Related papers: A conceptual breakthrough in sphere packing
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…
The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in…
The Cohn-Elkies linear program for sphere packing, which was used to solve the 8 and 24 dimensional cases, is conjectured to not be sharp in any other dimension $d>2$. By mapping feasible points of this infinite-dimensional linear program…
Symbolic and graphical tools, such as Mathematica, enable precise visualization and analysis of void spaces in sphere packings. In the cubic close packing (CCP, or face-centred cubic packing; FCC) arrangement these voids can be partitioned…
Although Kirby and Siebenmann showed that there are manifolds that do not admit PL structures, the possibility remained that all manifolds could be triangulated. In the late seventies Galewski and Stern and independently Matumoto showed…
We prove that if a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\widetilde M$ belonging…
A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance…
We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings, Kleinian packings exist in all…
We prove a size-sensitive version of Haussler's Packing lemma~\cite{Haussler92spherepacking} for set-systems with bounded primal shatter dimension, which have an additional {\em size-sensitive property}. This answers a question asked by…
The paper addresses the existence of multi-bubble solutions for the well-known Brezis-Nirenberg problem. Although there is extensive literature on the subject, the existence of solutions that blow up at multiple points in a 4D bounded…
The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as automobile, textile, food, and chemical industries. The so-called circle packing…
The packing of hard spheres (HS) of diameter $\sigma$ in a cylinder has been used to model experimental systems, such as fullerenes in nanotubes and colloidal wire assembly. Finding the densest packings of HS under this type of confinement,…
The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take…
We consider embeddings of a finite complex in a sphere. We give a homotopy theoretic classification of such embeddings in a wide range.
In this paper, we explore the geometry and the arithmetic of a family of polytopal sphere packings induced by regular polytopes in any dimension. We prove that every integral polytope is crystallographic, and we show that there are 11…
In Chapter 4 of [25] Triebel proved two theorems concerning pointwise multipliers and diffeomorphisms in function spaces of Besov and Triebel-Lizorkin type. In each case he presented two approaches, one via atoms and one via local means.…
We extend the classical theory of sphere theorems to the transverse geometry of Riemannian foliations. In this setting, we establish transverse analogues of the Grove-Shiohama diameter sphere theorem and of the Berger-Klingenberg…
Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behaviour at larger…
By applying an inverse Kaluza-Klein procedure, we provide explicit coordinate expressions for Riemannian metrics on two homeomorphic but not diffeomorphic spheres in seven dimensions. We identify Milnor's bundles, among which ten out of the…
In the present paper, multiscale systems of polynomial wavelets on an n-dimensional sphere are constructed. Scaling functions and wavelets are investigated,and their reproducing and localization properties and positive definiteness are…