Discrete Dynamical System Framework for Construction of Connections between Critical Regions in Lattice Height Data

Abstract

The concept of discrete multivalued dynamical systems for noncommutative CW complexes is developed. Stable and unstable manifolds are introduced and their role in geometric and topological configurations of noncommutative CW complexes is studied. Our technique is illustrated by an example on the noncommutative CW complex decomposition of the algebra of continuous functions on two dimensional torus.2010 MSC: 46L85, 55U10, 54H20, 34D35.

Multivalued maps as a tool in modeling and rigorous numerics

“…In Section 5, we outline our most recent contributions [2,3] to topological analysis of digital images and multidimensional data sets, inspired by discrete multivalued dynamical systems. The object of study is a scalar function f : X → R on a multidimensional data set X .…”

A Global Reduction Based Algorithm for Computing Homology of Chain Complexes

“…In digital image analysis, topological invariants are useful in shape description, indexation, and classification. Among shape descriptors based on homology theory, there are the Morse shape descriptor [1] [2], the Morse Connection Graph [3], and the persistence barcodes for shape [4]. The necessity of improved algorithms appears evident as new applications of the homology computation arise in research for very large data sets.…”

“…For cubical sets embedded in 3  , Mrozek et al proposed a method [12] based on the computation of acyclic subspaces by using lookup tables. Experimentally, this method performed in linear time.…”

“…Experimentally, this method performed in linear time. Another method running in quasi linear time for finite simplicial complexes embedded in 3  was proposed by Delfinado and Edelsbrunner in [13]. For problems dealing with higher dimensions, Mrozek and Batko developed an algorithm [14] based on the concept of coreduction which showed interesting performance results on cubical sets.…”

Discrete dynamics on noncommutative CW complexes

“…Section 3 is devoted to a review from [12] on the basics of modified Morse theory on noncommutative CW complexes. Discrete multivalued dynamical systems have been introduced in [1,9]. In section 4 we develop discrete multivalued dynamical systems on noncommutative CW complexes and provide tools to relate a dynamical picture to the topology and geometry of noncommutative CW complexes.…”