On Vietoris-Rips Complexes of Ellipses
| Autor: | Samadwara Reddy, Henry Adams, Michal Adamaszek |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Vietoris–Rips complex
05E45 55U10 68R05 Computer Science::Computational Geometry Ellipse 01 natural sciences Combinatorics Simplicial complex Mathematics - Metric Geometry Mathematics::K-Theory and Homology FOS: Mathematics Computer Science::General Literature 0101 mathematics Mathematics - General Topology Mathematics Persistent homology Simplex Computer Science::Information Retrieval Homotopy 010102 general mathematics homotopy General Topology (math.GN) Astrophysics::Instrumentation and Methods for Astrophysics Metric Geometry (math.MG) Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ellipses persistent homology 010101 applied mathematics clique complex Geometry and Topology Eccentricity (mathematics) Analysis Clique complex |
| Zdroj: | Adamaszek, M, Adams, H & Reddy, S 2019, ' On Vietoris–Rips complexes of ellipses ', Journal of Topology and Analysis, vol. 11, no. 3, pp. 661-690 . https://doi.org/10.1142/S1793525319500274 |
| DOI: | 10.14760/owp-2017-11 |
| Popis: | For (Formula presented.) a metric space and (Formula presented.) a scale parameter, the Vietoris–Rips simplicial complex (Formula presented.) (resp. (Formula presented.)) has (Formula presented.) as its vertex set, and a finite subset (Formula presented.) as a simplex whenever the diameter of (Formula presented.) is less than (Formula presented.) (resp. at most (Formula presented.)). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev 13, 16, they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses (Formula presented.) of small eccentricity, meaning (Formula presented.). Indeed, we show that there are constants (Formula presented.) such that for all (Formula presented.), we have (Formula presented.) and (Formula presented.), though only one of the two-spheres in (Formula presented.) is persistent. Furthermore, we show that for any scale parameter (Formula presented.), there are arbitrarily dense subsets of the ellipse such that the Vietoris–Rips complex of the subset is not homotopy equivalent to the Vietoris–Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs. |
| Databáze: | OpenAIRE |
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