Weighted distances in scale-free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear in Random Structures and Algorithm

Ultrametric Logarithm Laws, II

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock-Margulis and obtain related results in Metric Diophantine Approximation.Comment: submitted to Montasefte Fur Mathemati

Tunable-filter imaging of quasar fields at z~1. I. A cluster around MRC B0450-221

Using a combination of multicolour broad- and narrow-band imaging techniques and follow-up spectroscopy, we have detected an overdensity of galaxies in the field of quasar MRC B0450-221, whose properties are consistent with a cluster at the quasar redshift z=0.9. An excess of red galaxies (V-I>2.2, I-K'>3.8) is evident within 1' of the quasar, with the colours expected for galaxies at z=0.9 that have evolved passively for 3 Gyr or more. A number of line-emitting galaxies (nine candidates with equivalent widths EW>70A) are also detected in the field using the TAURUS Tunable Filter (TTF). Three have been confirmed spectroscopically to indeed lie at z=0.9. The TTF candidates with the strongest [O II] line emission cluster in a group which lies 200-700 kpc away from the quasar and the red galaxy excess, and therefore most likely on the outskirts of the cluster. These observations are the first in a series probing quasar environments at z~1 with TTF.Comment: Accepted for publication in AJ. 25 pages, 24 figs (large files in jpg or gif format), uses emulateapj.st

Exact solution of a two-type branching process: Clone size distribution in cell division kinetics

We study a two-type branching process which provides excellent description of experimental data on cell dynamics in skin tissue (Clayton et al., 2007). The model involves only a single type of progenitor cell, and does not require support from a self-renewed population of stem cells. The progenitor cells divide and may differentiate into post-mitotic cells. We derive an exact solution of this model in terms of generating functions for the total number of cells, and for the number of cells of different types. We also deduce large time asymptotic behaviors drawing on our exact results, and on an independent diffusion approximation.Comment: 16 page

Renormalization group approach to multiscale modelling in materials science

Dendritic growth, and the formation of material microstructure in general, necessarily involves a wide range of length scales from the atomic up to sample dimensions. The phase field approach of Langer, enhanced by optimal asymptotic methods and adaptive mesh refinement, copes with this range of scales, and provides an effective way to move phase boundaries. However, it fails to preserve memory of the underlying crystallographic anisotropy, and thus is ill-suited for problems involving defects or elasticity. The phase field crystal (PFC) equation-- a conserving analogue of the Hohenberg-Swift equation --is a phase field equation with periodic solutions that represent the atomic density. It can natively model elasticity, the formation of solid phases, and accurately reproduces the nonequilibrium dynamics of phase transitions in real materials. However, the PFC models matter at the atomic scale, rendering it unsuitable for coping with the range of length scales in problems of serious interest. Here, we show that a computationally-efficient multiscale approach to the PFC can be developed systematically by using the renormalization group or equivalent techniques to derive appropriate coarse-grained coupled phase and amplitude equations, which are suitable for solution by adaptive mesh refinement algorithms

CLTs and asymptotic variance of time-sampled Markov chains

For a Markov transition kernel P and a probability distribution μ on nonnegative integers, a time-sampled Markov chain evolves according to the transition kernel Pμ = Σkμ(k)Pk. In this note we obtain CLT conditions for time-sampled Markov chains and derive a spectral formula for the asymptotic variance. Using these results we compare efficiency of Barker's and Metropolis algorithms in terms of asymptotic variance

Superconductivity in the SU(N) Anderson Lattice at U=\infty

We present a mean-field study of superconductivity in a generalized N-channel cubic Anderson lattice at U=\infty taking into account the effect of a nearest-neighbor attraction J. The condition U=\infty is implemented within the slave-boson formalism considering the slave bosons to be condensed. We consider the $f$-level occupancy ranging from the mixed valence regime to the Kondo limit and study the dependence of the critical temperature on the various model parameters for each of three possible Cooper pairing symmetries (extended s, d-wave and p-wave pairing) and find interesting crossovers. It is found that the d- and p- wave order parameters have, in general, very similar critical temperatures. The extended s-wave pairing seems to be relatively more stable for electronic densities per channel close to one and for large values of the superconducting interaction J.Comment: Seven Figures; one appendix. Accepted for publication in Phys. Rev.

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos corrected; improvements on the statements and comments suggested by a referee. Keywords: singular flows, singular-hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm la

On the Spectral Index of Distant Radio Galaxies

The problems of using the spectral index of radio galaxies in various tests, in particular, in selecting distant radio sources are considered. The history of the question of choosing a criterion of searching for distant radio galaxies based on the spectral index is presented. For a new catalog of 2442 radio galaxies constructed from NED, SDSS, and CATS data, an analytical form of the sp ectral index.redshift relation has been determined for the first time. The spectral index.angular size and spectral index.flux density diagrams have also been constructed. Peculiarities of the distribution of sources on these diagrams are discussed.Comment: 7 pages,5 figure

VLT spectropolarimetry of two powerful radio galaxies at z~1.4: UV continuum, emission-line properties and the nature of high-redshift dust

(Abridged) Deep VLT spectropolarimetric observations are presented for two powerful radio galaxies, 0850-206 and 1303+091. These observations cover the rest-frame wavelength range ~ 1450 - 3750 A. New radio observations and continuum images of the same sources are also presented. These galaxies are the first two observed from a complete sample of nine radio sources with redshifts in the range 1.3 < z < 1.5 (selected from the equatorial sample of powerful radio sources of Best, Rottgering & Lehnert), as part of a project aimed to investigate the multi-component nature of the UV continuum in radio galaxies and any variations of the continuum properties with the radio source age. The larger radio source of the two, 0850-206, presents a high continuum fractional polarization, averaging 17% across the observed wavelength range and reaching 24% at rest-frame wavelengths of <2000 A. The smaller radio source, 1303+091, shows a lower continuum polarization, averaging 8% and rising to 11% for rest-frame wavelengths >3000 A. For both galaxies, the position angle of the electric vector is generally constant with wavelength and within ~15 degrees of perpendicular to the radio axis. Both their total flux spectra and polarized flux spectra reveal the 2200 A dust feature, and comparison with dust scattering models suggests that the composition of the dust in these galaxies is similar to that of Galactic dust. In 0850-206, scattered quasar radiation dominates the UV continuum emission, with the nebular continuum accounting for no more than ~22% and no requirement for any additional emission component such as emission from young stars. By contrast, in 1303+091, unpolarized radiation could be a major constituent of the UV continuum emission, with starlight accounting for up to ~50% and the nebular continuum accounting for ~11%.Comment: 20 pages, including 14 figures. MNRAS accepte