Combinatorial structures of the space of Hamiltonian vector fields on compact surfaces

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-05-27 DOI:10.1016/j.topol.2025.109439

Tomoo Yokoyama

引用次数: 0

Abstract

In the time evolution of fluids, the topologies of fluids can be changed by the creations and annihilations of singular points and by switching combinatorial structures of separatrices. In this paper, we construct foundations of descriptions of the time evaluations of fluid phenomena (e.g. Euler equations, Navier-Stokes equations). In particular, we study the combinatorial structure of the “moduli space” of Hamiltonian vector fields. In fact, under the conditions of the non-existence of creations and annihilations of singular points, the space of topological equivalence classes of such Hamiltonian vector fields on compact surfaces has non-contractible connected components and is a disjoint union of finite abstract cell complexes such that the codimension of a cell corresponds to the instability of a Hamiltonian vector field by using combinatorics and simple homotopy theory.

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紧曲面上哈密顿向量场空间的组合结构

在流体的时间演化中，奇异点的产生和湮灭以及分离点组合结构的切换可以改变流体的拓扑结构。本文建立了流体现象时间评价的描述基础（如欧拉方程、纳维-斯托克斯方程）。特别地，我们研究了哈密顿向量场的“模空间”的组合结构。事实上，在不存在奇点产生和消灭的条件下，紧致曲面上这种哈密顿向量场的拓扑等价类空间具有不可收缩的连通分量，并且是有限抽象细胞复合体的不相交并，利用组合学和简单同伦理论，使得一个细胞的余维数对应于哈密顿向量场的不稳定性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.