Related papers: The Sphere-Packing Problem
We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond…
We present a simple argument to account for crystallization of hard spheres under the action of a gravitational field. The paper attempts to bridge the gap between two communities of scientists, one working on granular materials and the…
Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…
We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is…
We survey recent developments in the theory and applications of the broken ray transforms. Furthermore, we discuss some open problems.
We present the first space-filling bearing in three dimensions. It is shown that a packing which contains only loops with even number of spheres can be constructed in a self-similar way and that it can act as a three dimensional bearing in…
This paper collects some problems that I have encountered during the years, have puzzled me and which, to the best of my knowledge, are still open. Most of them are well-known and have been first stated by other authors. In this sad season…
Spatially ordered systems confined to surfaces such as spheres exhibit interesting topological structures because of curvature induced frustration in orientational as well as translational order. The study of these structures is important…
A Short Course on Frame Theory.
This is the second 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…
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…
We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite…
I present a brief update on the transverse polarization distributions, focusing on model calculations and phenomenological perspectives.
We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…
The eigensolutions of many-body quantum systems are always difficult to compute. The envelope theory is a method to easily obtain approximate, but reliable, solutions in the case of identical particles. It is extended here to treat systems…
The aim of this paper is to highlight recent progress in using conic optimization methods to study geometric packing problems. We will look at four geometric packing problems of different kinds: two on the unit sphere -- the kissing number…
Dense packing of particles has provided important models to study the structure of matter in various systems such as liquid, glassy and crystalline phase, etc. The simplest sphere packing models are able to represent and capture salient…
The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by…
By some new recursive algorithms, in this paper, we will give some improvements on Waring's problem.
In this paper, on envelopes created by circle families in the plane, all four basic problems (existence problem, representation problem, problem on the number of envelopes, problem on relationships of definitions) are solved.