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Article

Asymptotic Properties of Discrete Minimal s,logt-Energy Constants and Configurations

by 1,†,‡ and 1,2,*,‡
1
Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
2
Centre of Excellence in Mathematics, Commission on Higher Education, Ministry of Education, Si Ayutthaya Road, Bangkok 10400, Thailand
*
Author to whom correspondence should be addressed.
Results of this article constitute part of Nichakan Loesatapornpipit’s senior project under the mentorship of Nattapong Bosuwan at Mahidol University.
These authors contributed equally to this work.
Academic Editors: Yongwimon Lenbury, Ravi P. Agarwal and Elvin Moore
Received: 18 April 2021 / Revised: 18 May 2021 / Accepted: 20 May 2021 / Published: 24 May 2021
(This article belongs to the Special Issue Modelling and Simulation of Natural Phenomena of Current Interest)
We investigated the energy of N points on an infinite compact metric space (A,d) of a diameter less than 1 that interact through the potential (1/ds)(log1/d)t, where s,t0 and d is the metric distance. With Elogts(A,N) denoting the minimal energy for such N-point configurations, we studied certain continuity and differentiability properties of Elogts(A,N) in the variable s. Then, we showed that in the limits, as s and as ss0>0,N-point configurations that minimize the s,logt-energy tends to an N-point best-packing configuration and an N-point configuration that minimizes the s0,logt-energy, respectively. Furthermore, we considered when A are circles in the Euclidean space R2. In particular, we proved the minimality of N distinct equally spaced points on circles in R2 for some certain s and t. The study on circles shows a possibility for the utilization of N points generated through such new potential to uniformly discretize on objects with very high symmetry. View Full-Text
Keywords: discrete minimal energy; best-packing; Riesz energy; logarithmic energy discrete minimal energy; best-packing; Riesz energy; logarithmic energy
MDPI and ACS Style

Loesatapornpipit, N.; Bosuwan, N. Asymptotic Properties of Discrete Minimal s,logt-Energy Constants and Configurations. Symmetry 2021, 13, 932. https://0-doi-org.brum.beds.ac.uk/10.3390/sym13060932

AMA Style

Loesatapornpipit N, Bosuwan N. Asymptotic Properties of Discrete Minimal s,logt-Energy Constants and Configurations. Symmetry. 2021; 13(6):932. https://0-doi-org.brum.beds.ac.uk/10.3390/sym13060932

Chicago/Turabian Style

Loesatapornpipit, Nichakan, and Nattapong Bosuwan. 2021. "Asymptotic Properties of Discrete Minimal s,logt-Energy Constants and Configurations" Symmetry 13, no. 6: 932. https://0-doi-org.brum.beds.ac.uk/10.3390/sym13060932

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