Discrete Modified Projection Methods for Urysohn Integral Equations with Green's Function Type Kernels

In the present paper we consider discrete versions of the modified projection methods for solving a Urysohn integral equation with a kernel of the type of Green's function. For $r \geq 0,$ a space of piecewise polynomials of degree $\leq r$ with respect to an uniform partition is chosen to be the approximating space. We define a discrete orthogonal projection onto this space and replace the Urysohn integral operator by a Nystr\"{o}m approximation. The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modified projection methods. Numerical results are given for a specific example.Comment: This is the the same paper with the arXiv identifier 1904.07895, but the shortened version. A bit change in the title als

Metallicity Evolution of Damped Lyman-Alpha Galaxies

We have reanalyzed the existing data on Zinc abundances in damped Ly-alpha (DLA) absorbers to investigate whether their mean metallicity evolves with time. Most models of cosmic chemical evolution predict that the mass- weighted mean interstellar metallicity of galaxies should rise with time from a low value ~ 1/30 solar at z ~ 3 to a nearly solar value at z ~ 0. However, several previous analyses have suggested that there is little or no evolution in the global metallicity of DLAs. We have used a variety of statistical techniques to quantify the global metallicity-redshift relation and its uncertainties, taking into account both measurement and sampling errors. Three new features of our analysis are: (a) an unbinned N(H I)-weighted nonlinear chi-square fit to an exponential relation; (b) survival analysis to treat the large number of limits in the existing data; and (c) a comparison of the data with several models of cosmic chemical evolution based on an unbinned N(H I)-weighted chi-square. We find that a wider range of evolutionary rates is allowed by the present data than claimed in previous studies. The slope of the exponential fit to the N(H I)-weighted mean Zn metallicity vs. redshift relation is -0.20 plus minus 0.11 counting limits as detections and -0.27 plus minus 0.12 counting limits as zeros. Similar results are also obtained if the data are binned in redshift, and if survival analysis is used. These slopes are marginally consistent with no evolution, but are also consistent with the rates predicted by several models of cosmic chemical evolution. Finally, we outline some future measurements necessary to improve the statistics of the global metallicity-redshift relation.Comment: 25 pages, 1 figure, accepted for publication in the Astrophysical Journa