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Peter Saveliev

Publications and source records attributed to Peter Saveliev.

8 recordsLinked to original sources
arXiv2011

Homology groups of filtrations

Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a measure of robustness of the homology classes of the filtration of the lower level sets of this function. In this paper we introduce the homology group of filtration as the product of the kernels of the homology maps of the inclusions. This group contains all possible homology classes in all elements of the filtration so that we can later pick the features that lie within the user's choice of the acceptable level of noise.

math.AT↗
arXiv2003

Removing coincidences of maps between manifolds of different dimensions

We consider sufficient conditions of local removability of coincidences of maps f,g:N->M, where M,N are manifolds with dimensions dimN>dimM. The coincidence index is the only obstruction to the removability for maps with fibers either acyclic or homeomorphic to spheres of certain dimensions. We also address the normalization property of the index and coincidence-producing maps.

math.AT↗
arXiv2003

Higher order Nielsen numbers

Suppose X,Y are manifolds, f,g:X->Y are maps. The well-known Coincidence Problem studies the coincidence set C={x:f(x)=g(x)}. The number m=dimX-dimY is called the codimension of the problem. More general is the Preimage Problem. For a map f:X->Z and a submanifold Y of Z, it studies the preimage set C={x:f(x) in Y}, and the codimension is m=dimX+dimY-dimZ. In case of codimension 0, the classical Nielsen number N(f,Y) is a lower estimate of the number of points in C changing under homotopies of f, and for an arbitrary codimension, of the number of components of C. We extend this theory to take into account other topological characteristics of C. The goal is to find a "lower estimate" of the bordism group Omega_{p}(C) of C. The answer is the Nielsen group S_{p}(f,Y) defined as follows. In the classical definition the Nielsen equivalence of points of C based on paths is replaced with an equivalence of singular submanifolds of C based on bordisms. We let S_{p}^{prime}(f,Y) be the quotient group of Omega_{p}(C) with respect to this equivalence relation, then the Nielsen group of order p is the part of this group preserved under homotopies of f. The Nielsen number N_{p}(f,Y) of order p is the rank of this group (then N(f,Y)=N_{0}(f,Y)). These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided.

math.GT↗
arXiv2000

Lomonosov's Invariant Subspace Theorem for Multivalued Linear Operators

The famous Lomonosov's invariant subspace theorem states that if a continuous linear operator T on an infinite-dimensional normed space E "commutes" with a compact nonzero operator K, i.e., TK=KT, then T has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators.

math.FA↗
arXiv2000

Lefschetz Coincidence Theory for Maps Between Spaces of Different Dimensions

For a given pair of maps f,g:X->M from an arbitrary topological space to an n-manifold, the Lefschetz homomorphism is a certain graded homomorphism L:H(X)->H(M) of degree (-n). We prove a Lefschetz-type coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x in X such that f(x)=g(x).

math.AT↗
arXiv1999

Fixed points and selections of multifunctions on spaces with convexity

We provide theorems containnig both Kakutani and Browder fixed points theorems as immediate corollaries, as well as Michael and Browder selection theorems. For this purpose we introduce convex structures more general than those of locally convex and non-locally convex topological vector spaces or generalized convexity structures due to Michael, Van de Vel and Horvath.

math.FA↗
arXiv1999

A Lefschetz type coincidence theorem

A Lefschetz-type coincidence theorem for two maps f,g:X->Y from an arbitrary topological space X to a manifold Y is given: I(f,g)=L(f,g), the coincidence index is equal to the Lefschetz number. It follows that if L(f,g) is not equal to zero then there is an x in X such that f(x)=g(x). In particular, the theorem contains some well-known coincidence results for (i) X,Y manifolds and (ii) f with acyclic fibers.

math.AT↗