Vovk’s randomness criterion characterizes sequences that are random relative to two distinct computable probability measures. The uniqueness of the criterion lies in the fact that, unlike the standard criterion based on the likelihood ratio test, it is expressed in terms of a geometrical quantity, the Hellinger distance, on the space of probability measures. In this paper, we generalize the ran...

Among engineers, risk is defined as a product of probability of the occurrence of an undesired event and the expected consequences in terms of human, economic, and environmental loss. These two components are equally important; therefore, the appropriate estimation of these values is a matter of great significance. This paper deals with one of these two components—the assessment of the probabil...

Stochastic simulations are used to characterize the knotting distributions of random ring polymers confined in spheres of various radii. The approach is based on the use of multiple Markov chains and reweighting techniques, combined with effective strategies for simplifying the geometrical complexity of ring conformations without altering their knot type. By these means we extend previous studi...

Over the years, problems like percolation and self-avoiding walks have provided important testing grounds for our understanding of the nature of the critical state. I describe some very recent ideas, as well as some older ones, which cast light both on these problems themselves and on the quantum field theories to which they correspond. These ideas come from conformal field theory, Coulomb gas ...

Over the years, problems like percolation and self-avoiding walks have provided important testing grounds for our understanding of the nature of the critical state. I describe some very recent ideas, as well as some older ones, which cast light both on these problems themselves and on the quantum field theories to which they correspond. These ideas come from conformal field theory, Coulomb gas ...

A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S−unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a consequence.

We present an interpretation of Dempster-Shafer theory based on the probability of provability. We present two forms of revision (conditioning) that lead to the geometrical rule of conditioning and to Dempster rule of conditioning, respectively.