43,524 research outputs found
On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body
The interface problem describing the scattering of time-harmonic
electromagnetic waves by a dielectric body is often formulated as a pair of
coupled boundary integral equations for the electric and magnetic current
densities on the interface . In this paper, following an idea developed
by Kleinman and Martin \cite{KlMa} for acoustic scattering problems, we
consider methods for solving the dielectric scattering problem using a single
integral equation over for a single unknown density. One knows that
such boundary integral formulations of the Maxwell equations are not uniquely
solvable when the exterior wave number is an eigenvalue of an associated
interior Maxwell boundary value problem. We obtain four different families of
integral equations for which we can show that by choosing some parameters in an
appropriate way, they become uniquely solvable for all real frequencies. We
analyze the well-posedness of the integral equations in the space of finite
energy on smooth and non-smooth boundaries
Shape derivatives of boundary integral operators in electromagnetic scattering. Part I: Shape differentiability of pseudo-homogeneous boundary integral operators
In this paper we study the shape differentiability properties of a class of
boundary integral operators and of potentials with weakly singular
pseudo-homogeneous kernels acting between classical Sobolev spaces, with
respect to smooth deformations of the boundary. We prove that the boundary
integral operators are infinitely differentiable without loss of regularity.
The potential operators are infinitely shape differentiable away from the
boundary, whereas their derivatives lose regularity near the boundary. We study
the shape differentiability of surface differential operators. The shape
differentiability properties of the usual strongly singular or hypersingular
boundary integral operators of interest in acoustic, elastodynamic or
electromagnetic potential theory can then be established by expressing them in
terms of integral operators with weakly singular kernels and of surface
differential operators
Shape derivatives of boundary integral operators in electromagnetic scattering. Part II : Application to scattering by a homogeneous dielectric obstacle
We develop the shape derivative analysis of solutions to the problem of
scattering of time-harmonic electromagnetic waves by a bounded penetrable
obstacle. Since boundary integral equations are a classical tool to solve
electromagnetic scattering problems, we study the shape differentiability
properties of the standard electromagnetic boundary integral operators. The
latter are typically bounded on the space of tangential vector fields of mixed
regularity TH\sp{-1/2}(\Div_{\Gamma},\Gamma). Using Helmholtz decomposition,
we can base their analysis on the study of pseudo-differential integral
operators in standard Sobolev spaces, but we then have to study the G\^ateaux
differentiability of surface differential operators. We prove that the
electromagnetic boundary integral operators are infinitely differentiable
without loss of regularity. We also give a characterization of the first shape
derivative of the solution of the dielectric scattering problem as a solution
of a new electromagnetic scattering problem.Comment: arXiv admin note: substantial text overlap with arXiv:1002.154
Shape derivatives of boundary integral operators in electromagnetic scattering
We develop the shape derivative analysis of solutions to the problem of
scattering of time-harmonic electromagnetic waves by a bounded penetrable
obstacle. Since boundary integral equations are a classical tool to solve
electromagnetic scattering problems, we study the shape differentiability
properties of the standard electromagnetic boundary integral operators. Using
Helmholtz decomposition, we can base their analysis on the study of scalar
integral operators in standard Sobolev spaces, but we then have to study the
G\^ateaux differentiability of surface differential operators. We prove that
the electromagnetic boundary integral operators are infinitely differentiable
without loss of regularity and that the solutions of the scattering problem are
infinitely shape differentiable away from the boundary of the obstacle, whereas
their derivatives lose regularity on the boundary. We also give a
characterization of the first shape derivative as a solution of a new
electromagnetic scattering problem
Asymptotic Exponential Arbitrage and Utility-based Asymptotic Arbitrage in Markovian Models of Financial Markets
Consider a discrete-time infinite horizon financial market model in which the
logarithm of the stock price is a time discretization of a stochastic
differential equation. Under conditions different from those given in a
previous paper of ours, we prove the existence of investment opportunities
producing an exponentially growing profit with probability tending to
geometrically fast. This is achieved using ergodic results on Markov chains and
tools of large deviations theory.
Furthermore, we discuss asymptotic arbitrage in the expected utility sense
and its relationship to the first part of the paper.Comment: Forthcoming in Acta Applicandae Mathematica
Manifolds admitting stable forms
In this note we give a direct method to classify all stable forms on
as well as to determine their automorphism groups. We show that in dimension
6,7,8 stable forms coincide with non-degnerate forms.
We present necessary conditions and sufficient conditions for a manifold to
admit a stable form. We also discuss rich properties of the geometry of such
manifolds.Comment: 19 page
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