New developments in the theory of Groebner bases and applications to formal verification

We present foundational work on standard bases over rings and on Boolean Groebner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Groebner basis in the polynomial ring over Z/2n while the bit-level model leads to Boolean Groebner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Groebner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of Pure and Applied Algebr

A simple but rigorous micro-macro transition

This paper is devoted to a case study of micro-macro transitions. The main objective is the mathematically rigorous description of the macroscopic behavior of highly oscillating microscopic variables. In particular, we show that the theory of Young measures provides an elegant approach to this problem. A nontrivial application of the results is given in WIAS-Preprint 724

On the approximation of periodic traveling waves for the nonlinear atomic chain

We study a scheme from \cite{FV99}, which allows to approximate periodic traveling waves in the nonlinear atomic chain with nearest neighbour interactions. We prove a compactness result for this scheme, and derive some generalizations. Moreover, we discuss the thermodynamic properties of traveling waves

Towards rigorous micro-macro transitions: The microscopic oscillator motion

The atomic chain whose dynamics evolve according to Newton's equations of motion serves as a simple microscopic many-particle system for an analysis of the macroscopic or thermodynamic limit. If the interaction potential has a sufficient strong repulsive part, it is possible to create a special microscopic motion, that we call oscillator motion, which is simpler as the classical thermal motion. However, also the oscillator motion leads to temperature, a Gibbs equation and an entropy. In the current paper we derive the thermodynamics for the oscillator motion without the subtle replacement of the many-particle sytem by a single equation of motion as it is done in [4]. Furthermore we introduce a different mathematical setting for micro-macro transitions as in [4], which is better suited for a rigorous treatment

Numerical experiments on the modulation theory for the nonlinear atomic chain

Modulation theory with periodic traveling waves is a powerful, but not rigorous tool to derive a thermodynamic description for the atomic chain. We investigate the validity of this theory by means of several numerical experiments

A gas-liquid solid phase peptide and protein sequenator

A new miniaturized protein and peptide sequenator has been constructed which uses gas phase reagents at the coupling and cleavage steps of the Edman degradation. The sample is embedded in a matrix of Polybrene dried onto a porous glass fiber disc located in a small cartridge-style reaction cell. The protein or peptide, though not covalently attached to the support, is essentially immobile throughout the degradative cycle, since only relatively apolar, liquid phase solvents pass through the cell. This instrument can give useful sequence data on as little as 5 pmol or protein, can perform extended sequence runs (greater than 30 residues) on subnanomole quantities of proteins purified by sodium dodecyl sulfate-polyacrylamide gel electrophoresis, and can sequence hydrophobic peptides to completion. The sequenator is characterized by a high repetitive yield during the degradation, low reagent consumption, low maintenance requirements, and a degradative cycle time of only 50 min using a complete double cleavage program