The concept of "random loose packing" (RLP) has evolved through extensive study of loose packings of spheres, which has resulted in an accepted definition as the loosest packing that can be obtained by pouring grains. We extend this consideration to packings of nonspherical grains (ellipsoids) formed by slow settling in a viscous liquid, and perform a detailed analysis of the structural propert...

A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B : B ≤T A} contains no 1-random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We...

In this paper we study the generating functionals of several random packing processes: the classical Matérn hard-core model; its extensions, the k-Matérn models and the ∞-Matérn model, which is an example of random sequential packing process. We first give a sufficient condition for the ∞-Matérn model to be well-defined (unlike the other two, the latter may not be well-defined on unbounded spac...

We measure the two-point density correlations and Voronoi cell distributions of cyclically sheared granular spheres obtained with a fluorescence technique and compare them with random packing of frictionless spheres. We find that the radial distribution function g(r) is captured by the Percus-Yevick equation for initial volume fraction ϕ=0.59. However, small but systematic deviations are observ...

A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B : B ≤T A} contains no 1-random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We...

There are many ways to pack, and arrange, filaments in space. The packing can be quantified in terms of how much filament length is contained in a typical unit of volume. How something is “arranged” is harder to define, but one can think in terms of systems that are random as opposed to those that have some kinds of regularity. We identify three fundamental regimes of packing (linear, laminar, ...

Packing of monosized spheres in a cylindrical container of a fixed diameter is a frequently discussed subject in recent studies. It is motivated by the high applicability of these models, particularly by the advances in nanomaterial science and engineering, associated with the development of hierarchically ordered matters of specific structures and properties. Their features strongly depend on ...

For a graph G = (V (G), E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χs(G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n, p) was considered in [3], [4], [12], and [16]. In this paper, we consider χs(G) for a related class of graphs ...

O’Hern, Silbert, Liu and Nagel [Phys. Rev. E. 68 , 011306 (2003)] (OSLN) claim that a special point J of a ”jamming phase diagram” (in density, temperature, stress space) is related to random close packing of hard spheres, and that it represents, for their suggested definitions of jammed and random, the recently introduced maximally random jammed state. We point out several difficulties with th...