Some colouring problems for Paley graphs

The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved

A possible observational evidence for θ−2\theta^{-2} angular distribution of opening half-angle of GRB jets

We propose a method to estimate the pseudo jet opening half-angle of GRBs using the spectral peak energy (\Ep)--peak luminosity relation (so called Yonetoku relation) as well as the \Ep--collimation-corrected $\gamma$-ray energy relation (so called Ghirlanda relation). For bursts with known jet break times and redshifts, we compared the pseudo jet opening half-angle with the standard one and found that the differences are within a factor 2. We apply the method to 689 long GRBS. We found that the distribution function of the pseudo jet opening half-angle obeys $f(\theta_j)\propto\theta_j^{-2.2 \pm 0.2}$ with possible cutoffs for $\theta_j 0.3$ although the log-normal fit is also possible. $\theta^{-2}$ distribution is compatible with the structured jet model. From the distribution function we found that the beaming correction for the rate of GRBs is $\sim 340$, which means $\sim 10^{-5}$ yr$^{-1}$ galaxy$^{-1}$ or only one in $10^2$ type Ib/c supernovae. We also found the evolution of the distribution function as a function of the redshift.Comment: 5 pages, 5 figures, submitted to MNRA

Isochronicity Correction in the CR Storage Ring

A challenge for nuclear physics is to measure masses of exotic nuclei up to the limits of nuclear existence which are characterized by low production cross sections and short half-lives. The large acceptance Collector Ring (CR) at FAIR tuned in the isochronous ion-optical mode offers unique possibilities for measuring short-lived and very exotic nuclides. However, in a ring designed for maximal acceptance, many factors limit the resolution. One point is a limit in time resolution inversely proportional to the transverse emittance. But most of the time aberrations can be corrected and others become small for large number of turns. We show the relations of the time correction to the corresponding transverse focusing and that the main correction for large emittance corresponds directly to the chromaticity correction for transverse focusing of the beam. With the help of Monte-Carlo simulations for the full acceptance we demonstrate how to correct the revolution times so that in principle resolutions of dm/m=1E-6 can be achieved. In these calculations the influence of magnet inhomogeneities and extended fringe fields are considered and a calibration scheme also for ions with different mass-to-charge ratio is presented.Comment: 6 figures, recised version May 201

Coherent imaging of extended objects

When used with coherent light, optical imaging systems, even diffraction-limited, are inherently unable to reproduce both the amplitude and the phase of a two-dimensional field distribution because their impulse response function varies slowly from point to point (a property known as non-isoplanatism). For sufficiently small objects, this usually results in a phase distortion and has no impact on the measured intensity. Here, we show that the intensity distribution can also be dramatically distorted when objects of large extension or of special shapes are imaged. We illustrate the problem using two simple examples: the pinhole camera and the aberration-free thin lens. The effects predicted by our theorical analysis are also confirmed by experimental observations.Comment: 10 pages, 9 figures, submitted to Optics Communication

Atmospheric extinction properties above Mauna Kea from the Nearby Supernova Factory spectro-photometric data set

We present a new atmospheric extinction curve for Mauna Kea spanning 3200--9700 \AA. It is the most comprehensive to date, being based on some 4285 standard star spectra obtained on 478 nights spread over a period of 7 years obtained by the Nearby SuperNova Factory using the SuperNova Integral Field Spectrograph. This mean curve and its dispersion can be used as an aid in calibrating spectroscopic or imaging data from Mauna Kea, and in estimating the calibration uncertainty associated with the use of a mean extinction curve. Our method for decomposing the extinction curve into physical components, and the ability to determine the chromatic portion of the extinction even on cloudy nights, is described and verified over the wide range of conditions sampled by our large dataset. We demonstrate good agreement with atmospheric science data obtain at nearby Mauna Loa Observatory, and with previously published measurements of the extinction above Mauna Kea.Comment: 22 pages, 24 figures, 6 table