The goal of this paper is to give a new and short proof of the local Atiyah-Singer index theorem by using approximations of heat semigroups. The heat equation approach to index theorems is not new: It was suggested by Atiyah-Bott [1] and McKean-Singer [18], and first carried out by Patodi [21] and Gilkey [13]. Bismut in [8] introduces stochastic methods based on Feynman-Kac formula. For probabi...

Consider the equation (∗) 1 2 ∆u− V u = 0 in R, for d ≥ 3. For certain classes of potentials V , we use probabilistic tools to study the bounded solutions and the positive solutions for (*). A primary motivation is to offer probabilistic intuition for the results.

Class of cellular automata (CA) for modeling reaction diffusion systems has been presented. The construction of the CA is general enough to be applicable to large class of reaction diffusion equation s. Tne automata are based on running average procedure and on probabilistic table look up to implement diffusion reactions. The evolution of probabilistic CA simulates a spatially distributed proce...

In recent years, several approaches to generate probabilistic counterexamples have been proposed. The interpretation of probabilistic counterexamples, however, continues to be problematic since they have to be represented as sets of paths, and the number of paths in this set may be very large. Fault trees (FTs) are a well-established industrial technique to represent causalities for possible sy...

A probabilistic method to predict macrocrack initiation due to fatigue damage is presented in this paper. Acoustic non-linearity is used to quantify pre-macrocrack initiation damage. This data is then used in a probabilistic analysis of fatigue damage. The probabilistic fatigue damage analysis consists of a suitably chosen damage evolution equation to model accumulated damage coupled with a pro...