## Fachbereich Mathematik

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- MP Prototype Specification (1997)

- Relating Rewriting Techniques on Monoids and Rings: Congruences on Monoids and Ideals in Monoid Rings (1997)
- A first explicit connection between finitely presented commutative monoids and ideals in polynomial rings was used 1958 by Emelichev yielding a solution tothe word problem in commutative monoids by deciding the ideal membership problem. The aim of this paper is to show in a similar fashion how congruenceson monoids and groups can be characterized by ideals in respective monoid and group rings. These characterizations enable to transfer well known resultsfrom the theory of string rewriting systems for presenting monoids and groups to the algebraic setting of subalgebras and ideals in monoid respectively grouprings. Moreover, natural one-sided congruences defined by subgroups of a group are connected to one-sided ideals in the respective group ring and hencethe subgroup problem and the ideal membership problem are directly related. For several classes of finitely presented groups we show explicitly howGröbner basis methods are related to existing solutions of the subgroup problem by rewriting methods. For the case of general monoids and submonoidsweaker results are presented. In fact it becomes clear that string rewriting methods for monoids and groups can be lifted in a natural fashion to definereduction relations in monoid and group rings.

- Splitting algorithm for vector bundles (1997)
- A new criteria for indecomposability of vector bundles on projective varieties is presented. It is deduced from a new finite algorithm computing direct sumdecompositions of graded modules over graded algebras. This algorithm applies as well to modules over local complete algebras over a field.

- String Rewriting and Gröbner Bases - A General Approach to Monoid and Group Rings (1997)
- The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review somefundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. Thetechniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Gröbnerbases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some non-commutative cases. Severalresults on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced formonoid rings having a finite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, theexistence of finite Gröbner bases for finitely generated right ideals is shown and a procedure to compute them is given.

- An algorithm for constructing isomorphisms of modules (1997)
- This paper is a continuation of a joint paper with B. Martin [MS] dealing with the problem of direct sum decompositions. The techniques of that paper areused to decide wether two modules are isomorphic or not. An positive answer to this question has many applications - for example for the classification ofmaximal Cohen-Macaulay module over local algebras as well as for the study of projective modules. Up to now computer algebra is normally dealing withequality of ideals or modules which depends on chosen embeddings. The present algorithm allows to switch to isomorphism classes which is more natural inthe sense of commutative algebra and algebraic geometry.

- A multiple objective planar location problem with a line barrier (1997)
- The Multiple Objective Median Problem involves locating a new facility so that a vector of performance criteria is optimized over a given set of existing facilities. A variation of this problem is obtained if the existing facilities are situated on two sides of a linear barrier. Such barriers like rivers, highways, borders, or mountain ranges are frequently encountered in practice. In this paper, theory of the Multiple Objective Median Problem with line barriers is developped. As this problem is nonconvex but specially-structured, a reduction to a series of convex optimization problems is proposed. The general results lead to a polynomial algorithm for finding the set of efficient solutions. The algorithm is proposed for bi-criteria problems with different measures of distance.

- Some Personal Views on the Current State and the Future of Locational Analysis (1997)
- In this paper a group of participants of the 12th European Summer Institute which took place in Tenerifa, Spain in June 1995 present their views on the state of the art and the future trends in Locational Analysis. The issue discussed includes modelling aspects in discrete, network and continuous location, heuristic techniques, the state of technology and undesirable facility location. Some general questions are stated reagrding the applicability of location models, promising research directions and the way technology affects the development of solution techniques.

- Wavelet Thresholding in Anisotropic Function Classes and Application to Adaptive Estimation of Evolutionary Spectra (1997)
- We derive minimax rates for estimation in anisotropic smoothness classes. This rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present.; As an important application we introduce a new adaptive wavelet estimator of the time-dependent spectrum of a locally stationary time series. Using this model which was resently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, i.e. how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.

- Regularization Wavelets and Multiresolution (1997)
- Many problems arising in (geo)physics and technology can be formulated as compact operator equations of the first kind \(A F = G\). Due to the ill-posedness of the equation a variety of regularization methods are in discussion for an approximate solution, where particular emphasize must be put on balancing the data and the approximation error. In doing so one is interested in optimal parameter choice strategies. In this paper our interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular value decomposition as a wavelet analysis. This enables us to perform, e.g., Tikhonov-Philips regularization as multiresolution. In other words, we are able to pass over from one regularized solution to another one by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are efficiently applicable to a future problem in satellite geodesy, viz. satellite gravity gradiometry.

- An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit (1997)
- An asymptotic-induced scheme for nonstationary transport equations with thediffusion scaling is developed. The scheme works uniformly for all ranges ofmean free paths. It is based on the asymptotic analysis of the diffusion limit ofthe transport equation. A theoretical investigation of the behaviour of thescheme in the diffusion limit is given and an approximation property is proven.Moreover, numerical results for different physical situations are shown and atheuniform convergence of the scheme is established numerically.