论文标题
某种类别的球形设计的潜力的绝对最小值
Absolute Minima of Potentials of a Certain Class of Spherical Designs
论文作者
论文摘要
我们使用线性编程技术在单位球体上找到绝对最小的点$ s^{d} $中的$ \ Mathbb r^{d+1} $，这是点配置的总潜力$ ch $ω_n\ subset s^{d} $，这是一个球形$（2M-1）$ - $（2M-1）$ - 均为$ m $ m m $ m m $ m m $ m m $ m m $ m m $ m。点之间的相互作用由内核$ k（{\ bf x}，{\ bf y}）= f（\ left | weft | {\ bf x} - {\ bf y} \ right |^2）$，$ \ \ \ \ \ \ \ \ cdot \ \！ r^{D+1} $。潜在函数$ f $假定具有凸导数$ f^{（2m-2）} $。最低点不取决于$ f $，并且是那些恰好形成$ m $ difters Dot产品的$ f $的点。该定理的证明是在2022年1月在ESI的一个研讨会上介绍的。使用此结果，我们在高维球体上找到了某些六个配置的通用最小值集。
We use linear programming techniques to find points of absolute minimum over the unit sphere $S^{d}$ in $\mathbb R^{d+1}$ of the total potential of a point configuration $ω_N\subset S^{d}$ which is a spherical $(2m-1)$-design contained in the union of some $m$ parallel hyperplanes. The interaction between points is described by the kernel $K({\bf x},{\bf y})=f(\left|{\bf x}-{\bf y}\right|^2)$, where $\left|\ \!\cdot\ \!\right|$ is the Euclidean norm in $\mathbb R^{d+1}$. The potential function $f$ is assumed to have a convex derivative $f^{(2m-2)}$. Points of minimum do not depend on $f$ and are those and only those which form exactly $m$ distinct dot products with points of $ω_N$. The proof of this theorem was presented at a workshop at ESI in January 2022. Using this result, we find sets of universal minima of certain six configurations on higher-dimensional spheres.
