Continuous and Discrete Nonlocal Evolution Equations and Applications

该项目涉及空间离散和连续但非局部进化系统的动力学。特别地，我们关注的是相互作用和反应系统中表现出阈值行为的空间模式的存在、稳定性和多样性。不仅是固定的模式，还有那些以可预测的方式进化的模式。方程系统出现在计算神经网络领域，用于执行诸如自动图像识别之类的特殊任务。它们也出现在群体动力学、相变模型、大脑初级视觉皮层模型以及各种连续过程的计算机模拟中，其中离散化是标准方法。该项目的结果将对计算实验的理解和许多物理现象的数学模型所依赖的假设产生直接影响。从一个角度来看，该项目探索了当今材料科学的一些基本挑战：对导致并伴随物质的相或晶体变化的过程进行建模。新的数学模型不断被提出，希望能深入了解这些过程的基本物理原理。成功的建模可以为开发新的高性能材料提供帮助，这对经济和其他国家利益至关重要。从另一个角度来看，神经网络的理论研究将导致自动模式识别的发展，在文本分析、目标识别和图像增强等广泛的领域都很有用。

9975451ChmajThis 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.