Browsing by Author "Bondarenko, A.V."

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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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    On concentrators and related approximation constants
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Bondarenko, A.V.; Prymak, A.; Radchenko, D.
    Pippenger ([Pip77]) showed the existence of (6m; 4m; 3m; 6)-concentrator for each positive integer m using a probabilistic method. We generalize his approach and prove existence of (6m; 4m; 3m; 5:05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from 44:5 (established by Kalton and Roberts in [KR83]) to 39. We show a more direct connection of the latter problem to the Whitney type estimate for approximation of continuous functions on a cube in Rd by linear functions, and improve the estimate of this Whitney constant from 802 (proved by Brudnyi and Kalton in [BK00]) to 73.