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

MusinDMS-0807640 BargDMS-0807411该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真实的球体上的点分配和相关配置。所研究的点集的性质包括限制点集上任何一对不同点之间的最小角距或它所支持的容积公式的次数.这些问题的最新进展包括研究者解决了四维接吻数问题，三维接吻数问题的新证明，以及利用半定规划求解码的界的新方法，是施里弗巴霍克和调查员的功劳在x维空间中，接吻问题是指在x维空间中有多少个单位球面可以接触（接吻）一个单位球面。在这些工作中开发的新思想铺平了道路，为进一步的进步在thebounds的代码的大小和在一些其他问题的距离几何。 该项目要解决的主要问题是当球面上的点集具有某种性质时，其最优点集的大小的边界问题，以及边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点之间具有超过给定值的最小角间距（这与信号处理问题有关），并且支持给定次数的球谐函数的精确体积公式。所研究的点集的应用包括通信理论、数值分析、网格划分问题、数据表示和传感器网络定位。

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.