Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userKissing number in hyperbolic space
109
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in $mathbb{H}^n$, for $ngeq 2$. For that purpose, the kissing number is replaced by the kissing function $kappa(n, r)$ which depends on the radius $r$. After we obtain some theoretical lower and upper bounds for $kappa(n, r)$, we study their asymptotic behaviour and show, in particular, that $lim_{rto infty} frac{log kappa(n,r)}{r} = n-1$. Finally, we compare them with the numeric upper bounds obtained by solving a suitable semidefinite program.
rate research
Read More
This paper investigates the behaviour of the kissing number $kappa(n, r)$ of congruent radius $r > 0$ spheres in $mathbb{S}^n$, for $ngeq 2$. Such a quantity depends on the radius $r$, and we plot the approximate graph of $kappa(n, r)$ with relatively high accuracy by using new upper and lower bounds that are produced via semidefinite programming and by using spherical codes, respectively.
In this note, we give a short solution of the kissing number problem in dimension three.
The average kissing number of $mathbb{R}^n$ is the supremum of the average degrees of contact graphs of packings of finitely many balls (of any radii) in $mathbb{R}^n$. We provide an upper bound for the average kissing number based on semidefinite programming that improves previous bounds in dimensions $3, ldots, 9$. A very simple upper bound for the average kissing number is twice the kissing number; in dimensions $6, ldots, 9$ our new bound is the first to improve on this simple upper bound.
This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in hyperbolic $mathbb{H}^n$ and spherical $mathbb{S}^n$ spaces, for $ngeq 2$. For that purpose, the kissing number is replaced by the kissing function $kappa_H(n, r)$, resp. $kappa_S(n, r)$, which depends on the dimension $n$ and the radius $r$. After we obtain some theoretical upper and lower bounds for $kappa_H(n, r)$, we study their asymptotic behaviour and show, in particular, that $kappa_H(n,r) sim (n-1) cdot d_{n-1} cdot B(frac{n-1}{2}, frac{1}{2}) cdot e^{(n-1) r}$, where $d_n$ is the sphere packing density in $mathbb{R}^n$, and $B$ is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of $kappa_S(n, r)$, for $n= 3,, 4$, over subintervals in $[0, pi]$ with relatively high accuracy.
In 1969, Fejes Toth conjectured that in Euclidean 3-space any packing of equal balls such that each ball is touched by twelve others consists of hexagonal layers. This article verifies this conjecture.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
