Truth and Consequences

International audienc

Imaging polarizable dipoles

We present a method for imaging the polarization vector of an electric dipole distribution in a homogeneous medium from measurements of the electric field made at a passive array. We study an electromagnetic version of Kirchhoff imaging and prove, in the Fraunhofer asymptotic regime, that range and cross-range resolution estimates are identical to those in acoustics. Our asymptotic analysis provides error estimates for the cross-range dipole orientation reconstruction and shows that the range component of the dipole orientation is lost in this regime. A naive generalization of the Kirchhoff imaging function is afflicted by oscillatory artifacts in range, that we characterize and correct. We also consider the active imaging problem which consists in imaging both the position and polarizability tensors of small scatterers in the medium using an array of collocated sources and receivers. As in the passive array case, we provide resolution estimates that are consistent with the acoustic case and give error estimates for the cross-range entries of the polarizability tensor. Our theoretical results are illustrated by numerical experiments.Comment: 35 pages, 18 figure

Mathematical models for dispersive electromagnetic waves: an overview

In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notion of non-dissipativity and passivity. We consider successively the case of so-called local media and general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background.Comment: 46 pages, 16 figure

Solving moment problems by dimensional extension

The first part of this paper is devoted to an analysis of moment problems in R^n with supports contained in a closed set defined by finitely many polynomial inequalities. The second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the first part, for decomposing a polynomial which is positive on such a semi-algebraic set into a canonical sum of squares of rational functions times explicit multipliers.Comment: 21 pages, published version, abstract added in migratio

EXTENDED SPECTRUM AND EXTENDED EIGENSPACES OF QUASI-NORMAL OPERATORS

International audienceWe say that a complex number λ is an extended eigenvalueof a bounded linear operator T on a Hilbert space H if there exists anonzero bounded linear operator X acting on H, called extended eigen-vector associated to λ, and satisfying the equation T X = λXT . In thispaper we describe the sets of extended eigenvalues and extended eigen-vectors for the product of a positive and a self-adjoint operator whichare both injective. We also treat the case of normal operators

Extended Spectrum, Extended Eigenspaces and Normal Operators

International audienceWe say that a complex number λ is an extended eigenvalue of a bounded linear operator T on a Hilbert space H if there exists a nonzero bounded linear operator X acting on H, called extended eigen- vector associated to λ, and satisfying the equation T X = λXT . In this paper we describe the sets of extended eigenvalues and extended eigen- vectors for the product of a positive and a self-adjoint operator which are both injective. We also treat the case of normal operators