On Computability and Triviality of Well Groups
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
On Computability and Triviality of Well Groups
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Result continuities
Project
<a href="/en/project/GA15-14484S" target="_blank" >GA15-14484S: Simulation-Based Computation of Robust Invariants of Hybrid Dynamical Systems</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Discrete & Computational Geometry
ISSN
0179-5376
e-ISSN
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Volume of the periodical
56
Issue of the periodical within the volume
1
Country of publishing house
US - UNITED STATES
Number of pages
39
Pages from-to
126-164
UT code for WoS article
000377722100005
EID of the result in the Scopus database
2-s2.0-84973161597