Related papers: Optimal Packing Behavior of some 2-block Patterns
The problem of finding the most efficient way to pack spheres has an illustrious history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal in the 60's. This problem finds applications…
We generate non-lattice packings of spheres in up to 22 dimensions using the geometrical constraint satisfaction algorithm RRR. Our aggregated data suggest that it is easy to double the density of Ball's lower bound, and more tentatively,…
Subspace codes are collections of subspaces of a projective space such that any two subspaces satisfy a pairwise minimum distance criterion. Recent results have shown that it is possible to construct optimal $(5,3)$ subspace codes from…
This is the eighth and final paper in a series giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is…
This is the second in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
A two-dimensional granular packing under horizontally circular shaking exhibits various collective motion modes depending on the strength of the oscillation and the global packing density. For intermediate packing density and oscillation…
The class of complex random vectors whose covariance matrix is linearly parameterized by a basis of Hermitian Toeplitz (HT) matrices is considered, and the maximum compression ratios that preserve all second-order information are derived…
We investigate the two-dimensional packing of extremely prolate (aspect ratio $\alpha=L/D>10$) granular materials, comparing experiments with Monte-Carlo simulations. The average packing fraction of particles with aspect ratio $\alpha=12$…
In this paper, upper bounds for the densities of the densest translative tetrahedron packings and the densest translative cubooctahedron packings are obtained.
Random sequential adsorption (RSA) of various two dimensional objects is studied in order to find a shape which maximizes the saturated packing fraction. This investigation was begun in our previous paper [Cie\'sla et al., Phys. Chem. Chem.…
Let $L \subset {\Bbb R}^3$ be the union of unit balls, whose centres lie on the $z$-axis, and are equidistant with distance $2d \in [2, 2\sqrt{2}]$. Then a packing of unit balls in ${\Bbb R}^3$ consisting of translates of $L$ has a density…
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…
The problem of optimal node density for ad hoc sensor networks deployed for making inferences about two dimensional correlated random fields is considered. Using a symmetric first order conditional autoregressive Gauss-Markov random field…
Given a convex disk $K$ and a positive integer $j$, let $\delta_L^j(K)$ and $\vartheta_L^j(K)$ denote the $j$-fold lattice packing density and the $j$-fold lattice covering density of $K$, respectively. I will prove that for every triangle…
Connectedness percolation phenomena in two-dimensional packings of elongated particles (discorectangles) were studied numerically. The packings were produced using random sequential adsorption (RSA) off-lattice model with preferential…
Dietzfelbinger and Weidling [DW07] proposed a natural variation of cuckoo hashing where each of $cn$ objects is assigned $k = 2$ intervals of size $\ell$ in a linear (or cyclic) hash table of size $n$ and both start points are chosen…
In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We…
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…
The sintering behavior of close packed spheres is investigated using a numerical model. The investigated systems are the body centered cubic (BCC), face centered cubic (FCC) and hexagonal closed packed spheres (HCP). The sintering behavior…
The densest packings of N unit squares in a torus are studied using analytical methods as well as simulated annealing. A rich array of dense packing solutions are found: density-one packings when N is the sum of two square integers; a…