Spectral gap for some invariant log-concave probability measures

We show that the conjecture of Kannan, Lov\'{a}sz, and Simonovits on isoperimetric properties of convex bodies and log-concave measures, is true for log-concave measures of the form $\rho(|x|_B)dx$ on $\mathbb{R}^n$ and $\rho(t,|x|_B) dx$ on $\mathbb{R}^{1+n}$, where $|x|_B$ is the norm associated to any convex body $B$ already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.Comment: To appear in Mathematika. This version can differ from the one published in Mathematik

Concavity and Efficient Points of Discrete Distributions in Probabilistic Programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples

ANALYSE DES SÉRIES DES DONNÉS DES DÉBITS DES COURS D'EAU DANS LE DOMAIN DES ÉTIAGES

The discharges of the water courses are generated by the procedure of the precipitation - which is an easily analysable stochastic procedure. The procedure of the discharges is transformed by the boundary conditions of the water system. Consequently the procedure is a very complicated structured stochastic procedure. The Cramer-Leadbetter crossing method was constructed for analysis of such complicated stochastic procedures. For the water management purposes it is needed to complete the random variables introduced by Cramer and Leadbetter with the sum and extrem values of these parameters. Finally the stochastic procedure will be characterised by the series of condition al probability distribution of 17 random variables defined by the different crossing levels. To complete the information the Poissonian structure of the highwaters and the exponential character of the recession curves were used. A special tank model was used to deduce analytical formulas for the distribution functions of these 17 random variables. Two more sophisticated models were constructed - these models the condition al prob- ability functions can be estimated by numerical algorithms with help of the adequate software