用于追逃的正亚历山大对偶
代数拓扑
2016-06-02 v2
摘要
考虑一类追逃博弈，其中逃避者试图避免被探测。此类博弈可表述为在欧氏空间中随时间轴在覆盖区域补集上寻找截面。先前的结果给出了一般情形下逃避的同调判据，但这些判据既非必要也非充分。本文给出了一般情形下逃避的一个必要且充分的正上同调判据。主要工具为：(1) 覆盖区域的 Čech 上同调的细化，引入编码空间方向的正锥；(2) 覆盖空隙的 Borel-Moore 同调的细化，引入编码时间方向的正锥；(3) Alexander 对偶的一个正变体。正上同调分解为时间轴上局部正上同调层的总体截面；我们展示了这一分解如何使正上同调可计算为一个线性规划。
引用
@article{arxiv.1507.04741,
title = {Positive Alexander Duality for Pursuit and Evasion},
author = {Robert Ghrist and Sanjeevi Krishnan},
journal= {arXiv preprint arXiv:1507.04741},
year = {2016}
}
备注
19 pages, 6 figures; improvements made throughout: e.g. positive (co)homology generalized to arbitrary degrees; Positive Alexander Duality generalized from homological degrees 0,1; Morse and smoothness conditions generalized; illustrations of positive homology added. minor corrections in proofs, notation, organization, and language made throughout. variant of Borel-Moore homology now used