Asymptotic Properties of Discrete Minimal s,log^t-Energy Constants and Configurations

2021/04/26 by Loesatapornpipit, Nichakan, Bosuwan, Nattapong
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2104.12607

Abstract

Combining the ideas of Riesz s-energy and log-energy, we introduce the so-called s,log^t-energy. In this paper, we investigate the asymptotic behaviors for N,t fixed and s varying of minimal N-point s,log^t-energy constants and configurations of an infinite compact metric space of diameter less than 1. In particular, we study certain continuity and differentiability properties of minimal N-point s,log^t-energy constants in the variable s and we show that in the limits as s→ ∞ and as s→ s₀>0, minimal N-point s,log^t-energy configurations tend to an N-point best-packing configuration and a minimal N-point s₀,log^t-energy configuration, respectively. Furthermore, the optimality of N distinct equally spaced points on circles in ℝ² for some certain s,log^t energy problems was proved.

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