Related papers: Metric axioms: a structural study

We develop a general framework for reasoning about distances between transition systems with quantitative information. Taking as starting point an arbitrary distance on system traces, we show how this leads to natural definitions of a…

We propose a unified theory of generalized weights for linear codes endowed with an arbitrary distance. Instead of relying on supports or anticodes, the weights of a code are defined via the intersections of the code with a chosen family of…

This paper investigates the distribution of rational and algebraic points of bounded weighted height in weighted projective spaces over number fields. For a weighted projective space with weights q over a number field k of degree m, we…

These are notes from talks given at ICMS, Edinburgh, 4/2007 ("Geometry and Algorithms workshop") and at Bernoulli Center, Lausanne 5/2007 ("Limits of graphs in group theory and computer science"). We survey the following type of dichotomies…

This paper presents a distance function between sets based on an average of distances between their elements. The distance function is a metric if the sets are non-empty finite subsets of a metric space. It can be applied to produce various…

We study the metric structure of walks on graphs, understood as Lipschitz sequences. To this end, a weighted metric is introduced to handle sequences, enabling the definition of distances between walks based on stepwise vertex distances and…

Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…

We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…

We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…

Given a finite directed graph with $n$ vertices, we define a metric $d_G$ on $\mathbb{F}_q^n$, where $\mathbb{F}_q$ is the finite field with $q$ elements. The weight of a word is defined as the number of vertices that can be reached by a…

We study questions inspired by Erd\H os' celebrated distance problems with dot products in lieu of distances, and for more than a single pair of points. In particular, we study point configurations present in large finite point sets in the…

Given a metric pair $(X,A)$, i.e. a metric space $X$ and a distinguished closed set $A \subset X$, one may construct in a functorial way a pointed pseudometric space $\mathcal{D}_\infty(X,A)$ of persistence diagrams equipped with the…

Recent decades have seen the discovery of numerous complex materials. At the root of the complexity underlying many of these materials lies a large number of possible contending atomic- and larger-scale configurations and the intricate…

Recall that Janson showed that if the edges of the complete graph $K_n$ are assigned exponentially distributed independent random weights, then the expected length of a shortest path between a fixed pair of vertices is asymptotically equal…

In this paper we prove equivalence of sets of axioms for non-discrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends…

This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on…

This paper explores alternative statements of the axioms for lattice gluing, focusing on lattices that are modular, locally finite, and have finite covers, but may have infinite height. We give a set of "maximal" axioms that maximize what…

A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…

This paper challenges some of the common assumptions underlying the mathematics used to describe the physical world. We start by reviewing many of the assumptions underlying the concepts of real, physical, rigid bodies and the translational…

We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding…