Related papers: Order and disorder in energy minimization
Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…
Why are simple, regular, and symmetric shapes common in nature? Many natural shapes arise as solutions to energy minimisation or other optimisation problems, but is there a general relation between optima and simple, regular shapes and…
We revisit the classical problem of optimal experimental design (OED) under a new mathematical model grounded in a geometric motivation. Specifically, we introduce models based on elementary symmetric polynomials; these polynomials capture…
During the last few years several new results on packing problems were obtained using a blend of tools from semidefinite optimization, polynomial optimization, and harmonic analysis. We survey some of these results and the techniques…
We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous…
Given an undirected graph representing similarities between a set of items and an additive measure evaluating the items, we treat the position of a special subset of items in an ordinal ranking through a collection of combinatorial…
We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function…
We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples of aperiodic tilings in Euclidean space…
The Bin Packing Problem involves efficiently packing items into a limited number of bins without exceeding their capacity. In this paper, we try to answer a specific question in this field. Mathematically the combinatorial optimization…
Three-graded root systems can be arranged into nested sequences. One exceptional sequence provides a natural means to recover some structures and symmetries familiar in the context of particle physics.
We formulate explicit predictions concerning the symmetry of optimal codes in compact metric spaces. This motivates the study of optimal codes in various spaces where these predictions can be tested.
A class of algorithms for the solution of discrete material optimization problems in electromagnetic applications is discussed. The idea behind the algorithm is similar to that of the sequential programming. However, in each major iteration…
Black holes are thought to describe the geometry of massive, dark compact objects in the universe. To further support and quantify this long-held belief requires knowledge of possible, if exotic alternatives. Here, we wish to understand how…
Distributing points on a (possibly high-dimensional) sphere with minimal energy is a long-standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and…
In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the…
We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In…
We discuss the optimal presentations of mathematical objects under well defined symbol libraries. We shall examine what light our chosen symbol libraries and syntax shed upon the objects they represent. A major part of this work will focus…
Many discrete minimization problems, including various versions of the shortest path problem, can be efficiently solved by dynamic programming (DP) algorithms that are "pure" in that they only perform basic operations, as min, max, +, but…
We analyze the asymptotic properties of a Euclidean optimization problem on the plane. Specifically, we consider a network with three bins and $n$ objects spatially uniformly distributed, each object being allocated to a bin at a cost…
We propose a general-purpose method for finding high-quality solutions to hard optimization problems, inspired by self-organizing processes often found in nature. The method, called Extremal Optimization, successively eliminates extremely…