On the presentation of pointed Hopf algebras

We give a presentation in terms of generators and relations of Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. As a consequence we get an analog presentation of Nichols algebras of diagonal type

The Lens-Redshift Test Revisited

Kochanek (1992) suggested that the redshifts of gravitational lens galaxies rule out a large cosmological constant. This result was questioned by Helbig & Kayser (1996), who pointed out that selection effects related to the brightness of the lens can bias the results of this test against a high lambda value; however, we did not claim that the observations favoured a high lambda value, merely that current observational data were not sufficient to say either way, using the test as proposed by Kochanek (1992) but corrected for selection effects. Kochanek (1996) pointed out that additional information (fraction of measured lens redshifts) provides additional information which restores the sensitivity of the test to the cosmological model, at least somewhat. Here, I consider three aspects. First, I examine the accuracy of the correction to the test proposed by Kochanek (1996). Second, I compare the slightly different statistical methods which have been used in connection with this test. Third, I discuss what results can be obtained today now that more and better-defined observations are available.Comment: 4 pages, LaTeX, 2 included PostScript files; to appear in the proceedings of the XXXVth Rencontres de Moriond, "L2K - Cosmological Physics with Gravitational Lensing", J.-P. Kneib, Y. Mellier, M. Moniez & J. Tran Thanh Van (eds.); see also http://terapix.iap.fr/L2K/l2k_one.htm

The mm-zz relation for Type Ia supernovae: safety in numbers or safely without worry?

The $m$-$z$ relation for Type Ia supernovae is compatible with the cosmological concordance model if one assumes that the Universe is homogeneous, at least with respect to light propagation. This could be due to the density along each line of sight being equal to the overall cosmological density, or to `safety in numbers', with variation in the density along all lines of sight averaging out if the sample is large enough. Statistical correlations (or lack thereof) between redshifts, residuals (differences between the observed distance moduli and those calculated from the best-fitting cosmological model), and observational uncertainties suggest that the former scenario is the better description, so that one can use the traditional formula for the luminosity distance safely without worry.Comment: 5 pages, 6 figures, accepted for publication in Monthly Notices of the Royal Astronomical Society (includes small changes made while checking the proofs). Related information available at http://www.multivax.de:8000/helbig/research/publications/info/etasnia2.htm

The Current Status of CLASS

I give a brief overview of the current status of some aspects of the Cosmic Lens All-Sky Survey (CLASS): description of the survey, current list of lens systems, cosmological parameters from lensing statistics, H_0 from time delays, dark lenses, wide-separation lenses.Comment: 4 pages, LaTeX; to appear in the proceedings of the XXXVth Rencontres de Moriond, "L2K - Cosmological Physics with Gravitational Lensing", J.-P. Kneib, Y. Mellier, M. Moniez & J. Tran Thanh Van (eds.); see also http://terapix.iap.fr/L2K/l2k_one.html . Of course, such a proceedings contribution is just a snapshot of the status at the time; for more up-to-date information, follow the links from http://gladia.astro.rug.nl:8000/ceres/ceres.htm

The mm-zz relation for type Ia supernovae, locally inhomogeneous cosmological models, and the nature of dark matter

The $m$-$z$ relation for type Ia supernovae is one of the key pieces of evidence supporting the cosmological `concordance model' with $\lambda_0 \approx 0.7$ and $\Omega_0 \approx 0.3$. However, it is well known that the $m$-$z$ relation depends not only on $\lambda_0$ and $\Omega_0$ (with $H_0$ as a scale factor) but also on the density of matter along the line of sight, which is not necessarily the same as the large-scale density. I investigate to what extent the measurement of $\lambda_0$ and $\Omega_0$ depends on this density when it is characterized by the parameter $\eta$ ($0 \le \eta \le 1$), which describes the ratio of density along the line of sight to the overall density. I also discuss what constraints can be placed on $\eta$, both with and without constraints on $\lambda_0$ and $\Omega_0$ in addition to those from the $m$-$z$ relation for type~Ia supernovae.Comment: 11 pages, 17 figures, published in Monthly Notices of the Royal Astronomical Society. This version contains minor changes made while correcting proofs in order to correspond as closely as practical to the offical version. No changes in content. Related information available at http://www.astro.multivax.de:8000/helbig/research/publications/info/etasnia.htm

Cosmological parameters and the redshift distribution of gravitational lenses

For known gravitational lens systems the redshift distribution of the lenses is compared with theoretical expectations for $10^{4}$~Friedmann-Lema\^\i tre~cosmological models, which more than cover the range of possible cases. The comparison is used for assigning a relative probability to each of the models. The entire procedure is repeated for different values of the inhomogeneity parameter~$\eta$ and the limiting spectroscopic magnitude~$m_{\rm lim}$, which is important for selection effects. The dependence on these two parameters is examined in more detail for the special cases~$\lambda_{0}=0$ and $k=0$. Previous results that this method is a better probe for~$\lambda_{0}$ than $\Omega_{0}$ are confirmed, but it appears that the low probability of models with large~$\lambda_{0}$~values found using similar methods is due to a selection effect. The power of this method to discriminate between cosmological models can of course be improved if more gravitational lens systems are found. However, our numerical simulations indicate that a reasonable number of observed systems cannot deliver interesting constraints on the cosmological parameters.Comment: One self-unpacking uuencoded compressed PostScript file (created with uufiles) containing the complete text with all figures included (here 140kb gzipped, unpacked 975kb). 10 pages. Should be printable without loss on US-size paper. Paper version available on request. Changes--nothing removed, typos corrected, the following additions: one figure, one table and some text explaining the structure in the plots in more detail; more detailed explanation of the statistics used. Accepted by Astronomy and Astrophysic

Structure of the first order Reduced Density Matrix in three electron systems: A Generalized Pauli Constraints assisted study

We investigate the structure of the one-body Reduced Density Matrix (1RDM) of three electron systems, i.e. doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use Configuration Interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e. the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in Reduced Density Matrix Functional Theory minimization schemes when Generalized Pauli Constraints are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost "pinned"