Browsing by Author "Saff, E.B."

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    Mesh ratios for best-packing and limits of minimal energy configurations
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Bondarenko, A.V.; Hardin, D.P.; Saff, E.B.
    For N-point best-packing configurations ωN on a compact metric space (A,ρ), we obtain estimates for the mesh-separation ratio γ(ρN,A), which is the quotient of the covering radius of ωN relative to A and the minimum pairwise distance between points in ωN . For best-packing configurations ωN that arise as limits of minimal Riesz s-energy configurations as s→∞, we prove that γ(ωN,A)≤1 and this bound can be attained even for the sphere. In the particular case when N=5 on S1 with ρ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid ω∗5, that is the limit (as s→∞) of 5-point s-energy minimizing configurations. Moreover, γ(ω∗5,S2)=1.
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    Minimal Riesz energy on the sphere for axis-supported external fields
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Brauchart, J.S.; Dragnev, P.D.; Saff, E.B.
    We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere Sd in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y|−s with d−2 s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on Sd is determined. The special case s = d − 2 yields interesting phenomena, which we investigate in detail. A weak∗ asymptotic analysis is provided as s ! (d − 2)+.