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Mykhailo Zarichnyi

Publications and source records attributed to Mykhailo Zarichnyi.

12 recordsLinked to original sources
arXiv2026

Invariant idempotent $\ast$-measures for generalized iterated function systems

The notion of $\ast$-measure on a compact Hausdorff space can be defined for arbitrary continuous triangular norm $\ast$. The well-known Hutchinson-Barnsley theory deals with the iterated function systems (IFSs) of probability measures and establishes existence and uniqueness of invariant measures. In the previous paper, IFSs of $\ast$-measures were considered. In the present paper we deal with generalized invariant function systems (GIFSs) of $\ast$-measures, which are counterparts of GIFSs in the sense of Mihail and Miculescu. The notion of invariant $\ast$-measure is introduced for such GIFSs and we prove existence and uniqueness of such elements.

math.GN↗
arXiv2026

90+ years of the Scottish Book

Inspired by the recent 90th anniversary of the Scottish Book we present some reflections about its impact. First we discuss new areas of mathematics it helped launch. Then we argue that it was actively used in stimulating the interests and results of junior mathematicians and students. Also, we summarize the progress during the decade that has passed since the publication of [55], which contained a review of solved problems from the Scottish Book. We also provide an overview of collections of open problems related in one way or another to the Scottish Book. All formulations of the Scottish Book problems in English are cited here from Mauldin, Richard Daniel (ed.) 2015: The Scottish Book. Mathematics from the Scottish Café. With selected problems from the New Scottish Book. 2nd updated and enlarged edition. Cham: Birkhäuser/Springer

math.HO↗
arXiv2023

Invariant idempotent $\ast$-measures

The notion of $\ast$-idempotent measure is a modification of the notion of idempotent measure defined for every triangular norm $\ast$. We prove existence and uniqueness of invariant $\ast$-idempotent measures for iterated function systems on compact metric spaces.

math.DS↗
arXiv2020

Strong topology on the set of persistence diagrams

We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described. Also, we prove that the space of persistence diagrams with the bottleneck metric has infinite asymptotic dimension in the sense of Gromov.

math.GN↗
arXiv2019

Spaces of max-min measures on compact Hausdorff spaces

The notion of max-min measure is a counterpart of the notion of max-plus measure (Maslov measure or idempotent measure). In this paper we consider the spaces of max-min measures on the compact Hausdorff spaces. It is proved that the obtained functor of max-min measures is isomorphic to the functor of max-plus (idempotent) measures considered by the second-named author. However, it turns out that the monads generated by these functors are not isomorphic.

math.GN↗
arXiv2017

On convergence in the subpower Higson corona of metric spaces

The subpower Higson corona of a proper metric space is defined in \cite{KZ}. We prove that, unlikely to the Higson corona, the closure of a $σ$-compact subset of the subpower Higson corona of a proper unbounded metric space does not necessarily coincide with its Stone-Čech corona.

math.GN↗
arXiv2013

Max-min measures on ultrametric spaces

The ultrametrization of the set of all probability measures of compact support on the ultrametric spaces was first defined by Hartog and de Vink. In this paper we consider a similar construction for the so called max-min measures on the ultrametric spaces. In particular, we prove that the functors max-min measures and idempotent measures are isomorphic. However, we show that this is not the case for the monads generated by these functors.

math.GN↗
arXiv2012

Nonexistence of linear operators extending Lipschitz (pseudo)metric

We present an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear extension operators for Lipschitz functions.

math.GN↗
arXiv2012

Hyperspaces of max-plus convex subsets of powers of the real line

The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers $\mathbb R^τ$ of the real line. We show that the corresponding spaces are AR's if and only if $τ\leω_1$.

math.GN↗
arXiv2011

Convex hyperspaces of probability measures and extensors in the asymptotic category

The objects of the Dranishnikov asymptotic category are proper metric spaces and the morphisms are asymptotically Lipschitz maps. In this paper we provide an example of an asymptotically zero-dimensional space (in the sense of Gromov) whose space of compact convex subsets of probability measures is not an absolute extensor in the asymptotic category in the sense of Dranishnikov.

math.GN↗
arXiv2011

On asymptotic extension dimension

The aim of this paper is to introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes a relation between the asymptotic extensional dimension of a proper metric space and extension dimension of its Higson corona.

math.GT↗