Convergence of normal form transformations: The role of symmetries

We discuss the convergence problem for coordinate transformations which take a given vector field into Poincar\'e-Dulac normal form. We show that the presence of linear or nonlinear Lie point symmetries can guaranteee convergence of these normalizing transformations, in a number of scenarios. As an application, we consider a class of bifurcation problems.Comment: 20 pages, no figure

Dimension increase and splitting for Poincare'-Dulac normal forms

Integration of nonlinear dynamical systems is usually seen as associated to a symmetry reduction, e.g. via momentum map. In Lax integrable systems, as pointed out by Kazhdan, Kostant and Sternberg in discussing the Calogero system, one proceeds in the opposite way, enlarging the nonlinear system to a system of greater dimension. We discuss how this approach is also fruitful in studying non integrable systems, focusing on systems in normal form.Comment: 16 page

Side conditions for ordinary differential equations

We specialize Olver's and Rosenau's side condition heuristics for the determination of particular invariant sets of ordinary differential equations. It turns out that side conditions of so-called LaSalle type are of special interest. Moreover we put side condition properties of symmetric and partially symmetric equations in a wider context. In the final section we present an application to parameter-dependent systems, in particular to quasi-steady state for chemical reactions.Comment: To appear in J. of Lie Theor

About the morphology of dwarf spheroidal galaxies and their dark matter content

The morphological properties of the Carina, Sculptor and Fornax dwarfs are investigated using new wide field data with a total area of 29 square degrees. The stellar density maps are derived, hinting that Sculptor possesses tidal tails indicating interaction with the Milky Way. Contrary to previous studies we cannot find any sign of breaks in the density profiles for the Carina and Fornax dwarfs. The possible existence of tidal tails in Sculptor and of King limiting radii in Fornax and Carina are used to derive global M/L ratios, without using kinematic data. By matching those M/L ratios to kinematically derived values we are able to constrain the orbital parameters of the three dwarfs. Fornax cannot have M/L smaller than 3 and must be close to its perigalacticon now. The other extreme is Sculptor that needs to be on an orbit with an eccentricity bigger than 0.5 to be able to form tidal tails despite its kinematic M/L.Comment: 9 pages, 7 figures, accepted by A&

Polynomial Structure of the (Open) Topological String Partition Function

In this paper we show that the polynomial structure of the topological string partition function found by Yamaguchi and Yau for the quintic holds for an arbitrary Calabi-Yau manifold with any number of moduli. Furthermore, we generalize these results to the open topological string partition function as discussed recently by Walcher and reproduce his results for the real quintic.Comment: 15 page

Status of Nucleon Resonances with Masses M<MN+MπM<M_N+M_{\pi}

We discuss different interpretations of peaks observed a few years ago by Tatischeff et al. in missing mass spectra of the reaction $pp->pi^+pX$, which were declared as new exited nucleon states with small masses. A study of the possible production of such states in the process $\gamma p->pi^+N^*->pi^+ +\gamma\gamma n$ by analyzing the invariant mass spectrum of $\gamma\gamma n$ is proposed. It is shown that the data, obtained recently at MAMI-B, can allow to analyze this process and to get information about an existence of exited nucleon states with small masses.Comment: 4 pages, 2 figures, LaTeX with ws-p8-50x6-00.cls. Talk presented at the NSTAR2001 Workshop, Mainz, Germany, March 7-10, 200

Abundance patterns in early-type galaxies: is there a 'knee' in the [Fe/H] vs. [alpha/Fe] relation?

Early-type galaxies (ETGs) are known to be enhanced in alpha elements, in accordance with their old ages and short formation timescales. In this contribution we aim to resolve the enrichment histories of ETGs. This means we study the abundance of Fe ([Fe/H]) and the alpha-element groups ([alpha/Fe]) separately for stars older than 9.5 Gyr ([Fe/H]o, [alpha/Fe]o) and for stars between 1.5 and 9.5 Gyr ([Fe/H]i, [alpha/Fe]i). Through extensive simulation we show that we can indeed recover the enrichment history per galaxy. We then analyze a spectroscopic sample of 2286 early-type galaxies from the SDSS selected to be ETGs. We separate out those galaxies for which the abundance of iron in stars grows throughout the lifetime of the galaxy, i.e. in which [Fe/H]o < [Fe/H]i. We confirm earlier work where the [Fe/H] and [alpha/Fe] parameters are correlated with the mass and velocity dispersion of ETGs. We emphasize that the strongest relation is between [alpha/Fe] and age. This relation falls into two regimes, one with a steep slope for old galaxies and one with a shallow slope for younger ETGs. The vast majority of ETGs in our sample do not show the 'knee' in the plot of [Fe/H] vs. [alpha/Fe] commonly observed in local group galaxies. This implies that for the vast majority of ETGs, the stars younger than 9.5 Gyrs are likely to have been accreted or formed from accreted gas. The properties of the intermediate-age stars in accretion-dominated ETGs indicate that mass growth through late (minor) mergers in ETGs is dominated by galaxies with low [Fe/H] and low [alpha/Fe]. The method of reconstructing the stellar enrichment histories of ETGs introduced in this paper promises to constrain the star formation and mass assembly histories of large samples of galaxies in a unique way.Comment: 22 pages, 25 figures, accepted for publication by A&

Differential stellar population models: how to reliably measure [Fe/H] and [alpha/Fe] in galaxies

We present differential stellar population models, which allow improved determinations of the ages, iron and alpha-element abundances of old stellar populations from spectral fitting. These new models are calibrated at solar abundances using the predictions from classical, semi-empirical stellar population models. We then use the predictive power of fully synthetic models to compute predictions for different [Fe/H] and [alpha/Fe]. We show that these new differential models provide remarkably accurate fits to the integrated optical spectra of the bulge globular clusters NGC6528 and NGC6553, and that the inferred [Fe/H] and [alpha/Fe] agree with values derived elsewhere from stellar photometry and spectroscopy. The analysis of a small sample of SDSS early-type galaxies further confirms that our alpha-enhanced models provide a better fit to the spectra of massive ellipticals than the solar-scaled ones. Our approach opens new opportunities for precision measurements of abundance ratios in galaxies.Comment: 5 pages, 5 figures, MNRAS in pres