## Fachbereich Mathematik

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- Monomial Representations for Gröbner Bases Computations (1998)
- Monomial representations and operations for Gröbner bases computations are investigated from an implementation point of view. The technique ofvectorized monomial operations is introduced and it is shown how it expedites computations of Gröbner bases. Furthermore, a rank-based monomialrepresentation and comparison technique is examined and it is concluded that this technique does not yield an additional speedup over vectorizedcomparisons. Extensive benchmark tests with the Computer Algebra System SINGULAR are used to evaluate these concepts.

- A Note on Nielsen Reduction and Coset Enumeration (1998)
- Groups can be studied using methods from different fields such as combinatorial group theory or string rewriting. Recently techniques from Gröbner basis theory for free monoid rings (non-commutative polynomial rings) respectively free group rings have been added to the set of methods due to the fact that monoid and group presentations (in terms of string rewriting systems) can be linked to special polynomials called binomials. In the same mood, the aim of this paper is to discuss the relation between Nielsen reduced sets of generators and the Todd-Coxeter coset enumeration procedure on the one side and the Gröbner basis theory for free group rings on the other. While it is well-known that there is a strong relationship between Buchberger's algorithm and the Knuth-Bendix completion procedure, and there are interpretations of the Todd-Coxeter coset enumeration procedure using the Knuth-Bendix procedure for special cases, our aim is to show how a verbatim interpretation of the Todd-Coxeter procedure can be obtained by linking recent Gröbner techniques like prefix Gröbner bases and the FGLM algorithm as a tool to study the duality of ideals. As a side product our procedure computes Nielsen reduced generating sets for subgroups in finitely generated free groups.

- MRC - A System for Computing Gröbner Bases in Monoid and Group Rings (1998)
- Gröbner bases and Buchberger's algorithm have been generalized to monoid and group rings. In this paper we summarize procedures from this field and present a description of their implementation in the system Mrc V 1.0.

- Reinitialization for Level Set Methods (1998)
- In the following an introduction to the level set method will be givenso that one becomes aware of the arising problems, which lead to the needof reinitialization. The problems concerning reinitialization itself will be analysed more detailed and a solution for area loss will be proposed. This solution consists in a combination of the commonly used PDE for reinitialization and extrapolation around the zero level set. Numericalexperiments show rather satisfactory results as far as area loss and computation of curvature are concerned.

- Bicriteria cost versus service analysis of the distribution network of a chemical company (1998)
- In order to improve the distribution system for the Nordic countries the BASF AG considered 13 alternative scenarios to the existing system. These involved the construction of warehouses at various locations. For every scenario the transportation, storage, and handling cost incurred was to be as low as possible, where restrictions on the delivery time were given. The scenarios were evaluated according to (minimal) total cost and weighted average delivery time. The results led to a restriction to only three cases, involving only one new warehouse each. For these a more accurate model for the cost was developped and evaluated, yielding results similar to a simple linear model. Since there were no clear preferences between cost and delivery time, the final decision was chosen to represent a compromise between the two criteria.

- Robust facility location (1998)
- Let A be a nonempty finite subset of R^2 representing the geographical coordinates of a set of demand points (towns, ...), to be served by a facility, whose location within a given region S is sought. Assuming that the unit cost for a in A if the facility is located at x in S is proportional to dist(x,a) - the distance from x to a - and that demand of point a is given by w_a, minimizing the total trnsportation cost TC(w,x) amounts to solving the Weber problem. In practice, it may be the case, however, that the demand vector w is not known, and only an estimator {hat w} can be provided. Moreover the errors in sich estimation process may be non-negligible. We propose a new model for this situation: select a threshold valus B 0 representing the highest admissible transportation cost. Define the robustness p of a location x as the minimum increase in demand needed to become inadmissible, i.e. p(x) = min{||w^*-{hat w}|| : TC(w^*,x) B, w^* = 0} and solve then the optimization problem max_{x in S} p(x) to get the most robust location.

- Bootstrapping neural networks (1998)
- Knowledge about the distribution of a statistical estimator is important for various purposes like, for example, the construction of confidence intervals for model parameters or the determiation of critical values of tests. A widely used method to estimate this distribution is the so-called bootstrap which is based on an imitation of the probabilistic structure of the data generating process on the basis of the information provided by a given set of random observations. In this paper we investigate this classical method in the context of artificial neural networks used for estimating a mapping from input to output space. We establish consistency results for bootstrap estimates of the distribution of parameter estimates.

- A steady-state particle method for the Boltzmann equation (1998)
- We present a particle method for the numerical simulation of boundary value problems for the steady-state Boltzmann equation. Referring to some recent results concerning steady-state schemes, the current approach may be used for multi-dimensional problems, where the collision scattering kernel is not restricted to Maxwellian molecules. The efficiency of the new approach is demonstrated by some numerical results obtained from simulations for the (two-dimensional) BEnard's instability in a rarefied gas flow.

- An adaptive domain decomposition procedure for Boltzmann and Euler equations (1998)
- In this paper we present a domain decomposition approach for the coupling of Boltzmann and Euler equations. Particle methods are used for both equations. This leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains. Adaptive time and space discretizations and a direct coupling procedure leads to considerable gains in CPU time compared to a solution of the full Boltzmann equation. Several test cases involving a large range of Knudsen numbers are numerically investigated.