标题： 点在球面上配置的下界和上界 显示英文标题

标题： Lower and upper bounds for configurations of points on a sphere

摘要： 我们提出了一种新的证明（基于谱分解）了 Sidelnikov~ 之前证明的关于单位球面上的框架势$\sum_{ij} \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell $的界，该单位球面位于$d$维空间中。 Sidelnikov 的界是加权和$\sum_{ij} f_i f_j \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell$的下界的特殊情况，其中$f_i>0$是与球面上每个点相关联的标量量，我们也使用谱分解证明了这一点。 此外，在三维空间中，再次使用谱分解，我们找到了$\sum_{ijk}^N \left[ \left( {\bf P}_i \times {\bf P}_j\right) \cdot {\bf P}_k \right]^2$的精确上界。 我们探讨了这些界的两个应用：首先，我们研究了对应于 Thomson 问题对于$N=972$的局部极小值点的点配置；其次，我们分析了三维体积内的各种点分布，其中定义了一个合适的加权和以满足特定的界。 显示英文摘要

摘要： We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov~\, for the frame potentials $\sum_{ij} \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell $ on a unit--sphere in $d$ dimensions. Sidelnikov's bound is a special case of the lower bound for the weighted sums $\sum_{ij} f_i f_j \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell$, where $f_i>0$ are scalar quantities associated to each point on the sphere, which we also prove using spectral decomposition. Moreover, in three dimensions, again using spectral decomposition, we find a sharp upper bound for $\sum_{ijk}^N \left[ \left( {\bf P}_i \times {\bf P}_j\right) \cdot {\bf P}_k \right]^2$. We explore two applications of these bounds: first, we examine configurations of points corresponding to the local minima of the Thomson problem for $N=972$; second, we analyze various distributions of points within a three-dimensional volume, where a suitable weighted sum is defined to satisfy a specific bound.