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

Published online by Cambridge University Press:  20 November 2018

S. V. Borodachov
S. V. Borodachov*
Affiliation:
Department of Mathematics, Towson University, 8000 York Road, Towson, MD, 21252, USA email: sborodachov@towson.edu
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Abstract

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We consider the problem of minimizing the energy of $N$ points repelling each other on curves in ${{\mathbb{R}}^{d}}$ with the potential ${{\left| x\,-\,y \right|}^{-s}},\,s\,\ge \,1$, where $\left| \cdot \right|$ 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 $s\,\ge \,2$, the minimal pairwise distance in optimal configurations asymptotically equals $L/N,\,N\to \,\infty $, where $L$ is the length of the curve.

Keywords

31C2065D17minimal discrete Riesz energylower order termpower law potentialseparation radius
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 1 , 01 February 2012 , pp. 24 - 43
Copyright
Copyright © Canadian Mathematical Society 2012

References

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