Related papers: Metric axioms: a structural study
A new sequential approach to investigations of structure of metric spaces at infinity is proposed. Criteria for finiteness and boundedness of metric spaces at infinity are found.
The centuries-long practice of the teaching turned mechanics into an academic construct detached from its underlying science, the physics of macroscopic bodies. In particular, the regularities that delineate the scope of validity of…
Networked structures arise in a wide array of different contexts such as technological and transportation infrastructures, social phenomena, and biological systems. These highly interconnected systems have recently been the focus of a great…
We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of finding…
Given a distance $D$ on a finite set $X$ with $n$ elements, it is interesting to understand how the ranking $R_x = z_1,z_2,\dots,z_n$ obtained by ordering the elements in $X$ according to increasing distance $D(x,z_i)$ from $x$, varies with…
This paper argues that mathematical objects are constructions and that constructions introduce a flexibility in the ways that mathematical objects are represented (as sets of binary sequences for example) and presented (in a particular…
We consider weighted structures, which extend ordinary relational structures by assigning weights, i.e. elements from a particular group or ring, to tuples present in the structure. We introduce an extension of first-order logic that allows…
When network and graph theory are used in the study of complex systems, a typically finite set of nodes of the network under consideration is frequently either explicitly or implicitly considered representative of a much larger finite or…
We define coarse proximity structures, which are an analog of small-scale proximity spaces in the large-scale context. We show that metric spaces induce coarse proximity structures, and we construct a natural small-scale proximity…
A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid - i.e. whether every other suitably close framework with the same…
This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…
Self-assembly processes are widespread in nature, and lie at the heart of many biological and physical phenomena. The characteristics of self-assembly building blocks determine the structures that they form. Two crucial properties are the…
We prove an existence and uniqueness theorem for fixed points of contraction maps in the framework of quantum metric spaces, where distinguishability is defined by the $L^2$ norm: $d_Q(\psi_1,\psi_2) = \|\psi_1 - \psi_2\|$. The result…
Let $X=X(n,q)$ be the set of $n\times n$ Hermitian matrices over $\mathbb{F}_{q^2}$. It is well known that $X$ gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study $d$-codes in this…
Assortativity measures the tendency of a vertex in a network being connected by other vertexes with respect to some vertex-specific features. Classical assortativity coefficients are defined for unweighted and undirected networks with…
The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…
The concept of quasi-partial b-metric-like spaces is being introduced and studied with the help of topology. Examples are also discussed to support the results. Some fixed point theorems are proved in the setting of quasi-partial…
This article provides examples of distal metric structures. One source of examples are metric valued fields. By analyzing indiscernible sequences, we show that real closed metric valued fields are distal, and conclude that algebraically…
With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions…
Fix an integer $r\geq 3$. We consider metric spaces on $n$ points such that the distance between any two points lies in $\{1,..., r\}$. Our main result describes their approximate structure for large $n$. As a consequence, we show that the…