Related papers: Kissing number in hyperbolic space

We show that the temperature of a diffusing fluid with the diffusion constant \kappa^{2} in an expanding universe approaches a constant limit T=\kappa^{2}/H in its final de Sitter stage characterized by the horizon 1/H determined by the…

Let $X(D,1) =\Gamma(D,1) \backslash \mathbb{H}$ denote the Shimura curve of level $N=1$ arising from an indefinite quaternion algebra of fixed discriminant $D$. We study the discrete average of the error term in the hyperbolic circle…

We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0…

In this note, we give a short solution of the kissing number problem in dimension three.

An asymmetric binary covering code of length n and radius R is a subset C of the n-cube Q_n such that every vector x in Q_n can be obtained from some vector c in C by changing at most R 1's of c to 0's, where R is as small as possible.…

Let $\varepsilon > 0$. We prove that there are constants $m_0=m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon) > 0$ for which the following holds: For every integer $m > m_0$ and every nontrivial Dirichlet character modulo $m$, there are…

We study non-paraSasakian paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$ (equivalent to $h^2=0$ but $h\neq0$). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of…

Let $\psi $ be a conformal map on $\mathbb{D}$ with $ \psi \left( 0 \right)=0$ and let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$ for $\alpha >0$. Denote by ${H^p}\left( \mathbb{D}…

Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then…

We first consider {\it deterministic} immersions of the $d$-dimensional sphere into high dimensional Euclidean spaces, where the immersion is via spherical harmonics of level $n$. The main result of the article is the, a priori unexpected,…

In 2009, Roeglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multi-criteria optimization problems is polynomially bounded in the number $n$ of variables and the maximum density $\phi$ of the semi-random…

It is provided an overview of existed results concerning classification of contact metric, almost cosymplectic and almost Kenmotsu $(\kappa,\mu)$-manifolds. In the case of dimension three it is described in full details structure of contact…

If a real harmonic function inside the open unit disk $B(0,1) \subset \mathbb{R}^2$ has its level set $\left\{x: u(x) = u(0)\right\}$ diffeomorphic to an interval, then we prove the sharp bound $\kappa \leq 8$ on the curvature of the level…

Given any $d$-dimensional Lipschitz Riemannian manifold $(M,g)$ with heat kernel $\mathsf{p}$, we establish uniform upper bounds on $\mathsf{p}$ which can always be decoupled in space and time. More precisely, we prove the existence of a…

This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact…

Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on \textit{at most} a bits? Let 0\leq a\leq n-1. We show that any deterministic protocol, as well as any error-free quantum protocol (C*…

The problem of computing the cardinality of the intersection of multiple balls in the Hamming space has attracted a lot of attention recently due to their applications in the list reconstruction problem and information retrieval in…

We study the approximation of halfspaces $h:\{0,1\}^n\to\{0,1\}$ in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the "hardest" halfspace, for which we prove…

Let $\mathbb{R}^m$ be endowed with the Euclidean metric. The covering radius of a lattice $\Lambda \subset \mathbb{R}^m$ is the least distance $r$ such that, given any point of $\mathbb{R}^m$, the distance from that point to $\Lambda$ is…

It is known that the infimum of the sectional curvatures (on the regular part) of orbit spaces of isometric actions on unit spheres in bounded above by $4$. We show that the infimum is $1$ for "most" actions, and determine the cases in…