An analogue of Ryser's Theorem for partial Sudoku squares

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can be completed if and only if $R$ satisfies Hall's Condition for partial latin squares. We define Hall's Condition for partial Sudoku squares and show that Hall's Condition for partial Sudoku squares gives a criterion for the completion of partial Sudoku rectangles that is both necessary and sufficient. In the particular case where $n=pq$, $p|r$, $q|s$, the result is especially simple, as we show that any $r \times s$ partial $(p,q)$-Sudoku rectangle can be completed (no further condition being necessary).Comment: 19 pages, 10 figure

A softer look at MCG--6-30-15 with XMM-Newton

We present analysis and results from the Reflection Grating Spectrometer during the 320 ks XMM observation of the Seyfert 1 galaxy MCG-6-30-15. The spectrum is marked by a sharp drop in flux at 0.7 keV which has been interpreted by Branduardi-Raymont et al. as the blue wing of a relativistic OVIII emission line and by Lee at al. as a dusty warm absorber. We find that the drop is well explained by the FeI L2,3 absorption edges and obtain reasonable fits over the 0.32-1.7 keV band using a multizone, dusty warm absorber model. Some residuals remain which could be due to emission from a relativistic disc, but at a much weaker level than from any model relying on relativistic emission lines alone. A model based on such emission lines can be made to fit if sufficient (warm) absorption is added, although the line strengths exceed those expected. The EPIC pn difference spectrum between the highest and lowest flux states of the source indicates that this is a power-law in the 3-10 keV band which, if extrapolated to lower energies, reveals the absorption function acting on the intrinsic spectrum, provided that any emission lines do not scale exactly with the continuum. We find that this function matches our dusty warm absorber model well. The soft X-ray spectrum is therefore dominated by absorption structures, with the equivalent width of any individual emission lines in the residuals being below about 30 eV. (abridged)Comment: 8 pages, 10 figures, submitted to MNRA

The Impact of Lidar Detection Sensitivity on Assessing Aerosol Direct Radiative Effects

Spaceborne lidar observations have great potential to provide accurate global estimates of the aerosol direct radiative effect (DRE) in both clear and cloudy conditions. However, comparisons between observations from the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite (CALIPSO) and multiple years of Atmospheric Radiation Measurement (ARM) programs ground-based Raman lidars (RL) show that CALIPSO does not detect all radiatively significant aerosol, i.e. aerosol that directly modifies the Earths radiation budget. We estimated that using CALIPSO observations results in an underestimate of the magnitude of the global mean aerosol DRE by up to 54%. The ARM RL datasets along with NASA Langley airborne high spectral resolution lidar (HSRL) data from multiple field campaigns are used to compute the detection sensitivity required to accurately resolve the aerosol DRE. This shows that a lidar with a backscatter coefficient detection sensitivity of about 12x10(exp -4)km(exp -1)sr(exp -1) at 532nm would resolve all the aerosol needed to derive the DRE to within 1%

Rapid X-ray Variability of Seyfert 1 Galaxies

The rapid and seemingly random fluctuations in X-ray luminosity of Seyfert galaxies provided early support for the standard model in which Seyferts are powered by a supermassive black hole fed from an accretion disc. However, since EXOSAT there has been little opportunity to advance our understanding of the most rapid X-ray variability. Observations with XMM-Newton have changed this. We discuss some recent results obtained from XMM-Newton observations of Seyfert 1 galaxies. Particular attention will be given to the remarkable similarity found between the timing properties of Seyferts and black hole X-ray binaries, including the power spectrum and the cross spectrum (time delays and coherence), and their implications for the physical processes at work in Seyferts.Comment: To appear in From X-ray Binaries to Quasars: Black Hole Accretion on All Mass Scales, ed. T. J. Maccarone, R. P. Fender, and L. C. Ho (Dordrecht: Kluwer

Thermally stable laminating resins Final report

Polyimide resin and resin reinforced composite