Soit$T$un courant positif fermé sur un voisinage de 0 dans$C$^{n}. Nous montrons que$T$admet un cône tangent (limite de la famille de ses homothétiques), dès que les masses projectives$v$_{T}(r) convergent assez vite vers$V$_{T}(0) pour que ($v$_{T}(r)−$v$_{T}(0))/$r$soit localement sommable en$r$=0. Cette condition suffisante est optimale: nous construisons des courants de bidegré (1, 1) n'ayant pas de cône tangent, tels que l'intégrale de ($v$_{T}(r)−$v$_{T}(0))/$r$soit aussi peu divergente qu'on le souhaite. Lorsque$T$est le courant d'intégration sur un ensemble analytique, on vérifie que$v$_{T}(r)−$v$_{T}(0)=0$r$^{ε}), ce qui redonne le théorème de Thie-King sur l'existence du cône tangent.

In this paper, we consider the time dependent Maxwell’s equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell’s equations when dispersive media are involved.

We propose a simple and efficient finite-difference method for the incompressible MHD equation. The numerical method combines the advantage of the MAC scheme for the Navier–Stokes equation and Yee's scheme for the Maxwell equation. In particular, the semi-discrete version of our scheme introduces no numerical dissipation and preserves the energy identity exactly.

For Banach lattices X and Y, let X | | Y and X | | Y denote the positive projective and injective tensor products of X and Y, respectively. In this paper, we characterize the inheritance of reflexivity and containment of copies of c 0, 1, by X | | Y and X | | Y from X and Y, when one of them is an atomic Banach lattice. In this case, we also give an affirmative answer to an open question of Jeurnink.

We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of (N−12)×(N−12) inside triangles with vertices (0,0),(0,λβ−−√) and (λ/β−−√,0) with respect to β>0, for fixed λ≥0. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice (N+σ)×(N+τ), for σ,τ>−1. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, σ=τ=−12, is a critical case.