Related papers: The Phase Transition in 5 Point Energy Minimizatio…
This work rigorously verifies the phase transition in 5-point energy minimization first observed by Melnyk-Knop-Smith in 1977. More precisely, we prove that there is a constant S = [15+24/512,15+25/512] such that the triangular bi-pyramid…
Combining a brilliant obserbation of A. Tumanov with our computational approach to Thomson's 5-electron problem, we prove that the triangular bi-pyramid is the unique global minimizer for the Rieze potential R_s(r) = sign(s) r^{-s} amongst…
We give a rigorous computer-assisted proof that the triangular bi-pyramid is the unique configuration of 5 points on the 2-sphere that globally minimizes the Coulomb (1/r) potential. We also prove the same result for the (1/r^2) potential.…
We use moment techniques to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate (from below) the infinite dimensional optimization problems in this…
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where…
We study probability measures that minimize the Riesz energy with respect to the geodesic distance $\vartheta (x,y)$ on projective spaces $\mathbb{FP}^d$ (such energies arise from the 1959 conjecture of Fejes T\'oth about sums of non-obtuse…
Consider n points on the unit 2-sphere. The potential of the interaction of two points is a function f(r) of the distance r between the points. The total energy E of n points is the sum of the pairwise energies. The question is how to place…
Given $N$ points $X=\{x_k\}_{k=1}^N$ on the unit circle in $\mathbb{R}^2$ and a number $0\leq p \leq \infty$ we investigate the minimizers of the functional $\sum_{k, \ell =1}^N |\langle x_k, x_\ell\rangle|^p$. While it is known that each…
For the unit sphere S^d in Euclidean space R^(d+1), we show that for d-1<s<d and any N>1, discrete N-point minimal Riesz s-energy configurations are well separated in the sense that the minimal distance between any pair of distinct points…
We study a variational problem for the Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions arising in 2D micromagnetics, focusing on the Bogomol'nyi regime. We first determine the minimal energy for arbitrary topological…
In this paper, we study the relaxed energy for biharmonic maps from a $m$-dimensional domain into spheres. By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer…
The Riesz $s$-energy of an $N$-point configuration in the Euclidean space $\mathbb{R}^{p}$ is defined as the sum of reciprocal $s$-powers of all mutual distances in this system. In the limit $s\to0$ the Riesz $s$-potential $1/r^s$ ($r$ the…
We study sets of $N$ points on the $d-$dimensional torus $\mathbb{T}^d$ minimizing interaction functionals of the type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i - x_j)}. \] The main result states that for a class of functions $f$ that…
We are interested in studying the stationary solutions and phase transitions of aggregation equations with degenerate diffusion of porous medium-type, with exponent $1 < m < \infty$. We first prove the existence of possibly infinitely many…
Combining the ideas of Riesz $s$-energy and $\log$-energy, we introduce the so-called $s,\log^t$-energy. In this paper, we investigate the asymptotic behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point $s,\log^t$-energy constants…
We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S^d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that…
In this paper we study the existence of minimizers for interaction energies with the presence of external potentials. We consider a class of subharmonic interaction potentials, which include the Riesz potentials $|{\bf…
Smale's 7-th problem concerns N-point configurations on the 2-dim sphere which minimize the logarithmic pair-energy V_0(r) = -ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a…
The construction of an effective 3D theory at high temperatures for the MSSM as a model of electroweak baryogenesis is discussed. The analysis for a single light scalar field shows, that given the experimental constraints, there is no value…
Considering the quintom model with arbitrary potential, it is shown that there always exists a solution which evolves from w > -1 region to w < -1 region. The problem is restricted to the slowly varying potentials, i.e. the slow-roll…