On certain optimal diffeomorphisms between closed curves
type: Item
instance of: scholarly article
title: On certain optimal diffeomorphisms between closed curves
zbMATH Open document ID: 1312.57039
author: Andrea Cerri
author: Barbara Di Fabio
published in: Forum Mathematicum
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0906.4648
full work available at URL: CerriDiFabioAmsActa.pdf
reviewed by: Massimo Ferri
Mathematics Subject Classification ID: 57S05
Mathematics Subject Classification ID: 57S10
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 68U05
Mathematics Subject Classification ID: 68T10
Mathematics Subject Classification ID: 58C05
zbMATH DE Number: 6368603
zbMATH Keywords: natural pseudodistance
zbMATH Keywords: measuring function
zbMATH Keywords: size pair
zbMATH Keywords: Morse function
MaRDI profile type: Publication
OpenAlex ID: W1986465913
arXiv ID: 0906.4648
DOI: 10.1515/FORM.2011.172
Recommended article: Q3514524
Recommended article: Stability of persistence diagrams
Recommended article: Persistent Homology: Theory and Practice
Recommended article: Q4825375
Recommended article: Geometric Approaches to Persistent Homology
Recommended article: Necessary conditions for discontinuities of multidimensional persistent Betti numbers
Recommended article: Persistent homology transform for modeling shapes and surfaces
Recommended article: One-dimensional reduction of multidimensional persistent homology
review text: In \textit{size theory}, the pattern recognition side of persistent homology, shape is formalized not as a space, but as a \textit{size pair} \((X, \varphi)\) of a topological space and a map defined on it, mostly (and here in particular) with \(\mathbb R\) as a range [\textit{S. Biasotti}, \textit{L. De Floriani}, \textit{B. Falcidieno}, \textit{P. Frosini}, \textit{D. Giorgi}, \textit{C. Landi}, \textit{L. Papaleo} and \textit{M. Spagnuolo}, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys, 40(4), 12:1-12:87 (2008)].NEWLINENEWLINEA pseudodistance between size pairs \((X, \varphi)\), \((Y, \psi)\), where \(X\) and \(Y\) are homeomorphic, is defined as follows. For any homeomorphism \(f: X \to Y\) let \(\Theta(f) = max_{x\in X} |\varphi(x)-\psi(f(x))|\). Then the \textit{natural pseudodistance} \(\delta \big( (X, \varphi), (Y, \psi)\big)\) is defined as the infimum of \(\{\Theta(f)\}\) for \(f\) in the set of all such homeomorphisms \textit{P. Frosini} and \textit{M. Mulazzani} [Bull. Belg. Math. Soc. - Simon Stevin 6, No. 3, 455--464 (1999; Zbl 0937.55010)]. \(f: X \to Y\) is said to be an \textit{optimal} homeomorphism if \(\Theta(f) = \delta \big( (X, \varphi), (Y, \psi)\big)\).NEWLINENEWLINEDoes an optimal homeomorphism always exist? Yes, and it is of class \({\mathcal C}^2\), if the two spaces are curves, the two maps are Morse functions and the pseudodistance vanishes (Theorem 3.4). No, if any of these three conditions is missing, as the authors show by smart examples.
description: scientific article; zbMATH DE number 6368603