A note on the growth factor in Gaussian elimination for generalized Higham matrices

The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite and $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth factor in Gaussian elimination is less than 3. In this paper, based on the previous results, a new bound of the growth factor is obtained by using the maximum of the condition numbers of matrixes B and C for the generalized Higham matrix A, which strengthens this bound to 2 and proves the Higham's conjecture.Comment: 8 pages, 2 figures; Submitted to MOC on Dec. 22 201

Chandra Observation of a Weak Shock in the Galaxy Cluster A2556

Based on a 21.5 ks \chandra\ observation of A2556, we identify an edge on the surface brightness profile (SBP) at about 160$h_{71}^{-1}$ kpc northeast of the cluster center, and it corresponds to a shock front whose Mach number $\mathcal{M}$ is calculated to be $1.25_{-0.03}^{+0.02}$. No prominent substructure, such as sub-cluster, is found in either optical or X-ray band that can be associated with the edge, suggesting that the conventional super-sonic motion mechanism may not work in this case. As an alternative solution, we propose that the nonlinear steepening of acoustic wave, which is induced by the turbulence of the ICM at the core of the cluster, can be used to explain the origin of the shock front. Although nonlinear steepening weak shock is expected to occur frequently in clusters, why it is rarely observed still remains a question that requires further investigation, including both deeper X-ray observation and extensive theoretical studies.Comment: 15 pages, 4 figures, accepted by Ap

Radio Sources in the NCP Region Observed with the 21 Centimeter Array

We present a catalog of 624 radio sources detected around the North Celestial Pole (NCP) with the 21 Centimeter Array (21CMA), a radio interferometer dedicated to the statistical measurement of the epoch of reionization (EoR). The data are taken from a 12 h observation made on 2013 April 13, with a frequency coverage from 75 to 175 MHz and an angular resolution of ~ 4 arcmin. The catalog includes flux densities at eight sub-bands across the 21CMA bandwidth and provides the in-band spectral indices for the detected sources. To reduce the complexity of interferometric imaging from the so-called "w" term and ionospheric effects, the present analysis are restricted to the east-west baselines within 1500 m only. The 624 radio sources are found within 5 degrees around the NCP down to ~ 0.1 Jy. Our source counts are compared, and also exhibit a good agreement, with deep low-frequency observations made recently with the GMRT and MWA. In particular, for fainter radio sources below ~ 1 Jy, we find a flattening trend of source counts towards lower frequencies. While the thermal noise (~0.4 mJy) is well controlled to below the confusion limit, the dynamical range (~10^4) and sensitivity of current 21CMA imaging is largely limited by calibration and deconvolution errors, especially the grating lobes of very bright sources, such as 3C061.1, in the NCP field which result from the regular spacings of the 21CMA. We note that particular attention should be paid to the extended sources, and their modeling and removals may constitute a large technical challenge for current EoR experiments. Our analysis may serve as a useful guide to design of next generation low-frequency interferometers like the Square Kilometre Array.Comment: 16 pages, 21 figures, 7 tables, 1 machine readable table, accepted for publication in Ap