Related papers: Log-optimal configurations on the sphere

In this paper, we study the existence of an optimal strategy for the stochastic control of diffusion in general case and a saddle-point for zero-sum stochastic differential games. The problem is formulated as an extended BSDE with…

Consider a population of $N$ individuals, each having $d\geq 1$ different traits, and an additive measure, called dispersion, which rewards large pairwise separations between traits. The goal is to select $M\leq N$ individuals such that…

Let $U_1,\ldots,U_n$ be independent random vectors uniformly distributed on the unit sphere $\mathbb S^{d-1}\subseteq\mathbb R^d$, where $n\ge d$, and consider the random polyhedral cone \[ \mathcal W_{n,d}:=\mathop{\mathrm{pos}}…

Using a global optimization approach that directly searches for the composition of greatest stability, we have been able to find the particularly stable structures for binary Lennard-Jones clusters with up to 100 atoms for a range of…

Given $N$ unit points charges on the surface of a unit conducting sphere, what configuration of charges minimizes the Coulombic energy $\sum_{i>j=1}^N 1/r_{ij}$? Due to an exponential rise in good local minima, finding global minima for…

We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap $\mathbb{L}_2$-discrepancy of any $N$ points on the unit sphere $\mathbb S^d$ in $\mathbb{R}^{d+1}$,…

Networked discrete dynamical systems are often used to model the spread of contagions and decision-making by agents in coordination games. Fixed points of such dynamical systems represent configurations to which the system converges. In the…

Spherical t-designs are Chebyshev-type averaging sets on the d-sphere S^d which are exact for polynomials of degree at most t. This concept was introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the minimum possible size…

We present a generalization of a family of points on $\mathbb{S}^2$, the Diamond ensemble, containing collections of $N$ points on $\mathbb{S}^2$ with very small logarithmic energy for all $N\in\mathbb{N}$. We extend this construction to…

A solution manifold is the collection of points in a $d$-dimensional space satisfying a system of $s$ equations with $s<d$. Solution manifolds occur in several statistical problems including hypothesis testing, curved-exponential families,…

We revisit the moving least squares (MLS) approximation scheme on the sphere $\mathbb S^{d-1} \subset \mathbb R^d$, where $d>1$. It is well known that using the spherical harmonics up to degree $L \in \mathbb N$ as ansatz space yields for…

In this paper we study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev spaces…

In this paper, we provide $R$-estimators of the location of a rotationally symmetric distribution on the unit sphere of $\R^k$. In order to do so we first prove the local asymptotic normality property of a sequence of rotationally symmetric…

We use a dictionary between lattice point counting inside dilated d-dimensional ellipsoids (Euclidean counting) and counting of lifts of a closed horosphere that intersect a ball of increasing radius, to obtain two types of results.…

This paper develops a systematic and geometric theory of optimal quantization on the unit sphere $\mathbb S^2$, focusing on finite uniform probability distributions supported on the spherical surface - rather than on lower-dimensional…

We consider the determination of the optimal stationary singular stochastic control of a linear diffusion for a class of average cumulative cost minimization problems arising in various financial and economic applications of stochastic…

We consider optimal designs for general multinomial logistic models, which cover baseline-category, cumulative, adjacent-categories, and continuation-ratio logit models, with proportional odds, non-proportional odds, or partial proportional…

Selecting cost-effective optimal sensor configurations for subsequent inference of parameters in black-box stochastic systems faces significant computational barriers. We propose a novel and robust approach, modelling the joint distribution…

We investigate multiple scattering of scalar waves by an ensemble of $N$ resonant point scatterers in three dimensions. For up to $N = 21$ scatterers, we numerically optimize the positions of the individual scatterers, such as to maximize…

We use methods of approximation theory to find the absolute minima on the sphere of the potential of spherical $(2m-3)$-designs with a non-trivial index $2m$ that are contained in a union of $m$ parallel hyperplanes, $m\geq 2$, whose…