Prépublications du Laboratoire de Mathématiques

Rational real algebraic models of topological surfaces

Résumé :
Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or a sphere or a torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic model. We prove that they admit exactly one rational real algebraic model. This was known earlier only for the sphere, torus, projective plane and the Klein bottle.

Principal bundles over a smooth real projective curve of genus zero

Résumé :

We study principal bundles over a smooth projective real algebraic curve of genus 0. Let X be such a curve. We show that there is a real algebraic group H and a principal H-bundle F over X with the following property. For any connected reductive linear algebraic group G and for any principal G-bundle E over X, there is a morphism f from H into G such that E is isomorphic to the principal G-bundle obtained from F by extension of the structure group through f.

Reducible curves over a finite field and geometric group laws

Résumé :
We present an explicit geometric description of the group law on an algebraic curve over a finite field of arbitrary arithmetic genus, all of whose irreducible components are rational. This allows algorithms to be written that are based on the arithmetics of the group law. Our aim is to transform the formal description to an optimized logic circuit description.

The Shiffman Conjecture for Moving Hypersurfaces

Résumé :

In 1979, B. Shiffman conjectured that if f is an algebraically nondegenerate holomorphic map of C into P^n and D_1,...,D_q are hypersurfaces in P^n in general position, then the sum of the defects is at most n+1. This conjecture was proved by M. Ru in 2004.
In this paper, the Shiffman conjecture will be proved more generally in the case of slowly moving hypersurfaces. Moreover, we introduce a truncation in the corresponding Second Main Theorem.

Stochastic differential games with asymmetric information

Résumé :

We investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual solutions of some second order Hamilton-Jacobi equation.

Approachability Theory, Discriminating Domain and Differential Games

Résumé :

We study the notion of approachability in a repeated game G with vector payoffs from a new point of view using recently developed differential game techniques. Namely : we relate the sufficient condition for approachability (\bB-set) to the notion of discriminating domain for a suitably chosen differential game \Gamma and we introduce an alternative repeated game G^*. We prove that a closed set is *approachable if and only if it contains a nonempty \bB-set, hence approachability and *approchability coincide. Finally, we define maps between the strategies in the differential game and in the repeated game preserving approachability properties.

Linear programming analysis of deterministic infinite horizon optimal control problems
(discounting and time averaging cases)

Résumé :
We investigate relationships between infinite time horizon optimal control problems with discounting and time average criteria and certain infinite dimensional linear programming problems (IDLPPs). We establish duality results for these IDLPPs and use these results for construction of necessary and sufficient optimality conditions and for characterization of viability kernels. The consideration is based on results from the theory of viscosity solutions and non-smooth analysis.

Minimizing movements for dislocation dynamics with a mean curvature term

Résumé :
We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.

The group of algebraic diffeomorphisms of a real rational surface is n-transitive

Résumé :

We prove that the group of algebraic diffeomorphisms of a real rational surface acts n-transitively for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are algebraically diffeomorphic if and only if they are homeomorphic as topological surfaces.

Planar Lorentz gas walk in a random scenery

Résumé :
We consider the planar Lorentz gas with finite horizon. The random scenery is given by a sequence of independent, identically distributed, centered and square integrable random variables. To each obstacle, we associate one of these random variables. We suppose that each time the particule hits an obstacle, it wins the amount given by the random variable associated to the obstacle. We prove a convergence in distribution to a Wiener process for the amount won by the particle when the time goes to infinity.

Résumé :
The recurrence for the planar Lorentz gas with finite horizon comes from a criteria of Conze and of Schmidt. Total ergodicity follows from these results (Simanyi, Pène). In this paper we answer a question of Szasz about the asymptotic behaviour of the number of visited cells when the time goes to infinity.

It is not more difficult to study the asymptotic of the number of obstacles hit by the particle when the time goes to infinity. We give an estimate for the mean and a result of almost sure convergence. For the simple random walk in ${\mathbb Z}^2$, this question has been studied by Dvoretzky and Erdös in \cite{DE}. We adapt the proof of Dvoretzky and Erdös. The lack of independence is compensated by a strong decorrelation result due to Chernov and by some extensions of the local limit theorem proved by Szasz and Varju.

A simple proof for the equivalence between invariance for stochastic and deterministic Systems

Résumé :
We provide a short and elementary proof for the recently proved result by G. da Prato and H. Frankowska that a closed set is stochastically invariant if and only if it is deterministically invariant.

Existence of Equilibria of Set-valued Maps on Bounded Epilipschitz Domains in Hilbert Spaces without Invariance Conditions

Résumé :
In the paper, we provide a new result of existence of equilibria for set-valued maps on bounded closed subsets $K$ of Hilbert spaces. We do not impose neither convexity nor compactness assumptions on $K$ but we assume that $K$ is {\em epilipschitz} i.e. it is locally the epigraph of lipschitz functions. In finite dimensional spaces, the famous Brouwer theorem asserts existence of fixed point for a continuous function from a compact convex set $K$ to itself. Our result could be view has a kind of generalization of this classical result in the context of Hilbert spaces and when the function (or the set-valued map) does not necessarily map $K$ into itself ( $K$ is not {\em invariant } by the map). Our approach is based firstly on degree theory for compact and for condensing set-valued maps and second on flows generated by trajectories of differential inclusions.

Viability with probabilistic knowledge of initial condition, application to optimal control

Résumé :
In this paper we provide an extension of the Viability and Invariance Theorems in the Wasserstein metric space of probability measures with finite quadratic moments in $\R^{d}$ for controlled systems of which the dynamic is bounded and Lipschitz. Then we characterize the viability and invariance kernels as the largest viability (resp. invariance) domains. As application of our result we consider an optimal control problem of Mayer type with lower semicontinuous cost function for the same controlled system with uncertainty on the initial state modeled by a probability measure. Following Frankowska, we prove using the epigraphical viability approach that the value function is the unique lower semicontinuous proximal episolution of a suitable Hamilton Jacobi equation.

Résumé :
The purpose of this article is to show uniqueness theorems for meromorphic mappings of C^m to CP^n with few hyperplanes H_j, j=1,...,q. It is well known that uniqueness theorems hold for q \geq 3n+2. In this paper we show that for every nonnegative integer c there exists a positive integer N(c), depending only on c in an explicit way, such that uniqueness theorems hold if q\geq (3n+2 -c) and n\geq N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n>>0. At the same time, a big number of recent uniqueness theorems are generalized considerably.

Résumé :
We provide a short and elementary proof for the recently proved result by G. da Prato and H. Frankowska that -under minimal assumptions - a closed set is invariant with respect to a stochastic control system if and only if it is invariant with respect to the (associated) deterministic control system.