Discrete and continuous nonlocal evolution equations and applications

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99743430BatesThis project concerns the dynamics of both spatially discrete and continuous but non-local evolutionary systems. In particular, we are concerned with the existence, stability, and variety of spatial patterns in interacting and reacting systems which exhibit threshold behavior. Not just stationary patterns but also those which evolve in a predictable way. The systems of equations arise in the field of computational neural networks designed to perform specialized tasks such as automatic image recognition. They also arise in population dynamics, in models of phase transitions, in models of the primary visual cortex of the brain, and in computer simulation of a wide variety of continuum processes where discretization is the standard approach. Results from the project will have a direct impact on the understanding of computational experiments and on the assumptions on which the mathematical models of many physical phenomena rest.From one perspective, the project explores some of the basic challenges of present day material science: The modeling of processes which cause and accompany change of phases or crystalline variants of a substance. New mathematical models are being continually proposed, with the hope of shedding insight into the basic physics of these processes. Successful modeling can offer help in the development of new high performance materials, which is vital to economic and other national interests. From another perspective, the theoretical study of neural networks will lead to the development of automatic pattern recognition, useful in such widely diverse areas as text analysis, target identification, and image enhancement.