Collaborative Research: Positive definite functions in distance geometry and combinatorics

合作研究：距离几何和组合学中的正定函数

• 批准号: 1101687
• 负责人: Alexander Barg
• 金额: $ 8万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101687&HistoricalAwards=false
• 关键词: Collaborative; Research; Positive; definite; functions

The project is devoted to the study of finite point configurations in metric spaces. Classical results in harmonic analysis and group representations imply the existence of functions that satisfy certain positivity constraints when evaluated on such configurations. These constraints give a set of necessary conditions for the existence of the configuration. The project studies the extent to which the positivity constraints are also sufficient in that they imply that a configuration with desired properties exists. A related topic studied in the project is the maximum size of point sets with few distances in metric spaces. The context for the development of the project is related to the recently established semidefinite programming bounds on codes in homogeneous spaces. The project also pursues a link between uniformly distributed sets of points known as spherical and Euclidean designs and a more general concept of cubature formulas with the aim to use methods of algebraic combinatorics to study cubature formulas in metric and functional spaces. One of the goals is to establish new universal bounds on cubature formulas in homogeneous spaces. Finite collections of points in space find applications in reliable and numerical analysis. Studying the structure of point configurations creates insights into construction of optimal signal transmission schemes and of optimal nets for Monte-Carlo integration.

该项目致力于研究度量空间中的有限点配置。调和分析和群表示中的经典结果意味着在对此类配置进行评估时，存在满足某些正性约束的函数。这些约束给出了配置存在的一组必要条件。该项目研究了正约束在多大程度上也足够，因为它们意味着存在具有所需属性的配置。该项目研究的一个相关主题是度量空间中距离很少的点集的最大尺寸。该项目的开发背景与最近在同质空间中建立的代码半定编程界限有关。该项目还寻求均匀分布的点集（球面和欧几里得设计）与更一般的体积公式概念之间的联系，旨在使用代数组合方法来研究度量和函数空间中的体积公式。目标之一是在齐次空间中的体积公式上建立新的通用界限。空间中点的有限集合可用于可靠的数值分析。研究点配置的结构可以深入了解构建最佳信号传输方案和蒙特卡罗积分的最佳网络。