The zeros of random polynomials cluster uniformly near the unit circle

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdos and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.Comment: Corrects some typos and strengthens Theorem

Radiative transfer of synchrotron radiation through a compressed random magnetic field

The generally low polarization levels present in extragalactic radio sources suggest magnetic fields that are largely turbulent, presumably maintained by turbulence motion of the underlying plasma. The polarization of radio emission is often generated as the plasma is compressed by shock waves, and this type of configuration has been widely applied, from the cores of AGN to the lobes of classical double radio sources. Here we note that, in the absence of Faraday rotation, the synchrotron radiative transfer equations for such a configuration, separate, leading to relatively simple solutions for opaque or partially opaque sources. Expressions for the emission and absorption coefficients are derived, and, although in general, these must be evaluated numerically, the process is much simpler than a full numerical solution of the transfer equations

Mock-Gaussian Behaviour for Linear Statistics of Classical Compact Groups

We consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups. We compute their moments and find that the first few moments are Gaussian, whereas the limiting distribution is not. The precise number of Gaussian moments depends upon the particular statistic considered

Managerial Discretion and Takeover Performance

We investigate the relation between long run takeover performance and board share ownership in the acquiring company for a sample of 142 UK takeovers completed between 1985-95. We find evidence of a non-linear relationship both between board ownership and takeover profitability, and between board ownership and post-takeover share returns. We cast the analysis in a simultaneous equations framework using non- linear two-stage least squares, and find that our results are robust to this alternative specification. The results are therefore consistent with a managerial alignment / entrenchment trade-off.Corporate takeovers; board ownership; profitability; long run share returns

On the Radio Polarization Signature of Efficient and Inefficient Particle Acceleration in Supernova Remnant SN 1006

We present a radio polarization study of SN 1006, based on combined VLA and ATCA observations at 20 cm that resulted in sensitive images with an angular resolution of 10 arcsec. The fractional polarization in the two bright radio and X-ray lobes of the SNR is measured to be 0.17, while in the southeastern sector, where the radio and non-thermal X-ray emission are much weaker, the polarization fraction reaches a value of 0.6 +- 0.2, close to the theoretical limit of 0.7. We interpret this result as evidence of a disordered, turbulent magnetic field in the lobes, where particle acceleration is believed to be efficient, and a highly ordered field in the southeast, where the acceleration efficiency has been shown to be very low. Utilizing the frequency coverage of our observations, an average rotation measure of ~12 rad/m2 is determined from the combined data set, which is then used to obtain the intrinsic direction of the magnetic field vectors. While the orientation of magnetic field vectors across the SNR shell appear radial, a large fraction of the magnetic vectors lie parallel to the Galactic Plane. Along the highly polarized southeastern rim, the field is aligned tangent to the shock, and therefore also nearly parallel to the Galactic Plane. These results strongly suggest that the ambient field surrounding SN 1006 is aligned with this direction (i.e., from northeast to southwest) and that the bright lobes are due to a polar cap geometry. Our study establishes that the most efficient particle acceleration and generation of magnetic turbulence in SN 1006 is attained for shocks in which the magnetic field direction and shock normal are quasi-parallel, while inefficient acceleration and little to no generation of magnetic turbulence obtains for the quasi-perpendicular case.Comment: Accepted for publication in Astronomical Journa