Lower and upper bounds for configurations of points on a sphere
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909419374116864 |
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| author | Amore, Paolo Sáenz, Ricardo A. |
| author_facet | Amore, Paolo Sáenz, Ricardo A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05800 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower and upper bounds for configurations of points on a sphere Amore, Paolo Sáenz, Ricardo A. Mathematical Physics Soft Condensed Matter 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. |
| title | Lower and upper bounds for configurations of points on a sphere |
| topic | Mathematical Physics Soft Condensed Matter |
| url | https://arxiv.org/abs/2412.05800 |