距离函数、临界点与随机Čech复形的拓扑
Distance Functions, Critical Points, and the Topology of Random Čech Complexes
摘要：
对于$R^d$中的点$P$的有限集合，函数$d_P: R^d \to R^+$测量到集合$P$的欧几里得距离。研究了当$P$为泊松过程时，$d_P$的临界点的个数.特别是，我们研究的极限行为的$N_k$-$d_P$的临界点的数量与莫尔斯指数$k$ -作为点的密度的增长。我们提出了明确的计算规范化，限制，期望和方差的$N_k$，以及分布极限定理。我们将这些结果与最近的结果联系起来，在这些结果中，基于$P$的随机Čech复形的Betti数进行了研究。
英文摘要：
For a finite set of points $P$ in $R^d$, the function $d_P: R^d \to R^+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is a Poisson process. In particular, we study the limit behavior of $N_k$ - the number of critical points of $d_P$ with Morse index $k$ - as the density of points grows. We present explicit computations for the normalized, limiting, expectations and variances of the $N_k$, as well as distributional limit theorems. We link these results to recent results in which the Betti numbers of the random Čech complex based on $P$ were studied.
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