Related papers: Log-optimal (d+2)-configurations in d-dimensions

In the Minimum $d$-Dimensional Arrangement Problem (d-dimAP) we are given a graph with edge weights, and the goal is to find a 1-1 map of the vertices into $\mathbb{Z}^d$ (for some fixed dimension $d\geq 1$) minimizing the total weighted…

Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…

Motivated by the construction of confidence intervals in statistics, we study optimal configurations of $2^d-1$ lines in real projective space $RP^{d-1}$. For small $d$, we determine line sets that numerically minimize a wide variety of…

Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of $n$ points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltr\'an and…

We prove existence and uniqueness of a solution to the problem of minimizing the logarithmic energy of vector potentials associated to a $d$-tuple of positive measures supported on closed subsets of the complex plane. The assumptions we…

An ordinary hypersphere of a set of points in real $d$-space, where no $d+1$ points lie on a $(d-2)$-sphere or a $(d-2)$-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly $d+1$ points of the set.…

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of…

Planar central configurations can be seen as critical points of the reduced potential or solutions of a system of equations. By the homogeneity and invariance of the potential with respect to SO(2), it is possible to see that the…

This paper discusses the regularity of multiple-valued Dirichlet minimizing maps into the sphere. It shows that even at branched point, as long as the normalized energy is small enough, we have the energy decay estimate. Combined with the…

For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Gr\"unbaum, for these values of $m$.…

We consider a variational model for periodic partitions of the upper half-space into three regions, where two of them have prescribed volume and are subject to the geometrical constraint that their union is the subgraph of a function, whose…

We explore universal critical behavior in models with two competing order parameters, and an O(N)+O(M) symmetry for dimensions $d \leq 3$. In d=3, there is always exactly one stable Renormalization Group fixed point, corresponding to…

In this note, we prove that for homogeneous polynomial optimization on the sphere, if the objective $f$ is generic in the input space, all feasible points satisfying the first order and second order necessary optimality conditions are local…

We consider the problem of finding an $N$-point configuration on the sphere $S^d\subset \RR^{d+1}$ with the smallest absolute maximum value over $S^d$ of its total potential. The potential induced by each point ${\bf y}$ in a given…

We show that in all homotopy classes of mappings from complex projective space to Riemannian manifolds, the infimum of the energy is proportional to the infimal area in the homotopy class of mappings of the 2-sphere which represents the…

Let $R$ be a regular ring of dimension $d$ and $L$ be a $c$-divisible monoid. If ${K}_1{Sp}(R)$ is trivial and $k \geq d+2,$ then we prove that the symplectic group ${Sp}_{2k}(R[L])$ is generated by elementary symplectic matrices over…

We establish a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (abbreviated as a $d$-polytope) with $2d+2$ vertices, extending the previously known case for $k=1$. We identify all…

We are interested in algebraic properties of empty lattice simplices $\Delta$, that is, $d$-dimensional lattice polytopes containing exactly $d+1$ points of the integer lattice $\mathbb{Z}^d$. The cyclicity rank of $\Delta$ is the minimal…

We consider the semilinear problem \[ \Delta u = \lambda_+ \left(-\log u^+\right) 1_{\{u > 0\}} - \lambda_- \left(-\log u^- \right) 1_{\{u < 0\}} \qquad \hbox{ in } B_1, \] where $B_1$ is the unit ball in $\mathbb{R}^n$ and assume…

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…