We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this ...process is nonstationary and its probability distribution exhibits rich features. In a finite domain, we define a nontrivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.
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Explicit formulas for the expected volume and expected number of facets of the convex hull of several multidimensional Gaussian random walks are derived in terms of the Gaussian persistence ...probabilities. Special cases include the already known results about the convex hull of a single Gaussian random walk and the d-dimensional Gaussian polytope with or without origin.
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 fortioripour 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.
We introduce a multidimensional, neural network approach to reveal and measure urban segregation phenomena, based on the self-organizing map algorithm (SOM). The multidimensionality of SOM allows one ...to apprehend a large number of variables simultaneously, defined on census blocks or other types of statistical blocks, and to perform clustering along them. Levels of segregation are then measured through correlations between distances on the neural network and distances on the actual geographical map. Further, the stochasticity of SOM enables one to quantify levels of heterogeneity across census blocks. We illustrate this new method on data available for the city of Paris.
The paper provides numerical measures and visualizations of urban segregation based on a new index, the Distortion coefficient. Distortion coefficients are derived from trajectories of contact with ...the city’s population as an individual will encounter an increasing number of persons in a growing distance from their original location. They can be interpreted as measures of how different, or in technical terms, how distorted the view of the city is from any one location. In this paper, we present the theoretical rationale and the procedure leading to the computation of the Distortion coefficients. Through a detailed illustration using Chicago as a case study, we provide the general framework for analyzing and visualizing Distortion. We show that these measures are able to capture complex demographic changes over time and paint a more complete picture of segregation than indices based on imposed scales.
Fine-scale data is particularly important for the analysis of multiscalar segregation phenomena. Using dis-aggregated data from an EU data challenge, we show here how to apply a recently developed ...method that measures segregation at multiple scales and provides a visualization of the levels of segregation across scale and space. We illustrate the technique with results for two groups of citizen migrants in the city of Paris.
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