发表机构
Delft University of Technology（代尔夫特理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本论文通过完全正函数框架，将球面码和避距集的半定规划层级扩展到球堆积问题，证明收敛性，并应用于Witsenhausen问题和$t$-几乎等角集问题，给出解析解和最优构造。
AI 中文摘要
完全正函数是完全正矩阵的推广。已知它们可以刻画最大球面码和$\mathbb{R}^n$及某些紧致度量空间中最大密度的避距子集。本论文将此框架扩展到有限测度空间中的相关问题和球堆积问题。对于后者，本文进一步表明最优球堆积密度可以用Schwartz函数逼近。已知基于完全正函数和图上的Lovász theta数的半定规划界存在收敛的层级，用于估计最优球面码的大小。本论文将这些层级扩展到避距集和类似问题以及球堆积问题，并证明它们收敛到最大密度。对于避距集，还引入了其他层级，如矩层级，并证明它们比完全正层级更强，因此它们也收敛。还研究了紧致堆积问题的相关层级。这些界被应用于Witsenhausen问题，该问题询问$n$维单位球面上可被一个避免正交对的集合覆盖的最大比例$\alpha_n$；以及$t$-几乎等角集问题：寻找$n$维单位球面的一个子集的最大大小$\alpha(n,t)$，其中每个三元组包含一对内积为$t \in [-1,1)$。该界的解析解为$n=2$和$3$且$t \geq 0$时提供了最优构造的枚举。
英文摘要
Completely positive functions are an extension of completely positive matrices. They are known to characterize maximal spherical codes and maximum-density distance-avoiding subsets of $\mathbb{R}^n$ and certain compact metric spaces. This thesis expands this framework to related classes of problems in finite measure spaces and to the sphere-packing problem. For the latter, this is sharpened to show that the optimal sphere-packing density can be approximated using Schwartz functions. Converging hierarchies of semidefinite programming bounds on the size of optimal spherical codes are known, based on approximations of completely positive functions and the Lovász theta number of a graph. This thesis extends these hierarchies to distance-avoiding sets and similar problems and to the sphere-packing problem, and proves their convergence to the maximum density. For distance-avoiding sets, additional hierarchies, such as the moment hierarchy, are introduced and shown to be stronger than the completely positive hierarchy, hence they also converge. Related hierarchies for compact packing problems are also investigate. These bounds are implemented for Witsenhausen's problem, which asks for the maximum fraction $α_n$ of the $n$-dimensional unit sphere that is coverable by a set avoiding orthogonal pairs; and for the $t$-almost-equiangular-set problem: finding the maximum size $α(n,t)$ of a subset of the $n$-dimensional unit sphere in which every triple contains a pair with inner product $t \in [-1,1)$. An analytic solution to this bound yields an enumeration of optimal constructions for $n = 2$ and $3$ when $t \geq 0$.
Comments197 pages, 19 figures
