Symmetry induced by economy

AbstractVarious extremum problems are presented which lead to highly symmetric geometrical configurations

Toward the Jamming Threshold of Sphere Packings: Tunneled Crystals

We have discovered a new family of three-dimensional crystal sphere packings that are strictly jammed (i.e., mechanically stable) and yet possess an anomalously low density. This family constitutes an uncountably infinite number of crystal packings that are subpackings of the densest crystal packings and are characterized by a high concentration of self-avoiding "tunnels" (chains of vacancies) that permeate the structures. The fundamental geometric characteristics of these tunneled crystals command interest in their own right and are described here in some detail. These include the lattice vectors (that specify the packing configurations), coordination structure, Voronoi cells, and density fluctuations. The tunneled crystals are not only candidate structures for achieving the jamming threshold (lowest-density rigid packing), but may have substantially broader significance for condensed matter physics and materials science.Comment: 19 pages, 5 figure

NAP1 Modulates Binding of Linker Histone H1 to Chromatin and Induces an Extended Chromatin Fiber Conformation

NAP1 (nucleosome assembly protein 1) is a histone chaperone that has been described to bind predominantly to the histone H2A·H2B dimer in the cell during shuttling of histones into the nucleus, nucleosome assembly/remodeling, and transcription. Here it was examined how NAP1 interacts with chromatin fibers isolated from HeLa cells. NAP1 induced a reversible change toward an extended fiber conformation as demonstrated by sedimentation velocity ultracentrifugation experiments. This transition was due to the removal of the linker histone H1. The H2A·H2B dimer remained stably bound to the native fiber fragments and to fibers devoid of linker histone H1. This was in contrast to mononucleosome substrates, which displayed a NAP1-induced removal of a single H2A·H2B dimer from the core particle. The effect of NAP1 on the chromatin fiber structure was examined by scanning/atomic force microscopy. A quantitative image analysis of ∼36,000 nucleosomes revealed an increase of the average internucleosomal distance from 22.3 ± 0.4 to 27.6 ± 0.6 nm, whereas the overall fiber structure was preserved. This change reflects the disintegration of the chromatosome due to binding of H1 to NAP1 as chromatin fibers stripped from H1 showed an average nucleosome distance of 27.4 ± 0.8 nm. The findings suggest a possible role of NAP1 in chromatin remodeling processes involved in transcription and replication by modulating the local linker histone content

Equidistribution of the Fekete points on the sphere

The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently

Equidistribution of the Fekete points on the sphere

The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently

The strong thirteen spheres problem

The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag

Fluid/solid transition in a hard-core system

We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid/solid phase transition