The “thirteen spheres problem”, also known as the “Gregory-Newton problem” is to determine the maximum number of three-dimensional spheres that can simultaneously touch a given sphere, where all the spheres have the same radius. The history of the problem goes back to a disagreement between Isaac Newton and David Gregory in 1694. Using a combination … Read more
This paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry. The goal is to find a configuration of a cluster or a molecule with minimum total energy. The minimization problems in this class of geometry molecular … Read more
We present large-scale optimization techniques to model the energy function that underlies the folding process of proteins. Linear Programming is used to identify parameters in the energy function model, the objective being that the model predict the structure of known proteins correctly. Such trained functions can then be used either for {\em ab-initio} prediction or … Read more
Let $S\subset[-1,1)$. A finite set $C=\{x_i\}_{i=1}^M\subset\Re^n$ is called a {\em spherical S-code} if $||x_i||=1$ for each $i$, and $x_i^T x_j\in S$, $i\ne j$. For $S=[-1,.5]$ maximizing $M=|C|$ is commonly referred to as the {\em kissing number} problem. A well-known technique based on harmonic analysis and linear programming can be used to bound $M$. We consider … Read more