Hardy spaces and boundary conditions from the Ising model

Functions in Hardy spaces on multiply-connected domains in the plane are given an explicit characterization in terms of a boundary condition inspired by the two-dimensional Ising model. The key underlying property is the positivity of a certain operator constructed inductively on the number of components of the boundary.     

On a class of degenerate elliptic operators arising from Fleming-Viot processes

We are dealing with the solvability of an elliptic problem related to a class of degenerate second order operators which arise from the theory of Fleming-Viot processes in population genetics. In the one dimensional case the problem is solved in the space of continuous functions. In higher dimension we study the problem in spaces with respect to an explicit measure which, under suitable assumptions, can be taken invariant and symmetrizing for the operators. We prove the existence and uniqueness of weak solutions and we show that the closure of the operator in such spaces generates an analytic -semigroup. Received December 4, 2000; accepted December 9, 2000.     

Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow

A time-dependent model corresponding to an Oldroyd-B viscoelastic fluid is considered, the convective terms being disregarded. Global existence in time is proved in Banach spaces provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using again an implicit function theorem. Supported by the Swiss National Science Foundation. Fellowship PBEL2–114311.     

Random repeated quantum interactions and random invariant states

We consider a generalized model of repeated quantum interactions, where a system ${\mathcal{H}}$ is interacting in a random way with a sequence of independent quantum systems ${\mathcal{K}_n, n \geq 1}$ . Two types of randomness are studied in detail. One is provided by considering Haar-distributed unitaries to describe each interaction between ${\mathcal{H}}$ and ${\mathcal{K}_n}$ . The other involves random quantum states describing each copy ${\mathcal{K}_n}$ . In the limit of a large number of interactions, we present convergence results for the asymptotic state of ${\mathcal{H}}$ . This is achieved by studying spectral properties of (random) quantum channels which guarantee the existence of unique invariant states. Finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble.     

Growth rate of the Schrödinger group on Zhidkov spaces

We give an upper bound on the growth rate of the Schrödinger group on Zhidkov spaces. In dimension 1, we prove that this bound is sharp. To cite this article: C. Gallo, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Approximation of Markov semigroups in total variation distance under an irregular setting: An application to the CIR process

In this paper, we propose a method to prove the total variation convergence of approximation of Markov semigroups with singularities. In particular our approach is adapted to the study of numerical schemes for Stochastic Differential Equation (SDE) with simply locally smooth coefficients. First we present this method and then, we apply it to the CIR process. In particular, we consider the weak second order scheme introduced in Alfonsi (2010) and we prove that it also converges towards the CIR diffusion process for the total variation distance. This convergence occurs with almost order two.     

A survey of dual-feasible and superadditive functions

Dual-feasible functions are valuable tools that can be used to compute both lower bounds for different combinatorial problems and valid inequalities for integer programs. Several families of functions have been used in the literature. Some of them were defined explicitly, and others not. One of the main objectives of this paper is to survey these functions, and to state results concerning their quality. We clearly identify dominant subsets of functions, i.e. those which may lead to better bounds or stronger cuts. We also describe different frameworks that can be used to create dual-feasible functions. With these frameworks, one can get a dominant function based on other ones. Two new families of dual-feasible functions obtained by applying these methods are proposed in this paper.