AGN Feedback in groups and clusters of galaxies

The lack of very cool gas at the cores of groups and clusters of galaxies, even where the cooling time is significantly shorter than the Hubble time, has been interpreted as evidence of sources that re-heat the intergalactic medium. Most studies of rich clusters adopt AGN feedback to be this source of heating. From ongoing GMRT projects involving clusters and groups, we demonstrate how low-frequency GMRT radio observations, together with Chandra/XMM-Newton X-ray data, present a unique insight into the nature of feedback, and of the energy transfer between the AGN and the IGM.Comment: 5 pages, 3 figures, To appear in ASP Conference Series, Vol. 407, The Low-Frequency Radio Universe, Eds: D. J. Saikia, D. A. Green, Y. Gupta and T. Venturi (Invited talk, conference held at NCRA-TIFR, Pune, INDIA, 8-12 December, 2008

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

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

Localised HI 21-cm absorption towards a double-lobed z=0.24 radio galaxy

We present the results of a mini-survey for associated HI 21-cm absorption at z < 0.42 with the Giant Metrewave Radio Telescope. Our targets are radio galaxies, selected on the basis that the 1216 Angstrom luminosities are below 10^23 W/Hz, above which there has never been a detection of 21-cm absorption. Of the three sources for which we obtained good data, two are unclassified active galactic nuclei (AGN) and one is type-2. Being a non-detection, the type-2 object is consistent with our previous result that 21-cm absorption in radio sources is not dictated by unified schemes of AGN. In the case of the detection, the absorption only occurs towards one of the two resolved radio lobes in PKS 1649-062. If the absorption is due to an another intervening galaxy, or cool HI gas in the intergalactic medium, covering only the south-west lobe, then, being at the same redshift, this is likely to be gravitationally bound to the optical object identified as PKS 1649-062. If the absorption is due to an inclined disk centred between the lobes, intervening the SW lobe while being located behind the NE lobe, by assuming that it covers the emission peak at 150 kpc from the nucleus, we estimate a dynamical mass of ~3 x 10^12 solar masses for the disk.Comment: 5 pages accepted by MNRAS Letter

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

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

The statistical geometry of scale-free random trees

The properties of scale-free random trees are investigated using both preconditioning on non-extinction and fixed size averages, in order to study the thermodynamic limit. The scaling form of volume probability is found, the connectivity dimensions are determined and compared with other exponents which describe the growth. The (local) spectral dimension is also determined, through the study of the massless limit of the Gaussian model on such trees.Comment: 21 pages, 2 figures, revtex4, minor changes (published version

Length functions on currents and applications to dynamics and counting

The aim of this (mostly expository) article is twofold. We first explore a variety of length functions on the space of currents, and we survey recent work regarding applications of length functions to counting problems. Secondly, we use length functions to provide a proof of a folklore theorem which states that pseudo-Anosov homeomorphisms of closed hyperbolic surfaces act on the space of projective geodesic currents with uniform north-south dynamics.Comment: 35pp, 2 figures, comments welcome! Second version: minor corrections. To appear as a chapter in the forthcoming book "In the tradition of Thurston" edited by V. Alberge, K. Ohshika and A. Papadopoulo