arXiv subjects

Vitalij A. Chatyrko

Publications and source records attributed to Vitalij A. Chatyrko.

6 recordsLinked to original sources

arXiv2025

Simple polynomial equations over (mxm)-matrices

Let $m$ be any integer $\geq 3$. We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(m \times m)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb R$ for each $i$ and $n \geq 1$. We discuss its solution set $S$ supplied with the natural Euclidean topology. In particular, we describe the solution set $S$ for $m=3$ and calculate its dimension.

math.RA↗

arXiv2025

Simple polynomial equations over $(2 \times 2)$-matrices

We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(2 \times 2)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb R$ for each $i$ and $n \geq 2$. We discuss its solution set $S$ supplied with the natural Euclidean topology. We completely describe $S$. We also show that $\dim S =2.$

math.RA↗

arXiv2024

On a modification of finite-dimensional Niemytzki spaces

In this paper we extend the construction of the Niemytzki plane to dimension $n \geq 3$. Further, we consider a poset of topologies on the closed $n$-dimensional Euclidean half-space similar to one from \cite{AAK} which is related to the Niemytzki plane topology. Then we explore properties of topologies from the poset.

math.GN↗

arXiv2024

From homogeneity to discrete homogeneity

This is a survey of recent and classical results concerning various types of homogeneity, such as n-homogeneity, discrete homogeneity, and countable dense homogeneity. Some new results are also presented, and several problems are posed.

math.GN↗

arXiv2023

On $π$-compatible topologies and their special cases

Topologies $τ, σ$ on a set $X$ are called $π$-compatible if $τ$ is a $π$-network for $σ$, and vice versa. If topologies $τ, σ$ on a set $X$ are $π$-compatible then the families of nowhere dense sets (resp. meager sets or sets possessing the Baire property) of the spaces $(X, τ)$ and $(X, σ)$ coincide. A topology $σ$ on a set $X$ is called an admissible extension of a topology $τ$ on $X$ if $τ\subseteq σ$ and $τ$ is a $π$-network for $σ$. It turns out that examples of admissible extensions were occurred in literature several times. In the paper we provide some new facts about the $π$-compatibility and the admissible extension as well as about their particular cases.

math.GN↗

arXiv2023

Discrete homogeneity and ends of manifolds

It is shown that a connected non-compact metrizable manifold of dimension $\ge 2$ is strongly discrete homogeneous if and only if it has one end (in the sense of Freudenthal compactification).

math.GN↗