T. E. Harris and branching processes

T. E. Harris was a pioneer par excellence in many fields of probability theory. In this paper, we give a brief survey of the many fundamental contributions of Harris to the theory of branching processes, starting with his doctoral work at Princeton in the late forties and culminating in his fundamental book "The Theory of Branching Processes," published in 1963.Comment: Published in at http://dx.doi.org/10.1214/10-AOP599 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Credit exclusion in quantitative models of bankruptcy: does it matter?

Bankruptcy ; Credit

Growth of preferential attachment random graphs via continuous-time branching processes

A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time branching scheme and, using the branching process apparatus, several results on the graph model asymptotics are obtained, some extending previous results, such as growth rates for a typical degree and the maximal degree, behavior of the vertex where the maximal degree is attained, and a law of large numbers for the empirical distribution of degrees which shows certain ``scale-free'' or ``power-law'' behaviors.Comment: 20 page

Limit theorems for semi-Markov processes

A new construction of regeneration times is exploited to prove ergodic and renewal theorems for semi-Markov processes on general state spaces. This work extends results of the authors in Ann. Probability (6 (1978), 788-797)

Ergodic theorems for transient one-dimensional diffusions

For one-dimensional diffusions X that drift off to + ∞ we give conditions on a set B and the drift and diffusion coefficients of X for (1/t)∫t0 IB(X(u))du to converge w.p.l as t → ∞

On the role of confinement on solidification in pure materials and binary alloys

We use a phase-field model to study the effect of confinement on dendritic growth, in a pure material solidifying in an undercooled melt, and in the directional solidification of a dilute binary alloy. Specifically, we observe the effect of varying the vertical domain extent ($\delta$) on tip selection, by quantifying the dendrite tip velocity and curvature as a function of $\delta$, and other process parameters. As $\delta$ decreases, we find that the operating state of the dendrite tips becomes significantly affected by the presence of finite boundaries. For particular boundary conditions, we observe a switching of the growth state from 3-D to 2-D at very small $\delta$, in both the pure material and alloy. We demonstrate that results from the alloy model compare favorably with those from an experimental study investigating this effect.Comment: 13 pages, 9 figures, 3 table

Scale-free random branching tree in supercritical phase

We study the size and the lifetime distributions of scale-free random branching tree in which $k$ branches are generated from a node at each time step with probability $q_k\sim k^{-\gamma}$. In particular, we focus on finite-size trees in a supercritical phase, where the mean branching number $C=\sum_k k q_k$ is larger than 1. The tree-size distribution $p(s)$ exhibits a crossover behavior when $2 < \gamma < 3$; A characteristic tree size $s_c$ exists such that for $s \ll s_c$, $p(s)\sim s^{-\gamma/(\gamma-1)}$ and for $s \gg s_c$, $p(s)\sim s^{-3/2}\exp(-s/s_c)$, where $s_c$ scales as $\sim (C-1)^{-(\gamma-1)/(\gamma-2)}$. For $\gamma > 3$, it follows the conventional mean-field solution, $p(s)\sim s^{-3/2}\exp(-s/s_c)$ with $s_c\sim (C-1)^{-2}$. The lifetime distribution is also derived. It behaves as $\ell(t)\sim t^{-(\gamma-1)/(\gamma-2)}$ for $2 < \gamma < 3$, and $\sim t^{-2}$ for $\gamma > 3$ when branching step $t \ll t_c \sim (C-1)^{-1}$, and $\ell(t)\sim \exp(-t/t_c)$ for all $\gamma > 2$ when $t \gg t_c$. The analytic solutions are corroborated by numerical results.Comment: 6 pages, 6 figure

Ultraluminous Star-forming Galaxies and Extremely Luminous Warm Molecular Hydrogen Emission at z = 2.16 in the PKS 1138–26 Radio Galaxy Protocluster

A deep Spitzer Infrared Spectrograph map of the PKS 1138–26 galaxy protocluster reveals ultraluminous polycyclic aromatic hydrocarbon (PAH) emission from obscured star formation in three protocluster galaxies, including Hα-emitter (HAE) 229, HAE 131, and the central Spiderweb Galaxy. Star formation rates of ~500-1100 M_☉ yr^(–1) are estimated from the 7.7 μm PAH feature. At such prodigious formation rates, the galaxy stellar masses will double in 0.6-1.1 Gyr. We are viewing the peak epoch of star formation for these protocluster galaxies. However, it appears that extinction of Hα is much greater (up to a factor of 40) in the two ULIRG HAEs compared to the Spiderweb. This may be attributed to different spatial distributions of star formation-nuclear star formation in the HAEs versus extended star formation in accreting satellite galaxies in the Spiderweb. We find extremely luminous mid-IR rotational line emission from warm molecular hydrogen in the Spiderweb Galaxy, with L(H_2 0-0 S(3)) = 1.4 × 10^(44) erg s^(–1) (3.7 × 10^(10) L_☉), ~20 times more luminous than any previously known H2 emission galaxy (MOHEG). Depending on the temperature, this corresponds to a very large mass of >9 × 10^(6)-2 × 10^9 M_☉ of T > 300 K molecular gas, which may be heated by the PKS 1138–26 radio jet, acting to quench nuclear star formation. There is >8 times more warm H_2 at these temperatures in the Spiderweb than what has been seen in low-redshift (z < 0.2) radio galaxies, indicating that the Spiderweb may have a larger reservoir of molecular gas than more evolved radio galaxies. This is the highest redshift galaxy yet in which warm molecular hydrogen has been directly detected

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

A Global Probe of Cosmic Magnetic Fields to High Redshifts

Faraday rotation (RM) probes of magnetic fields in the universe are sensitive to cosmological and evolutionary effects as $z$ increases beyond $\sim$1 because of the scalings of electron density and magnetic fields, and the growth in the number of expected intersections with galaxy-scale intervenors, $d$N/$dz$. In this new global analysis of an unprecedented large sample of RM's of high latitude quasars extending out to $z\sim$3.7 we find that the distribution of RM broadens with redshift in the 20 $-$ 80 rad m$^{-2}$ range range, despite the (1 +$z$)$^{-2}$ wavelength dilution expected in the observed Faraday rotation. Our results indicate that the Universe becomes increasingly ``Faraday-opaque'' to sources beyond $z \sim$ 2, that is, as $z$ increases progressively fewer sources are found with a ``small'' RM in the observer's frame. This is in contrast to sources at z \la1. They suggest that the environments of galaxies were significantly magnetized at high redshifts, with magnetic field strengths that were at least as strong within a few Gyr of the Big Bang as at the current epoch. We separately investigate a simple unevolving toy model in which the RM is produced by MgII absorber systems, and find that it can approximately reproduce the observed trend with redshift. An additional possibility is that the intrinsic RM associated with the radio sources was much higher in the past, and we show that this is not a trivial consequence of the higher radio luminosities of the high redshift sources.Comment: 10 pages, 8 figures Astrophysical Jounrnal in press, March 200