Carathéodory convex integrand operators and probability theory

Preliminaries : Convex Analysis and Probability Theory

In this lecture, we will continue our discussion on vector spaces and convex analysis and look at results such as the Riesz representation theorem and convex projections. In addition, we will discuss some fundamentals of Probability Theory including σ-fields, probability measures, notion of independence, expectations etc. We will present a formal definition of random variables and distribution ...

Multiscale Homogenization of Convex Functionals with Discontinuous Integrand

This article is devoted to obtain the Γ-limit, as ε tends to zero, of the family of functionals u 7→ ∫ Ω f ( x, x ε , . . . , x ε ,∇u(x) ) dx, where f = f(x, y, . . . , yn, z) is periodic in y, . . . , yn, convex in z and satisfies a very weak regularity assumption with respect to x, y, . . . , yn. We approach the problem using the multiscale Young measures.

Probability Operators

This is a study in the meaning of natural language probability operators, sentential operators such as probably and likely. We ask what sort of formal structure is required to model the logic and semantics of these operators. Along the way we investigate their deep connections to indicative conditionals and epistemic modals, probe their scalar structure, observe their sensitivity to contextuall...

Optimal Inequalities in Probability Theory: A Convex Optimization Approach

We address the problem of deriving optimal inequalities for P(X E S), for a multivariate random variable X that has a given collection of moments, and S is an arbitrary set. Our goal in this paper is twofold: First, to present the beautiful interplay of probability and optimization related to moment inequalities, from a modern, optimization based, perspective. Second, to understand the complexi...

Optimal Inequalities in Probability Theory: A Convex Optimization