Includes bibliographical references and index. (alk. paper) 1. Potential theory (Mathematics) I.

he problem of distributing a large number of points uniformly over the surface of a tsphere has not only inspired mathematical researchers, it has the attracted attention ~f:/t/~176 ::t:u;hifi;l:: ::/i~:l :i~t;~ ogous problem is simply that of uniformly distributing points on the circumference of a disk, and equally spaced points provide an obvious answer. So we are faced with this question: What sets of points on the sphere imitate the role of the roots of unity on the unit circle? One way such points can be generated is via optimization with respect to a suitable criterion such as "generalized energy." Although there is a large and growing literature concerning such optimal spherical configurations of N points when N is "small," here we shall focus on this question from an asymptotic perspective (N--~ ~).The discovery of stable carbon-60 molecules (Kroto, et al., 1985)* with atoms arranged in a spherical (soccer ball) pattern has had a considerable influence on current scientific pursuits. The study of this C60 buckminsterfifllerene also has an elegant mathematical component, revealed by F.R.K. Chung, B. Kostant, and S. Sternberg [5]. Now the search is on for much larger stable carbon molecules! Although such molecules are not expected to have a strictly spherical structure (due to bonding constraints), the construction of large stable configurations of spherical points is of interest here, as an initial step in hypothesizing more complicated molecular net structures.In electrostatics, locating identical point charges on the sphere so that they are in equilibrium with respect to a Coulomb potential law is a challenging problem, sometimes referred to as the dual problem for stable molecules.Certainly, uniformly distributing many points on the sphere has important applications to the field of computation. Indeed, quadrature formulas rely on appropriately chosen sampled data-points in order to approximate area integrals by taking averages in these points. Another example arises in the study of computational complexity, where M. Shub and S. Smale [22] encountered the problem of determining spherical points that maximize the product of their mutual distances.These various points of view clearly lead to different extremal conditions imposed on the distribution of N points. Except for some special values of N (e.g., N = 2, 3, 6, 12, 24) these various conditions yield different optimal configurations. However, and this is the main theme of this article, the general pattern for optimal configurations is the same: points (for N large) appear to arrange themselves according to a hexagonal pattern that is slightly perturbed in order to fit on the sphere.To make this more precise, we introduce some notation. We denote by S 2 the unit sphere in the Euclidean space R3: S 2 = {x E R 3 : Jxt = 1}.Lebesgue (area) measure on S 2 is denoted by ~, so that *

Abstract. We investigate the energy of arrangements of N points on the surface of a sphere in R 3 , interacting through a power law potential V = r α , −2 < α < 2, where r is Euclidean distance. For α = 0, we take V = log(1/r). An area-regular partitioning scheme of the sphere is devised for the purpose of obtaining bounds for the extremal (equilibrium) energy for such points. For α = 0, finer estimates are obtained for the dominant terms in the minimal energy by considering stereographical projections on the plane and analyzing certain logarithmic potentials. A general conjecture on the asymptotic form (as N → ∞) of the extremal energy, along with its supporting numerical evidence, is presented. Also we introduce explicit sets of points, called "generalized spiral points", that yield good estimates for the extremal energy. At least for N ≤ 12, 000 these points provide a reasonable solution to a problem of M. Shub and S. Smale arising in complexity theory.

Abstract. We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space H s (S d ) with smoothness parameter s > d/2 defined over the unit sphere S d in R d+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 of QMC designs: these are sequences of N -point configurations X N on S d such that the worst-case error satisfieswith an implied constant that depends on theHere σ d is the normalized surface measure on S d . We provide methods for generation and numerical testing of QMC designs. An essential tool is an expression for the worst-case error in terms of a reproducing kernel for the space H s (S d ) with s > d/2. As a consequence of this and a recent result of Bondarenko et al. on the existence of spherical designs with appropriate number of points, we show that minimizers of the N -point energy for this kernel form a sequence of QMC designs for H s (S d ). Furthermore, without appealing to the Bondarenko et al. result, we prove 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 the interval (d/2, d/2 + 1). For such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order Quasi Monte Carlo integration.Numerical experiments suggest that many familiar sequences of point sets on the sphere (equal area points, spiral points, minimal [Coulomb or logarithmic] energy points, and Fekete points) are QMC designs for appropriate values of s. For comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute QMC designs for any s > d/2.If (X N ) is a sequence of QMC designs for H s (S d ), we prove that it is also a sequence of QMC designs for H s ′ (S d ) for all s ′ ∈ (d/2, s). This leads to the question of determining the supremum of such s, for which we provide estimates based on computations for the aforementioned sequences.

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Abstract. We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V = 1/r s , where s > 0 and r is Euclidean distance. With E d (s, N ) denoting the minimal energy for such N -point arrangements we obtain bounds (valid for all N ) for E d (s, N ) in the cases when 0 < s < d and 2 ≤ d < s. For s = d, we determine the precise asymptotic behavior ofAs a corollary, lower bounds are given for the separation of any pair of points in an N -point minimal energy configuration, when s ≥ d ≥ 2.For the unit sphere in R 3 (d = 2), we present two conjectures concerning the asymptotic expansion of E 2 (s, N ) that relate to the zeta function ζ L (s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of ζ L (s) when 0 < s < 2 (the divergent case).

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Abstract. Given a closed d-rectifiable set A embedded in Euclidean space, we investigate minimal weighted Riesz energy points on A; that is, N points constrained to A and interacting via the weighted power law potential V = w(x, y) |x − y| −s , where s > 0 is a fixed parameter and w is an admissible weight. (In the unweighted case (w ≡ 1) such points for N fixed tend to the solution of the best-packing problem on A as the parameter s → ∞.) Our main results concern the asymptotic behavior as N → ∞ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, our results provide a method for generating N -point configurations on A that are "well-separated" and have asymptotic distribution ρ(x) as N → ∞.