Energy bounds for weighted spherical codes and designs via linear programming
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912160954712064 |
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| author | Borodachov, Sergiy Boyvalenkov, Peter Dragnev, Peter Hardin, Douglas Saff, Edward Stoyanova, Maya |
| author_facet | Borodachov, Sergiy Boyvalenkov, Peter Dragnev, Peter Hardin, Douglas Saff, Edward Stoyanova, Maya |
| contents | Universal bounds for the potential energy of weighted spherical codes are obtained by linear programming. The universality is in the sense of Cohn-Kumar -- every attaining code is optimal with respect to a large class of potential functions (absolutely monotone), in the sense of Levenshtein -- there is a bound for every weighted code, and in the sense of parameters (nodes and weights) -- they are independent of the potential function. We derive a necessary condition for optimality (in the linear programming framework) of our lower bounds which is also shown to be sufficient when the potential is strictly absolutely monotone. Bounds are also obtained for the weighted energy of weighted spherical designs. We explore our bounds for several previously studied weighted spherical codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Energy bounds for weighted spherical codes and designs via linear programming Borodachov, Sergiy Boyvalenkov, Peter Dragnev, Peter Hardin, Douglas Saff, Edward Stoyanova, Maya Metric Geometry Combinatorics 52C17, 31C20 Universal bounds for the potential energy of weighted spherical codes are obtained by linear programming. The universality is in the sense of Cohn-Kumar -- every attaining code is optimal with respect to a large class of potential functions (absolutely monotone), in the sense of Levenshtein -- there is a bound for every weighted code, and in the sense of parameters (nodes and weights) -- they are independent of the potential function. We derive a necessary condition for optimality (in the linear programming framework) of our lower bounds which is also shown to be sufficient when the potential is strictly absolutely monotone. Bounds are also obtained for the weighted energy of weighted spherical designs. We explore our bounds for several previously studied weighted spherical codes. |
| title | Energy bounds for weighted spherical codes and designs via linear programming |
| topic | Metric Geometry Combinatorics 52C17, 31C20 |
| url | https://arxiv.org/abs/2403.07457 |