Beatrice Amrhein

A case study of multi-threaded Gröbner basis completion

We investigate sources of parallelism in the Grobner Basis algorithm for their practical use on the desk-top. Our execution environment is a standard multi-processor workstation, and our parallel programming environment is PARSAC-2 on top of a multi-t breaded operating system. We invest igate the performance of two main variants of our master parallel algorithm on a standard set of examples. The first version exploits only work parallelism in a strategy compliant way. The second version investigates search parallelism in addkion, where large super-linear speedups can be obtained. These speedups are due to improved S-polynomial selection behavior and therefore camy over to single processor machines. Since we obtain our parallel variants by a controlled variation of only a few parameters in the master algorithm, we obtain new insights into the way in which different sources of parallelism interact in Grobner Basis completion.

On the walk

Abstract The Grobner Walk is a basis conversion method proposed by Collart, Kalkbrener, and Mall. It converts a given Grobner basis G of a (possibly positive dimensional) polynomial ideal I to a Grobner basis G′ of I with respect to another term order. The target Grobner basis is approached in several steps (the Walk), each performing a simpler Grobner basis computation. We address a host of questions associated with this method: alternative ways of presenting the main algorithm, algorithmic variations and refinements, implementation techniques, promising applications, and its practical performance, including a comparison with the FGLM conversion method. Our results show that the Walk has the potential to become a key tool for computing and manipulating ideal bases and solving systems of equations.

Visualizations for mathematics courses based on a computer algebra system

Abstract This paper discusses the introduction of a computer-algebra system to perform visualizations in mathematics education. With the Illustrated Mathematics project, we provide a comprehensive collection of graphics and animations for various topics in mathematics, which can directly be used for teaching. Because the programs (written in Mathematica ) we used for the creation of this collection are included, it is easy to design new examples by modifying parameters in existing examples. Therefore, Illustrated Mathematics can be used as a first step of the introduction of a computer-algebra system in mathematics education. In the second part, we report on the development of a workbench and learning environment, with which the students can discover the mathematical concepts by themselves. For that, the given examples have to be didactically enriched and processed. By creating their own visualizations of increasing complexity, the students improve their knowledge about the underlying computer-algebra system.

Parallel Completion Techniques

We survey and categorize techniques for the parallelization of completion procedures. We cover both Knuth-Bendix term completion and Buchberger’s algorithm for Grobner Basis completion. The survey includes a discussion of parallel installations of these algorithms in our own systems PaReDuX and GB/PARSAC, running on parallel desktop workstations.

Birkhoff's HSP-Theorem for Cumulative Logic Programs

Birkhoffs HSP theorem is that the models of a set of algebraic equations form a variety, i.e. a category of algebras which admits homomorphic images, subalgebras and products. We show here first, that every equational set of retract structures in combinatory logic is a variety, and second, that every set of combinators, closed under certain operations, is equational. It follows that the models of cumulative logic programs form an equational variety.

Analysis Alive—an Integrated Learning Environment for Higher Mathematics

We give an overview of Analysis Alive, a learning environment for Analysis at the level of a first year course at a German university. Analysis Alive is a traditional textbook in combination with a CD-ROM containing graphics and animations realized with Maple worksheets. The goals of our project are to introduce the component of experimentation with visual feedback into the teaching and learning of Analysis. Although Analysis Alive is based on the use of computers and Computer Algebra Systems, it requires only minimal knowledge of computers and computer algebra systems.