Related papers: Magnitude function determines generic finite metri…

Magnitude is an isometric invariant of metric spaces introduced by Leinster. Since its inception, it has inspired active research into its connections with integral geometry, geometric measure theory, fractal dimensions, persistent…

We set up a model for reasoning about metric spaces with belief theoretic measures. The uncertainty in these spaces stems from both probability and metric. To represent both aspect of uncertainty, we choose an expected distance function as…

We study the geometric significance of Leinster's notion of magnitude for a smooth manifold with boundary of arbitrary dimension, motivated by open questions for the unit disk in $\mathbb{R}^2$. For a large class of distance functions,…

We prove general results about separation and weak$^\#$-convergence of boundedly finite measures on separable metric spaces and Souslin spaces. More precisely, we consider an algebra of bounded real-valued, or more generally a $*$-algebra…

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…

Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact spaces, in an equivalent but more natural…

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…

Let M be a complete metric space. It is proved that if the space or scalar-valued bounded continuous functions on M admits an isometric shift, then M is separable.

We investigate how effectively finite metric spaces can be distinguished by distance-based invariants. As model spaces, we consider regular polygons, cycle graphs, and their generalization, circular metric spaces, and as invariants we…

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in $Q$-polynomial distance-regular…

Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel…

Magnitude is a measure of size defined for certain classes of metric spaces; it arose from ideas in category theory. In particular, magnitude is defined for compact subsets of Euclidean space and, in arXiv:1507.02502, Barcel\'o and Carbery…

In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We…

Metric space magnitude, an active subject of research in algebraic topology, originally arose in the context of biology, where it was used to represent the effective number of distinct species in an environment. In a more general setting,…

We classify the metric spaces that can be approximated by finite homogeneous ones.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of…

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 prove that, for any topological space $X$ and any metric space $(Y,d)$, the fine topology on the space of continuous functions from $X$ into $Y$ is independent of the metric $d$.

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…

In practical purposes for some geometrical problems in computer science we have as information the coordinates of some finite points in surface instead of the whole body of a surface. The problem arised here is: "How to define a distance…