Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets

Authors:	S V Borodachov D P Hardin E B Saff

Institution:	Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240 ; Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240 ; Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240

Abstract:	Given a closed -rectifiable set embedded in Euclidean space, we investigate minimal weighted Riesz energy points on ; that is, points constrained to and interacting via the weighted power law potential $V=w(x,y)\left\vert x-y\right\vert^{-s}$ , where is a fixed parameter and is an admissible weight. (In the unweighted case ( $w\equiv 1$ ) such points for fixed tend to the solution of the best-packing problem on as the parameter $s\to \infty$ .) Our main results concern the asymptotic behavior as $N\to \infty$ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution $\rho(x)$ with respect to -dimensional Hausdorff measure on , our results provide a method for generating -point configurations on that are ``well-separated' and have asymptotic distribution $\rho (x)$ as $N\to \infty$ .

Keywords:	Minimal discrete Riesz energy best-packing Hausdorff measure rectifiable sets non-uniform distribution of points power law potential separation radius

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