Discrete Minimal Energy Configurations and Related Problems in Potential Theory

势理论中的离散最小能量配置及相关问题

• 批准号: 0603828
• 负责人: Edward Saff
• 金额: --
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-15 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603828&HistoricalAwards=false
• 关键词: Discrete; Minimal; Energy; Configurations; Related

The research will primarily focus on three areas: (i) the analysis of minimal energy point configurations; (ii) the noninvasive detection of faults (cracks) in homogeneous media; and (iii) limiting behaviors of certain basis functions over planar regions. Regarding (i), we shall analyze certain equilibrium (or "ground state") configurations for N particles interacting via a pairwise repulsive interaction V and confined to a fixed set (such a sphere or torus). Such configurations are useful for purposes of sampling data, computer graphics, best-packing, and understanding the physics of self-assembling materials. Concerning (ii), we shall continue with the development of rational approximation methods that can determine the existence and location of cracks inside a conducting material. Regarding (iii), our focus will be on polynomials orthogonal over planar regions and how their behavior can be used to solve problems related to tomography (the recovery of the underlying planar region).

研究将主要集中在三个方面：(i)最小能量点配置的分析；（ii）均匀介质中故障（裂缝）的非侵入性检测；(3)平面区域上某些基函数的极限行为。关于(i)，我们将分析N个粒子通过配对排斥相互作用V相互作用并局限于固定集合（如球体或环面）的某些平衡（或“基态”）构型。这样的配置对于采样数据、计算机图形、最佳包装和理解自组装材料的物理特性非常有用。关于（ii），我们将继续发展合理的近似方法，以确定导电材料内部裂纹的存在和位置。关于（iii），我们的重点将放在平面区域上的正交多项式以及如何使用它们的行为来解决与断层扫描相关的问题（底层平面区域的恢复）。