Free Turbulence on R^3 and T^3

The hydrodynamics of Newtonian fluids has been the subject of a tremendous amount of work over the past eighty years, both in physics and mathematics. Sadly, however, a mutual feeling of incomprehension has often hindered scientific contacts. This article provides a dictionary that allows mathematicians to define and study the spectral properties of Kolmogorov-Obukov turbulence in a simple deterministic manner. In other words, this approach fits turbulence into the mathematical framework of studying the qualitative properties of solutions of PDEs, independently from any a-priori model of the structure of the flow. To check that this new approach is correct, this article proves some of the classical statements that can be found in physics textbooks. This is followed by an investigation of the compatibility between turbulence and the smoothness of solutions of Navier-Stokes in 3D, which was the initial motivation of this study.Comment: 47 pages, 6 figure

On the localization of the magnetic and the velocity fields in the equations of magnetohydrodynamics

We study the behavior at infinity, with respect to the space variable, of solutions to the magnetohydrodynamics equations in ${\bf R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free nonstationnary Navier--Stokes equations when $|x|\to +\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are boundedness criteria for convolution operators in weighted spaces.Comment: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics (to appear) (0000) --xx-

New Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System

We show that solutions $u(x,t)$ of the non-stationnary incompressible Navier--Stokes system in $\R^d$ ($d\geq2$) starting from mild decaying data $a$ behave as $|x|\to\infty$ as a potential field: u(x,t) = e^{t\Delta}a(x) + \gamma_d\nabla_x(\sum_{h,k} \frac{\delta_{h,k}|x|^2 - d x_h x_k}{d|x|^{d+2}} K_{h,k}(t))+\mathfrak{o}(\frac{1}{|x|^{d+1}}) where $\gamma_d$ is a constant and $K_{h,k}=\int_0^t(u_h| u_k)_{L^2}$ is the energy matrix of the flow. We deduce that, for well localized data, and for small $t$ and large enough $|x|$, c t |x|^{-(d+1)} \le |u(x,t)|\le c' t |x|^{-(d+1)}, where the lower bound holds on the complementary of a set of directions, of arbitrary small measure on $\mathbb{S}^{d-1}$. We also obtain new lower bounds for the large time decay of the weighted-$L^p$ norms, extending previous results of Schonbek, Miyakawa, Bae and Jin.Comment: 26 pages, article to appear in Journal de Math\'ematiques Pures et Appliqu\'ee

Non-reciprocal optical reflection from a bidimensional array of subwavelength holes in a metallic film

Using simulations and theoretical arguments we investigate the specular reflection of a perforated gold film deposited on a glass substrate. A square lattice of cylindrical holes is assumed to produce the periodic lateral corrugation needed to hybridize the surface plasmons with radiative states. It is shown that, contrasting transmission approaches, a knowledge of the reflection on either side of the film provides separate information on the gold-vacuum surface plasmons and on the gold-glass interface plasmons. Recent experimental data on a specific implementation of this system are reexamined; these show a good agreement between the measured reflections and the simulations in both directions of incident wave probes. This confirms the importance of taking into account the reflection asymmetry in the far-field assessment of surface plasmons properties.Comment: 4 pages, 3 figures. Published versio

Indispensable binomials in semigroup ideals

In this paper, we deal with the problem of uniqueness of minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for uniqueness of such minimal system of generators. These conditions come from the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal.Comment: 11 pages. This paper was initially presented at the II Iberian Mathematical Meeting (http://imm2.unex.es). To appear in the Proc. Amer. Math. So

The short resolution of a semigroup algebra

This work generalizes the short resolution given in Proc. Amer. Math. Soc. \textbf{131}, 4, (2003), 1081--1091, to any affine semigroup. Moreover, a characterization of Ap\'{e}ry sets is given. This characterization lets compute Ap\'{e}ry sets of affine semigroups and the Frobenius number of a numerical semigroup in a simple way. We also exhibit a new characterization of the Cohen-Macaulay property for simplicial affine semigroups.Comment: 12 pages. In this new version, some proofs have been detailed, the references on the computatation of the Frobenius number of a numerical semigroup have been updated and some typpos have been correcte

Bounded modes to the rescue of optical transmission

This paper presents a brief survey of the evolution of knowledge about diffraction gratings. After recalling some basic facts, historically and physically, we introduce the concept of Wood anomalies. Next, we present some recent works in order to introduce the role of bounded modes in transmission gratings. The consequences of these recent results are then introduced. This paper is a secondary publication, published in Europhysics News (EPN 38, 3 (2007) 27-31). In the present version, some additional notes have been added with related references.Comment: 6 pages, 6 figures. Secondary publication. Brief revie

Bootstrap for neural model selection

Bootstrap techniques (also called resampling computation techniques) have introduced new advances in modeling and model evaluation. Using resampling methods to construct a series of new samples which are based on the original data set, allows to estimate the stability of the parameters. Properties such as convergence and asymptotic normality can be checked for any particular observed data set. In most cases, the statistics computed on the generated data sets give a good idea of the confidence regions of the estimates. In this paper, we debate on the contribution of such methods for model selection, in the case of feedforward neural networks. The method is described and compared with the leave-one-out resampling method. The effectiveness of the bootstrap method, versus the leave-one-out methode, is checked through a number of examples.Comment: A la suite de la conf\'{e}rence ESANN 200