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Let X be a class of metric spaces and let Px be the set of all f: [0, oo) [0, oo) preserving X, i.e., (/, f o p) e X whenever (/, p) e X. For arbitrary subset A of the set of all metric preserving functions, we show that the equality Px = A has a solution if A is a monoid with respect to the operation of function composition. In particular, for the set SI of all amenable subadditive increasing functions, there is a class X of metric spaces such that Px = SI holds. 2020 Mathematics Subject Classification: Primary 26A30, Secondary 54E35, 20M20 [ABSTRACT FROM AUTHOR]

The present article deals with new fixed point theorems by means of GG-strongly contractive maps. The research findings are demonstrated in a spherically complete ultrametric space with a graph for single-valued mappings. The special cases of the results that extend the current ones are offered, along with some examples that illustrate our results. Besides, an application utilized in dynamic programming that endorses the acquired observations is also provided.

We prove the following results on enumerating or counting furthest pairs given an ultra- metric space with n elements: * There is a deterministic O(F + n log n)-time algorithm for enumerating all furthest pairs, where F denotes the total number of furthest pairs. * There is a Monte Carlo O(n/ε²)-time algorithm that estimates the number of furthest pairs to within a multiplicative factor in ( 1 - ε, 1 + ε), where ε > 0. Furthermore, the time complexity of O(n/ ε²) cannot be improved to o(n⋅f/(ε)) for any f(⋅). [ABSTRACT FROM AUTHOR]

Let X be a class of metric spaces and let PX be the set of all f : [0, ∞) → [0, ∞) preserving X, i.e., (Y, f ∘ ρ) ∈ X whenever (Y, ρ) ∈ X. For arbitrary subset A of the set of all metric preserving functions, we show that the equality PX = A has a solution if A is a monoid with respect to the operation of function composition. In particular, for the set SI of all amenable subadditive increasing functions, there is a class X of metric spaces such that PX = SI holds.2020 Mathematics Subject ClassificationPrimary 26A30, Secondary 54E35, 20M20

We investigate compact ultrametric measure spaces which form a subset U w of the collection of all metric measure spaces M w . In analogy with the notion of the ultrametric Gromov–Hausdorff distance on the collection of ultrametric spaces U , we define ultrametric versions of two metrics on U w , namely of Sturm's Gromov–Wasserstein distance of order p and of the Gromov–Wasserstein distance of order p. We study the basic topological and geometric properties of these distances as well as their relation and derive for p = ∞ a polynomial time algorithm for their calculation. Further, several lower bounds for both distances are derived and some of our results are generalized to the case of finite ultra-dissimilarity spaces. Finally, we study the relation between the Gromov–Wasserstein distance and its ultrametric version (as well as the relation between the corresponding lower bounds) in simulations and apply our findings for phylogenetic tree shape comparisons. [ABSTRACT FROM AUTHOR]

We consider the operator that is a perturbation of the Taibleson–Vladimirov operator by a potential , where and . We prove that the operator is closable and its minimal closure is a nonnegative definite self-adjoint operator (where the critical value depends on ). While the operator is nonnegative definite, the potential may well take negative values as for all . The equation admits a Green function , that is, the integral kernel of the operator . We obtain sharp lower and upper bounds on the ratio of the Green functions and . [ABSTRACT FROM AUTHOR]

The aim of this study is to prove the existence and uniqueness of fixed point and common fixed point theorems for self-mappings in modular ultrametric spaces. These theorems are proved under varying contractive circumstances and without the property of spherical completeness. As a consequence, the examples of fixed point and common fixed point problems are correctly formulated. As an application, the well-posedness of a common fixed point problem is proved. This study expands on prior research in modular ultrametric space to provide a more comprehensive understanding of such spaces using generalized contraction. [ABSTRACT FROM AUTHOR]

In this paper, using the rational contraction we prove certain fixed-point theorems (FPTs) via p-adic distance over partially ordered ultrametric spaces. Further, these results are explored with suitable examples. [ABSTRACT FROM AUTHOR]

Let G be a graph with a vertex set V. The graph G is path-proximinal if there is a semimetric d : V × V → [0, ∞[ and disjoint proximinal subsets of the semimetric space (V, d) such that V = A ∪ B. The vertices u, v ∈ V are adjacent iff d u v ⩽ inf d x y : x ∈ A y ∈ B , and, for every p ∈ V , there is a path connecting A and B in G, and passing through p. It has been shown that a graph is path-proximinal if and only if all of its vertices are not isolated. It has also been shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of each of its vertices is equal to 1. [ABSTRACT FROM AUTHOR]

The functor of $\ast$-measures of compact support on the category of ultrametric spaces and non-expanding maps is introduced in the previous publication of the authors. In the present note, we prove that this functor determines a monad on this category. The monad structure allows us to define the tensor product of $\ast$-measures. We consider some applications of this notion to equilibrium theory.

The present paper deals with new fixed point theorems by means of F−contraction. The results are present in spherically complete ultrametric space for single valued rational type mappings. The special cases of the results which reduce the existing ones are presented and some examples are given to illustrating our results.

The aim of this study is to prove the existence and uniqueness of fixed point and common fixed point theorems for self-mappings in modular ultrametric spaces. These theorems are proved under varying contractive circumstances and without the property of spherical completeness. As a consequence, the examples of fixed point and common fixed point problems are correctly formulated. As an application, the well-posedness of a common fixed point problem is proved. This study expands on prior research in modular ultrametric space to provide a more comprehensive understanding of such spaces using generalized contraction.

For each given we investigate certain sub-family of the collection of all compact metric spaces which are characterized by the satisfaction of a strengthened form of the triangle inequality which encompasses, for example, the strong triangle inequality satisfied by ultrametric spaces. We identify a one parameter family of Gromov-Hausdorff like distances on and study geometric and topological properties of these distances as well as the stability of certain canonical projections . For the collection of all compact ultrametric spaces, which corresponds to the case of the family , we explore a one parameter family of interleaving-type distances and reveal their relationship with . [ABSTRACT FROM AUTHOR]

We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space. [ABSTRACT FROM AUTHOR]

The functor of ∗-measures of compact support on the category of ultrametric spaces and non-expanding maps is introduced in the previous publication of the authors. In the present note, we prove that this functor determines a monad on this category. The monad structure allows us to define the tensor product of ∗-measures. We consider some applications of this notion to equilibrium theory. [ABSTRACT FROM AUTHOR]

We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that if A and B are compact metric spaces and the Hausdorff dimension of A is bigger than the upper box dimension of B , then there exist a compact set A ′ ⊂ A and a Lipschitz onto map f : A ′ → B. As a corollary we prove that any 'natural' dimension in R n must be between the Hausdorff and upper box dimensions. We show that if A and B are self-similar sets with the strong separation condition with equal Hausdorff dimension and A is homogeneous, then A can be mapped onto B by a Lipschitz map if and only if A and B are bilipschitz equivalent. For given α > 0 we also give a characterization of those compact metric spaces that can be obtained as an α -Hölder image of a compact subset of R. The quantity we introduce for this turns out to be closely related to the upper box dimension. [ABSTRACT FROM AUTHOR]

Given a metric space (X,d), a set S ⊆ X is called a k-metric generator for X if any pair of different points of X is distinguished by at least k elements of S. A k-metric basis is a k-metric generator of the minimum cardinality in X. We prove that ultrametric spaces do not have finite k-metric bases for k >2. We also characterize when the metric and 2-metric bases of an ultrametric space are finite and, when they are finite, we characterize them. Finally, we prove that an ultrametric space can be easily recovered knowing only the metric basis and the coordinates of the points in it. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence of best proximity pair in ultrametric spaces. We show, under suitable assumptions, that the proximinal pair 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. [ABSTRACT FROM AUTHOR]

In this paper, using the idea of ultrametrization of metric spaces we introduce ultradiversification of diversities. We show that every diversity has an ultradiversification which is the greatest nonexpansive ultra-diversity image of it. We also investigate a Hausdorff-Bayod type problem in the setting of diversities, namely, determining what diversities admit a subdominant ultradiversity. This gives a description of all diversities which can be mapped onto ultradiversities by an injective nonexpansive map. The given results generalize similar results in the setting of metric spaces.

The p-adic completion ℚp of the rational numbers induces a different absolute value |⋅|p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ+ → ℝ+, for which the composition with the p-adic metric dp generated by |⋅|p is still a metric on ℚp, with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ dp is still an ultrametric on ℚp if and only if there is a function g such that f ∘ dp = g ∘ dp and g ∘ d is still an ultrametric for every ultrametric d. Some general variants of the last statement are also proved. [ABSTRACT FROM AUTHOR]

The notion of ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the Arens-Eells isometric embedding theorem of metric spaces, the Hausdorff extension theorem of metrics, the Niemytzki-Tychonoff characterization theorem of the compactness, and the author's interpolation theorem of metrics and theorems on dense subsets of spaces of metrics. [ABSTRACT FROM AUTHOR]

Diversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective. [ABSTRACT FROM AUTHOR]

Let (X , d) be a locally compact separable ultrametric space. Given a measure m on X and a function C (B) defined on the set of all non-singleton balls B of X , we consider the hierarchical Laplacian L = L C . The operator L acts in ℒ 2 (X , m) , is essentially self-adjoint and has a purely point spectrum. Choosing a sequence { 𝜀 (B) } of i.i.d. random variables, we consider the perturbed function C (B , ω) and the perturbed hierarchical Laplacian L ω = L C (ω). Under certain conditions, the density of states 𝔪 exists and it is a continuous function. We choose a point λ such that 𝔪 (λ) > 0 and build a sequence of point processes defined by the eigenvalues of L ω located in the vicinity of λ. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity 𝔪 (λ). [ABSTRACT FROM AUTHOR]

By introducing new deformations on symbolic Cantor sets and ultrametric spaces, we prove that doubling ultrametric spaces admit bilipschitz embedding into Cantor sets. If in addition the spaces are uniformly perfect, we show that they are quasisymmetrically equivalent to Cantor sets. Moreover, we provide a new proof for a recent work of Heer regarding quasimöbius uniformization of Cantor set. [ABSTRACT FROM AUTHOR]

In this paper, we give some necessary and sufficient conditions under which the topology generated by a partial metric is equivalent to the topology generated by a suitably defined metric. Next, we study some new extensions of the Generalized Banach Contraction Principle to partial metric spaces. Moreover, we draw a particular attention to the space of all sequences showing, in particular, that some well-known fixed point theorems for ultrametric spaces, can be used for operators acting in that space. We illustrate our considerations by suitable examples and counterexamples. [ABSTRACT FROM AUTHOR]

In this paper, we establish some hyperstability results concerning the following Cauchy - Jensen functional equation f ( x + y/2 + z ) + f ( x - y/2 + z ) = f(x) + 2f(z) in ultrametric Banach spaces. [ABSTRACT FROM AUTHOR]

Given an ultrametric space (X , d) and its related distance δ d between points and subsets of X , we construct an extension (Z , δ ζ) , where Z is the underlying set of the Smirnov compactification of a suitable proximity on X and δ ζ is a distance between points and subsets of Z. The underlying topological space defined by the closure p ∈ cl ζ A if and only if δ ζ (p , A) = 0 , for a point p ∈ Z and a subset A ⊆ Z , is a Hausdorff zero-dimensional compactification of the underlying topological space of (X , d) and the distance δ ζ is an extension of the distance δ d. The construction is performed in the larger category ZDApp of Hausdorff zero-dimensional approach spaces with as objects sets structured with a distance, subject to suitable axioms and with contractions as morphisms. Both the categories UMet of ultrametric spaces with non-expansive maps and ZDim of zero-dimensional topological spaces and continuous maps are fully embedded in ZDApp. In this broader setting we obtain an approach counterpart for the Banaschewski compactification which is a reflection ZDApp 2 → kZDApp 2 , from Hausdorff zero-dimensional approach spaces to Hausdorff compact zero-dimensional approach spaces. In the topological case the construction coincides with the topological Banaschewski compactification. In the ultrametric case the space (Z , δ ζ) retains numerical information from (X , d) , whereas the topological Banaschewski compactification applied to the underlying topological space of (X , d) is generally not (ultra) metrisable. [ABSTRACT FROM AUTHOR]