Searcharxiv⌕ Search

arXiv subjects

Oleksiy Dovgoshey

Publications and source records attributed to Oleksiy Dovgoshey.

At least 37 records · Page 2Linked to original sources
arXiv2024

On monoids of metric preserving functions

Let $\mathbf{X}$ be a class of metric spaces and let $\mathbf{P}_{\mathbf{X}}$ be the set of all $f:[0, \infty)\to [0, \infty)$ preserving $\mathbf{X},$ $(Y, f\circρ)\in\mathbf{X}$ whenever $(Y, ρ)\in\mathbf{X}.$ For arbitrary subset $\mathbf{A}$ of the set of all metric preserving functions we show that the equality $\mathbf{P}_{\mathbf{X}}=\mathbf{A}$ has a solution iff $\mathbf{A}$ is a monoid with respect to the operation of function composition. In particular, for the set $\mathbf{SI}$ of all amenable subadditive increasing functions there is a class $\mathbf{X}$ of metric spaces such that $\mathbf{P}_{\mathbf{X}}=\mathbf{SI}$ holds, which gives a positive answer to the question of paper [1].

math.GN↗
arXiv2024

Strongly ultrametric preserving functions

An ultrametric preserving function $f$ is said to be strongly ultrametric preserving if ultrametrics $d$ and $f \circ d$ define the same topology on $X$ for each ultrametric space $(X,d)$. The set of all strongly ultrametric preserving functions is characterized by several distinctive features. In particular, it is shown that an ultrametric preserving $f$ belongs to this set iff $f$ preserves the property to be compact.

math.GN↗
arXiv2023

Pseudometric spaces. From minimality to maximality in the groups of combinatorial self-similarities

The group of combinatorial self-similarities of a pseudometric space $(X, d)$ is the maximal subgroup of the symmetric group $\mathbf{Sym} (X)$ whose elements preserve the four-point equality $d(x,y)=d(u,v)$. Let us denote by $\mathcal{IP}$ the class of all pseudometric spaces $(X, d)$ for which every combinatorial self-similarity $Φ\colon~X~\to~X$ satisfies the equality $d(x, Φ(x))=0,$ but all permutations of metric reflection of $(X, d)$ are combinatorial self-similarities of this reflection. The structure of $\mathcal{IP}$ spaces is fully described.

math.MG↗
arXiv2023

Locally finite ultrametric spaces and labeled trees

It is shown that a locally finite ultrametric space $(X, d)$ is generated by labeled tree if and only if, for every open ball $B \subseteq X$, there is a point $c \in B$ such that $d(x, c) = \operatorname{diam} B$ whenever $x \in B$ and $x \neq c$. For every finite ultrametric space $Y$ we construct an ultrametric space $Z$ having the smallest possible number of points such that $Z$ is generated by labeled tree and $Y$ is isometric to a subspace of $Z$. It is proved that for a given $Y$, such a space $Z$ is unique up to isometry.

math.GN↗
arXiv2023

How much can we extend the Assouad embedding theorem?

The celebrated Assouad embedding theorem has been known for over 40 years. It states that for any doubling metric space (with doubling constant $C_0$) there exists an integer $N$, such that for any $α\in (0.5,1)$ there exists a positive constant $C(α,C_0)$ and an injective function $F:X\to \mathbb{R}^N$ such that \[ \forall x,y\in X \quad C^{-1} d(x,y)^α \leq \|F(x)-F(y) \| \leq Cd(x,y)^α \] In the paper we use the remetrization techniques to extend the said theorem to a broad subclass of semimetric spaces. We also present the limitations of this extension -- in particular, we prove that in any semimetric space which satisfies the claim of the Assouad theorem, the relaxed triangle condition holds as well, which means that there exists $K\geq 1$ such that \[ \forall x,y,z\in X \quad d(x,z)\leq K\left( d(x,y)+d(y,z)\right). \]

math.GN↗
arXiv2023

Proximinal sets and connectedness in graphs

Let $G$ be a graph with a vertex set $V$. The graph $G$ is path-proximinal if there are a semimetric $d \colon V \times V \to [0, \infty[$ and disjoint proximinal subsets of the semimetric space $(V, d)$ such that $V = A \cup B$, and vertices $u$, $v \in V$ are adjacent iff \[ d(u, v) \leqslant \inf \{d(x, y) \colon x \in A, y \in B\}, \] and, for every $p \in V$, there is a path connecting $A$ and $B$ in $G$, and passing through $p$. It is shown that a graph is path-proximinal if and only if all its vertices are not isolated. It is also shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of every its vertex is equal to $1$.

math.GN↗
arXiv2022

Completeness, Closedness and Metric Reflections of Pseudometric Spaces

It is well-known that a metric space $(X, d)$ is complete iff the set $X$ is closed in every metric superspace of $(X, d)$. For a given pseudometric space $(Y, ρ)$, we describe the maximal class $\mathbf{CEC}(Y, ρ)$ of superspaces of $(Y, ρ)$ such that $(Y, ρ)$ is complete if and only if $Y$ is closed in every $(Z, Δ) \in \mathbf{CEC}(Y, ρ)$. We also introduce the concept of pseudoisometric spaces and prove that spaces are pseudoisometric iff their metric reflections are isometric. The last result implies that a pseudometric space is complete if and only if this space is pseudoisometric to a complete pseudometric space.

math.GN↗
arXiv2022

When all Permutations are Combinatorial Similarities

Let $(X, d)$ be a semimetric space. A permutation $Φ$ of the set $X$ is a combinatorial self similarity of $(X, d)$ if there is a bijective function $f \colon d(X^2) \to d(X^2)$ such that $$ d(x, y) = f(d(Φ(x), Φ(y))) $$ for all $x$, $y \in X$. We describe the set of all semimetrics $ρ$ on an arbitrary nonempty set $Y$ for which every permutation of $Y$ is a combinatorial self similarity of $(Y, ρ)$.

math.CO↗
arXiv2022

Uniqueness of best proximity pairs and rigidity of semimetric spaces

For arbitrary semimetric space $(X, d)$ and disjoint proximinal subsets $A$, $B$ of $X$ we define the proximinal graph as a bipartite graph with parts $A$ and $B$ whose edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A, B)$. We characterize the semimetric spaces whose proximinal graphs have at most one edge and the semimetric spaces whose proximinal graphs have the vertices with degree at most $1$ only. This allows us to describe the necessary and sufficient conditions for uniqueness of the best proximity pairs and best approximations.

math.GN↗
arXiv2022

Labeled trees generating complete, compact, and discrete ultrametric spaces

We investigate the interrelations between labeled trees and ultrametric spaces generated by these trees. The labeled trees, which generate complete ultrametrics, totally bounded ultrametrics, and discrete ones, are characterized up to isomorphism. As corollary, we obtain a characterization of labeled trees generating compact ultrametrics, and discrete totally bounded ultrametrics. It is also shown that every ultrametric space generated by labeled tree contains a dense discrete subspace.

math.CO↗
arXiv2022

Bipartite graphs and best proximity pairs

We say that a bipartite graph $G(A, B)$ with fixed parts $A$, $B$ is proximinal if there is a semimetric space $(X, d)$ such that $A$ and $B$ are disjoint proximinal subsets of $X$ and all edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A, B)$. It is proved that a bipartite graph $G$ is not isomorphic to any proximinal graph iff $G$ is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph $G$ is a disjoint union of complete bipartite graphs iff $G$ is isomorphic to a nonempty proximinal graph for an ultrametric space.

math.CO↗
arXiv2021

On equivalence of unbounded metric spaces at infinity

Let $(X, d)$ be an unbounded metric space. To investigate the asymptotic behavior of $(X, d)$ at infinity, one can consider a sequence of rescaling metric spaces $(X, \frac{1}{r_n} d)$ generated by given sequence $(r_n)_{n \in \mathbb N}$ of positive reals with $r_n \to \infty$. Metric spaces that are limit points of the sequence $(X, \frac{1}{r_n} d)_{n \in \mathbb N}$ will be called pretangent spaces to $(X, d)$ at infinity. We found the necessary and sufficient conditions under which two given unbounded subspaces of $(X, d)$ have the same pretangent spaces at infinity. In the case when $(X, d)$ is the real line with Euclidean metric, we also describe all unbounded subspaces of $(X, d)$ isometric to their pretangent spaces.

math.MG↗
arXiv2021

Ultrametrics and complete multipartite graphs

We describe the class of graphs for which all metric spaces with diametrical graphs belonging to this class are ultrametric. It is shown that a metric space $(X, d)$ is ultrametric iff the diametrical graph of the metric $d_{\varepsilon}(x, y) = \max\{d(x, y), \varepsilon\}$ is either empty or complete multipartite for every $\varepsilon > 0$. A refinement of the last result is obtained for totally bounded spaces. Moreover, using complete multipartite graphs we characterize the compact ultrametrizable topological spaces. The bounded ultrametric spaces, which are weakly similar to unbounded ones, are also characterized via complete multipartite graphs.

math.MG↗
arXiv2021

The range of ultrametrics, compactness, and separability

We describe the order type of range sets of compact ultrametrics and show that an ultrametrizable infinite topological space $(X, τ)$ is compact iff the range sets are order isomorphic for any two ultrametrics compatible with the topology $τ$. It is also shown that an ultrametrizable topology is separable iff every compatible with this topology ultrametric has at most countable range set.

math.GN↗
arXiv2020

Ultrametric preserving functions and weak similarities of ultrametric spaces

Let $WS(X, d)$ be the class of ultrametric spaces which are weakly similar to ultrametric space $(X, d)$. The main results of the paper completely describe the ultrametric spaces $(X, d)$ for which the equality $$ ρ(x, y) = f(d(Φ(x), Φ(y))) $$ holds for every $(Y, ρ) \in WS(X, d)$, every weak similarity $Φ\colon Y \to X$, and all $x$, $y \in Y$ with some ultrametric (pseudoultrametric) preserving function $f$ depending on $Φ$.

math.GN↗
arXiv2020

Isomorphism of trees and isometry of ultrametric spaces

We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated by labelings given on the vertices of trees. The obtained results generalized some facts previously known for phylogenetic trees and for Gurvich-Vyalyi monotone trees.

math.MG↗
arXiv2019

On ultrametric-preserving functions

Characterizations of pseudoultrametric-preserving functions and semimetric-preserving functions are found. The structural properties of pseudoultrametrics which can be represented as a composition of an ultrametric and ultrametric-pseudoultrametric-preserving function are found. A dual form of Pongsriiam-Termwuttipong characterization of the ultrametric-preserving functions is described. We also introduce a concept of $k$-separating family of functions and use it to characterize the ultrametric spaces.

math.MG↗