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Logarithmic Potentials with External Fields
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
Asymptotics for minimal discrete energy on the sphere
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
QMC designs: Optimal order Quasi Monte Carlo integration schemes on the sphere
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 $s\in(d/2, d/2+1)$.
Constrained energy problems with applications to orthogonal polynomials of a discrete variable
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
Minimal Discrete Energy on the Sphere
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
On univalent functions convex in one direction
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 <
Discretizing Manifolds via Minimum Energy Points
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,
Extremal problems for polynomials with exponential weights
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