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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets
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by S. V. Borodachov, D. P. Hardin and E. B. Saff PDF
Trans. Amer. Math. Soc. 360 (2008), 1559-1580 Request permission

Abstract:

Given a closed $d$-rectifiable set $A$ embedded in Euclidean space, we investigate minimal weighted Riesz energy points on $A$; that is, $N$ points constrained to $A$ and interacting via the weighted power law potential $V=w(x,y)\left |x-y\right |^{-s}$, where $s>0$ is a fixed parameter and $w$ is an admissible weight. (In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ 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 $d$-dimensional Hausdorff measure on $A$, our results provide a method for generating $N$-point configurations on $A$ that are “well-separated” and have asymptotic distribution $\rho (x)$ as $N\to \infty$.
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Additional Information
  • S. V. Borodachov
  • Affiliation: Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240
  • Address at time of publication: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
  • MR Author ID: 656604
  • Email: sergiy.v.borodachov@vanderbilt.edu, borodasv@math.gatech.edu
  • D. P. Hardin
  • Affiliation: Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240
  • MR Author ID: 81245
  • ORCID: 0000-0003-0867-2146
  • Email: doug.hardin@vanderbilt.edu
  • E. B. Saff
  • Affiliation: Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240
  • MR Author ID: 152845
  • Email: edward.b.saff@vanderbilt.edu
  • Received by editor(s): February 10, 2006
  • Published electronically: October 17, 2007
  • Additional Notes: The research of the first author was conducted as a graduate student under the supervision of E.B. Saff and D. P. Hardin at Vanderbilt University.
    The research of the second author was supported, in part, by the U. S. National Science Foundation under grants DMS-0505756 and DMS-0532154.
    The research of the third author was supported, in part, by the U. S. National Science Foundation under grant DMS-0532154.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 1559-1580
  • MSC (2000): Primary 11K41, 70F10, 28A78; Secondary 78A30, 52A40
  • DOI: https://doi.org/10.1090/S0002-9947-07-04416-9
  • MathSciNet review: 2357705