D. P. Hardin

Periodic discrete energy for long-range potentials

We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to define the periodic energy corresponding to a large class of long-range potentials. Two particularly interesting examples are the logarithmic potential and the Riesz potential when the Riesz parameter is smaller than the dimension of the space. For these examples, we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the Epstein Hurwitz Zeta function. We apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large. We conclude with some detailed calculations in the case of one dimension, which shows the utility of this approach.

The support of the limit distribution of optimal Riesz energy points on sets of revolution in R3

Let A be a compact point set in the right half of the xy plane and Γ(A) the set in R3 obtained by rotating A about the y axis. We investigate the support of the limit distribution of minimal energy point charges on Γ(A) that interact according to the Riesz potential 1∕rs, 0<s<1, where r is the Euclidean distance between points. Potential theory yields that this limit distribution coincides with the equilibrium measure on Γ(A) which is supported on the outer boundary of Γ(A). We show that there are sets of revolution Γ(A) such that the support of the equilibrium measure on Γ(A) is not the complete outer boundary, in contrast to the Coulomb case s=1. However, the support of the limit distribution on the set of revolution Γ(R+A) as R goes to infinity is the full outer boundary for certain sets A, in contrast to the logarithmic case (s=0).

On spherical codes with inner products in a prescribed interval

We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval

Mesh ratios for best-packing and limits of minimal energy configurations

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Constrained minimum Riesz energy problems for a condenser with intersecting plates

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Asymptotics of weighted best-packing on rectifiable sets

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