Related papers: The comb representation of compact ultrametric spa…

Let f be a continuous map of a complete separable metric space E onto the irrationals. We show that if a complete separable metric space M contains isometric copies of every closed relatively discrete set in E, then M contains also an…

Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like…

We consider groups $\mathbb{I}$ of isometries of ultrametric Urysohn spaces $\mathbb{U}$. Such spaces $\mathbb{U}$ admit transparent realizations as boundaries of certain $R$-trees and the groups $\mathbb{I}$ are groups of automorphisms of…

Given two ordinal $\lambda$ and $\gamma$, let $f:[0,\lambda) \rightarrow [0,\gamma)$ be a function such that, for each $\alpha<\gamma$, $\sup\{f(t): t\in[0, \alpha]\}<\gamma.$ We define a mapping $d_{f}: [0,\lambda)\times [0,\lambda)…

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of…

Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a…

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining…

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…

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree ($T$, $d$) is a metric space such that between any…

Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for…

Let $T$ be a $C_0$--contraction on a separable Hilbert space. We assume that $I_H-T^*T$ is compact. For a function $f$ holomorphic in the unit disk $\DD$ and continuous on $\bar\DD$, we show that $f(T)$ is compact if and only if $f$…

The Hausdorff distance, the Gromov-Hausdorff, the Fr\'echet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as $\inf_\rho…

In this paper we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We show that every S-extension can be identified with an algebraic object. We use this algebraic representation to give a…

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,…

By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance.…

Using the wedge sum of metric spaces, for all compact metrizable spaces, we construct a topological embedding of the compact metrizable space into the set of all metric trees in the Gromov--Hausdorff space with finite prescribed values. As…

We introduce the Hermitian-invariant group $\Gamma_f$ of a proper rational map $f$ between the unit ball in complex Euclidean space and a generalized ball in a space of typically higher dimension. We use properties of the groups to define…

We describe the order type of range sets of compact ultrametrics and show that an ultrametrizable infinite topological space $(X, \tau)$ is compact iff the range sets are order isomorphic for any two ultrametrics compatible with the…

A well known result in the analysis of finite metric spaces due to Gromov says that given any $(X,d_X)$ there exists a \emph{tree metric} $t_X$ on $X$ such that $\|d_X-t_X\|_\infty$ is bounded above by twice $\mathrm{hyp}(X)\cdot…

We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum…