This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to… Expand

We investigate the energy of arrangements of N points on the surface of the unit sphere Sd in Rd+1 that interact through a power law potential V = 1/rs, where s > 0 and r is Euclidean distance. With… Expand

It is proved that point sets that maximize the sum of suitable powers of the Euclidean distance between pairs of points form a sequence of QMC designs for H^s(S^d) with $s\in(d/2, d/2+1)$.Expand

Given a positive measure Σ with gs > 1, we write Με ℳΣ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight… Expand

We investigate the energy of arrangements of N points on the surface of a sphere in R3, interacting through a power law potential V = rα, −2 < α < 2, where r is Euclidean distance. For α = 0, we take… Expand

Let f(z) = z + "L2akzk be analytic and univalent in the unit disk E: \z\ < 1 and map the disk onto a domain which is convex in the direction of the imaginary axis. We show by example that for V2 -1 <… Expand

1186 NOTICES OF THE AMS VOLUME 51, NUMBER 10 T here are a variety of needs for the discretization of a manifold—statistical sampling, quadrature rules, starting points for Newton’s method,… Expand

For the extremal problem: E„r(a):= min||exp(-W«)(x-+ ■■■)\\L„ a > 0, where U (0 < r < oo) denotes the usual integral norm over R, and the minimum is taken over all monic polynomials of degree n, we… Expand