Collaborative Research: Multivariate positive definite polynomials and their applications via SDP

MUSIN DMS-0807640 BargDMS-0807411本项目致力于利用距离几何、编码理论和半定规划的方法研究实球面上的点分配及其相关构形。所研究的点集的性质包括限制集合中任何一对不同的点之间的最小角距或它所支持的容度公式的程度。这些问题的最新进展包括4维吻接数问题的调查者的解，3维吻接数问题的新证明，以及由于Schrijver，Bac和调查者的半定编程对码的界的新方法。X维的接吻问题是一个单位球在x维上可以接触(接吻)多少个单位球的问题。这些工作中提出的新思想为进一步发展码大小的界限问题和距离几何中的其他一些问题铺平了道路。本项目要解决的主要问题是当球面上的最优点集具有一定性质时点集的大小的定界问题，以及定界问题与多元正定多项式之间的联系。点集的有趣性质包括在集合中的任何两个点之间具有超过给定值的最小角度间隔(这与信号处理问题相关)，并支持给定次数的球谐函数的精确立方公式。所研究的点集的应用包括通信理论、数值分析、网格问题、数据表示以及在传感器网络中的定位。

MusinDMS-0807640BargDMS-0807411 This project is devoted to the study of point allocations onthe real sphere and related configurations using methods ofdistance geometry, coding theory, and semidefinite programming. The properties of the point sets studied include restricting theminimum angular separation between any pair of distinct points inthe set or the degree of the cubature formula supported by it. Recent advances in these problems include a solution by theinvestigator of the kissing number problem in 4 dimensions, a newproof for the kissing number problem in 3 dimensions, and a newapproach to bounds on codes using semidefinite programming, dueto Schrijver, Bachoc, and the investigator. The kissing problemin x dimensions is the question of how many unit spheres cantouch (kiss) a unit sphere in x dimensions. The new ideasdeveloped in these works pave the way for further advances in theproblems of bounding the size of codes and in a number of otherproblems in distance geometry. The main problems to be addressed in the project are relatedto bounding the size of optimal sets of points on a sphere whenthe sets have a certain property, and the links between thebounding problem and multivariate positive definite polynomials. Interesting properties of the point set include having a minimumangular separation that exceeds a given value between any twopoints in the set (this is relevant for signal processingproblems), and supporting an exact cubature formula for sphericalharmonic functions of a given degree. Applications of the pointsets studied include communication theory, numerical analysis,the meshing problem, data representation, and localization insensor networks.