共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper, we study the expectation values of topological invariants of the Vietoris–Rips complex and ?ech complex for a finite set of sample points on a Riemannian manifold. We show that the Betti number and Euler characteristic of the complexes are Lipschitz functions of the scale parameter and that there is an interval such that the Betti curve converges to the Betti number of the underlying manifold.
相似文献2.
Matthew Kahle 《Discrete and Computational Geometry》2011,45(3):553-573
We study the expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝ
d
. We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs.
However, higher homology H
k
is not monotone when k>0. 相似文献
3.
We compute the homology of random ?ech complexes over a homogeneous Poisson process on the d‐dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erd?s ‐Rényi phase transition, where the ?ech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14–51, 2017 相似文献
4.
We consider different types of topologies on the set of functions between two ?ech closure spaces and investigate some of their properties. 相似文献
5.
《Quaestiones Mathematicae》2013,36(1-2):39-47
Abstract In this paper we generalize the well-known Vietoris-Begle theorem to uniform spaces. We formulate two uniform versions: one for the ?ech cohomology based on all finite uniform coverings and one for the ?ech cohomology based on all uniform coverings. 相似文献
6.
We calculate the singular homology and Čech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this
space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda’s proof that the first singular
homology groups of these spaces are isomorphic. 相似文献
7.
In this paper we study the homology of a random ?ech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of 7 , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one. 相似文献
8.
《Quaestiones Mathematicae》2013,36(1-2):137-162
Abstract In this paper we obtain classification and extension theorems for uniform spaces, using the ?ech cohomology theory based on the finite uniform coverings, and study the associated cohomological dimension theory. In particular, we extend results for the cohomological dimension theory on compact Hausdorff spaces or compact metric spaces to those for our cohomological dimension theory on uniform spaces. 相似文献
9.
Given a point set that samples a shape, we formulate conditions under which the Rips complex of the point set at some scale reflects the homotopy type of the shape. For this, we associate with each compact set X of two real-valued functions and defined on which provide two measures of how much the set X fails to be convex at a given scale. First, we show that, when P is a finite point set, an upper bound on entails that the Rips complex of P at scale r collapses to the ?ech complex of P at scale r for some suitable values of the parameters t and r. Second, we prove that, when P samples a compact set X, an upper bound on over some interval guarantees a topologically correct reconstruction of the shape X either with a ?ech complex of P or with a Rips complex of P. Regarding the reconstruction with ?ech complexes, our work compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive μ-reach. Most importantly, our work shows that Rips complexes can also be used to provide shape reconstructions having the correct homotopy type. This may be of some computational interest in high dimensions. 相似文献
10.
Raymond F. Gittings 《Topology and its Applications》1980,11(2):149-159
In this paper we obtain characterizations of metrizable spaces, paracompact M-spaces, Moore spaces and semimetrizable spaces in terms of the way those spaces are embedded in their Stone-?ech compactification. In addition, we give an internal characterization of paracompact M-spaces which we use in the proof of the embedding characterization. 相似文献
11.
One of the main properties of Hochschild homology of the algebra of smooth functions on a smooth manifold is its local character. In this paper, we consider subalgebras of smooth functions which are significant for singular spaces such that simplicial complexes or cones over smooth manifolds. We compute their Hochschild homology and investigate the local character. Our computations show that, in opposition with the smooth case, the (local part of the) Hochschild homology is not always isomorphic to the corresponding de Rham complex of differential forms. The method we use is a slight modification of the localization procedure introduced in by Sullivan. 相似文献
12.
We continue investigations of ?ech closure spaces and their hyperspaces started in [M. Mrševi?, M. Jeli?, Selection principles and hyperspace topologies in closure spaces, J. Korean Math. Soc. 43 (2006) 1099-1114] and [M. Mrševi?, M. Jeli?, Selection principles, γ-sets and αi-properties in ?ech closure spaces, Topology Appl., in press], focusing on generalized upper and lower Vietoris topologies. 相似文献
13.
M. Cencelj 《Topology and its Applications》2012,159(3):646-658
Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. This embedding can be realized by either Rips complexes or analogs of Roe?s anti-?ech approximations of spaces.In this model coarse n-connectedness of K={K1→K2→?} means that for each k there is m>k such that the bonding map from Kk to Km induces trivial homomorphisms of all homotopy groups up to and including n.The asymptotic dimension being at most n means that for each k there is m>k such that the bonding map from Kk to Km factors (up to contiguity) through an n-dimensional complex.Property A of G. Yu is equivalent to the condition that for each k and for each ?>0 there is m>k such that the bonding map from |Kk| to |Km| has a contiguous approximation g:|Kk|→|Km| which sends simplices of |Kk| to sets of diameter at most ?. 相似文献
14.
15.
Michael Goff 《Discrete and Computational Geometry》2011,46(1):132-155
Upper bounds on the Betti numbers over an arbitrary field of Vietoris–Rips complexes are established, and examples of such
complexes with large Betti numbers are given. 相似文献
16.
In [8] and [9] Moisil has introduced the resemblance relations. Following [9] we associate to every resemblance relation an extensive operator which commutes with arbitrary unions of sets. We are leading to consider spaces endowed with such closure operators; we shall call these spaces total ?ech spaces (TC-spaces).TC-spaces are in one-to-one, onto correspondence with reflexive relations. TC-spaces generated by transitive relations are in one-to-one, onto correspondence with the total topological spaces of W. Hartnett (which are called total Kuratowski spaces, TK-spaces).We study the category of TC-spaces and its full subcategory determined by TK-spaces. Both categories are Cartesian closed, but they are not elementary toposes. 相似文献
17.
Marian Mrozek 《Discrete and Computational Geometry》2010,44(3):546-576
A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is
a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology
and the Nerve Theorem. In many situations, this change of the representation of the input may help in bypassing the problems
with the complexity of the standard homology algorithms by reducing the size of necessary input. We show that the approach
is particularly advantageous in the case of homology map algorithms. 相似文献
18.
Umberto Cerruti 《Fuzzy Sets and Systems》1981,6(2):197-204
In this paper we obtain a reflective subcategory C of the category FTS of fuzzy topological spaces. The associated reflection has properties similar to those of the ‘Stone-?ech’ compactification β and, in effect, is an extension of it. We study relations between β and in particular subcategories of FTS; is completely determined in the case of fuzzy topological spaces topologically generated. 相似文献
19.
A corollary of the main result of this paper is the following Theorem. Suppose f: X → Y is a closed surjection of metrizable spaces whose point inverses are LCn + 1-divisors (n ? 1). If Y is complete and f is homology n-stable, then Y is LCn + 1provided X is LCn + 1.Intuitively, f is homology n-stable if the ?ech homology groups of its point inverses are locally constant up to dimension n. In addition, we obtain sufficient conditions for the Freudenthal compactification to be LCn. 相似文献
20.
We start by introducing a Čech homology with compact supports which we then use in order to construct an infinite-dimensional
homology theory. Next we show that under appropriate conditions on the nonlinearity there exists a ground state solution for
a semilinear Schr?dinger equation with strongly indefinite linear part. To this solution there corresponds a nontrivial critical
group, defined in terms of the infinite-dimensional homology mentioned above. Finally, we employ this fact in order to construct
solutions of multibump type. Although our main purpose is to survey certain homological methods in critical point theory,
we also include some new results.
Dedicated to Felix Browder on the occasion of his 80th birthday 相似文献