164 research outputs found
Mesh ratios for best-packing and limits of minimal energy configurations
For -point best-packing configurations on a compact metric
space , we obtain estimates for the mesh-separation ratio
, which is the quotient of the covering radius of
relative to and the minimum pairwise distance between points in
. For best-packing configurations that arise as limits of
minimal Riesz -energy configurations as , we prove that
and this bound can be attained even for the sphere.
In the particular case when N=5 on with 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
, that is the limit (as ) of 5-point -energy
minimizing configurations. Moreover,
Quasi-uniformity of Minimal Weighted Energy Points on Compact Metric Spaces
For a closed subset of a compact metric space possessing an
-regular measure with , we prove that whenever
, any sequence of weighted minimal Riesz -energy configurations
on (for `nice' weights) is
quasi-uniform in the sense that the ratios of its mesh norm to separation
distance remain bounded as grows large. Furthermore, if is an
-rectifiable compact subset of Euclidean space ( an integer)
with positive and finite -dimensional Hausdorff measure, it is possible
to generate such a quasi-uniform sequence of configurations that also has (as
) a prescribed positive continuous limit distribution with respect
to -dimensional Hausdorff measure. As a consequence of our energy
related results for the unweighted case, we deduce that if is a compact
manifold without boundary, then there exists a sequence of -point
best-packing configurations on whose mesh-separation ratios have limit
superior (as ) at most 2
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Mesh ratios for best-packing and limits of minimal energy configurations
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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Optimal and Near Optimal Configurations on Lattices and Manifolds
Optimal configurations of points arise in many contexts, for example classical ground states for interacting particle systems, Euclidean packings of convex bodies, as well as minimal discrete and continuous energy problems for general kernels. Relevant questions in this area include the understanding of asymptotic optimal configurations, of lattice and periodic configurations, the development of algorithmic constructions of near optimal configurations, and the application of methods in convex optimization such as linear and semidefinite programming
Riesz external field problems on the hypersphere and optimal point separation
We consider the minimal energy problem on the unit sphere in
the Euclidean space in the presence of an external field
, where the energy arises from the Riesz potential (where is the
Euclidean distance and is the Riesz parameter) or the logarithmic potential
. Characterization theorems of Frostman-type for the associated
extremal measure, previously obtained by the last two authors, are extended to
the range The proof uses a maximum principle for measures
supported on . When is the Riesz -potential of a signed
measure and , our results lead to explicit point-separation
estimates for -Fekete points, which are -point configurations
minimizing the Riesz -energy on with external field . In
the hyper-singular case , the short-range pair-interaction enforces
well-separation even in the presence of more general external fields. As a
further application, we determine the extremal and signed equilibria when the
external field is due to a negative point charge outside a positively charged
isolated sphere. Moreover, we provide a rigorous analysis of the three point
external field problem and numerical results for the four point problem.Comment: 35 pages, 4 figure
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