We study approximation theorems for the Euler characteristic of Vietoris-Rips and Čech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in critical regime. apply our results to smooth bootstrap determine its rate convergence Kantorovich-Wasserstein distance Kolmogorov distance.

This paper concerns the shape theory for triads of spaces which was introduced by the author. More precisely, in the first part, the shape dimension for triads of spaces (X; X0,X1) is introduced, and its upper and lower bounds are given in terms of the shape dimensions of X0, X1, X0 ∩ X1 and X. In the second part, a Whitehead type theorem for triads of spaces and a Mayer-Vietoris type theorem c...

We show that when X = P 1 k , the projective line over the eld k, with the open covering fX 0 = Spec(kx]); X 1 = Spec(ky])g, the Mayer-Vietoris sequence of the K-theory of the schemes with endo-morphisms for the covering is not exact for all K i when i = 2n + 1 and n 1. This leads to the conclusion that the K-theory of schemes with endomorphisms is a global theory.

In this paper, the authors study further properties and applications of weighted homology persistent homology. The Mayer-Vietoris sequence generalized Bockstein spectral for are introduced. For applications, show an algorithm to construct a filtration simplicial complexes from network. They also prove theorem calculate mod p2 provided with some information on p

We use a Mayer–Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen– Macaulay-type condition. As a result, we recover kn...

If H is a finitely generated subgroup of the finitely generated group G, then H is undistorted in G for their word metrics iff the induced homomorphism on the zero-dimensional l1-homology group is injective. The Mayer-Vietoris exact sequence for l1-homology is established and applications are given to hyperbolic groups and relative hyperbolicity. The notion of relative distortion of two maps is...