球面上的覆盖过程与线性规划的条件数
Coverage processes on spheres and condition numbers for linear programming
摘要：
本文件有两个议程。首先，我们展示了覆盖过程的新结果。令$p(n,m,α)$为$S^m$中角半径为$α$的$n$球冠不覆盖整个球体$S^m$的概率。我们给出了$p(n,m,α)$在$α\in[π/2,π]$情形下的精确公式和$p(n,m,α)$在$α\in [0,π/2]$情形下的上界，当$α\toπ/2$时，它趋向于$p(n,m,π/2)$。在$α\in[0,π/2]$的情况下，这产生了覆盖$S^m$所需的半径为$α$的球冠的预期数量的上限。其次，我们研究了线性规划可行性问题$\exists x\in\mathbb{R}^{m+1}Ax\le0,x\ne0$的条件数${\mathscr{C}}(A)$，其中$A\in\mathbb{R}^{n\times(m+1)}$是按标准正态分布随机选取的。我们精确地确定了$A$可行的条件下的${\mathscr{C}}(A)$的分布，并在不可行的情况下提供了分布函数的上界。使用这些结果，我们表明，$\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$的所有$n>m$，最尖锐的约束，这种期望的今天。这两个议程通过在覆盖率和条件之间转换的结果而相互关联。
英文摘要：
This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,α)$ be the probability that $n$ spherical caps of angular radius $α$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,α)$ in the case $α\in[π/2,π]$ and an upper bound for $p(n,m,α)$ in the case $α\in [0,π/2]$ which tends to $p(n,m,π/2)$ when $α\toπ/2$. In the case $α\in[0,π/2]$ this yields upper bounds for the expected number of spherical caps of radius $α$ that are needed to cover $S^m$. Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem $\exists x\in\mathbb{R}^{m+1}Ax\le0,x\ne0$ where $A\in\mathbb{R}^{n\times(m+1)}$ is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to $A$ being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all $n>m$, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
