Random line tessellations of the plane: statistical properties of many-sided cells

We consider a family of random line tessellations of the Euclidean plane introduced in a much more formal context by Hug and Schneider [Geom. Funct. Anal. 17, 156 (2007)] and described by a parameter \alpha\geq 1. For \alpha=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for \alpha=2 it coincides with the typical Poisson-Voronoi cell. Let p_n(\alpha) be the probability for the zero-cell to have n sides. By the methods of statistical mechanics we construct the asymptotic expansion of \log p_n(\alpha) up to terms that vanish as n\to\infty. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when \alpha>1, but gets delocalized for the Crofton cell, \alpha=1, which is a singular point of the parameter range. The large-n expansion of \log p_n(1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the {\it typical} n-sided cell of a Poisson line tessellation.Comment: 26 pages, 3 figure

Three-dimensional random Voronoi tessellations: From cubic crystal lattices to Poisson point processes

We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian noise whose adimensional strength is controlled by the parameter a, and analyze the topological and metrical properties of the resulting Voronoi Tessellations (VT). The topological properties of the VT of the SC and FCC crystals are unstable with respect to the introduction of noise, because the corresponding polyhedra are geometrically degenerate, whereas the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals VT increases quadratically with a. In the case of perturbed SCC crystals, there is an optimal amount of noise that minimizes the mean area of the cells. Already for a moderate noise (a>0.5), the properties of the three perturbed VT are indistinguishable, and for intense noise (a>2), results converge to the Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent model for the empirical of of all considered properties. The VT of the perturbed BCC and FCC structures are local maxima for the isoperimetric quotient, which measures the degre of sphericity of the cells, among space filling VT. In the BCC case, this suggests a weaker form of the recentluy disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells, anomalous scalings with exponents >3/2 is observed between the area and the volumes of the cells, and, except for the FCC case, also for a->0. In the Poisson-VT limit, the exponent is about 1.67. As the number of faces is positively correlated with the sphericity of the cells, the anomalous scaling is heavily reduced when we perform powerlaw fits separately on cells with a specific number of faces

The Hamiltonian formulation of General Relativity: myths and reality

A conventional wisdom often perpetuated in the literature states that: (i) a 3+1 decomposition of space-time into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [Proc. Roy. Soc. A 246 (1958) 333] and of ADM [Arnowitt, Deser and Misner, in "Gravitation: An Introduction to Current Research" (1962) 227] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i-iii) have been shown to be incorrect in [Kiriushcheva et al., Phys. Lett. A 372 (2008) 5101] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. We show that points (i-iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric $g_{\mu\nu}$ to lapse and shift functions and the three-metric $g_{km}$, which is not canonical. This proves that point (iv) is incorrect. Points (i-iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein's theory itself.Comment: References are added and updated, Introduction is extended, Subsection 3.5 is added, 83 pages; corresponds to the published versio

Improved imputation quality of low-frequency and rare variants in European samples using the 'Genome of the Netherlands'

Although genome-wide association studies (GWAS) have identified many common variants associated with complex traits, low-frequency and rare variants have not been interrogated in a comprehensive manner. Imputation from dense reference panels, such as the 1000 Genomes Project (1000G), enables testing of ungenotyped variants for association. Here we present the results of imputation using a large, new population-specific panel: the Genome of The Netherlands (GoNL). We benchmarked the performance of the 1000G and GoNL reference sets by comparing imputation genotypes with 'true' genotypes typed on ImmunoChip in three European populations (Dutch, British, and Italian). GoNL showed significant improvement in the imputation quality for rare variants (MAF 0.05-0.5%) compared with 1000G. In Dutch samples, the mean observed Pearson correlation, r 2, increased from 0.61 to 0.71. W

Prompt K_short production in pp collisions at sqrt(s)=0.9 TeV

The production of K_short mesons in pp collisions at a centre-of-mass energy of 0.9 TeV is studied with the LHCb detector at the Large Hadron Collider. The luminosity of the analysed sample is determined using a novel technique, involving measurements of the beam currents, sizes and positions, and is found to be 6.8 +/- 1.0 microbarn^-1. The differential prompt K_short production cross-section is measured as a function of the K_short transverse momentum and rapidity in the region 0 < pT < 1.6 GeV/c and 2.5 < y < 4.0. The data are found to be in reasonable agreement with previous measurements and generator expectations.Comment: 6+18 pages, 6 figures, updated author lis