The Hall algebra and the composition monoid

Let Q be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid, as introduced by M. Reineke, and the generic composition algebra, as introduced by C. M. Ringel, specialised at q=0. In this thesis we continue their work. We show that if Q is a Dynkin quiver or an oriented cycle, then the composition algebra at q=0 is isomorphic to the monoid algebra of the composition monoid. Moreover, if Q is an acyclic, extended Dynkin quiver, we show that there exists an epimorphism from the composition algebra at q=0 to the monoid algebra of the composition monoid, and we describe its non-trivial kernel. Our main tool is a geometric version of BGP reflection functors on quiver Grassmannians and quiver flags, that is varieties consisting of filtrations of a fixed representation by subrepresentations of fixed dimension vectors. These functors enable us to calculate various structure constants of the composition algebra. Moreover, we investigate geometric properties of quiver flags and quiver Grassmannians, and show that under certain conditions, quiver flags are irreducible and smooth. If, in addition, we have a counting polynomial, these properties imply the positivity of the Euler characteristic of the quiver flag.Comment: 111 pages, doctoral thesis University of Paderborn (2009

MASS TRANSPORT THROUGH CHARGED MEMBRANES Dieter Bothe and Jan Pruss

A modern technique for desalination or softening of water is the so-called nano ltration by means of membranes which carry a xed electric charge. For the mathematical modeling of such processes, two features are particularly important. Firstly, the distribution of ionic species generates an electric eld which in turn aects the uxes of these ions. Therefore, besides diusion and convection, electromigration has to be taken into account. This also leads to a strong coupling of the concentrations of all charged species, which can often be adequately incorporated into the model via the assumption of electroneutrality. Secondly, electrical double-layers (Donnan potential) build up at the surface of the membrane, which cause discontinuities in the ionic concentration pro les. We deduce a mathematical model for such a nano ltration process. This leads to a strongly coupled quasilinear parabolic system with nonlinear transmission and dynamical boundary conditions. By means of degree theory we obtain existence of stationary solutions, while L -maximal regularity is employed to get local strong wellposedness of this model. AMS Classi cations Primary: 34B60, 35K50, 92E20 Secondary: 35K60, 35Q80, 47H11, 78A35 Key Words and Phrases diusion-migration system, electroneutrality, quasilinear parabolic system, dynamical boundary condition, degree theory, maximal regularity, nano ltration Running Head: Mass Transport Through Charged Membranes 1

EVA Abschlussbericht des Projekts 'Effective Validation'

TIB: RO 802 (1987,2) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Zentrum für Technomathematik

Error bounds on a semi-discrete finite element approximation of the weak solution to a one phase moving-boundary system describing concrete carbonatio