Special Capacities, the Hunt-Stein Theorem and Transformation Groups

The Restriction Theorem of Tomas and Stein

Varadhan’s theorem for capacities

Varadhan’s integration theorem, one of the corner stones of large-deviation theory, is generalized to the context of capacities. The theorem appears valid for any integral that obeys four linearity properties. We introduce a collection of integrals that have these properties. Of one of them, known as the Choquet integral, some continuity properties are established as well.

A structure theorem for product sets in extra special groups

Hegyvári and Hennecart showed that if B is a sufficiently large brick of a Heisenberg group, then the product set B ·B contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.

Language capacities: Is grammar special?

Some of the more heated debates in the cognitive sciences revolve around the nature of human language. Is there innate linguistic knowledge? Does language require the ability to learn and use symbolic rules, or can it be explained solely through associative capacities? And how autonomous is language — is the ability to learn to speak the result of more general social, cognitive and perceptual a...

Bayes' Theorem for Choquet Capacities

We give an upper bound for the posterior probability of a measurable set A when the prior lies in a class of probability measures 9. The bound is a rational function of two Choquet integrals. If g; is weakly compact and is closed with respect to majorization, then the bound is sharp if and only if the upper prior probability is 2-alternating. The result is used to compute-bounds for several set...