On Solution Sets of Information Inequalities

We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce important properties of Bayesian networks, which is important within causal inference.

Numerical Methods for the Unsteady Compressible Navier-Stokes Equations

We consider numerical methods for the compressible time dependent Navier-Stokes equations, discussing the spatial discretization by Finite Volume and Discontinuous Galerkin methods, the time integration by time adaptive implicit Runge-Kutta and Rosenbrock methods and the solution of the appearing nonlinear and linear equations systems by preconditioned Jacobian-Free Newton-Krylov, as well as Multigrid methods. As applications, thermal Fluid structure interaction and other unsteady flow problems are considered. The text is aimed at both mathematicians and engineers.

Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [Weinstein1991] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

On Parallel Communicating Grammar Systems and Correctness Preserving Restarting Automata

This paper contributes to the study of Freely Rewriting Restarting Automata (FRR-automata) and Parallel Communicating Grammar Systems (PCGS), which both are useful models in computational linguistics. For PCGSs we study two complexity measures called 'generation complexity' and 'distribution complexity', and we prove that a PCGS Pi, for which the generation complexity and the distribution complexity are both bounded by constants, can be transformed into a freely rewriting restarting automaton of a very restricted form. From this characterization it follows that the language L(Pi) generated by Pi is semi-linear, that its characteristic analysis is of polynomial size, and that this analysis can be computed in polynomial time.

On CD-systems of stateless deterministic R-automata with window size one

We study cooperating distributed systems (CD-systems) of restarting automata that are very restricted: they are deterministic, they cannot rewrite, but only delete symbols, they restart immediately after performing a delete operation, they are stateless, and they have a read/write window of size 1 only, that is, these are stateless deterministic R(1)-automata. We study the expressive power of these systems by relating the class of languages that they accept by mode =1 computations to other well-studied language classes, showing in particular that this class only contains semi-linear languages, and that it includes all rational trace languages. In addition, we investigate the closure and non-closure properties of this class of languages and some of its algorithmic properties.

On Globally Deterministic CD-Systems of Stateless R-Automata with Window Size One

It is known that cooperating distributed systems (CD-systems) of stateless deterministic restarting automata with window size 1 accept a class of semi-linear languages that properly includes all rational trace languages. Although the component automata of such a CD-system are all deterministic, in general the CD-system itself is not, as in each of its computations, the initial component and the successor components are still chosen nondeterministically. Here we study CD-systems of stateless deterministic restarting automata with window size 1 that are themselves completely deterministic. In fact, we consider two such types of CD-systems, the strictly deterministic systems and the globally deterministic systems.