arXiv subjects
Publications and source records attributed to O. Dovgoshey.
In the present paper, we study the existence of best proximity pairs in ultrametric spaces. We show, under suitable assumptions, that the proximinal pair $(A,B)$ has a best proximity pair. As a consequence we generalize a well known best approximation result and we derive some fixed point theorems. Moreover, we provide examples to illustrate the obtained results.
Let $X$, $Y$ be sets and let $Φ$, $Ψ$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $Φ$ is combinatorially similar to $Ψ$ if there are bijections $f \colon Φ(X^2) \to Ψ(Y^{2})$ and $g \colon Y \to X$ such that $Ψ(x, y) = f(Φ(g(x), g(y)))$ for all $x$, $y \in Y$. It is shown that the semigroups of binary relations generated by sets $\{Φ^{-1}(a) \colon a \in Φ(X^{2})\}$ and $\{Ψ^{-1}(b) \colon b \in Ψ(Y^{2})\}$ are isomorphic for combinatorially similar $Φ$ and $Ψ$. The necessary and sufficient conditions under which a given mapping is combinatorially similar to a pseudometric, or strongly rigid pseudometric, or discrete pseudometric are found. The algebraic structure of semigroups generated by $\{d^{-1}(r) \colon r \in d(X^{2})\}$ is completely described for nondiscrete, strongly rigid pseudometrics and, also, for discrete pseudometrics $d \colon X^{2} \to \mathbb{R}$.
Let $X$, $Y$ be sets and let $Φ$, $Ψ$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $Φ$ and $Ψ$ are combinatorially similar if there are bijections $f \colon Φ(X^2) \to Ψ(Y^{2})$ and $g \colon Y \to X$ such that $Ψ(x, y) = f(Φ(g(x), g(y)))$ for all $x$, $y \in Y$. Conditions under which a given mapping is combinatorially similar to an ultrametric or a pseudoultrametric are found. Combinatorial characterizations are also obtained for poset-valued ultrametric distances recently defined by Priess-Crampe and Ribenboim.
The necessary and sufficient conditions under which a given family $\mathcal{F}$ of subsets of finite set $X$ coincides with the family $\mathbf{B}_X$ of all balls generated by some ultrametric $d$ on $X$ are found. It is shown that the representing tree of the ultrametric space $(\mathbf{B}_{X}, d_H)$ with the Hausdorff distance $d_H$ can be obtained from the representing tree $T_X$ of ultrametric space $(X, d)$ by adding a leaf to every internal vertex of $T_X$.
Let $X$ be a nonempty set and $X^{2}$ be the Cartesian square of $X$. Some semigroups of binary relations generated partitions of $X^2$ are studied. In particular, the algebraic structure of semigroups generated by the finest partition of $X^{2}$ and, respectively, by the finest symmetric partition of $X^{2}$ are described.
We study extremal properties of finite ultrametric spaces $X$ and related properties of representing trees $T_X$. The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is shown that the finite rooted trees are isomorphic to the rooted trees of nonsingular balls of special finite ultrametric spaces. We also found conditions under which the isomorphism of representing trees $T_X$ and $T_Y$ implies the isometricity of ultrametric spaces $X$ and $Y$.
We recall the definition of quasinearly subharmonic functions, point out that this function class includes, among others, subharmonic functions, quasisubharmonic functions, nearly subharmonic functions and essentially almost subharmonic functions. It is shown that the sum of two quasinearly subharmonic functions may not be quasinearly subharmonic. Moreover, we characterize the harmonicity via quasinearly subharmonicity.
A metric space $X$ is rigid if the isometry group of $X$ is trivial. The finite ultrametric spaces $X$ with $|X| \geq 2$ are not rigid since for every such $X$ there is a self-isometry having exactly $|X|-2$ fixed points. Using the representing trees we characterize the finite ultrametric spaces $X$ for which every self-isometry has at least $|X|-2$ fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.
Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions under which, for given metric space $X$, there is a class $\mathfrak{M}$ of metric spaces such that $X$ is minimal $\mathfrak{M}$-universal. We generalize the notion of minimal $\mathfrak{M}$-universal metric space to notion of minimal $\mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class $\mathfrak{M}$ is minimal $\mathfrak{M}$-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces $X$ which possesses the following property. Among every three distinct points of $X$ there is one point lying between the other two points.
For subsets of $\mathbb{R}^+$ we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function $μ:\mathbb{N} \to \mathbb{R}^+$ provides an extension of the concept of porosity numbers on subsets of $\mathbb{N}$. The main results describe interconnections between porosity numbers of a set, features of the scaling functions and the geometry of so-called pretangent spaces to this set.
Let $(X,d)$ be a finite ultrametric space. In 1961 E.C. Gomory and T.C. Hu proved the inequality $|Sp(X)|\leqslant |X|$ where $Sp(X)=\{d(x,y)\colon x,y \in X\}$. Using weighted Hamiltonian cycles and weighted Hamiltonian paths we give new necessary and sufficient conditions under which the Gomory-Hu inequality becomes an equality. We find the number of non-isometric $(X,d)$ satisfying the equality $|Sp(X)|=|X|$ for given $Sp(X)$. Moreover it is shown that every finite semimetric space $Z$ is an image under a composition of mappings $f\colon X\to Y$ and $g\colon Y\to Z$ such that $X$ and $Y$ are finite ultrametric space, $X$ satisfies the above equality, $f$ is an $\varepsilon$-isometry with an arbitrary $\varepsilon>0$, and $g$ is a ball-preserving map.
Let $SP$ be the set of upper strongly porous at $0$ subsets of $\mathbb R^{+}$ and let $\hat I(SP)$ be the intersection of maximal ideals $I \subseteq SP$. Some characteristic properties of sets $E\in\hat I(SP)$ are obtained. It is shown that the ideal generated by the so-called completely strongly porous at $0$ subsets of $\mathbb R^{+}$ is a proper subideal of $\hat I(SP).$
Let $X$ be a nonempty set and $\mathcal{F}(X)$ be the set of nonempty finite subsets of $X$. The paper deals with the extended metrics $τ:\mathcal{F}(X)\to\mathbb{R}$ recently introduced by Peter Balk. Balk's metrics and their restriction to the family of sets $A$ with $|A|\leqslant n$ make possible to consider "distance functions" with $n$ variables and related them quantities. In particular, we study such type generalized diameters $\diam_{τ^n}$ and find conditions under which $B\mapsto\diam_{τ^n}B$ is a Balk's metric. We prove the necessary and sufficient conditions under which the restriction $τ$ to the set of $A\in\mathcal{F}(X)$ with $|A|\leqslant 3$ is a symmetric $G$-metric. An infinitesimal analog for extended by Balk metrics is constructed.
It was proved by Gomori and Hu in 1961 that for every finite nonempty ultrametric space $(X,d)$ the following inequality $|\Sp(X)|\leqslant |X|-1$ holds with $\Sp(X)=\{d(x,y):x,y \in X, x\neq y\}$. We characterize the spaces $X$, for which the equality in this inequality is attained by the structural properties of some graphs and show that the set of isometric types of such $X$ is dense in the Gromov-Hausdorff space of the compact ultrametric spaces.
We define and study the completely strongly porous at 0 subsets of R^{+}. Several characterizations of these subsets are obtained, among them the description via an universal property and structural one.
We study the statistical convergence of metric valued sequences and of their subsequences. The interplay between the statistical and usual convergences in metric spaces is also studied.
Let $\mathbb{R}_+=[0,\infty)$ and let $A\subseteq\mathbb{R}^n_+$. We have found the necessary and sufficient conditions under which a function $Φ:A\to\mathbb{R}_+$ has an isotone subadditive continuation on $\mathbb{R}^n_+$. It allows us to describe the metrics, defined on the Cartesian product $X_1\times...\times X_n$ of given metric spaces $(X_1,d_{X_1}),...,(X_n,...,d_{X_n})$, generated by the isotone metric preserving functions on $\mathbb{R}^n_+$. It also shows that the isotone metric preserving functions $Φ:\mathbb{R}^n_+\to\mathbb{R}_+$ coincide with the first moduli of continuity of the nonconstant bornologous functions $g:\mathbb{R}^n_+\to\mathbb{R}_+$. We discuss some algebraic properties of sets $X\subseteq \mathbb{R}$ providing the existence of isometric embeddings $f:B\to X$ for every three-point $B\subseteq \mathbb{R}$. In particular, we prove that every finite subset of $\mathbb{R}$ is isometric to some subset of transcendental real numbers.
Let (G,w) be a weighted graph. The necessary and sufficient conditions under which a weight w : E(G)-->R^+ can be extended to a pseudoultrametric on V(G) are found. A criterion of the uniqueness of this extension is also obtained. It is proved that G is complete k-partite with k >= 2 if and only if, for every pseudoultrametrizable weight w, there exists the smallest pseudoultrametric agreed with w. We characterize the structure of graphs for which the subdominant pseudoultrametric is an ultrametric for every strictly positive pseudoultrametrizable weight.