Mathématiques | EPFL Graph Search

thumb|upright|Raisonnement mathématique sur un tableau. Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets divers tels que les ensembles mathématiques, les nombres, les formes, les structures, les transformations ; ainsi qu'aux relations et opérations mathématiques qui existent entre ces objets. Elles sont aussi le domaine de recherche développant ces connaissances, ainsi que la discipline qui les enseigne. Elles possèdent plusieurs branches telles que : l'arithmétique, l'algèbre, l'analyse, la géométrie, la logique mathématique, les probabilités Il existe également une certaine séparation entre les mathématiques pures et les mathématiques appliquées. Les mathématiques se distinguent des autres sciences par un rapport particulier au réel car l'observation et l'expérience ne s'y portent pas sur des objets physiques ; les mathématiques ne sont pas une science empirique. Elles sont de nature ent

Geographic concentration of economic activities: on the validation of a distance-based mathematical index to identify optimal locations

Olivier Monod

The present study proposes a validation of a mathematical index Q able to identify optimal geographic places for economic activities, solely based on the location variable. This research work takes its roots in the 1970s with the statistical analysis of spatial patterns, or analysis of point processes, whose main goal is to understand if a resulting spatial distribution of points is due to chance or not. Indeed point objects are commonplace (towns in regions, plants in the landscape, galaxies in space, shops in towns) and the development of specific mathematical tools are useful to understand their own location processes. Spatial point deviations from purely random configurations may be analyzed either by quadrat or by distance methods. An interesting method of the second category – the cumulative function M – was developed recently for evaluating the relative geographic concentration and co-location of industries in a nonhomogeneous spatial framework. On this basis, and having quantified retail store interactions, The French physicist Pablo Jensen elaborated the Q-index to automatically detect promising locations. To test the relevance of this quality index, Jensen used location data from 2003 and 2005 for bakeries in the city of Lyon and discovered that between these two years, shops having closed were located on significantly lower quality sites. Here, using bankruptcy data provided by the Registrar of companies of the State of Valais in Switzerland and by the City Council of Glasgow in Scotland, we implemented a method based on univariate logistic regressions to systematically test for the relevance of the Q-index on the many commercial categories available. We show that the Q-index is reliable, although significance tests did not reach stringent levels. Access to trustable bankruptcy data remains a difficult task.

2011

Le "Fragmentum de inventionibus scientiarum" de Diego Palomino

Nouara Chekhar

The objective of this PhD thesis is the translation of, and the mathematical commentary on, a 16th-century Latin book. Its author, Diego Palomino is not well known. With a background in theology, he was a prior. In order to obtain his PhD at the University of Alcala (Madrid), he submitted a work, De mutations æris, in which he included a collection of what seems to be readings notes, entitled Fragmentum de inventionibus scientiarum. His readings have been drawn from various famous mathematicians of his time and ancient ones. The originality of his work relies mostly on his inventiveness and his style —which can be sarcastic— making the reading of it quite interesting and lively. His work consists for the main part in explaining some unclear demonstrations, or bringing new methods of solution ; he also innovates in solving pairs of indeterminate equations by providing the complete set of integral solutions. Before him, only one mathematician (Abu ̄ K ̄amil, at the end of the 10th century) did so, the other mathematicians restricting themselves to giving only one solution or a pair. Palomino did not hesitate in criticizing a well-established theory in ancient mathematics, namely Archimedes —although his critics seem to rely on a faulty edition. His book is entitled to have a significant place in the history of mathematics, for it both maintains the rigor of Greek classical mathematics and announces innovation, as did the 17th century, culminating with the discovery of the infinitesimal calculus.

EPFL2012

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