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We describe the canonical correspondence between set of all finite metric spaces and set of special symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those…
In this paper, we prove that the topology induced by algebraic cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e. any algebraic cone metric space is metrizable. Furthermore,…
We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a…
We examine unification of gauge couplings in four dimensional renormalizable gauge theories inspired by the latticized (deconstructed) SM or MSSM in five dimensions. The models are based on replicated gauge groups, spontaneously broken to…
We provide some properties and characterizations of homologically $UV^n$-maps and $lc^n_G$-spaces. We show that there is a parallel between recently introduced by Cauty algebraic $ANR$'s and homologically $lc^n_G$-metric spaces, and this…
Urysohn constructed a separable complete universal metric space homogeneous for all finite subspaces, which is today called the Urysohn universal metric space. Some authors have recently investigated an ultrametric analogue of this space.…
The present paper is a sequel to our paper "Metric characterization of isometries and of unital operator spaces and systems". We characterize certain common objects in the theory of operator spaces (unitaries, unital operator spaces,…
A little-known and highly economical characterization of the real interval [0, 1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such.…
Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results…
In this paper, generalized metrics mean metrics taking values in general linearly ordered Abelian groups. Using the Hahn fields, we first prove that for every generalized metric space, if the set of the Archimedean equivalence classes of…
We define linear and semilinear isometry for general subspace codes, used for random network coding. Furthermore, some results on isometry classes and automorphism groups of known constant dimension code constructions are derived.
We develop a unified categorical framework for gauging both continuous and finite symmetries in arbitrary spacetime dimensions. Our construction applies to geometric categories i.e. categories internal to stacks. This generalizes the…
In this article, we expand upon the concepts introduced by David Spivak about the relationship between the category $\mathbf{UM}$ of uber metric spaces and the category $\mathbf{sFuz}$ of fuzzy simplicial sets. We show that fuzzy simplicial…
We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
We study hierarchical clusterings of metric spaces that change over time. This is a natural geometric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent…
The main purpose of this paper is to study the characterizations of John spaces. We obtain five equivalence characteristics for length John spaces. As an application, we establish a dimension-free quasisymmetric invariance of length John…
In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the…
We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.