We discuss the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces. We prove our results in the axiomatic framework of [16].

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.

Choquet capacities are a generalization of probability measures that arise in robustness, decision theory and game theory. Many capacities that arise in robustness are symmetric or can be transformed into symmetric capacities. We characterize the extreme points of the set of upper distribution functions corresponding to coherent, symmetric Choquet capacities on [0,1]. We also show that the set ...

Consider a network N = (V; E; c) with an integer valued capacity function c: V V ! Z + , and let k be a positive integer. What is the minimal total increase by which the individual capacities must be increased in order that the value of the maximal ow between all nodes is at least k? Frank FRANK92] gives a min-max formula for which is proved using Maders MADER82] splitting theorem. In order to ...

I am going to talk about symplectic topology, a field that has seen a remarkable development in the past 15 years. Let’s begin with what was classically known – which was rather little. We start with a symplectic form ω, that is, a closed 2-form which is nondegenerate. This last condition means that ω is defined on an even dimensional manifold M and that its wedge with itself n times is a top d...

The theory of diffusion is one of the prevailing theories on the explanation of the quality of dissemination of innovations, expansion of ideas and thoughts in the area of culture.The present research is an attempt to respond to this question: Is it possible to disseminate the Islamic Revolution values and ideals through the window of Abrahamic Hajj into the world of Islam? And if the res...