Related papers: Random Lattices and Random Sphere Packings: Typica…

Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…

Let $\{U_n\}_{n \geq 0}$ and $\{V_m\}_{m \geq 0}$ be two linear recurrence sequences. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as differences $ U_n - V_m$. In…

We introduce and carefully study a natural probability measure over the numerical range of a complex matrix $A \in M_n(\C)$. This numerical measure $\mu_A$ can be defined as the law of the random variable $<AX,X> \in \C$ when the vector $X…

The possibility is considered for the formation in optical lattices of a heterogeneous state characterized by a spontaneous mesoscopic separation of the system into the spatial regions with different atomic densities. It is shown that such…

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of…

We have mapped over 50 massive, dense clumps with four dense gas tracers: HCN J=1-0 and 3-2; and CS J=2-1 and 7-6 transitions. Spectral lines of optically thin H^{13}CN 3-2 and C^{34}S 5-4 were also obtained towards the map centers. These…

Statistical properties of ensembles of random density matrices are investigated. We compute traces and von Neumann entropies averaged over ensembles of random density matrices distributed according to the Bures measure. The eigenvalues of…

Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where its entropy is crucial for many physical properties. Due to their inherent complex topological structure even the number…

We calculate the probability distribution of the local density of states $\nu$ in a disordered one-dimensional conductor or single-mode waveguide, attached at one end to an electron or photon reservoir. We show that this distribution does…

We find an analytical solution for the minimal matter density of a void, its central density. It turns out that the voids are not so empty: most of the voids have the central underdensity $\Delta_c \sim -50\%$ (which means that the matter…

The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of…

We establish various qualitative properties of liquid Lane-Emden stars in $\mathbb{R}^d$, including bounds for its density profile $\rho$ and radius $R$. Using them we prove that against radial perturbations, the liquid Lane-Emden stars are…

We investigate the densities of the sets of abundant numbers and of covering numbers, integers $n$ for which there exists a distinct covering system where every modulus divides $n$. We establish that the set $\mathcal{C}$ of covering…

We calculate the mean density profiles for luminous and dark matter on distance scales $D \sim(1 - 100)$ Mpc around us using recent all-sky catalogs of galaxy groups. Within the Local Volume $( D < 11 ~\rm Mpc)$ we derived the mean stellar…

Given an Apollonian Circle Packing $\mathcal{P}$ and a circle $C_0 = \partial B(z_0, r_0)$ in $\mathcal{P}$, color the set of disks in $\mathcal{P}$ tangent to $C_0$ red. What proportion of the concentric circle $C_{\epsilon} = \partial…

The average mass density profile measured in the CNOC cluster survey is well described with the analytic form rho(r)=A/[r(r+a_rho)^2], as advocated on the basis on n-body simulations by Navarro, Frenk & White. The predicted core radii are…

We present the densest known packing of regular tetrahedra with density phi = 4000/4671 = 0.856347... Like the recently discovered packings of Kallus et al. [arXiv:0910.5226] and Torquato-Jiao [arXiv:0912.4210], our packing is crystalline…

In SU(2) lattice gauge theory in maximal center gauge, we investigate the dependence of center-projected Creutz ratios and the vortex density on lattice size and the number of gauge copies. The dependence on the number of copies is rather…

For a given boundary set consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface spanning the boundary.

We develop an ab initio analytic theory of random lasing in an ensemble of atoms that both scatter and amplify light. The theory applies all the way from low to high density of atoms. The properties of the random laser are controlled by an…