arXiv 1507.047412015-07math.AT

正的Alexander对偶

Positive Alexander Duality for Pursuit and Evasion

Robert Ghrist, Sanjeevi Krishnan

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

摘要：

考虑的是一类追逃对策，其中逃避者试图避免被发现。这样的游戏可以被公式化为在时间轴上的欧几里得空间中搜索覆盖区域的补充部分。先前的结果给出了一般情况下逃避的同调标准，但这些标准是不必要和不充分的。本文在一般情况下给出了回避的一个充分必要的正上同调判据。主要的工具是（1）一个覆盖区域的Cech上同调与正锥编码空间方向的细化，（2）一个覆盖间隙的Borel-Moore同调与正锥编码时间方向的细化，和（3）亚历山大对偶的正变体。正上同调分解的整体瓣的一层局部正上同调的时间轴，我们展示了如何分解正上同调可计算的线性规划。

英文摘要：

Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the complement of a coverage region in a Euclidean space over a timeline. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in a general case. The principal tools are (1) a refinement of the Cech cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore homology of the coverage gaps with a positive cone encoding time orientation, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable as a linear program.

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。