Hasil untuk "math.MG"

T. Hales

This is the eighth and final paper in a series giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the oldest problem in discrete geometry and is an important part of Hilbert's 18th problem. An example of a packing achieving this density is the face-centered cubic packing. This paper completes the fourth step of the program outlined in math.MG/9811073: A proof that if some standard region has more than four sides, then the star scores less than $8 \pt$.

Sebastiano Nicolussi Golo, Ye Zhang

In this paper we characterize the geodesic dimension $N_{GEO}$ and give a new lower bound to the curvature exponent $N_{CE}$ on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which $N_{CE} > N_{GEO}$: this answers a question posed by Rizzi in arXiv:1510.05960v4 [math.MG].

Dirk Frettlöh, Alexey Garber, Lorenzo Sadun

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many. A very similar result is proven in arXiv:2011.00106 [math.MG].

O. Karpenkov

In this small paper we bring together some open problems related to the study of the configuration spaces of tensegrities, i.e. graphs with stresses on edges. These problems were announced in Doray et al. (Discrete Comput Geom 43:436–466, 2010), Karpenkov et al. (ARS Math Contemp 6:305–322, 2013), Karpenkov (The combinatorial geometry of stresses in frameworks. arXiv:1512.02563 [math.MG], 2017), and Karpenkov (Geometric Conditions of Rigidity in Nongeneric settings, 2016) (by F. Doray, J. Schepers, B. Servatius, and the author), for more details we refer to the mentioned articles.

J. Koivisto

A. Sokolov

This paper is to serve as a key to the projective (homogeneous) model developed by Charles Gunn (arXiv:1101.4542 [math.MG]). The goal is to explain the underlying concepts in a simple language and give plenty of examples. It is targeted to physicists and engineers and the emphasis is on explanation rather than rigorous proof. The projective model is based on projective geometry and Clifford algebra. It supplements and enhances vector and matrix algebras. It also subsumes complex numbers and quaternions. Projective geometry augmented with Clifford algebra provides a unified algebraic framework for describing points, lines, planes, etc, and their transformations, such as rotations, reflections, projections, and translations. The model is relevant not only to Euclidean space but to a variety of homogeneous metric spaces.

A. Sokolov

I apply the algebraic framework introduced in arXiv:1101.4542v3[math.MG] to Minkowski (pseudo-Euclidean) spaces in 2, 3, and 4 dimensions. The exposition follows the template established in arXiv:1307.2917[math.MG] for Euclidean spaces. The emphasis is on geometric structures, but some contact with special relativity is made by considering relativistic addition of velocities and Lorentz transformations, both of which can be seen as rotation applied to points and to lines. The language used in the paper reflects the emphasis on geometry, rather than applications to special relativity. The use of Clifford algebra greatly simplifies the study of Minkowski spaces, since unintuitive synthetic techniques are replaced by algebraic calculations.

Peter Nickolas, Reinhard Wolf

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by $I(mu) = \int_X \int_X d(x,y) dμ(x) dμ(y)$, and set $M(X) = \sup I(mu)$, where $μ$ ranges over the collection of measures in $\mathcal{M}(X)$ of total mass 1. The space $(X, d)$ is \emph{quasihypermetric} if $I(μ) \leq 0$ for all measures $μ$ in $\mathcal{M}(X)$ of total mass 0 and is \emph{strictly quasihypermetric} if in addition the equality $I(μ) = 0$ holds amongst measures $μ$ of mass 0 only for the zero measure. This paper explores the constant $M(X)$ and other geometric aspects of $X$ in the case when the space $X$ is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are $L^1$-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [Peter Nickolas and Reinhard Wolf, \emph{Distance geometry in quasihypermetric spaces. I}, \emph{II} and \emph{III}].

T. Hales

This is the first in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the oldest problem in discrete geometry and is an important part of Hilbert's 18th problem. An example of a packing achieving this density is the face-centered cubic packing. This paper has a historical overview and a synopsis of the rest of the series. The other papers in the series are math.MG/9811072, math.MG/9811073, math.MG/9811074, math.MG/9811075, math.MG/9811076, math.MG/9811077, and math.MG/9811078.

Peter Nickolas, Reinhard Wolf

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(μ) = \int_X \int_X d(x,y) dμ(x) dμ(y), \] and set $M(X) = \sup I(μ)$, where $μ$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. This paper, with two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\mathcal{M}(X)$ when equipped with a natural semi-inner product. Specifically, this paper explores links between the properties of $M(X)$ and metric embeddings of $X$, and the properties of $M(X)$ when $X$ is a finite metric space.

S. Alesker

This is the fourth part in the series of articles math.MG/0503397, math.MG/0503399, math.MG/0509512 where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations introduced in math.MG/0503399 is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations introduced in math.MG/0503399 is an automorphism of the algebra of valuations. Then an integration functional on valuations with compact support is introduced, and a property of selfduality of valuations is proved. Next a space of generalized valuations is defined, and some basic properties of it are proved. Finally a canonical imbedding of the space of constructible functions on a real analytic manifold into the space of generalized valuations is constructed, and various structures on valuations are compared with known structures on constructible functions.

M. D. Sikiric

Given a latticeL , a polytope P is called aDelaunay polytopeif the set of its vertices is S ∩ L with S being anempty sphereof the lattice. Extending our previous work (Available from http://il.arXiv.org/abs/math.MG/0108177 ) on the hypermetric cone HY P 7, we classify the six-dimensional Delaunay polytopes according to their combinatorial type. The list of 6241 combinatorial types is obtained by a study of the set of faces of the polyhedral cone HY P7. 1 © 2003 Elsevier Ltd. All rights reserved.

Samuel P. Ferguson

The Hales program to prove the Kepler conjecture on sphere packings consists of five steps, which if completed, will jointly comprise a proof of the conjecture. We carry out step five of the program [outlined in math.MG/9811073], a proof that the local density of a certain combinatorial arrangement, the pentahedral prism, is less than that of the face-centered cubic lattice packing. We prove various relations on the local density using computer-based interval arithmetic methods. Together, these relations imply the local density bound.

T. Hales

This is the fifth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the oldest problem in discrete geometry and is an important part of Hilbert's 18th problem. An example of a packing achieving this density is the face-centered cubic packing. This paper carries out the third step of the program outlined in math.MG/9811073: A proof that if all of the standard regions are triangles or quadrilaterals, then the total score is less than $8 \pt$ (excluding the case of pentagonal prisms).

T. Hales

This is the sixth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the oldest problem in discrete geometry and is an important part of Hilbert's 18th problem. An example of a packing achieving this density is the face-centered cubic packing. This paper completes part of the fourth step of the program outlined in math.MG/9811073: A proof that if some standard region has more than four sides, then the star scores less than $8 \pt$.

G. Kuperberg

The Mahler volume of a centrally symmetric convex body K is dened as M(K )=( Vol K)(VolK). Mahler conjectured that this volume is minimized when K is a cube. We introduce the bottleneck conjecture, which stipulates that a certain convex body K } KK has least volume when K is an ellip- soid. If true, the bottleneck conjecture would strengthen the best current lower bound on the Mahler volume due to Bourgain and Milman. We also generalize the bottleneck conjecture in the context of indenite orthogonal geometry and prove some special cases of the generalization. This article is in the xxx archive as: math.MG/9811119