Quasi-uniformity of Minimal Weighted Energy Points on Compact Metric Spaces
| Autor: | Douglas P. Hardin, Edward B. Saff, J. T. Whitehouse |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2011 |
| Předmět: |
Statistics and Probability
Control and Optimization General Mathematics Mesh–separation ratio FOS: Physical sciences 010103 numerical & computational mathematics 01 natural sciences Continuous functions on a compact Hausdorff space Combinatorics FOS: Mathematics Hausdorff measure Mathematics - Numerical Analysis 0101 mathematics Mathematical Physics 31C20 65N50 57N16 (Primary) 52A40 28A78 (Secondary) Mathematics Numerical Analysis Algebra and Number Theory Euclidean space Applied Mathematics 010102 general mathematics Mathematical Physics (math-ph) Numerical Analysis (math.NA) Minimal Riesz energy Covering radius Quasi-uniformity Metric space Compact space Hausdorff distance Norm (mathematics) Bounded function Separation distance Best-packing |
| Popis: | For a closed subset $K$ of a compact metric space $A$ possessing an $\alpha$-regular measure $\mu$ with $\mu(K)>0$, we prove that whenever $s>\alpha$, any sequence of weighted minimal Riesz $s$-energy configurations $\omega_N=\{x_{i,N}^{(s)}\}_{i=1}^N$ on $K$ (for `nice' weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as $N$ grows large. Furthermore, if $K$ is an $\alpha$-rectifiable compact subset of Euclidean space ($\alpha$ an integer) with positive and finite $\alpha$-dimensional Hausdorff measure, it is possible to generate such a quasi-uniform sequence of configurations that also has (as $N\to \infty$) a prescribed positive continuous limit distribution with respect to $\alpha$-dimensional Hausdorff measure. As a consequence of our energy related results for the unweighted case, we deduce that if $A$ is a compact $C^1$ manifold without boundary, then there exists a sequence of $N$-point best-packing configurations on $A$ whose mesh-separation ratios have limit superior (as $N\to \infty$) at most 2. |
| Databáze: | OpenAIRE |
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