Recovery of Immersions from Their Metric Tensors and Nonlinear Korn Inequalities: A Brief Survey

The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω  R~n→ E~n with prescribed metric tensor field(g ij) : Ω→ S~n, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.     

Géométries Lorentziennes de dimension 3 : classification et complétude

We classify three-dimensional Lorentz homogeneous spaces G/I having a compact manifold locally modeled on them. We prove a completeness result: any compact locally homogeneous Lorentz threefold M is isometric to a quotient of a Lorentz homogeneous space G/I by a discrete subgroup Γ of G acting properly and freely on G/I. Moreover, if I is noncompact, G/I is isometric to a Lie group L endowed with a left invariant Lorentz metric, where L is isomorphic to one of the following Lie groups:

Shape-preserving bivariate polynomial approximation inC([−1,1]×[−1,1])

In this paper we construct bivariate polynomials attached to a bivariate function, that approximate with Jackson-type rate involving a bivariate Ditzian-Totik ω2-modulus of smoothness and preserve some natural kinds of bivariate monotonicity and convexity of function.The result extends that in univariate case-of D. Leviatan in [5-6], improves that in bivariate case of the author in [3] and in some special cases, that in bivariate case of G. Anastassiou in [1].     

Approximation of Analytic Functions with an Arbitrary Order by Generalized Baskakov–Faber Operators in Compact Sets

By using a sequence \(\lambda _{n}>0\), \(n\in \mathbb {N}\) with the property that \(\lambda _{n}\rightarrow 0\) as fast we want, in this paper we obtain the approximation order \(O(\lambda _{n})\) for a generalized Baskakov–Faber operator attached to analytic functions of exponential growth in a continuum \(G\subset \mathbb {C}\). Several concrete examples of continuums G are given for which this operator can explicitly be constructed.     

Ergodicity for Nonlinear Stochastic Equations in Variational Formulation

This paper is concerned with nonlinear partial differential equations of the calculus of variation (see [13]) perturbed by noise. Well-posedness of the problem was proved by Pardoux in the seventies (see [14]), using monotonicity methods. The aim of the present work is to investigate the asymptotic behaviour of the corresponding transition semigroup P_t. We show existence and, under suitable assumptions, uniqueness of an ergodic invariant measure ν. Moreover, we solve the Kolmogorov equation and prove the so-called "identite du carre du champs". This will be used to study the Sobolev space W^1,2(H,ν) and to obtain information on the domain of the infinitesimal generator of P_t.     

An operator approach to zero-sum repeated games

We consider two person zero-sum stochastic games. The recursive formula for the valuesvλ (resp.v _n) of the discounted (resp. finitely repeated) version can be written in terms of a single basic operator Φ(α,f) where α is the weight on the present payoff andf the future payoff. We give sufficient conditions in terms of Φ(α,f) and its derivative at 0 for limv _n and limvλ to exist and to be equal. We apply these results to obtain such convergence properties for absorbing games with compact action spaces and incomplete information games.