Related papers: On the Minimax Spherical Designs
This survey discusses recent developments in the context of spherical designs and minimal energy point configurations on spheres. The recent solution of the long standing problem of the existence of spherical $t$-designs on $\mathbb{S}^d$…
In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions…
We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m…
Uniformly distributed point sets on the unit sphere with and without symmetry constraints have been found useful in many scientific and engineering applications. Here, a novel variant of the Thomson problem is proposed and formulated as an…
We study an optimal M-partition problem for the Yamabe equation on the round sphere, in the presence of some particular symmetries. We show that there is a correspondence between solutions to this problem and least-energy sign-changing…
We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their approximation properties within frame theory.…
In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In…
In the present paper, we consider the majorization theorem (also known as Karamata's inequality) and the respective minima of the majorization (the so-called M-sets) for f-energy potentials of $m$-point configurations on the unit sphere. In…
We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional $Q$-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of…
We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has…
This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point…
We describe several randomized collections of $3\times 3$ rotation matrices and analyze their associated logarithmic energy. The best one (i.e. the one attaining the lowest expected logarithmic energy) is constructed by choosing $r$…
We consider a free energy functional defined on probability densities on the unit sphere $\mathbb{S}^d$, and investigate its global minimizers. The energy consists of two components: an entropy and a nonlocal interaction energy, which…
In this paper, we mainly consider the problem of spherical distribution of 5 points, that is, how to configure 5 points on a sphere such that the mutual distance sum attains the maximum. It is conjectured that the sum of distances is…
The consideration of the N-body gravitational problem equations can give to us some class of boundary-value problems defined on the "beem's" construction. One can considere it as weak or so-called finite element method's approximation with…
This paper provides a survey of spherical designs and their applications, with a particular emphasis on the perspective of ``numerical analysis''. A set \(X_N\) of \(N\) points on the unit sphere \(\mathbb{S}^d\) is called a…
This paper presents a novel way to approximate a distribution governing a system of coupled particles with a product of independent distributions. The approach is an extension of mean field theory that allows the independent distributions…
We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner…
We consider the extremal pointset configuration problem of maximizing a kernel-based energy subject to the geometric constraints that the points are contained in a fixed set, the pairwise distances are bounded below, and that every closed…
We examined energy spectrums of some particular systems of binary spins. It is shown that the configuration space can be divided into classes, and in the limit the energy distributions in these classes can be approximated by the normal…