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Oleksiy Dovgoshey

Publications and source records attributed to Oleksiy Dovgoshey.

54 records · Page 3Linked to original sources
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↗
arXiv2017

Unions and ideals of locally strongly porous sets

For subsets of $\mathbb R^+ = [0,\infty)$ we introduce a notion of coherently porous sets as the sets for which the upper limit in the definition of porosity at a point is attained along the same sequence. We prove that the union of two strongly porous at $0$ sets is strongly porous if and only if these sets are coherently porous. This result leads to a characteristic property of the intersection of all maximal ideals containing in the family of strongly porous at $0$ subsets of $\mathbb R^+$. It is also shown that the union of a set $A \subseteq \mathbb R^+$ with arbitrary strongly porous at $0$ subset of $\mathbb R^+$ is porous at $0$ if and only if $A$ is lower porous at $0$.

math.CA↗
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↗
arXiv2016

Comparison theorems for hyperbolic type metrics

The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.

math.MG↗
arXiv2015

First Baire class functions in the pluri-fine topology

Let $B_{1}(Ω, \mathbb R)$ be the first Baire class of real functions in the pluri-fine topology on an open set $Ω\subseteq \mathbb C^{n}$ and let $H_{1}^{*}(Ω, \mathbb R)$ be the first functional Lebesgue class of real functions in the same topology. We prove the equality $B_{1}(Ω, \mathbb R)=H_{1}^{*}(Ω, \mathbb R)$ and show that for every $f\in B_{1}(Ω, \mathbb R)$ there is a separately continuous function $g: Ω^{2} \to\mathbb R$ in the pluri-fine topology on $Ω^2$ such that $f$ is the diagonal of $g.$

math.GN↗
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

Isotone extension of mappings

A set of necessary and sufficient conditions under which an isotone mapping from a subset of a poset X to a poset Y has an extension to an isotone mapping from X to Y are found.

math.CO↗
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↗
arXiv2012

Weak similarities of metric and semimetric spaces

Let (X,dX) and (Y,dY) be semimetric spaces with distance sets D(X) and, respectively, D(Y). A mapping F : X \to Y is a weak similarity if it is surjective and there exists a strictly increasing f : D(Y) \to D(X) such that dX = f \circ dY \circ F. It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F : X \to Y is an isometry if X and Y are ultrametric and compact with D(X) = D(Y). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.

math.MG↗
arXiv2012

Mean value type inequalities for quasinearly subharmonic functions

The mean value inequality is characteristic for upper semicontinuous functions to be subharmonic. Quasinearly subharmonic functions generalize subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the catheterization of nonnegative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalized mean value inequalities where, instead of balls, we consider arbitrary bounded sets which have nonvoid interiors and instead of the volume of ball some functions depending on the radius of this ball.

math.AP↗
arXiv2012

Tangent metric spaces to starlike sets on the plane

Let $A\subseteq\mathbb C$ be a starlike set with a center $a$. We prove that every tangent space to $A$ at the point $a$ is isometric to the smallest closed cone, with the vertex $a$, which includes $A$. A partial converse to this result is obtained. The tangent space to convex sets is also discussed.

math.MG↗
arXiv2011

Metrization of weighted graphs

We find a set of necessary and sufficient conditions under which the weight $w:E\to\mathbb R^+$ on the graph $G=(V,E)$ can be extended to a pseudometric $d:V\times V\to\mathbb R^+$. If these conditions hold and $G$ is a connected graph, then the set $\mathfrak M_w$ of all such extensions is nonvoid and the shortest-path pseudometric $d_w$ is the greatest element of $\mathfrak M_w$ with respect to the partial ordering $d_1 \leqslant d_2$ if and only if $d_1(u,v) \leqslant d_2(u,v)$ for all $u,v\in V$. It is shown that every nonvoid poset $(\mathfrak M_w,\leqslant)$ contains the least element $ρ_{0,w}$ if and only if $G$ is a complete $k$-partite graph with $k\geqslant 2$ and in this case the explicit formula for computation of $ρ_{0,w}$ is obtained.

math.CO↗
arXiv2009

Metric spaces with unique pretangent spaces

We find necessary and sufficient conditions under which an arbitrary metric space $X$ has a unique pretangent space at the marked point $a\in X$. Key words: Metric spaces; Tangent spaces to metric spaces; Uniqueness of tangent metric spaces; Tangent space to the Cantor set.

math.MG↗