On Computability and Triviality of Well Groups

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F16%3A00459972" target="_blank" >RIV/67985807:_____/16:00459972 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1007/s00454-016-9794-2" target="_blank" >http://dx.doi.org/10.1007/s00454-016-9794-2</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00454-016-9794-2" target="_blank" >10.1007/s00454-016-9794-2</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On Computability and Triviality of Well Groups

Popis výsledku v původním jazyce

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f: K -> R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within distance r from f for a given r>0 in the max-norm. The main drawback of the approach is that the computability of well groups was shown only when dim K=n or n=1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f,f’: K -> R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.

Název v anglickém jazyce

On Computability and Triviality of Well Groups

Popis výsledku anglicky

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f: K -> R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within distance r from f for a given r>0 in the max-norm. The main drawback of the approach is that the computability of well groups was shown only when dim K=n or n=1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f,f’: K -> R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

<a href="/cs/project/GA15-14484S" target="_blank" >GA15-14484S: Výpočet robustních invariantů hybridních dynamických systémů s využitím simulací</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2016

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Discrete & Computational Geometry

ISSN

0179-5376

e-ISSN

—

Svazek periodika

56

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

39

Strana od-do

126-164

Kód UT WoS článku

000377722100005

EID výsledku v databázi Scopus

2-s2.0-84973161597