相关论文: Topological Inference of the Conley Index

We explore a particular approach to the analysis of dynamical and geometrical properties of autonomous, Pfaffian non-holonomic systems in classical mechanics. The method is based on the construction of a certain auxiliary constrained…

Main theorem of this paper states that Floer cohomology groups in a Hilbert space are isomorphic to the cohomological Conley Index. It is also shown that calculating cohomological Conley Index does not require finite dimensional…

The aim of this paper is to explore the possibilities of Conley index techniques in the study of heteroclinic connections between finite and infinite invariant sets. For this, we remind the reader of the Poincar\'e compactification: this…

Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a…

If a complex analytic function, $f$, has a stratified isolated critical point, then it is known that the cohomology of the Milnor fibre of $f$ has a direct sum decomposition in terms of the normal Morse data to the strata. We use microlocal…

In this short note, we show that the distance function to any finite set $X\subset \mathbb{R}^n$ is a topological Morse function, regardless of whether $X$ is in general position. We also precisely characterize its topological critical…

This is the second paper in our sequence. Here, we apply our abstract Morse index formulation developed in the previous paper to study several optimization set-ups with constraints, including type I or/and type II considerations. A common…

We introduce the notion of a template for discrete Morse theory. Templates provide a memory efficient approach to the computation of homological invariants (e.g., homology, persistent homology, Conley complexes) of cell complexes. We…

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…

In this paper we investigate Deep Learning Models using topological dynamical systems, index theory, and computational homology. These mathematical machinery was invented initially by Henri Poincare around 1900 and developed over time to…

This paper gives the definition of the Conley index for a piecewise continuous map, which is only well defined on compatible isolating neighborhoods with Wazewski property slightly weaker than continuous situation.

We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the…

In this paper we study topological and dynamical features of isolated invariant continua of continuous flows defined on surfaces. We show that near an isolated invariant continuum the flow is topologically equivalent to a C1 flow. We deduce…

A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector…

We consider functions with isolated critical points on a closed surface. We prove that in a neighborhood of a critical point the function conjugates with Re$z^k$ for the some nonnegative integer k. The full topological invariant of such…

We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations -- ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems,…

We previously developed an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and the concept of virtual Morse-Bott indices for singular critical points of Hamiltonian functions for the…

Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient…

Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected noncompact semi-simple Lie group with finite center. The finest Morse decomposition of the…

In various situations in Floer theory, one extracts homological invariants from "Morse-Bott" data in which the "critical set" is a union of manifolds, and the moduli spaces of "flow lines" have evaluation maps taking values in the critical…