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Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

  Published:2011-09-15
 Printed: Feb 2012
  • S. V. Borodachov,
    Department of Mathematics, Towson University, 8000 York Road, Towson, MD, 21252, USA
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Abstract

We consider the problem of minimizing the energy of $N$ points repelling each other on curves in $\mathbb{R}^d$ with the potential $|x-y|^{-s}$, $s\geq 1$, where $|\, \cdot\, |$ is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal $s$-energy. On our way, we also prove that at least for $s\geq 2$, the minimal pairwise distance in optimal configurations asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length of the curve.
Keywords: minimal discrete Riesz energy, lower order term, power law potential, separation radius minimal discrete Riesz energy, lower order term, power law potential, separation radius
MSC Classifications: 31C20, 65D17 show english descriptions Discrete potential theory and numerical methods
Computer aided design (modeling of curves and surfaces) [See also 68U07]
31C20 - Discrete potential theory and numerical methods
65D17 - Computer aided design (modeling of curves and surfaces) [See also 68U07]
 

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