Let (St)t≥0 be the running maximum of a standard Brownian motion (Bt)t≥0 and Tm≔inf{t;mSt<t},m>0. In this note we calculate the joint distribution of Tm and BTm. The motivation for our work comes ...from a mathematical model for animal foraging. We also present results for Brownian motion with drift.
Résumé Rédigé par une historienne et un spécialiste de modélisation mathématique, cet article explore les enjeux épistémologiques de la collaboration interdisciplinaire à travers une étude de cas : ...l’épuration professionnelle du monde du spectacle à la Libération. Dans tout processus de justice, la question de l’équité, ou celle, équivalente, d’éventuelles discriminations, est difficile à instruire. A fortiori pour une épuration à caractère disciplinaire, où des artistes ont jugé leurs pairs. L’article montre que le formalisme mathématique, loin de se substituer à l’expertise historique, prolonge celle-ci par les moyens d’un autre langage, abstrait, enrichissant ainsi les modes d’accès au réel en faisant converger plusieurs dispositifs d’enquête. Progressant pas à pas dans la modélisation du problème et dans l’analyse des données, les deux chercheurs prennent soin d’expliciter les approches statistiques et mathématiques de plus en plus complexes qu’ils doivent mobiliser pour détecter des formes jurisprudentielles impossibles à capturer avec des outils classiques – jusqu’à l’idée originale de traiter un processus impliquant des décisions humaines comme un processus algorithmique complexe. Grâce au détournement d’une méthode d’inférence causale conçue pour étudier l’équité de certains processus algorithmiques de type « boîte noire », des résultats inédits, restés jusqu’alors totalement « cachés » dans les données, sont révélés et viennent, en retour, guider l’analyse historique.
Abstract Written by a historian and a mathematical scientist, this article explores the epistemological stakes of interdisciplinary collaboration by focusing on a specific case study: the purge of performing artists after the liberation of France. While the artists most compromised during the occupation were brought before purge tribunals, less serious cases were referred to specialized commissions comprised of their peers. In any legal proceeding, it can be hard to reach a verdict when it comes to questions of fairness or potential discrimination, and this was especially true for these purge commissions. The authors show how mathematical formalism, while obviously not replacing historical inquiry, can extend its reach, offering multiple ways to apprehend an elusive reality thanks to the versatility of an abstract language. Progressing step by step through the modelling of the question and the analysis of the data, they explain the increasingly complex statistical and mathematical approaches mobilized to observe forms of jurisprudence that escape more traditional analysis—arriving at the innovative proposal to treat a trial involving human decisions as a complex algorithmic process. Adapting a causal inference approach designed to evaluate the fairness of “black box” type algorithmic processes brings to light unprecedented results, hitherto hidden in the data. These findings, in turn, lead to new insights for the historian.
Segregation through the multiscalar lens Olteanu, Madalina; Randon-Furling, Julien; Clark, William A. V.
Proceedings of the National Academy of Sciences - PNAS,
06/2019, Volume:
116, Issue:
25
Journal Article
Peer reviewed
Open access
We introduce a mathematical framework that allows one to carry out multiscalar and multigroup spatial exploratory analysis across urban regions. By producing coefficients that integrate information ...across all scales and that are normalized with respect to theoretical maximally segregated configurations, this framework provides a practical and powerful tool for the comparative empirical analysis of urban segregation. We illustrate our method with a study of ethnic mixing in the Los Angeles metropolitan area.
How did written works evolve, disappear or survive down through the ages? In this paper, we propose a unified, formal framework for two fundamental questions in the study of the transmission of ...texts: how much was lost or preserved from all works of the past, and why do their genealogies (their ``phylogenetic trees'') present the very peculiar shapes that we observe or, more precisely, reconstruct? We argue here that these questions share similarities to those encountered in evolutionary biology, and can be described in terms of ``genetic'' drift and ``natural'' selection. Through agent-based models, we show that such properties as have been observed by philologists since the 1800s can be simulated, and confronted to data gathered for ancient and medieval texts across Europe, in order to obtain plausible estimations of the number of works and manuscripts that existed and were lost.
From urban segregation to spatial structure detection Randon-Furling, Julien; Olteanu, Madalina; Lucquiaud, Antoine
Environment and planning. B, Urban analytics and city science,
05/2020, Volume:
47, Issue:
4
Journal Article
Peer reviewed
We develop a ‘multifocal’ approach to reveal spatial dissimilarities in cities, from the most local scale to the metropolitan one. Think, for instance, of a statistical variable that may be measured ...at different scales, e.g. ethnic group proportions, social housing rate, income distribution, or public transportation network density. Then, to any point in the city there corresponds a sequence of values for the variable, as one zooms out around the starting point, all the way up to the whole city – as if with a varifocal camera lens. The sequences thus produced encode spatial dissimilarities in a precise manner: how much they differ from perfectly random sequences is indeed a signature of the underlying spatial structure. We introduce here a mathematical framework that allows to analyse this signature, and we provide a number of illustrative examples.
REPLY TO STEPINSKI AND DMOWSKA Olteanu, Madalina; Randon-Furling, Julien; Clark, William A. V.
Proceedings of the National Academy of Sciences - PNAS,
11/2019, Volume:
116, Issue:
45
Journal Article
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