Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift

We study the existence and the exponential ergodicity of a general interacting particle system, whose components are driven by independent diffusion processes with values in an open subset of \mathds{R}^d, $d\geq 1$. The interaction occurs when a particle hits the boundary: it jumps to a position chosen with respect to a probability measure depending on the position of the whole system. Then we study the behavior of such a system when the number of particles goes to infinity. This leads us to an approximation method for the Yaglom limit of multi-dimensional diffusion processes with unbounded drift defined on an unbounded open set. While most of known results on such limits are obtained by spectral theory arguments and are concerned with existence and uniqueness problems, our approximation method allows us to get numerical values of quasi-stationary distributions, which find applications to many disciplines. We end the paper with numerical illustrations of our approximation method for stochastic processes related to biological population models

Scaling limits for the uniform infinite quadrangulation

The uniform infinite planar quadrangulation is an infinite random graph embedded in the plane, which is the local limit of uniformly distributed finite quadrangulations with a fixed number of faces. We study asymptotic properties of this random graph. In particular, we investigate scaling limits of the profile of distances from the distinguished point called the root, and we get asymptotics for the volume of large balls. As a key technical tool, we first describe the scaling limit of the contour functions of the uniform infinite well-labeled tree, in terms of a pair of eternal conditioned Brownian snakes. Scaling limits for the uniform infinite quadrangulation can then be derived thanks to an extended version of Schaeffer's bijection between well-labeled trees and rooted quadrangulations.Comment: 36 page

Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison

It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance.Comment: Published in at http://dx.doi.org/10.1214/08-AAP517 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Galois Got his Gun

This paper appeals to the figure of \'Evariste Galois for investigating the gates between mathematics and their "publics." The figure of Galois draws some lines of/within mathematics for/from the outside of mathematics and these lines in turn sketch the silhouette of Galois as a historical figure. The present paper especially investigates the collective categories that have been used in various types of public discourses on Galois's work (e.g. equations, groups, algebra, analysis, France, Germany etc.). In a way, this paper aims at shedding light on the boundaries some individuals drew by getting Galois his gun. It is our aim to highlight the roles of authority some individuals (such as as Picard) took on in regard with the public figure of Galois as well as the roles such authorities assigned to other individuals (such as the mediating role assigned to Jordan as a mediator between Galois's "ideas" and the public). The boundary-works involved by most public references to Galois have underlying them a long-term tension between academic and public legitimacies in the definition of some models for mathematical lives (or mathematics personae

Patterns in the city: a mathematics project

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Actes du séminaire national de didactique des mathématiques 2012

Actes de la session 2012 du séminaire national de didactique des mathématiques. Le séminaire national de didactique des mathématiques est organisé par l'ARDM. Il a pour but de permettre la diffusion régulière des recherches nouvelles ou en cours, et de favoriser les échanges et débats au sein de la communauté francophone de didactique des mathématiques. Se trouvent également des textes correspondant à la fête des 30 ans de la revue RDM (Recherche en Didactique des Mathématiques) et au colloquium organisé conjointement par l'ARDM et la CFEM (Commission Française pour l'enseignement des mathématiques)

Integrability, Supersymmetry and Coherent States. A volume in honour of Professor Véronique Hussin

ISBN 978-3-030-20087-

On the concept of (homo)morphism : a key notion in the learning of abstract algebra

This article is dedicated to the investigation of difficulties involved in the understanding of the homomorphism concept. It doesn't restrict to group-theory but on the contrary raises the issue of developing teaching strategies aiming at gaining access to structuralist thinking. Emphasis is put on epistemological analysis and its interaction with didactics in an attempt to make Abstract Algebra more accessible