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An intuitive method for distributing N points on a manifold A ⊂ Rd is to consider minimal s-energy arrangements of points that interact through a power law (Riesz) potential V = 1/r, where s > 0 and r is Euclidean distance in R 0 . Under what conditions will these points be “uniformly” distributed on A for large N? In this talk I will present recent results… (More)

- Edward B. Saff
- 2007

Given a compact d-rectifiable set A embedded in Euclidean space and a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, we address the following question: how can one generate optimal configurations of N points on A that are “well-separated” and have asymptotic distribution ρ(x) as N → ∞? For this purpose we investigate minimal… (More)

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ Rd′ that interact through the power law (Riesz) potential V = 1/r, where s > 0 and r is Euclidean distance in R ′ . With Es(A, N) denoting the minimal energy for such N -point configurations, we determine the asymptotic behavior (as N → ∞) of Es(A, N) for each… (More)

- A. Mart́ınez-Finkelshtein, V. Maymeskul, E. A. Rakhmanov, Edward B. Saff
- 2002

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)

- A. Mart́ınez-Finkelshtein, Edward B. Saff
- 2005

We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic expansion for these polynomials, valid uniformly in the whole complex plane. As a consequence, we obtain some results about… (More)

- Andrei Martínez-Finkelshtein, Edward B. Saff
- Journal of Approximation Theory
- 2002

We study the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lamé differential equations, when their coefficients satisfy certain asymptotic conditions. The limit distribution is described by an equilibrium measure in presence of an external field, generated by charges at the singular points of the equation. Moreover, a… (More)

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 describe the asymptotic form of the error E„ r(a) (as n -» oo) as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The… (More)

- Johann S. Brauchart, Edward B. Saff, Ian H. Sloan, Robert S. Womersley
- Math. Comput.
- 2014

We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space Hs(Sd) with smoothness parameter s > d/2 defined over the unit sphere Sd in Rd+1. Focusing on N-point configurations that achieve optimal order QMC error bounds (as is the case for efficient spherical designs), we are led to introduce the concept… (More)

- Antonio J. Durán, P. Lopez-Rodriguez, Edward B. Saff
- 1999

Weak-star asymptotic results are obtained for the zeros of orthogonal matrix polynomials (i.e. the zeros of their determinants) on R from two di®erent assumptions: ̄rst from the convergence of matrix coe±cients occurring in the three-term recurrence for these polynomials and, second, from some conditions on the generating matrix measure. The matrix… (More)

Recently developed scanning magnetic microscopes measure the magnetic field in a plane above a thin-plate magnetization distribution. These instruments have broad applications in geoscience and materials science, but are limited by the requirement that the sample magnetization must be retrieved from measured field data, which is a generically nonunique… (More)