Finding the optimal random packing of non-spherical particles is an open problem with great significance in a broad range of scientific and engineering fields. So far, this search has been performed only empirically on a case-by-case basis, in particular, for shapes like dimers, spherocylinders and ellipsoids of revolution. Here we present a mean-field formalism to estimate the packing density ...

We extend the common theme of random Apollonian packing of circles to consider orientable grains with a noncircular shape. Systems of up to 10(6) grains are examined for a range of polygonal and elliptical shapes using both the random Apollonian packing model and the new rotational random Apollonian packing model which takes into account the extra rotational degree of freedom of noncircular gra...

We study the problem of packing arborescences in the random digraph D(n, p), where each possible arc is included uniformly at random with probability p = p(n). Let λ(D(n, p)) denote the largest integer λ ≥ 0 such that, for all 0 ≤ ` ≤ λ, we have ∑`−1 i=0(`− i)|{v : din(v) = i}| ≤ `. We show that the maximum number of arcdisjoint arborescences in D(n, p) is λ(D(n, p)) a.a.s. We also give tight e...

In 1611 Kepler proposed that the densest packing of spheres could be achieved by stacking close-packed planes of spheres. In such a packing, the spheres occupy π/ √ 18 ≈74.05% of space. The Kepler conjecture was (almost certainly) proved in 1998 by Thomas Hales. When we pour a large number of equal-sized spheres in a container and shake them down, we do not obtain the Kepler packing. Rather a d...

We consider the problem of packing fixed-length patterns into a permutation, and develop a connection between the number of large patterns and the number of bonds in a permutation. Improving upon a result of Kaplansky and Wolfowitz, we obtain exact values for the expectation and variance for the number of large patterns in a random permutation. Finally, we are able to generalize the idea of bon...

The Lovász Local Lemma is known to have an extension for cases where independence is missing but negative dependencies are under control. We show that this is often the case for random injections, and we provide easy-to-check conditions for the non-trivial task of verifying a negative dependency graph for random injections. As an application, we prove existence results for hypergraph packing an...

Let G = (V,E) be a d-regular graph of order n. Let Gp be the random subgraph of G for which each edge is selected from E(G) independently at random with probability p. For a fixed graph H, define m(H) := max{e(H ′)/(v(H ′)− 1) : H ′ ⊆ H}. We prove that n(m(H)−1)/m(H)/d is a threshold function for Gp to satisfy Ramsey, induced Ramsey, and canonical Ramsey properties with respect to vertex colori...