Mathematical Sciences: Topological Properties of Singularities and Solutions of Nonlinear Equations

数学科学：奇点的拓扑性质和非线性方程的解

• 批准号: 9400930
• 负责人: James Damon
• 金额: $ 12.9万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400930&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topological; Properties; Singularities

9400930 Damon Professor Damon has been developing analogues of infinitesimal methods introduced by Mather for answering questions about the structure of nonlinear mappings and the topological structure of special but important classes of highly nonisolated singularities, such as occur in discriminants of mappings. In quite different circumstances, he has also discovered how to modify these infinitesimal methods for understanding the generic behavior of "pixel intensity functions" under Gaussian blurring used in computer vision. He intends to build on his earlier results and derive general algebraic formulae for certain fundamental topological invariants for highly singular spaces. Second, he will further develop methods for determining the generic properties of solutions to partial differential equations and, in particular, apply them to recent nonlinear versions of Gaussian blurring. Third, he will apply the results on the topology of singular spaces to understand how the qualitative behavior of the solutions can be deduced from the singularities which occur. This research project will investigate certain qualitative properties of the set of solutions of systems of nonlinear equations. Such properties will be applied to generally appearing solutions of partial differential equations. Professor Damon has been developing methods for determining such qualitative properties. Moreover, he has begun applying them to understand the effects of Gaussian blurring applied to "pixel intensity functions" for analyzing images in computer vision. He intends to develop these methods further so that they apply to general partial differential equations. This will include the investigation of relations between certain invariants of the solutions and their qualitative properties. One particularapplication will be to the increasingly sophisticated nonlinear blurring schemes which are being developed for identifying essential features and properties of images suc h as edges, ridges, etc. ***

小行星9400930 达蒙教授一直在开发类似的无穷小方法介绍马瑟回答问题的结构非线性映射和拓扑结构的特殊但重要的类高度非孤立的奇点，如发生在判别式的映射。 在完全不同的情况下，他还发现了如何修改这些无穷小方法，以理解计算机视觉中使用的高斯模糊下“像素强度函数”的一般行为。 他打算建立在他以前的结果，并得出一般代数公式的某些基本拓扑不变量的高度奇异空间。 其次，他将进一步发展的方法来确定偏微分方程的解决方案的一般属性，特别是，将它们应用到最近的非线性版本的高斯模糊。 第三，他将应用奇异空间拓扑的结果来理解解的定性行为是如何从出现的奇异性推导出来的。 本研究计画将探讨非线性方程组解集的某些定性性质。 这样的性质将被应用到一般出现的偏微分方程的解决方案。 达蒙教授一直在开发确定这些定性性质的方法。 此外，他已经开始应用它们来理解高斯模糊应用于“像素强度函数”的影响，以分析计算机视觉中的图像。 他打算进一步发展这些方法，使它们适用于一般偏微分方程。 这将包括调查的某些不变量的解决方案和他们的定性性质之间的关系。 一个特殊的应用将是日益复杂的非线性模糊方案，这些方案正在开发中，用于识别图像的基本特征和属性，如边缘、脊等。