Related papers: A primer on Script Geometry

We study the properties of several proximity measures for the vertices of weighted multigraphs and multidigraphs. Unlike the classical distance for the vertices of connected graphs, these proximity measures are applicable to weighted…

: In studies of discrete structures, functions are frequently used that express proximity, but are not metrics. We consider a class of such functions that is characterized by a normalization condition and an inequality that plays the same…

In a previous paper, we showed how certain orientations of the edges of a graph G embedded in a closed oriented surface S can be understood as discrete spin structures on S. We then used this correspondence to give a geometric proof of the…

This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…

In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the…

The aim of this work is to study the geometry underlying mechanics and its application to describe autonomous and nonautonomous conservative dynamical systems of different types; as well as dissipative dynamical systems. We use different…

In this paper, we systematically develop the $m$-contiguity distance between simplicial maps as a discrete approximation framework for homotopical complexity in the category of simplicial complexes. We construct an increasing sequence of…

The aim of this article is to study certain categorical-algebraic frameworks for basic homological algebra, introduced in arXiv:2404.15896, with the aim of better understanding the differences between them. We focus on homological…

Discretization of general relativity is a promising route towards quantum gravity. Discrete geometries have a finite number of degrees of freedom and can mimic aspects of quantum geometry. However, selection of the correct discrete freedoms…

In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various…

The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…

Signatures, one of the key concepts of rough path theory, have recently gained prominence as a means to find appropriate feature sets in machine learning systems. In this paper, in order to compute signatures directly from discrete data…

We develop a unified framework for nonlinear subdivision schemes on complete metric spaces (CMS). We begin with CMS preliminaries and formalize refinement in CMS, retaining key structural properties, such as locality. We prove a convergence…

Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to…

In this work, we define the notion of unimodular random measured metric spaces as a common generalization of various other notions. This includes the discrete cases like unimodular graphs and stationary point processes, as well as the…

We assume that the points in volumes smaller than an elementary volume (which may have a Planck size) are indistinguishable in any physical experiment. This naturally leads to a picture of a discrete space with a finite number of degrees of…

Semantic matching aims to establish pixel-level correspondences between instances of the same category and represents a fundamental task in computer vision. Existing approaches suffer from two limitations: (i) Geometric Ambiguity: Their…

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…

We introduce a binary embedding framework, called Proximity Preserving Code (PPC), which learns similarity and dissimilarity between data points to create a compact and affinity-preserving binary code. This code can be used to apply fast…

The proposed theory of causally structured discrete fields studies integer values on directed edges of a self-similar graph with a propagation rule, which we define as a set of valid combinations of integer values and edge directions around…