Related papers: A combinatorial approach to discrete geometry

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…

We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of…

The Voronoi diagram is a geometric object which is widely used in many areas. Recently it has been shown that under mild conditions Voronoi diagrams have a certain continuity property: small perturbations of the sites yield small…

We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…

We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

A novel class of discrete integrable surfaces is recorded. This class of discrete O surfaces is shown to include discrete analogues of classical surfaces such as isothermic, `linear' Weingarten, Guichard and Petot surfaces. Moreover,…

A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give…

We consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for graphical models with both continuous and discrete variables that is amenable to…

In this paper, several discrete schemes for Gaussian curvature are surveyed. The convergence property of a modified discrete scheme for the Gaussian curvature is considered. Furthermore, a new discrete scheme for Gaussian curvature is…

We study the undirected divisibility graph in which the vertex set is a finite subset of consecutive natural numbers up to N.We derive analytical expressions for measures of the graph like degree, clustering, geodesic distance and…

Voronoi tessellation, also known as Voronoi diagram, is an important computational geometry technique that has applications in various scientific disciplines. It involves dividing a given space into regions based on the proximity to a set…

In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element…

Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many…

In this article we consider combinatorial maps approach to graphs on surfaces, and how between them can be establish terminological uniformity in favor of combinatorial maps in way rotations are set as base structural elements and all other…

We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…

We consider the popular and classical method of alternating projections for finding a point in the intersection of two closed sets. By situating the algorithm in a metric space, equipped only with well-behaved geodesics and angles (in the…

We introduce a general framework for undirected graphical models. It generalizes Gaussian graphical models to a wide range of continuous, discrete, and combinations of different types of data. The models in the framework, called exponential…