Approximate Čech Complex in Low and High Dimensions

2013 ◽  
pp. 666-676 ◽  
Author(s):  
Michael Kerber ◽  
R. Sharathkumar
Keyword(s):  
High Dimensions ◽  
Čech Complex ◽  
Cech Complex

scholarly journals Proregular sequences, local cohomology, and completion

2003 ◽  
Vol 92 (2) ◽  
pp. 161 ◽  
Author(s):  
Peter Schenzel
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Čech Complex ◽  
Regular Sequences ◽  
Local Cohomology Modules ◽  
Derived Functors ◽  
Adic Completion ◽  
Cech Complex ◽  
Cohomology Modules ◽  
The Ideal

As a certain generalization of regular sequences there is an investigation of weakly proregular sequences. Let $M$ denote an arbitrary $R$-module. As the main result it is shown that a system of elements $\underline x$ with bounded torsion is a weakly proregular sequence if and only if the cohomology of the Čech complex $\check C_{\underline x} \otimes M$ is naturally isomorphic to the local cohomology modules $H_{\mathfrak a}^i(M)$ and if and only if the homology of the co-Čech complex $\mathrm{RHom} (\check C_{\underline x}, M)$ is naturally isomorphic to $\mathrm{L}_i \Lambda^{\mathfrak a}(M),$ the left derived functors of the $\mathfrak a$-adic completion, where $\mathfrak a$ denotes the ideal generated by the elements $\underline x$. This extends results known in the case of $R$ a Noetherian ring, where any system of elements forms a weakly proregular sequence of bounded torsion. Moreover, these statements correct results previously known in the literature for proregular sequences.

Limit theorems for process-level Betti numbers for sparse and critical regimes

2020 ◽  
Vol 52 (1) ◽  
pp. 1-31
Author(s):  
Takashi Owada ◽  
Andrew M. Thomas
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Betti Numbers ◽  
Critical Regime ◽  
Dimensional Euclidean Space ◽  
Homogeneous Poisson Process ◽  
Čech Complex ◽  
Poisson Limit ◽  
Cech Complex ◽  
Process Level

AbstractThe objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ${\mathbb{R}}^d$ . We consider the case where the points of the Čech complex are generated by a Poisson process with intensity nf for a probability density f. We look at the cases where the behavior of the connectivity radius of the Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse regime, as well when the connectivity radius is of the order of $n^{-1/d}$ , the critical regime. We establish limit theorems in the aforementioned regimes: central limit theorems for the sparse and critical regimes, and a Poisson limit theorem for the sparse regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$ , i.e. the sparse regime, we can decompose the limiting processes into a time-changed Brownian motion or a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has a much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.

scholarly journals A Numerical Approach for the Filtered Generalized Čech Complex

Algorithms ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 11
Author(s):  
Jesús F. Espinoza ◽  
Rosalía Hernández-Amador ◽  
Héctor A. Hernández-Hernández ◽  
Beatriz Ramonetti-Valencia
Keyword(s):  
Numerical Approach ◽  
Scale Factor ◽  
Nonempty Intersection ◽  
Finite Collection ◽  
Čech Complex ◽  
Cech 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 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 d-dimensional disks in R d , and we show the performance of our algorithm.

scholarly journals Efficient construction of the Čech complex

2012 ◽  
Vol 36 (6) ◽  
pp. 708-713 ◽  
Author(s):  
Stefan Dantchev ◽  
Ioannis Ivrissimtzis
Keyword(s):  
Čech Complex ◽  
Efficient Construction ◽  
Cech Complex
Sign in / Sign up

Export Citation Format

Share Document