Integrable Systems, Analysis And Probability Seminar
Thursday, April 8, 2004, 4:00pm, 120 Physics
Ed Saff (Vanderbilt University)
Discretizing Manifolds via Minimum Energy Points
Abstract:
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a large class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere and are related to best-packing problems. The talk is aimed at a general audience and should also be of interest to chemists, physicists, as well as biologists.

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