论文标题
恒定截面曲率的非欧几里得空间中的亲吻号
Kissing number in non-Euclidean spaces of constant sectional curvature
论文作者
论文摘要
本文提供了一致半径$ r> 0 $ spheres y Mathbb {h}^n $和球形$ \ mathbb {s}^n $ space的接吻数量的上限和下限。为此，接吻号被接吻函数$κ_H（n，r）$，resp。 $κ_s（n，r）$，取决于尺寸$ n $和半径$ r $。在获得$κ_H（n，r）$的一些理论上和下限后，我们研究了它们的渐近行为，尤其表明$κ_h（n，r）\ sim（n，n-1）\ cdot d_ {n-1} {n-1} \ cdot b（\ cdot b（\ frac {n-1}} {2} {2} {2} {2} {2} r} $，其中$ d_n $是$ \ mathbb {r}^n $中的球体包装密度，而$ b $是beta功能。然后，我们通过求解合适的半限定程序以及来自混凝土球形代码的下限来产生数字上限。类似的方法使我们能够以相对较高的精度找到$ n = 3，\，4 $的$κ_s（n，r）$的值（n，r）$，$ n = 3，\，4 $。
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 $n\geq 2$. For that purpose, the kissing number is replaced by the kissing function $κ_H(n, r)$, resp. $κ_S(n, r)$, which depends on the dimension $n$ and the radius $r$. After we obtain some theoretical upper and lower bounds for $κ_H(n, r)$, we study their asymptotic behaviour and show, in particular, that $κ_H(n,r) \sim (n-1) \cdot d_{n-1} \cdot B(\frac{n-1}{2}, \frac{1}{2}) \cdot e^{(n-1) r}$, 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 $κ_S(n, r)$, for $n= 3,\, 4$, over subintervals in $[0, π]$ with relatively high accuracy.
