Discrete logarithmic energy on the sphere

Abstract

In this article we consider the problem posed by Whyte, about the distribution of N point charges on the unit sphere, whose mutual distances have maximal geometric mean. Some properties of the extremal points are discussed. In the case when N = 5 the optimal configuration is established rigorously, which solves an open problem communicated by Rakhmanov.

Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points

“…These points are referred to as either logarithmic points [11] or elliptic Fekete points [39], with mathematical origins in [16]. For small values of N , such points can be found by detailed geometrical constructions (see [11] for N = 5 and [28] for N = 6). In particular, for N = 5, it was proved in [11] that the optimal configuration for p(x 1 , .…”

“…For small values of N , such points can be found by detailed geometrical constructions (see [11] for N = 5 and [28] for N = 6). In particular, for N = 5, it was proved in [11] that the optimal configuration for p(x 1 , . .…”

“…[11], [39], [31], and [32]). In section 4 we also formulate a transcendental equation that effectively sums the infinite order expansion in powers of −1/ log(εa) for the principal eigenvalue of the Laplacian in the presence of N partially absorbing traps.…”

“…This optimal configuration has a trap at each pole together with three traps forming an equilateral triangle on the equatorial plane (cf. [11]). For the inhomogeneous source distribution M 2 in (5.1) the mean first passage time u is maximal in the two belt-like regions along the φ-direction (Figure 4(a)), while for the homogeneous distribution M 1 it is maximal in the three rectangular regions between each of the two equatorial traps …”

Optimal $$N$$ N -Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

“…By traditional methods (see [DLT02]) it can be shown that the triangular bi-pyramid consisting of two antipodal points at, say, the North and the South Pole, and three equally spaced points on the Equator, is the unique (up to orthogonal transformation) minimizer of the logarithmic average pair-energy. The proof that the same configuration maximizes the sum of distances (that is: assumes v −1 (5)) is computer-aided, exploiting interval methods and related techniques (see [HoSh11]).…”