In this paper we consider the φF category associated to the family of circle valued Morse functions defined on a closed manifold M . This number will be called the Morse-Smale characteristic of manifold M for circle-valued Morse functions and and it will be denoted by γs1(M). We present some basic notions and results concerning circle-valued Morse functions and we prove some properties for γs1(...

Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and MacPherson [17]. We describe the basics of this theory and prove fundamental theorems relating the topology of a general simplicial complex with the critical simplice...

An algebraic model based on Lie-algebraic techniques is applied to the analysis of thermodynamic vibrational properties of diatomic molecules. The local anharmonic effects are described by a Morse-like potential and corresponding anharmonic bosons associated with the U(2) algebra. A vibrational high temperature partition function and the related thermodynamic functions are derived in terms of t...

The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincaré-Hopf indices at critica...

Morse theory provides a powerful framework to study the topology of a manifold from a function de ned on it, but discrete constructions have remained elusive due to the di culty of translating smooth concepts to the discrete setting. Consider the problem of approximating the Morse-Smale (MS) complex of a Morse function from a point cloud and an associated nearest neighbor graph (NNG). While fol...

Morse decompositions have been proposed to compute and represent the topological structure of steady vector fields. Compared to the conventional differential topology, Morse decomposition and the resulting Morse Connection Graph (MCG) is numerically stable. However, the granularity of the original Morse decomposition is constrained by the resolution of the underlying spatial discretization, whi...

Morse decompositions have been proposed to compute and represent the topological structure of steady vector fields. Compared to the conventional differential topology, Morse decomposition and the resulting Morse Connection Graph (MCG) is numerically stable. However, the granularity of the original Morse decomposition is constrained by the resolution of the underlying spatial discretization, whi...