Testing surface area with arbitrary accuracy

Recently, Kothari et al.\ gave an algorithm for testing the surface area of an arbitrary set $A \subset [0, 1]^n$. Specifically, they gave a randomized algorithm such that if $A$'s surface area is less than $S$ then the algorithm will accept with high probability, and if the algorithm accepts with high probability then there is some perturbation of $A$ with surface area at most $\kappa_n S$. Here, $\kappa_n$ is a dimension-dependent constant which is strictly larger than 1 if $n \ge 2$, and grows to $4/\pi$ as $n \to \infty$. We give an improved analysis of Kothari et al.'s algorithm. In doing so, we replace the constant $\kappa_n$ with $1 + \eta$ for $\eta > 0$ arbitrary. We also extend the algorithm to more general measures on Riemannian manifolds.Comment: 5 page

Subelliptic Li-Yau estimates on three dimensional model spaces

We describe three elementary models in three dimensional subelliptic geometry which correspond to the three models of the Riemannian geometry (spheres, Euclidean spaces and Hyperbolic spaces) which are respectively the SU(2), Heisenberg and SL(2) groups. On those models, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation. We use for that the $\Gamma_{2}$ techniques that we adapt to those elementary model spaces. The important feature developed here is that although the usual notion of Ricci curvature is meaningless (or more precisely leads to bounds of the form $-\infty$ for the Ricci curvature), we describe a parameter $\rho$ which plays the same role as the lower bound on the Ricci curvature, and from which one deduces the same kind of results as one does in Riemannian geometry, like heat kernel upper bounds, Sobolev inequalities and diameter estimates

Log-Harnack Inequality for Stochastic Differential Equations in Hilbert Spaces and its Consequences

A logarithmic type Harnack inequality is established for the semigroup of solutions to a stochastic differential equation in Hilbert spaces with non-additive noise. As applications, the strong Feller property as well as the entropy-cost inequality for the semigroup are derived with respect to the corresponding distance (cost function)

String effects and the distribution of the glue in mesons at finite temperature

The distribution of the gluon action density in mesonic systems is investigated at finite temperature. The simulations are performed in quenched QCD for two temperatures below the deconfinment phase. Unlike the gluonic profiles displayed at T=0, the action density iso-surfaces display a prolate-spheroid like shape. The curved width profile of the flux-tube is found to be consistent with the prediction of the free Bosonic string model at large distances.Comment: 14 pages,10 figure

Dimension dependent hypercontractivity for Gaussian kernels

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck semigroup driven by a non-diffusive L\'evy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.Comment: 24 page

Uniform convergence to equilibrium for granular media

We study the long time asymptotics of a nonlinear, nonlocal equation used in the modelling of granular media. We prove a uniform exponential convergence to equilibrium for degenerately convex and non convex interaction or confinement potentials, improving in particular results by J. A. Carrillo, R. J. McCann and C. Villani. The method is based on studying the dissipation of the Wasserstein distance between a solution and the steady state

La sécurité alimentaire : perspectives d'amélioration des bananiers par voie biotechnologique

En complément des travaux d'amélioration génétique par hybridation et des nouvelles approches biotechnologiques, la connaissance des bananiers s'est considérablement accrue au cours de la dernière décennie. Les outils ainsi forgés ont déjà permis une meilleure diffusion du matériel végétal ainsi que la création de nouvelles variétés. L'ensemble de ces techniques contribue et contribuera sans nul doute à l'amélioration tant qualitative que quantitative de la production bananiere. En ce sens elles participent significativement au renforcement de la sécurité alimentaire de la planèt