We consider the s-energy E(Zn; s) = ∑ i6=j K(‖zi,n − zj,n‖; s) for point sets Zn = {zk,n : k = 0, . . . , n} on certain compact sets Γ in R having finite one-dimensional Hausdorff measure, where K(t; s) = { t−s, if s > 0, − ln t, if s = 0, is the Riesz kernel. Asymptotics for the minimum s-energy and the distribution of minimizing sequences of points is… (More)

We investigate bounds for point energies, separation radius, and mesh norm of certain arrangements of N points on sets A from a class A of d-dimensional compact sets embedded in R ′ , 1 ≤ d ≤ d′. We assume that these points interact through a Riesz potential V = | · |−s, where s > 0 and | · | is the Euclidean distance in R ′ . With δ∗ s (A,N) and ρ ∗ s(A,N)… (More)

In this paper, we announce and survey recent results on (1) point energies, scar defects, separation and mesh norm for optimal N ≥ 1 arrangments of points on a class of d-dimensional compact sets embedded in R, n ≥ 1, which interact through a Riesz potential, and (2) discrepancy estimates of numerical integration on the d-dimensional unit sphere S, d ≥ 2.