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 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.

Metrizability of Finite Dimensional Spaces with a Binary Convexity

1986 ◽  
Vol 38 (1) ◽  
pp. 1-18 ◽  
Author(s):  
M. Van de Vel
Keyword(s):  
Convex Hull ◽  
Weak Topology ◽  
Convex Sets ◽  
Nonempty Intersection ◽  
Finite Collection ◽  
Closed Sets ◽  
Finite Dimensional ◽  
The One ◽  
Image Position ◽  
Disjoint Convex

A convex structure consists of a set X, together with a collection of subsets of X, which is closed under intersection and under updirected union. The members of are called convex sets, and is a convexity on X. Fox A a subset of X, h (A) denotes the (convex) hull of A. If A is finite, then h(A) is called a polytope, is called a binary convexity if each finite collection of pairwise intersecting convex sets has a nonempty intersection. See [8], [21] for general references.If X is also equipped with a topology, then the corresponding weak topology is the one generated by the convex closed sets. It is usually assumed that at least all polytopes are closed. is called normal provided that for each two disjoint convex closed sets C, D there exist convex closed sets C′, D′, with

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 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

scholarly journals An iterative approach to a constrained least squares problem

2003 ◽  
Vol 2003 (8) ◽  
pp. 503-512 ◽  
Author(s):  
Simeon Reich ◽  
Hong-Kun Xu
Keyword(s):  
Least Squares ◽  
Regularization Method ◽  
Nonempty Intersection ◽  
Tikhonov Regularization Method ◽  
Finite Collection ◽  
Iterative Approach ◽  
Minimum Norm ◽  
Constrained Least Squares ◽  
Minimum Norm Solution ◽  
Least Squares Problem

A constrained least squares problem in a Hilbert spaceHis considered. The standard Tikhonov regularization method is used. In the case where the set of the constraints is the nonempty intersection of a finite collection of closed convex subsets ofH, an iterative algorithm is designed. The resulting sequence is shown to converge strongly to the unique solution of the regularized problem. The net of the solutions to the regularized problems strongly converges to the minimum norm solution of the least squares problem if its solution set is nonempty.

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