Lower and upper bounds for configurations of points on a sphere
| Autor: | Amore, Paolo, Sáenz, Ricardo A. |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 22 pages, 6 figures |
| Databáze: | arXiv |
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