Related papers: Ball packings in hyperbolic space
We prove the double bubble conjecture in the three-sphere $S^3$ and hyperbolic three-space $H^3$ in the cases where we can apply Hutchings theory: 1) in $S^3$, each enclosed volume and the complement occupy at least 10% of the volume of…
A nontrivial topology of the spatial section of the universe is an observable, which can be probed for all locally homogeneous and isotropic universes, without any assumption on the cosmological density parameters. We discuss how one can…
We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.
The aim of this paper is to study lattice-like coverings with congruent translation balls and the packings and coverings with a type of translation cylinders in Sol space related to the fundamental lattices. We introduce the notions of the…
We consider the covering of a ball in certain normed spaces by its congruent subsets and show that if the finite number of sets is not greater than the dimensionality of the space, then the centre of the ball either belongs to the interior…
In this note, we show that the half-plane capacity of a subset of the upper half-plane is comparable to a simple geometric quantity, namely the euclidean area of the hyperbolic neighborhood of radius one of this set. This is achieved by…
Recently, there has been a surge of interest in representation learning in hyperbolic spaces, driven by their ability to represent hierarchical data with significantly fewer dimensions than standard Euclidean spaces. However, the viability…
This article simply presents several coordinate systems for 2 and 3-dimensional hyperbolic spaces, describing the general solutions of Helmholtz equation in each one of these systems.
In this paper, upper bounds for the densities of the densest translative tetrahedron packings and the densest translative cubooctahedron packings are obtained.
According to the latest evidence, the Universe is entering an era of exponential expansion, where gravitationally bound structures will get disconnected from each other, forming isolated `island universes'. In this scenario, we present a…
We have previously explored cylindrical packings of disks and their relation to sphere packings. Here we extend the analytical treatment of disk packings, analysing the rules for phyllotactic indices of related structures and the variation…
We study the space $C(a_0,a_1,\dots,a_n)$ of hyperbolic 2-spheres with cone points of prescribed apex curvatures $2a_0,2a_1,\dots,2a_n\in]0,2\pi[$ and some related spaces. For $n=3$, we get a detailed description of such spaces. The…
A dimension reduction for the hyperbolic space is established. When points are far apart an embedding with bounded distortion into the hyperbolic plane is achieved.
We introduce and discuss the concept of Gaussian probability density function (pdf) for the n-dimensional hyperbolic space which has been proposed as an environment for coding and decoding signals. An upper bound for the error probability…
The variational capacity cap_p in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every subset E of R^n, inf_{x in R^n} (cap_p(E \cap B(x,r),B(x,2r)) / cap_p(B(x,r),B(x,2r))) is either zero or tends…
In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic…
Localization and dilation procedures are discussed for infinite dimensional $\alpha$-concave measures on abstract locally convex spaces (following Borell's hierarchy of hyperbolic measures).
Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an…
The article presents the mathematical sequences describing circle packing densities in four different geometric configurations involving a hexagonal lattice based equal circle packing in the Euclidian plane. The calculated sequences take…
This paper is devoted to confront two different approaches to the problem of dynam-ical perfect plasticity. Interpreting this model as a constrained boundary value Friedrichs' system enables one to derive admissible hyperbolic boundary…