A Numerical Approach for the Filtered Generalized Čech Complex
In this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a none...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-12-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/13/1/11 |
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| author | Jesús F. Espinoza Rosalía Hernández-Amador Héctor A. Hernández-Hernández Beatriz Ramonetti-Valencia |
| author_facet | Jesús F. Espinoza Rosalía Hernández-Amador Héctor A. Hernández-Hernández Beatriz Ramonetti-Valencia |
| author_sort | Jesús F. Espinoza |
| collection | DOAJ |
| description | In this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris−Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized Čech complex of a finite collection of <i>d</i>-dimensional disks in <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> </semantics> </math> </inline-formula>, and we show the performance of our algorithm. |
| first_indexed | 2024-12-12T21:12:42Z |
| format | Article |
| id | doaj.art-ed04379e5d21418d9ced20ead3a9882a |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-12-12T21:12:42Z |
| publishDate | 2019-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-ed04379e5d21418d9ced20ead3a9882a2022-12-22T00:11:51ZengMDPI AGAlgorithms1999-48932019-12-011311110.3390/a13010011a13010011A Numerical Approach for the Filtered Generalized Čech ComplexJesús F. Espinoza0Rosalía Hernández-Amador1Héctor A. Hernández-Hernández2Beatriz Ramonetti-Valencia3Departamento de Matemáticas, Universidad de Sonora, C.P. 83000, Hermosillo, MexicoDepartamento de Matemáticas, Universidad de Sonora, C.P. 83000, Hermosillo, MexicoDepartamento de Matemáticas, Universidad de Sonora, C.P. 83000, Hermosillo, MexicoDepartamento de Matemáticas, Universidad de Sonora, C.P. 83000, Hermosillo, MexicoIn this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris−Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized Čech complex of a finite collection of <i>d</i>-dimensional disks in <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> </semantics> </math> </inline-formula>, and we show the performance of our algorithm.https://www.mdpi.com/1999-4893/13/1/11disk systemgeneralized čech complexčech scalegeneralized vietoris–rips lemmaminiball problem |
| spellingShingle | Jesús F. Espinoza Rosalía Hernández-Amador Héctor A. Hernández-Hernández Beatriz Ramonetti-Valencia A Numerical Approach for the Filtered Generalized Čech Complex Algorithms disk system generalized čech complex čech scale generalized vietoris–rips lemma miniball problem |
| title | A Numerical Approach for the Filtered Generalized Čech Complex |
| title_full | A Numerical Approach for the Filtered Generalized Čech Complex |
| title_fullStr | A Numerical Approach for the Filtered Generalized Čech Complex |
| title_full_unstemmed | A Numerical Approach for the Filtered Generalized Čech Complex |
| title_short | A Numerical Approach for the Filtered Generalized Čech Complex |
| title_sort | numerical approach for the filtered generalized cech complex |
| topic | disk system generalized čech complex čech scale generalized vietoris–rips lemma miniball problem |
| url | https://www.mdpi.com/1999-4893/13/1/11 |
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