and depth of knowledge of the Sciencenter, its employees, the communities it serves, and its programs and initiatives. When coupled with her demonstrated commitment to and passion for the organization's mission and vision, she is an ideal leader for the Sciencenter."
"I am humbled and excited to be assuming leadership of the talented team at the Sciencenter," said Kortenaar. "Together we will engage our community by making science accessible to all. I am grateful to Dean for his generosity and look forward to guiding the Sciencenter in new directions while continuing to build on the strategic goals developed under his able leadership."
Kortenaar has been at the Sciencenter since 2010, serving as Director of Education, Senior Director of Engagement and Learning, and most recently as Vice President of Strategic Development. Kortenaar's career has focused on providing children with opportunities to explore science through open-ended experiences and discovery.Kortenaar has degrees in both science and education.Kortenaar's career began as a physics and biology teacher in Toronto, Canada and in West Orange, NJ. She served as chair of the math and science departments at her last school.Kortenaar was recruited as the founding director of a multi-department science center at a large summer camp in Milford, PA.Kortenaar has led the Sciencenter's work on a number of federal grants – including the Collaborative for Early Science Learning. She is currently the Principal Investigator of a National Science Foundation (NSF) funded grant – that creates opportunities for the Sciencenter to partner with Cornell University to engage public audiences in current NSF-funded STEM (Science, Technology, Engineering, and Math) research.Kortenaar serves on the extended leadership team of the National Informal STEM Education Network. She spearheaded the Sciencenter's partnership with many local institutions including Tompkins County Head Start. She has published in the field and often speaks at national conferences about informal science education and national partnerships.
v15i45
Copyright © 2005- by L-Star Publishing, Inc.


What are your Prices For Birthday Cakes?
Answer: Prices Vary. 1/4 sheet cakes and double layer 8 inch round cakes start at $35.00.
What are your prices for Wedding Cakes?
Answer: Wedding Cakes, Grooms Cakes and Novelty Cakes start at $4.00 per serving for butter cream and $5.00 per serving for Fondant.


With the addition of Goade's archery business, McCarthy's Outdoors is getting ready to move into a much bigger space.
Inventor Joe Goade shows off an archery device, which he holds the patent to, that instantly stops a compound bow string after it has fired an arrow at McCarthy's Outdoors and Gun Club, the outdoor shop he is co-owner of in Springfield.
A Cassville archery expert who patented a unique device to make bows quieter is teaming up with a Springfield gun shop to serve hunters and target shooters in both sports.
"I'm the bow guy. He's the gun guy," said inventor Joe Goade, who moved his archery business in Cassville to McCarthy's Outdoors and Gun Club in Springfield.
The co-owners have been busy refitting the McCarthy's Outdoors store at 1675 E. Seminole St. in Springfield.
Goade made waves in the archery world when he invented a way to instantly stop a compound bow string after it had fired an arrow.
"It takes a lot of vibration out of the bow and makes it quieter, which is important when you're hunting," Goade said. "A noisy bow will spook critters. I had an idea for attaching a rod to the bow with a cap and rubber stop on it. It worked! I didn't feel anything after I shot it. I thought, 'Hey I'm on to something here.' "
His Shock Terminator Suppressor stabilizer is now found on many compound bows, though he's had to battle some manufacturers who he believes infringed on his patent.
Goade said he loves to tune bows for customers to make them more accurate and improve the flight of arrows. Along with selling archery products, tuning bows is a skill he'll be bringing to McCarthy's Outdoors.
The shop will feature a 3-D video archery target range as well as a digital shooting range where customers can practice handgun and shotgun firing techniques.
Goade said he moved to Cassville to help with his ailing father.
But the Cassville area doesn't provide a large enough customer base to support his archery business. He met McCarthy's Outdoors owner Tom Younce when Younce brought him a bow to work on.
"I had been following him on Facebook for the longest time," Younce said. "I went to his shop to get my bows tuned and boy was it worth it."
Younce said bringing archery into his gun store was a no-brainer.
"You know, gun sales are good up until May, but then it's all archery," he said. "Gun sales pick up again in September. Once we get both guns and archery going together, we think this business is going to explode."
With the addition of Goade's archery business, McCarthy's Outdoors already is contemplating a move into a much bigger space.
Goade said McCarthy's Outdoors will be moving into the Dryer's Boots and Shoes store in the Plaza Shopping Center when the store closes in June. Dryer's Boots and Shoes has been a fixture in Springfield for 65 years.
The store will increase in size from 800 square feet now to 3,000 square feet, complete with digital target shooting range and a 20-yard archery range with shooting lanes for 10-12 people.
Younce said McCarthy's Outdoors is unusual in the outdoors industry business in that the store will finance purchases from $500 to $5,000, if a person qualifies.
The store also offers a variety of memberships. The top membership — $1,500 — offers gun cleaning and repair, firearm sighting in, bow repair and bow tuning for life.
"It's all about the customer's needs," Younce said. "We're taking customer service back 100 years."
Along with the new archery line, McCarthy's Outdoors sells new guns and used guns on consignment from private owners.
McCarthy's also has an online store where customers can buy a wide range of rifles, shotguns and handguns, camping gear, outdoor clothing, knives, ammo and more.
March 26, 2019, 1:04 a.m.
March 26, 2019, 12:29 a.m.
March 26, 2019, 4:32 p.m.
What's wrong with MSU baseball?
March 25, 2019, 2:02 p.m.