Discrete Potential Theory and Perturbations of Ground State Configurations

本文主要研究欧氏空间中紧致流形的最小能量离散化问题，特别是Riesz能量核的离散化问题。以前关于Riesz S能量(广义汤姆逊问题)的离散平衡组态的工作表明，研究行为对参数S的依赖是有用的。特别有趣的是当S等于流形的Hausdorff维度并且发生从长程相互作用到短程相互作用的转变时出现的临界值。这个项目将研究：(I)最小能量的更精细的渐近性及其与流形的曲率和光滑性的关系；(Ii)对于维度2，8和24，对于存在于这些维度中的特殊晶格，根据S的Zeta函数，确定(或估计)在最小能量展开中产生的常数；(Iii)自相似集上的能量的渐近结果；(Iv)“贪婪能量点”的行为，特别是在外场存在的情况下；(V)旋转表面上离散的远程最小能量构型的极限支撑度的确定；以及(Vi)流形上均匀分布点的快速生成算法的开发和分析。本研究项目主要研究曲面上的带电粒子在通过两粒子排斥相互作用时如何以稳定的构形排列。这种对物质有序性的研究将拓宽对膜和薄膜物理的理解，并将应用于设计具有新的光学和电学性质的新材料。该项目的一个相关方面是在曲面(如地球)上快速生成可用于测量各种物理特性的数据采样点。这种用于点生成的方法对于测试诸如雷达系统的检测设备也很有用。这项研究在几个不同的背景下解决了如何最好地从模拟转换到数字的基本问题。

This research project focuses on the study of discretizations of compact manifolds in Euclidean space via minimal energy points, with special emphasis on Riesz energy kernels. Previous work on the discrete equilibrium configurations for the Riesz s-energy (generalized Thomson problem) showed the utility of investigating the dependence of behavior on the parameter s. Of particular interest is the critical value that occurs when s equals the Hausdorff dimension of the manifold and a transition occurs from long range to short range interactions. This project will investigate: (i) finer asymptotics for the minimal energy and its connection with the curvature and smoothness properties of the manifold; (ii) for dimensions 2, 8, and 24, the determination (or estimation) of constants arising in the minimal energy expansion in terms of the zeta functions in s for special lattices existing in these dimensions; (iii) asymptotic results for energy on self-similar sets; (iv) the behavior of "greedy energy points," especially in the presence of an external field; (v) the determination of the limiting support of discrete long range minimal energy configurations on surfaces of revolution; and (vi) development and analysis of algorithms for the fast generation of uniformly distributed points on manifolds.This research project focuses on the mathematics of how charged particles on a curved surface arrange themselves in a stable configuration when interacting through two-particle repulsive interactions. This study of the ordering of matter will broaden the understanding of the physics of membranes and films and has applications to the design of new materials with novel optical and electronic properties. A related aspect of the project is the rapid generation of data sampling points on curved surfaces (such as the earth) which can be used to measure a variety of physical properties. Such methods for point generation are also useful for testing detection devices such as radar systems. This research addresses in several different contexts the fundamental problem of how best to convert from analog to digital.