Philosophie des mathématiques

La philosophie des mathématiques est la branche de la philosophie des sciences qui tente de répondre aux interrogations sur les fondements des mathématiques ainsi que sur leur usage. On y croise des questions telles que : « les mathématiques sont-elles nécessaires ? », « pourquoi les mathématiques sont-elles utiles ou efficaces pour décrire la nature ? », « dans quel(s) sens, peut-on dire que les entités mathématiques existent ? » ou « pourquoi et comment peut-on dire qu'une proposition mathématique est vraie ? ». Les différentes réponses possibles à ces questions s'organisent en différentes écoles de pensée, au nombre desquelles on compte, entre autres :

le réalisme mathématique ou réalisme platonicien ;
le formalisme ;
le logicisme ;
l'intuitionnisme ;
le structuralisme ;
le constructivisme (mathématiques). Ces pistes seront abordées dans la suite de l'article.

Thèmes récurrents Les thèmes récurrents comprennent : *Quel est le rôle de l'humanité dans le dév

Modeling an America's Cup Regatta as a Sequential Stochastic Game

Jean-Benoît Rossel

The main objective of this thesis is to model a regatta in the America’s Cup, and more precisely the first leg of the race, where the two competing sailboats have to move upwind. During the race, each crew attempts to be the first to reach the end of this leg, and is allowed to hinder its opponent as long as certain rules are respected. A boat which finishes this leg first is often able to manage its lead until the end of the regatta. An essential ingredient of the problem is the study of different maneuvers that the boats can execute during the race, as well as the effect generated by the presence of a boat on the wind around it. Indeed, a boat located just behind its opponent will receive less wind in its sails, and its speed will consequently be reduced. The boats considered in our model are of the type Class America, which were in used in the America’s Cup between 1982 and 2007. The race can hence be viewed as a game between two players which are assumed to be identical, in which each of them can make decisions sequentially. The goal of each player is to finish the first upwind leg with the largest lead possible over the opponent. The boats progress in an environment in which the wind fluctuates unpredictably. Thus, the game is a sequential stochastic game, in which each player will try to determine a sequence of actions which is the most favorable on average. A mathematical study of this kind of game will be done, and a strategy will be built by using tools of dynamic programming. A theorem will prove that this strategy is optimal, in the sense that no player has interest to deviate from this strategy. The last part of this thesis consists of applying the mathematical study of sequential stochastic games in the context of a sailing regatta. Given the current positions of the boats as well as the current state of the wind, a set of available actions and reactions for the boats can be defined. Each choice will bring the boats up to some new positions, where a new decision process will begin in a new state of the wind. Once the rules of the game are established, the objective is to define an algorithm which allows to build a strategy which will be an approximation of the optimal strategy. The implementation of this algorithm could produce a tool for decision support, that the crew could use on board during the regatta.

EPFL2013

Modeling and simulation of blood flow problems

Alfio Quarteroni

Medicine is one of the branches of scientific research in which numerical simulation can give a great support. In particular, a well mature branch like computational fluid dynamics (CFD) can provide a substantial help for the understanding of several pathologies of the cardiovascular system. The local behavior of blood flow may affect substantially the on-rise of such pathologies: a remarkable instance is atherosclerosis, which is unfortunately largely widespread in Western Countries. An accurate fluid dynamics analysis can better enlight these mechanisms, and CFD becomes necessary since physical data from

in vivo'' measures are troublesome to obtain. Moreover, allowing completely controlled simulations, CFD can eventually provide a useful paradigm for clinical practice, for instance suggesting optimal strategy for a surgeric operation. par However, the study of biological systems is a difficult task not only from the numerical viewpoint, but also in the modeling phase. Indeed, different systems interact each other at mechanical or biochemical level. Therefore, making a hierarchy of simplifications in the models considered is almost always mandatory. These notes illustrate some specific features of the blood flow problem, in the case of

large'' vessels, where the atherosclerotic plaques are localized. In particular, the interaction between blood and compliant vessel walls is of upmost interest. The purpose is to review shortly the existing literature on the field (limitedly to the most genuine mathematical aspects) and mention a first contribution from our side to the modeling of blood-wall interaction and its numerical simulation.

Bristeau, M.-O. (ed.) et al. Chichester: John Wiley & Sons.1997

Modeling an America's Cup Regatta as a Sequential Stochastic Game

Jean-Benoît Rossel

EPFL2013