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Upper Energy Bounds for Spherical Designs of Relatively Small Cardinalities

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Abstract

We derive upper bounds for the potential energy of spherical designs of cardinality close to the Delsarte–Goethals–Seidel bound. These bounds are obtained by linear programming with use of the Hermite interpolating polynomial of the potential function in suitable nodes. Numerical computations show that the results are quite close to certain lower energy bounds confirming that spherical designs are, in a sense, energy efficient.

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Notes

  1. Clearly, it is enough to have positive only the first three derivatives.

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Acknowledgements

The authors thank the anonymous referees for their useful remarks. Peter Boyvalenkov is also with Faculty of Engineering, South-Western University, Blagoevgrad, Bulgaria.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 G Bonchev Str., 1113, Sofia, Bulgaria

    Peter Boyvalenkov & Konstantin Delchev

  2. Département des sciences de l’ingénieur, École de Saint-Cyr Coëtquidan, 56381, Guer Cedex, France

    Matthieu Jourdain

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  1. Peter Boyvalenkov
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  2. Konstantin Delchev
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  3. Matthieu Jourdain
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Correspondence to Peter Boyvalenkov.

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The research of the first two authors was supported, in part, by Bulgarian NSF Contract DN02/2-2016.

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Boyvalenkov, P., Delchev, K. & Jourdain, M. Upper Energy Bounds for Spherical Designs of Relatively Small Cardinalities. Discrete Comput Geom 65, 244–260 (2021). https://doi.org/10.1007/s00454-019-00123-9

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  • DOI: https://doi.org/10.1007/s00454-019-00123-9

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