The Extended GMRT Radio Halo Survey II: Further results and analysis of the full sample

The intra-cluster medium contains cosmic rays and magnetic fields that are manifested through the large scale synchrotron sources, termed as radio halos, relics and mini-halos. The Extended Giant Metrewave Radio Telescope (GMRT) Radio Halo Survey (EGRHS) is an extension of the GMRT Radio Halo Survey (GRHS) designed to search for radio halos using GMRT 610/235 MHz observations. The GRHS+EGRHS consists of 64 clusters in the redshift range 0.2 -- 0.4 that have an X-ray luminosity larger than 5x10^44 erg/s in the 0.1 -- 2.4 keV band and with declinations > -31 deg in the REFLEX and eBCS X-ray cluster catalogues. In this second paper in the series, GMRT 610/235 MHz data on the last batch of 11 galaxy clusters and the statistical analysis of the full sample are presented. A new mini-halo in RXJ2129.6+0005 and candidate diffuse sources in Z5247, A2552 and Z1953 are discovered. A unique feature of this survey are the upper limits on the detections of 1 Mpc sized radio halos; 4 new are presented here making a total of 31 in the survey. Of the sample, 58 clusters that have adequately sensitive radio information were used to obtain the most accurate occurrence fractions so far. The occurrence of radio halos in our X-ray selected sample is ~22%, that of mini-halos is 13% and that of relics is ~5%. The radio power - X-ray luminosity diagrams for the radio halos and mini-halos with the detections and upper limits are presented. The morphological estimators namely, centroid shift (w), concentration parameter (c) and power ratios (P_3/P_0) derived from the Chandra X-ray images are used as proxies for the dynamical states of the GRHS+EGRHS clusters. The clusters with radio halos and mini-halos occupy distinct quadrants in the c-w, c-P_3/P_0 and w - P_3/P_0 planes, corresponding to the more and less morphological disturbance, respectively. The non-detections span both the quadrants.Comment: 24 pages, 5 tables, 25 figures, accepted for publication in A&

Optimal and Variational Multi-Parameter Quantum Metrology and Vector Field Sensing

We study multi-parameter sensing of 2D and 3D vector fields within the Bayesian framework for $SU(2)$ quantum interferometry. We establish a method to determine the optimal quantum sensor, which establishes the fundamental limit on the precision of simultaneously estimating multiple parameters with an $N$-atom sensor. Keeping current experimental platforms in mind, we present sensors that have limited entanglement capabilities, and yet, significantly outperform sensors that operate without entanglement and approach the optimal quantum sensor in terms of performance. Furthermore, we show how these sensors can be implemented on current programmable quantum sensors with variational quantum circuits by minimizing a metrological cost function. The resulting circuits prepare tailored entangled states and perform measurements in an appropriate entangled basis to realize the best possible quantum sensor given the native entangling resources available on a given sensor platform. Notable examples include a 2D and 3D quantum ``compass'' and a 2D sensor that provides a scalable improvement over unentangled sensors. Our results on optimal and variational multi-parameter quantum metrology are useful for advancing precision measurements in fundamental science and ensuring the stability of quantum computers, which can be achieved through the incorporation of optimal quantum sensors in a quantum feedback loop.Comment: 20 pages, 8 figure

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

Borel-Cantelli sequences

A sequence $\{x_{n}\}_1^\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if for all non-increasing sequences of positive real numbers $\{a_n\}$ with $\underset{i=1}{\overset{\infty}{\sum}}a_i=\infty$ the set $$\underset{k=1}{\overset{\infty}{\cap}} \underset{n=k}{\overset{\infty}{\cup}} B(x_n, a_n))=\{x\in[0,1)\mid |x_n-x|<a_n \text{for} \infty \text{many}n\geq1\}$$ has full Lebesgue measure. (To put it informally, BC sequences are sequences for which a natural converse to the Borel-Cantelli Theorem holds). The notion of BC sequences is motivated by the Monotone Shrinking Target Property for dynamical systems, but our approach is from a geometric rather than dynamical perspective. A sufficient condition, a necessary condition and a necessary and sufficient condition for a sequence to be BC are established. A number of examples of BC and not BC sequences are presented. The property of a sequence to be BC is a delicate diophantine property. For example, the orbits of a pseudo-Anosoff IET (interval exchange transformation) are BC while the orbits of a "generic" IET are not. The notion of BC sequences is extended to more general spaces.Comment: 20 pages. Some proofs clarifie

The upper critical field in superconducting MgB_2

The upper critical field Hc2(T) of sintered pellets of the recently discovered MgB_2 superconductor was investigated in magnetic fields up to 16 T. The upper critical field of the major fraction of the investigated sample was determined from ac susceptibility and resistance data and was found to increase up to Hc2(0) = 13 T at T = 0 corresponding to a coherence length of 5.0 nm. A small fraction of the sample exhibits higher upper critical fields which were measured both resistively and by dc magnetization measurements. The temperature dependence of the upper critical field, Hc2(T), shows a positive curvature near Tc and at intermediate temperatures. This positive curvature of Hc2(T) is similar to that found for the borocarbides YNi_2B_2C and LuNi_2B_2C indicating that MgB_2 is in the clean limit.Comment: 8 pages with 4 figure

The Hitting Times with Taboo for a Random Walk on an Integer Lattice

For a symmetric, homogeneous and irreducible random walk on d-dimensional integer lattice Z^d, having zero mean and a finite variance of jumps, we study the passage times (with possible infinite values) determined by the starting point x, the hitting state y and the taboo state z. We find the probability that these passages times are finite and analyze the tails of their cumulative distribution functions. In particular, it turns out that for the random walk on Z^d, except for a simple (nearest neighbor) random walk on Z, the order of the tail decrease is specified by dimension d only. In contrast, for a simple random walk on Z, the asymptotic properties of hitting times with taboo essentially depend on the mutual location of the points x, y and z. These problems originated in our recent study of branching random walk on Z^d with a single source of branching

Limit theorems for weakly subcritical branching processes in random environment

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, 'supercritical'. This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on non-extinction. Also a functional limit theorem is proven, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.Comment: 35 page

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