Mixed-type functional differential equations: A numerical approach

This is a PDF version of a preprint submitted to Elsevier. The definitive version was published in Journal of computational and applied mathematics and is available at www.elsevier.comThis preprint discusses mixed-type functional equations

Theory and numerics for multi-term periodic delay differential equations, small solutions and their detection

We summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step size is applied to approximate the solution to (†) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples

The XMM-Newton EPIC Background and the production of Background Blank Sky Event Files

We describe in detail the nature of XMM-Newton EPIC background and its various complex components, summarising the new findings of the XMM-Newton EPIC background working group, and provide XMM-Newton background blank sky event files for use in the data analysis of diffuse and extended sources. Blank sky event file data sets are produced from the stacking of data, taken from 189 observations resulting from the Second XMM-Newton Serendipitous Source Catalogue (2XMMp) reprocessing. The data underwent several filtering steps, using a revised and improved method over previous work, which we describe in detail. We investigate several properties of the final blank sky data sets. The user is directed to the location of the final data sets. There is a final data set for each EPIC instrument-filter-mode combination.Comment: Paper accepted by A&A 22 December 2006. 14 pages, 8 figures. Paper can also be found at http://www.star.le.ac.uk/~jac48/publications

In-orbit Vignetting Calibrations of XMM-Newton Telescopes

We describe measurements of the mirror vignetting in the XMM-Newton Observatory made in-orbit, using observations of SNR G21.5-09 and SNR 3C58 with the EPIC imaging cameras. The instrument features that complicate these measurements are briefly described. We show the spatial and energy dependences of measured vignetting, outlining assumptions made in deriving the eventual agreement between simulation and measurement. Alternate methods to confirm these are described, including an assessment of source elongation with off-axis angle, the surface brightness distribution of the diffuse X-ray background, and the consistency of Coma cluster emission at different position angles. A synthesis of these measurements leads to a change in the XMM calibration data base, for the optical axis of two of the three telescopes, by in excess of 1 arcminute. This has a small but measureable effect on the assumed spectral responses of the cameras for on-axis targets.Comment: Accepted by Experimental Astronomy. 26 pages, 18 figure

Constituents of the soft X-ray background

The X-ray background is generated by various classes of objects and variety of emission mechanisms. Relative contribution of individual components depends on energy. The goal is to assess the integral emission of the major components of the soft X-ray background (extragalactic discrete sources dominated by AGNs, galactic plasma, and the Warm/Hot Intergalactic Medium), investigating the angular structure of the background. Fluctuations of the background are measured using the auto-correlation function of the XRB determined in 5 energy bands between 0.3 and 4.5 keV. The investigation is based on the extensive observational data set selected from the XMM-Newton archives. Amplitudes of the auto-correlation functions calculated in three energy bands above ~1 keV are consistent with the conjecture that the background fluctuations result solely from clustering of sources which produce the background. At energies below 1 keV the relative fluctuation amplitude decreases indicating that a fraction of the soft XRB is associated with a smooth plasma emission in the Galaxy. It is shown, however, that the mean spectrum of extragalactic discrete sources steepens in the soft X-rays and is not well represented by a single power law in the energy range 0.3-4.5 keV. The WHIM contribution to the total background fluctuations is small and consistent with the WHIM properties derived from the cross-correlation of the XRB with galaxies.Comment: 7 pages, 2 figure

Existence theory for a class of evolutionary equations with time-lag, studied via integral equation formulations

In discussions of certain neutral delay differential equations in Hale’s form, the relationship of the original problem with an integrated form (an integral equation) proves to be helpful in considering existence and uniqueness of a solution and sensitivity to initial data. Although the theory is generally based on the assumption that a solution is continuous, natural solutions of neutral delay differential equations of the type considered may be discontinuous. This difficulty is resolved by relating the discontinuous solution to its restrictions on appropriate (half-open) subintervals where they are continuous and can be regarded as solutions of related integral equations. Existence and unicity theories then follow. Furthermore, it is seen that the discontinuous solutions can be regarded as solutions in the sense of Caratheodory (where this concept is adapted from the theory of ordinary differential equations, recast as integral equations)