We detail the rules and mathematical structure of Al-Jabar, a game invented by the authors based on intuitive concepts of color-mixing and ideas from abstract algebra. Game-play consists of manipulating colored game pieces; we discuss how these color
ed pieces form a group structure and how this structure, along with an operation used to combine the pieces, is used to create a game of strategy. We also consider extensions of the game rules to other group structures. Note: While this is an article for general readership originally published online by Gathering for Gardner in honor of Martin Gardners birthday (Oct. 2011), Al-Jabar has been played in university abstract algebra courses as a teaching tool, as well as by game enthusiasts, since its release. Moreover, the algebraic game structure described has sparked further work by other mathematicians and game designers. Thus, we submit this article to the ArXiV as a resource for educators as well as those interested in mathematical games.
PHL 1092 is a z~0.4 high-luminosity counterpart of the class of Narrow-Line Seyfert 1 galaxies. In 2008, PHL 1092 was found to be in a remarkably low X-ray flux state during an XMM-Newton observation. Its 2 keV flux density had dropped by a factor of
~260 with respect to a previous observation performed 4.5 yr earlier. The UV flux remained almost constant, resulting in a significant steepening of the optical-to-X-ray slope alpha_ox from -1.57 to -2.51, making PHL 1092 one of the most extreme X-ray weak quasars with no observed broad absorption lines (BALs) in the UV. We have monitored the source since 2008 with three further XMM-Newton observations, producing a simultaneous UV and X-ray database spanning almost 10 yr in total in the activity of the source. Our monitoring program demonstrates that the alpha_ox variability in PHL 1092 is entirely driven by long-term X-ray flux changes. We apply a series of physically-motivated models with the goal of explaining the UV-to-X-ray spectral energy distribution (SED) and the extreme X-ray and alpha_ox variability. We consider three possible models: i) A breathing corona scenario in which the size of the X-ray emitting corona is correlated with the X-ray flux. In this case, the lowest X-ray flux states of PHL 1092 are associated with an almost complete collapse of the X-ray corona down to the marginal stable orbit; ii) An absorption scenario in which the X-ray flux variability is entirely due to intervening absorption. If so, PHL 1092 is a quasar with standard X-ray output for its optical luminosity, appearing as X-ray weak at times due to absorption; iii) A disc-reflection-dominated scenario in which the X-ray emitting corona is confined within a few gravitational radii from the black hole at all times. In this case, the intrinsic variability of PHL 1092 only needs to be a factor of ~10 rather than the observed factor of ~260.
With the growing interest in and ability of using weak lensing studies to probe the non-Gaussian properties of the matter density field, there is an increasing need for the study of suitable statistical measures, e.g. shear three-point statistics. In
this paper we establish the relations between the three-point configuration space shear and convergence statistics, which are an important missing link between different weak lensing three-point statistics and provide an alternative way of relating observation and theory. The method we use also allows us to derive the relations between other two- and three-point correlation functions. We show the consistency of the relations obtained with already established results and demonstrate how they can be evaluated numerically. As a direct application, we use these relations to formulate the condition for E/B-mode decomposition of lensing three-point statistics, which is the basis for constructing new three-point statistics which allow for exact E/B-mode separation. Our work applies also to other two-dimensional polarization fields such as that of the Cosmic Microwave Background.
We present the fifth edition of the Sloan Digital Sky Survey (SDSS) Quasar Catalog, which is based upon the SDSS Seventh Data Release. The catalog, which contains 105,783 spectroscopically confirmed quasars, represents the conclusion of the SDSS-I an
d SDSS-II quasar survey. The catalog consists of the SDSS objects that have luminosities larger than M_i = -22.0 (in a cosmology with H_0 = 70 km/s/Mpc Omega_M = 0.3, and Omega_Lambda = 0.7) have at least one emission line with FWHM larger than 1000 km/s or have interesting/complex absorption features, are fainter than i > 15.0 and have highly reliable redshifts. The catalog covers an area of 9380 deg^2. The quasar redshifts range from 0.065 to 5.46, with a median value of 1.49; the catalog includes 1248 quasars at redshifts greater than four, of which 56 are at redshifts greater than five. The catalog contains 9210 quasars with i < 18; slightly over half of the entries have i< 19. For each object the catalog presents positions accurate to better than 0.1 rms per coordinate, five-band (ugriz) CCD-based photometry with typical accuracy of 0.03 mag, and information on the morphology and selection method. The catalog also contains radio, near-infrared, and X-ray emission properties of the quasars, when available, from other large-area surveys. The calibrated digital spectra cover the wavelength region 3800-9200 Ang. at a spectral resolution R = 2000 the spectra can be retrieved from the SDSS public database using the information provided in the catalog. Over 96% of the objects in the catalog were discovered by the SDSS. We also include a supplemental list of an additional 207 quasars with SDSS spectra whose archive photometric information is incomplete.
To render cosmic shear an astronomical tool of high precision, it is essential to eliminate systematic effects upon its signal, one of the most significant ones being correlations between the gravitational shear and the intrinsic ellipticity of sourc
e galaxies. Because of the crudeness of current models of intrinsic alignment, we have developed a model-independent, purely geometrical method to remove shear-ellipticity correlations. We eliminate the contributions to a tomographic cosmic shear signal that may be subject to contamination by shear-ellipticity correlations, making use of the characteristic dependence of these correlations on redshift. By introducing an appropriately chosen weight function to the lensing efficiency that nulls signals stemming from certain distances, new second-order measures of cosmic shear, free of intrinsic alignment, can be constructed. The loss of information induced by this nulling technique and the subsequent degradation of constraints on cosmological parameters is quantified in a likelihood analysis. Concerning the construction of optimal weight functions, good agreement is achieved between all approaches considered. For a survey divided into 20 redshift bins we find that the area of credible regions increases by 20% up to about 50% after the application of nulling, depending on the cosmological parameters considered. We demonstrate that due to an optimization of the weight functions nearly all information is contained in a small subset of the new second-order measures. The use of a significantly smaller number of redshift bins than 20 for the nulling considerably degrades parameter constraints under conservative assumptions, emphasizing the need for detailed redshift information.
In recent years cosmic shear, the weak gravitational lensing effect by the large-scale structure of the Universe, has proven to be one of the observational pillars on which the cosmological concordance model is founded. Several cosmic shear statistic
s have been developed in order to analyze data from surveys. For the covariances of the prevalent second-order measures we present simple and handy formulae, valid under the assumptions of Gaussian density fluctuations and a simple survey geometry. We also formulate these results in the context of shear tomography, i.e. the inclusion of redshift information, and generalize them to arbitrary data field geometries. We define estimators for the E- and B-mode projected power spectra and show them to be unbiased in the case of Gaussianity and a simple survey geometry. From the covariance of these estimators we demonstrate how to derive covariances of arbitrary combinations of second-order cosmic shear measures. We then recalculate the power spectrum covariance for general survey geometries and examine the bias thereby introduced on the estimators for exemplary configurations. Our results for the covariances are considerably simpler than and analytically shown to be equivalent to the real-space approach presented in the first paper of this series. We find good agreement with other numerical evaluations and confirm the general properties of the covariance matrices. The studies of the specific survey configurations suggest that our simplified covariances may be employed for realistic survey geometries to good approximation.
The weak gravitational lensing effect, small coherent distortions of galaxy images by means of a gravitational tidal field, can be used to study the relation between the matter and galaxy distribution. In this context, weak lensing has so far only be
en used for considering a second-order correlation function that relates the matter density and galaxy number density as a function of separation. We implement two new, third-order correlation functions that have recently been suggested in the literature, and apply them to the Red-Sequence Cluster Survey. We demonstrate that it is possible, even with already existing data, to make significant measurements of third-order lensing correlations. We develop an optimised computer code for the correlation functions. To test its reliability a set of tests are performed. The correlation functions are transformed to aperture statistics, which allow easy tests for remaining systematics in the data. In order to further verify the robustness of our measurement, the signal is shown to vanish when randomising the source ellipticities. Finally, the lensing signal is compared to crude predictions based on the halo-model. On angular scales between roughly 1 arcmin and 11 arcmin a significant third-order correlation between two lens positions and one source ellipticity is found. We discuss this correlation function as a novel tool to study the average matter environment of pairs of galaxies. Correlating two source ellipticities and one lens position yields a less significant but nevertheless detectable signal on a scale of 4 arcmin. Both signals lie roughly within the range expected by theory which supports their cosmological origin.[ABRIDGED]
We have measured the stellar velocity dispersions (sigma_*) and estimated the central black hole (BH) masses for over 900 broad-line active galactic nuclei (AGNs) observed with the Sloan Digital Sky Survey. The sample includes objects which have reds
hifts up to z=0.452, high quality spectra, and host galaxy spectra dominated by an early-type (bulge) component. The AGN and host galaxy spectral components were decomposed using an eigenspectrum technique. The BH masses (M_BH) were estimated from the AGN broad-line widths, and the velocity dispersions were measured from the stellar absorption spectra of the host galaxies. The range of black hole masses covered by the sample is approximately 10^6 < M_BH < 10^9 M_Sun. The host galaxy luminosity-velocity dispersion relationship follows the well-known Faber-Jackson relation for early-type galaxies, with a power-law slope 4.33+-0.21. The estimated BH masses are correlated with both the host luminosities (L_{H}) and the stellar velocity dispersions (sigma_*), similar to the relationships found for low-redshift, bulge-dominated galaxies. The intrinsic scatter in the correlations are large (~0.4 dex), but the very large sample size allows tight constraints to be placed on the mean relationships: M_BH ~ L_H^{0.73+-0.05} and M_BH ~ sigma_*^{3.34+-0.24}. The amplitude of the M_BH-sigma_* relation depends on the estimated Eddington ratio, such that objects with larger Eddington ratios have smaller black hole masses than expected at a given velocity dispersion.
We introduce an optimized data vector of cosmic shear measures (N). This data vector has high information content, is not sensitive against B-mode contamination and only shows small correlation between data points of different angular scales. We show
that a data vector of the two-point correlation function (2PCF) in general contains more information on cosmological parameters compared to a data vector of the aperture mass dispersion. Reason for this is the fact that <M_ap^2> lacks the information of the convergence power spectrum (P_kappa) on large angular scales, which is contained in the 2PCF data vector. Therefore we create a combined data vector N, which retains the advantages of <M_ap^2> and in addition is also sensitive to the large-scale information of P_kappa. We compare the information content of the three data vectors by performing a detailed likelihood analysis and use ray-tracing simulations to derive the covariance matrices. In the last part of the paper we contaminate all data vectors with B-modes on small angular scales and examine their robustness against this contamination.The combined data vector strongly improves constraints on cosmological parameters compared to <M_ap^2>. Although, in case of a pure E-mode signal the information content of the 2PCF is higher, in the more realistic case where B-modes are present the 2PCF data vector is strongly contaminated and yields biased cosmological parameter estimates. N shows to be robust against this contamination. Furthermore the individual data points of N show a much smaller correlation compared to the 2PCF leading to an almost diagonal covariance matrix.
We present the fourth edition of the Sloan Digital Sky Survey (SDSS) Quasar Catalog. The catalog contains 77,429 objects; this is an increase of over 30,000 entries since the previous edition. The catalog consists of the objects in the SDSS Fifth Dat
a Release that have luminosities larger than M_i = -22.0 (in a cosmology with H_0 = 70 km/s/Mpc, Omega_M = 0.3, and Omega_Lambda = 0.7) have at least one emission line with FWHM larger than 1000 km/s, or have interesting/complex absorption features, are fainter than i=15.0, and have highly reliable redshifts. The area covered by the catalog is 5740 sq. deg. The quasar redshifts range from 0.08 to 5.41, with a median value of 1.48; the catalog includes 891 quasars at redshifts greater than four, of which 36 are at redshifts greater than five. Approximately half of the catalog quasars have i < 19; nearly all have i < 21. For each object the catalog presents positions accurate to better than 0.2 arcsec. rms per coordinate, five-band (ugriz) CCD-based photometry with typical accuracy of 0.03 mag, and information on the morphology and selection method. The catalog also contains basic radio, near-infrared, and X-ray emission properties of the quasars, when available, from other large-area surveys. The calibrated digital spectra cover the wavelength region 3800--9200A at a spectral resolution of ~2000. The spectra can be retrieved from the public database using the information provided in the catalog. The average SDSS colors of quasars as a function of redshift, derived from the catalog entries, are presented in tabular form. Approximately 96% of the objects in the catalog were discovered by the SDSS.
