Related papers: Remarks on Ultrametrics and Metric-Preserving Func…

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…

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…

A number of topics involving metrics and measures are discussed, including some of the special structure associated with ultrametrics.

We consider metric ultraproducts of finite groups with respect to some classes of length functions. All sofic groups embed into these ultraproducts. We study embeddings of normed groups. We also show that in some natural situations such an…

Let $\mathbb{R}^{+}=[0, \infty)$ and let $\mathbf{End}_{\mathbb{R}^+}$ be the set of all endomorphisms of the monoid $(\mathbb{R}^+, \vee)$. The set $\mathbf{End}_{\mathbb{R}^+}$ is a monoid with respect to the operation of the function…

This paper addresses the natural question: ``How should frames be compared?'' We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a…

An ultrametric defined on a subset S of a metric space X can be extended to X while roughly preserving distances between pairs in S x X.

The concept of a quasi-metric space arises by relaxing the requirement of the symmetry axiom in the definition of a metric. This small variation alters several structural properties possessed by a standard metric space. This article aims to…

The $p$-adic completion $\mathbb{Q}_p$ of the rational numbers induces a different absolute value $|\cdot|_p$ than the typical $| \cdot |$ we have on the real numbers. In this paper we compare and contrast functions $f \colon \mathbb{R}^{+}…

Thermodynamically consistent measurements can either preserve statistics (unbiased) or preserve marginal states (non-invasive) but not both. Here we show the existence of metrological tasks which unequally favor each of the aforementioned…

Kirk and Shahzad have recently given fixed point theorems concerning local radial contractions and metric transforms. In this article, we replace the metric transforms by metric-preserving functions. This in turn gives several extensions of…

Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…

A function between two metric spaces is said to be totally bounded regular if it preserves totally bounded sets. These functions need not be continuous in general. Hence the purpose of this article is to study such functions vis-\'a-vis…

We describe some Cartesian products of metric spaces and find conditions under which products of ultrametric spaces are ultrametric.

S-metric and b-metric spaces are metrizable, but it is still quite impossible to get an explicit form of the concerned metric function. To overcome this, the notion of $\phi$-metric is developed by making a suitable modification in triangle…

We supplement the result of the first part of the work with estimates of the integrals of the difference of subharmonic functions in measure with some deterioration of the absolute constants, but these estimates have the form of a…

Metric-preserving functions (here, metric aggregation functions) offer a natural method for constructing metrics on Cartesian products of metric spaces or for aggregating multiple metrics defined on a common set. Strongly metric-preserving…

Let $\mathbb{R}_+=[0,\infty)$ and let $A\subseteq\mathbb{R}^n_+$. We have found the necessary and sufficient conditions under which a function $\Phi:A\to\mathbb{R}_+$ has an isotone subadditive continuation on $\mathbb{R}^n_+$. It allows us…

A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric…

Isometries of metric spaces $(X,d)$ preserve all level sets of $d$. We formulate and prove cases of a conjecture asserting if $X$ is a complete Riemannian manifold, then a function $f:X \rightarrow X$ preserving at least one level set…