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Viktoriia Bilet

Publications and source records attributed to Viktoriia Bilet.

15 recordsLinked 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↗
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↗
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↗
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↗
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↗
arXiv2018

Finite asymptotic clusters of metric spaces

Let $(X, d)$ be an unbounded metric space and let $\tilde r=(r_n)_{n\in\mathbb N}$ be a sequence of positive real numbers tending to infinity. A pretangent space $Ω_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity is a limit of the rescaling sequence $\left(X, \frac{1}{r_n}d\right).$ The set of all pretangent spaces $Ω_{\infty, \tilde r}^{X}$ is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph $(G_{X, \tilde r}, ρ_{X})$ whose maximal cliques coincide with $Ω_{\infty, \tilde r}^{X}$ and the weight $ρ_{X}$ is defined by metrics on $Ω_{\infty, \tilde r}^{X}$. We describe the structure of metric spaces having finite asymptotic clusters of pretangent spaces and characterize the finite weighted graphs which are isomorphic to these clusters.

math.MG↗
arXiv2017

Asymptotic structure of general metric spaces at infinity

Let $(X,d)$ be an unbounded metric space and $\tilde r=(r_n)_{n\in\mathbb N}$ be a scaling sequence of positive real numbers tending to infinity. We define the pretangent space $Ω_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity as a metric space whose points are equivalence classes of sequences $(x_n)_{n\in\mathbb N}\subset X$ which tend to infinity with the speed of $\tilde r$. It is proved that the pretangent spaces $Ω_{\infty, \tilde r}^{X}$ are complete for every unbounded metric space $(X, d)$ and every scaling sequence $\tilde r$. The finiteness conditions of $Ω_{\infty, \tilde r}^{X}$ are found.

math.MG↗
arXiv2016

Boundedness of Lebesgue constants and interpolating Faber bases

We investigate some conditions under which the Lebesgue constants or Lebesgue functions are bounded for the classical Lagrange polynomial interpolation on a compact subset of $\mathbb R$. In particular, relationships of such boundedness with uniform and pointwise convergence of Lagrange polynomials and with the existence of interpolating Faber bases are discussed.

math.CA↗
arXiv2014

Uniform boundedness of pretangent spaces, local constancy of metric derivatives and strong right upper porosity at a point

Let $(X,d,p)$ be a pointed metric space. A pretangent space to $X$ at $p$ is a metric space consisting of some equivalence classes of convergent to $p$ sequences $(x_n), x_n \in X,$ whose degree of convergence is comparable with a given scaling sequence $(r_n), r_n\downarrow 0.$ A scaling sequence $(r_n)$ is normal if this sequence is eventually decreasing and there is $(x_n)$ such that $\mid d(x_n,p)-r_n\mid=o(r_n)$ for $n\to\infty.$ Let $\mathbf{Ω_{p}^{X}(n)}$ be the set of pretangent spaces to $X$ at $p$ with normal scaling sequences. We prove that $\mathbf{Ω_{p}^{X}(n)}$ is uniformly bounded if and only if $\{d(x,p): x\in X\}$ is a so-called completely strongly porous set. It is also proved that the uniform boundedness of $\mathbf{Ω_{p}^{X}(n)}$ is an equivalent of the constancy of metric derivatives of all metrically differentiable mappings on $X$ in the open balls of a fixed radius centered at the marked points of pretangent spaces.

math.MG↗
arXiv2013

Uniform boundedness of pretangent spaces and local strong one-side porosity

Let (X,d,p) be a pointed metric space. A pretangent space to X at p is a metric space consisting of some equivalence classes of convergent to p sequences (x_n), x_n \in X, whose degree of convergence is comparable with a given scaling sequence (r_n), r_n\downarrow 0. We say that (r_n) is normal if there is (x_n) such that |d(x_n,p)-r_n|=o(r_n) for n\to\infty. Let Omega_{p}^{X}(n) be the set of pretangent spaces to X at p with normal scaling sequences. We prove that the spaces from Omega_{p}^{X}(n) are uniformly bounded if and only if {d(x,p:x\in X}is a so-called completely strongly porous set.

math.MG↗
arXiv2012

Boundedness of pretangent spaces to general metric spaces

Let (X,d,p) be a metric space with a metric d and a marked point p. We define the set of w-strongly porous at 0 subsets of [0,\infty) and prove that the distance set {d(x,p): x\in X} is w-strongly porous at 0 if and only if every pretangent space to X at p is bounded.

math.MG↗