Discrete gradient fields on infinite complexes
type: Item
instance of: scholarly article
title: Discrete gradient fields on infinite complexes
zbMATH Open document ID: 1230.55011
author: Rafael Ayala
author: Gregor Jerše
author: Neža Mramor Kosta
author: J. A. Vilches
published in: Discrete and Continuous Dynamical Systems
publication date: 2011-07-08T00:00:00Z
review text: Let \(M\) be a simplicial complex, and let us say that a pair of simplices \((\sigma,\tau)\) forms a face-pair if \(\sigma\) is a face of \(\tau\) and the difference in their dimensions is exactly one. A discrete vector field \(V\) is a collection of face-pairs such that each simplex in \(M\) belongs to at most one pair in \(V\). To depict this graphically, we draw an arrow from the center of \(\sigma\) to the center of \(\tau\). One way to construct a discrete vector field is from a discrete Morse function: a real-valued function on the set of all simplices of \(M\) with the property that for any simplex \(\sigma\), there is at most one face-pair \((\sigma,\tau)\) such that (1) \(f(\sigma)\geq f(\tau)\), and at most one face-pair \((\nu,\sigma)\) with (2) \(f(\nu)\geq f(\sigma)\). We obtain a discrete vector field on \(M\) from the set of all face-pairs that satisfy (1) or (2). A discrete vector field obtained in this way is said to be a discrete gradient vector field. It is a standard result that if \(M\) is a finite simplicial complex, a discrete vector field \(V\) is gradient if and only if there are no closed \(V\)-paths. Here, a \(V\)-path is a sequence of simplices \(\sigma_0,\tau_0,\sigma_1,\tau_1,\dots\) with \((\sigma_i,\tau_i)\) a face-pair from \(V\) and \((\sigma_{i+1},\tau_i)\) a face-pair not from \(V\). The authors of the paper extend this result to infinite complexes that are locally finite. In the special case when the complex is one-dimensional (i.e., a graph), it is shown that \(V\) having no closed \(V\)-paths is necessary and sufficient to guarantee that \(V\) is gradient. However in the general case, additional criteria must be placed on the \(V\)-paths, in particular on the incidence of two paths, in order to ensure that \(V\) is a discrete gradient field.
reviewed by: Jason Hanson
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 05E45
Mathematics Subject Classification ID: 68R10
zbMATH DE Number: 5919129
zbMATH Keywords: infinite simplicial complex
zbMATH Keywords: critical simplex
zbMATH Keywords: discrete gradient field
zbMATH Keywords: gradient path
zbMATH Keywords: discrete Morse theory
MaRDI profile type: Publication
full work available at URL: dcds.2011.30.623
OpenAlex ID: W2326391251
DOI: 10.3934/DCDS.2011.30.623
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description: scientific article; zbMATH DE number 5919129