El Món de les variables sense moments finits de tots els ordres: de la paradoxa de Sant Petersburg als processos de Lévy

Les variables aleatòries sense moments de tots els ordres no són actualment una patologia, sinó que ocupen una posició central en la teoria de la probabilitat. Des de la paradoxa de Sant Petersburg, de començaments del segle xviii, fins a les distribucions estables no gaussianes i als vols de Lévy, s'ha desenvolupat una teoria plena de bellesa, de la qual us pretenem presentar alguns aspectes en aquest article. Acabem amb unes aplicacions a distintes situacions, entre aquestes una nova mirada a la vella paradoxa de Sant Petersburg, i una descripció de l'evolució de la temperatura de la Terra en els darrers 250.000 anys, com a tast de la seva importància en la modelització matemàtica de la realitat.Random variables without finite moments of all orders are not at present a pathology, but a central subject in probability theory. From the Saint Petersburg paradox, dated at the beginning of the eighteenth century, until the non-Gaussian stable distributions and Lévy flights, a very beautiful theory has been developed, at which, in this paper, we will take a quick glance. We will finish with some applications to different situations as a taste of the importance of this theory in mathematical modelling. Specifically, we will consider asset prices modelling, the study of earthquakes, a new view to the classical Saint Petersburg paradox, and finally the evolution of the mean temperature in the North Atlantic sea over the last 250,000 years

Incertesa i probabilitat : un passeig per algunes paradoxes i problemes clàssics de la teoria de la probabilitat

Els humans sempre han temut la incertesa que governa les seves vides. En aquest treball, fem la descripció d'un passeig amable per les nostres reaccions davant de l'aleatorietat, des dels vells temps passatsquan va ser divinitzada , fins a la matematització que representa la teoria de la probabilitat, que, segons Laplace, no és més que el bon sentit reduït al càlcul. Considerarem algunes preguntes que ja apareixen a la correspondència entre Pascal i Fermat, en el naixement de la teoria, així com alguns dels problemes clàssics, com el famós problema de les tres portes, l'aposta a la martingala o el passeig aleatori amb barreres, seleccionats entre l'immens número de sorprenents resultats i paradoxes d'aquesta teoria.The humans always have feared the uncertainty that directs their lives. In this paper, we describe a kind walk along our behavior in front of randomness, from the old times, when it was divinized, until the present mathematical treatment with the Theory of Probability, that, as Laplace wrote, is only the common sense reduced to calculus.We will consider the first questions that appear in the correspondence among Pascal and Fermat, at the dawn of the Theory, and also some of the most classical problems, as for instance the martingale betting, and the random walk between barriers, selected in the so rich and enormous set of surprising results and paradoxes of this Theory

Incertesa i probabilitat : un passeig per algunes paradoxes i problemes clàssics de la teoria de la probabilitat

Els humans sempre han temut la incertesa que governa les seves vides. En aquest treball, fem la descripció d'un passeig amable per les nostres reaccions davant de l'aleatorietat, des dels vells temps passats—quan va ser divinitzada— , fins a la matematització que representa la teoria de la probabilitat, que, segons Laplace, no és més que el bon sentit reduït al càlcul. Considerarem algunes preguntes que ja apareixen a la correspondència entre Pascal i Fermat, en el naixement de la teoria, així com alguns dels problemes clàssics, com el famós problema de les tres portes, l'aposta a la martingala o el passeig aleatori amb barreres, seleccionats entre l'immens número de sorprenents resultats i paradoxes d'aquesta teoria.The humans always have feared the uncertainty that directs their lives. In this paper, we describe a kind walk along our behavior in front of randomness, from the old times, when it was divinized, until the present mathematical treatment with the Theory of Probability, that, as Laplace wrote, is only the common sense reduced to calculus.We will consider the first questions that appear in the correspondence among Pascal and Fermat, at the dawn of the Theory, and also some of the most classical problems, as for instance the martingale betting, and the random walk between barriers, selected in the so rich and enormous set of surprising results and paradoxes of this Theory

Performance of mixed formulations for the particle finite element method in soil mechanics problems

This paper presents a computational framework for the numerical analysis of fluid-saturated porous media at large strains. The proposal relies, on one hand, on the particle finite element method (PFEM), known for its capability to tackle large deformations and rapid changing boundaries, and, on the other hand, on constitutive descriptions well established in current geotechnical analyses (Darcy’s law; Modified Cam Clay; Houlsby hyperelasticity). An important feature of this kind of problem is that incompressibility may arise either from undrained conditions or as a consequence of material behaviour; incompressibility may lead to volumetric locking of the low-order elements that are typically used in PFEM. In this work, two different three-field mixed formulations for the coupled hydromechanical problem are presented, in which either the effective pressure or the Jacobian are considered as nodal variables, in addition to the solid skeleton displacement and water pressure. Additionally, several mixed formulations are described for the simplified single-phase problem due to its formal similitude to the poromechanical case and its relevance in geotechnics, since it may approximate the saturated soil behaviour under undrained conditions. In order to use equal-order interpolants in displacements and scalar fields, stabilization techniques are used in the mass conservation equation of the biphasic medium and in the rest of scalar equations. Finally, all mixed formulations are assessed in some benchmark problems and their performances are compared. It is found that mixed formulations that have the Jacobian as a nodal variable perform better.Peer ReviewedPostprint (author's final draft

Numerical simulation of penetration problems in geotechnical engineering with the particle finite element method (PFEM)

This paper highlights a computational framework for the numerical analysis of saturated soil-structure interaction problems. The variational equations of linear momentum and mass balance are obtained for the large deformation case. These equations are solved using the Particle Finite Element Method. The paper concludes with a benchmark test and the analysis of a penetration test

Coupled effective stress analysis of insertion problems in geotechnics with the Particle Finite Element Method

This paper describes a computational framework for the numerical analysis of quasi-static soil-structure insertion problems in water saturated media. The Particle Finite Element Method is used to solve the linear momentum and mass balance equations at large strains. Solid-fluid interaction is described by a simplified Biot formulation using pore pressure and skeleton displacements as basic field variables. The robustness and accuracy of the proposal is numerically demonstrated presenting results from two benchmark examples. The first one addresses the consolidation of a circular footing on a poroelastic soil. The second one is a parametric analysis of the cone penetration test (CPTu) in a material described by a Cam-clay hyperelastic model, in which the influence of permeability and contact roughness on test results is assessed.Peer ReviewedPostprint (author's final draft