Discrepancy estimates for sequences: new results and open problems

In this paper we give an overview of recent results on (upper and lower) discrepancy estimates for (concrete) sequences in the unit-cube. In particular we also give an overview of discrepancy estimates for certain classes of hybrid sequences. Here by a hybrid sequence we understand an $(s+t)$-dimensional sequence which is a combination of an $s$-dimensional sequence of a certain type (e.g. Kronecker-, Niederreiter-, Halton-,... type) and a $t$-dimensional sequence of another type. The analysis of the discrepancy of hybrid sequences (and of their components) is a rather current and vivid branch of research. We give a collection of some challenging open problems on this topic.Comment: 17 page

Lithium abundances of halo dwarfs based on excitation temperature. I. LTE

The discovery of the Spite plateau in the abundances of 7Li for metal-poor stars led to the determination of an observationally deduced primordial lithium abundance. However, after the success of the Wilkinson Microwave Anisotropy Probe (WMAP) in determining the baryon density, OmegaBh^2, there was a discrepancy between observationally determined and theoretically determined abundances in the case of 7Li. One of the most important uncertain factors in the calculation of the stellar 7Li abundance is the effective temperature, Teff. We use sixteen metal-poor halo dwarfs to calculate new Teff values using the excitation energy method. With this temperature scale we then calculate new Li abundances for this group of stars in an attempt to resolve the 7Li discrepancy. Using high signal-to-noise (S/N ~ 100) spectra of 16 metal-poor halo dwarfs, obtained with the UCLES spectrograph on the AAT, measurements of equivalent widths from a set of unblended FeI lines are made. These equivalent widths are then used to calculate new Teff values with the use of the single line radiative transfer program WIDTH6, where we have constrained the gravity using either theoretical isochrones or the Hipparcos parallax, rather than the ionization balance. The lithium abundances of the stars are calculated with these temperatures. The physical parameters are derived for the 16 programme stars, and two standards. These include Teff, log g, [Fe/H], microturbulence and 7Li abundances. A comparison between the temperature scale of this work and those adopted by others has been undertaken. We find good consistency with the temperatures derived from the Halpha line by Asplund et al. (2006), but not with the hotter scale of Melendez & Ramirez (2004).Comment: 13 pages, 9 figure

Discrepancy bounds for low-dimensional point sets

The class of $(t,m,s)$-nets and $(t,s)$-sequences, introduced in their most general form by Niederreiter, are important examples of point sets and sequences that are commonly used in quasi-Monte Carlo algorithms for integration and approximation. Low-dimensional versions of $(t,m,s)$-nets and $(t,s)$-sequences, such as Hammersley point sets and van der Corput sequences, form important sub-classes, as they are interesting mathematical objects from a theoretical point of view, and simultaneously serve as examples that make it easier to understand the structural properties of $(t,m,s)$-nets and $(t,s)$-sequences in arbitrary dimension. For these reasons, a considerable number of papers have been written on the properties of low-dimensional nets and sequences