Tame Flows
Tame Flows
摘要：
驯服流是在“好”空间上的“好”流。好集是Khovanski引入的pfteran集，如果pfteran集$X$上的流$Φ: \mathbb{R}\times X\to X$的图是$\mathbb{R}\times X\times X$的pfteran子集，则它是好集.任何紧的驯服集都允许大量的驯服流。证明了一般实解析函数关于一般实解析度量的梯度所决定的流是驯服的。典型的Tame梯度流满足Morse-Smale条件，证明了在Tame情形下Morse-Smale条件等价于非稳定流形的分层是Verdier和Whitney正则的.我们解释了如何根据不稳定流计算其孤立稳定点的Conley指数，然后使用这种技术产生一个关于偏序集的莫尔斯理论，推广了R。Forman的离散莫尔斯理论。最后，我们使用Harvey-Lawson有限体积流技术来产生一个同伦之间的DeRham复形的光滑流形和单纯的链复形与三角形。
英文摘要：
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $Φ: \mathbb{R}\times X\to X$ on pfaffian set $X$ is tame if the graph of $Φ$ is a pfaffian subset of $\mathbb{R}\times X\times X$. Any compact tame set admits plenty tame flows. We prove that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metric is tame. The typical tame gradient flow satisfies the Morse-Smale condition, and we prove that in the tame context the Morse-Smale condition is equivalent to the fact that the stratification by unstable manifolds is Verdier and Whitney regular. We explain how to compute the Conley indices of isolated stationary points of tame flows in terms of their unstable varieties, and then use this technology to produce a Morse theory on posets generalizing R. Forman's discrete Morse theory. Finally, we use the Harvey-Lawson finite volume flow technique to produce a homotopy between the DeRham complex of a smooth manifold and the simplicial chain complex associated to a triangulation.
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。
