Related papers: How dense can one pack spheres of arbitrary size d…
We call a packing of hyperspheres in $n$ dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the $n$-dimensional space; and every sphere in the packing is a member of a cluster of $n+2$…
In this paper we present a new algorithm for a layout optimization problem: this concerns the placement of weighted polygons inside a circular container, the two objectives being to minimize imbalance of mass and to minimize the radius of…
Various solutions of the kinetic equation for the equilibrium of a gravitating sphere of uniform density with a quadratic gravitational potential and a linear dependence of gravitational force on radius are examined. New analytic solutions…
The goal of random sequential adsorption (RSA), a time-dependent packing method, is to create a regular or asymmetric covering of an empty space that can fit in the allocated space without overlapping. The density of coverage tends to reach…
This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including…
Accurate quantification of local packing density and mixing in simulations of particulate systems is essential for many industrial applications. Traditional methods which simply count the number of particle centres within a given volume of…
In this paper we present a theoretical analysis of the deterministic on-line {\em Sum of Squares} algorithm ($SS$) for bin packing introduced and studied experimentally in \cite{CJK99}, along with several new variants. $SS$ is applicable to…
Unraveling the complexities of random packing in three dimensions has long puzzled physicists. While both experiments and simulations consistently show a maximum density of 64 percent for tightly packed random spheres, we still lack an…
We give an asymptotic approximation scheme (APTAS) for the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height $1+\gamma$, for some arbitrarily small…
We consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in $\mathbb{R}^d$, its reversible probability measure, and its projection on the subset of the large spheres. The main feature is…
An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of $R^3$ into polyhedra. The polyhedra are divided…
Motivated by the analysis of range queries in databases, we introduce the computation of the Depth Distribution of a set $\mathcal{B}$ of axis aligned boxes, whose computation generalizes that of the Klee's Measure and of the Maximum Depth.…
The structure of the densest crystal packings is determined for a variety of concave shapes in 2D constructed by the overlap of two or three disks. The maximum contact number per particle pair is defined and proposed as a useful means of…
We propose microscopic density functional theory for inhomogeneous star polymers. Our approach is based on fundamental measure theory for hard spheres, and on Wertheim's first- and second-order perturbation theory for the interparticle…
We consider circle packings in the plane with circles of sizes $1$, $r\simeq 0.834$ and $s\simeq 0.651$. These sizes are algebraic numbers which allow a compact packing, that is, a packing in which each hole is formed by three mutually…
Packings of identical objects have fascinated both scientists and laymen alike for centuries, in particular the sphere packings and the packings of identical regular tetrahedra. Mathematicians have tried for centuries to determine the…
Packing problems have been of great interest in many diverse contexts for many centuries. The optimal packing of identical objects has been often invoked to understand the nature of low temperature phases of matter. In celebrated work,…
The densest local packing (DLP) problem in d-dimensional Euclidean space Rd involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such that the greatest distance from the center of…
A sphere packing bound (SPB) with a prefactor that is polynomial in the block length $n$ is established for codes on a length $n$ product channel $W_{[1,n]}$ assuming that the maximum order $1/2$ Renyi capacity among the component channels,…
The exploration of the densest sphere packings is a fundamental problem in mathematics and a wide variety of sciences including materials science. We present our exhaustive computational exploration of the densest ternary sphere packings…