For every natural number m there exists a ring R with a completely prime ideal P so that there are exactly m non-isomorphic indecomposable injective right R-modules with P as associated prime ideal.

For two-slanted matrices, there is shown the close connection between their spectral properties and the zeros of their corresponding symbols. The results are applied to two-scale difference equations.

The present empirical study deals with the question of the (world) view of mathematics (i.e. the image of mathematics) held by university mathematics teachers in countries of German as a first lan-guage. The basis of this study is a voluntary survey by means of a closed questionnaire of 119 persons during an annual meeting of mathematicians. This questionnaire was on the whole employed for two other studies on mathematics teachers (N= 300) and pupils (N=1650). Four to five dimensions were defined by means of factor analysis and subsequently verified as relevant dimensions of the view of mathematics. These dimensions can be called the formalism aspect, the schema aspect, the process aspect, the application aspect and the Platonism aspect of mathematics. Attitudes towards these aspects differ on average, so that the "average" view of mathematics of university teachers is clearly accentuated in these five aspects. The process aspect acclaims the highest agreement, whereby the aspects formalism and application claim an average to above average assessment. In contrast, the Platonic aspect receives only weak to very weak agreement, and the schema aspect is on the whole rejected. Furthermore the structure of the view of mathematics resulting out of the relations between the dimensions is investigated. The part of the view of mathematics that is considered here contains two different viewpoints in content, namely the static view of mathematics as a system and the dynamic view of mathematics as a process. In contrast to the observa-tions in the other two populations the two viewpoints are not opposites. Mathematics is in the view of university teachers a comple-mentary toge-therness of both viewpoints. Their view of mathematics is insofar broader differentiated than the two other populations.

This paper describes a way to define Euclidean spaces with the help of relations. Directions and distances of the geometrie are regarded as relational groupings. Theorems about parallelograms and parrallelograms of distances are added as axioms. The axiomatic is then a description of an Euclidean geometrie.

In order to develop and evaluate characteristics of ideal-typical patterns of private additional lessons in mathematics, I refer to investigations on mathematics school lessons of Hans Freudenthal, Hans Werner Heymann, Alan H. Schoenfeld, and Heinrich Winter. Their ideal-typical patterns are equally represented in the six fundamental dimensions independent work, subject orientation, process and problem oriented interactions, structural thinking, discovering education processes, and application orientation. For private additional lessons, the content of the six dimensions is determined by case studies, consisting of the execution and analysis of remedial teaching, and evaluated by synopsis to the characteristics of school teaching. As an outlook I present a concept of school-organized remedial teaching.

In this thesis two classes of rational matrix differential resp. difference equations which contain the matrix Riccati equations of continuous- and discrete-time optimal control as particular cases are considered. Existence and convergence theorems for these equations are proved and conditions ensuring that the corresponding algebraic equations have a stabilizing solution are derived.

The presented dissertation is concerned with anisotropic curvature motion of two-dimensional parametric surfaces as well as their application in surface fairing and surface restauration. Mainly the so called anisotropic mean curvature motion (AMCM) and the anisotropic Willmore-flow are being treated. These flows are generalizations of the classical mean curvature flow and the classical Willmore-flow, respectively. The anisotropies are induced by positive, 1-homogenous and convex functions, which can be regarded as support functions of convex bodies, so called Wulff-shapes. Being a method of fourth order of differentiation in the surface coordinates, the anisotropic Willmore-flow allows the prescription of boundary values for the position of the boundary itself as well as for the surface normals on the boundary of a surface patch under consideration. Hence it is an appropriate method for the reconstruction of partially destroyed surfaces. In this work a numerical scheme for the anisotropic Willmore-flow is presented, which is based on an operator splitting of the fourth order evolution equation into two weak equations of first order of differentiation, which is discretized using linear finite elements in space. In particular the discretization of the AMCM turns out to be one of these equations. Based on the AMCM a method for the fairing of surfaces with crystalline edges is developed. Modifications of the discrete AMCM are also used for surface modeling purposes. Schemes for the artificial aging and for virtual engraving of surfaces are presented. Further on a subdivision scheme based on the isotropic mean curvature motion is introduced. Finally, the isotropic as well as the anisotropic Willmore-flow is employed for the restauration of partially destroyed triangulated surfaces.

This habilitation thesis is concerned with the behaviour of meromorphic functions in the plane in the neighbourhood of infinity. We say that two functions share limit values, if both functions converge to certain given values on the same sequences to infinity. Shared limit values lead to shared islands in a suitable sense, which makes it possible to apply Ahlfors's theory of covering surfaces. For example, we proof a (weak)analogue of Nevanlinna's theorem on five shared values. In the second chapter deficient rational functions and zeros of composite meromorphic functions are studied.

According to the TIMS-Study students in Germany show great deficits in problem solving abilities according to a lack of flexibility in thinking in mathematical networks. However, there exists hardly information about the exact lacks and the reasons of the deficits. The aim of the study in this dissertation thesis was to investigate how a mathematical network as it is presented in textbooks is transformed when carried over into students' minds during teaching and learning processes. The thesis subdivides into a theoretical part and the presentation of the empirical investigation. The theoretical part gives first an overview on the didactical discussion to the topic of connections in mathematics education. It follows a conceptual foundation of mathematical networks and their different aspects, pointing out that the underlying structure of a network is a graph, whose vertices represent mathematical objects and nonmathematical components related to these, and whose edges represent existing relations on them. According to the different sorts of relations, network categories with relevance for mathematics education in school are defined. Further, some theories and models describing aspects of genesis, memorizing and recalling connections are presented and information with regard to the different network categories defined above are brought out. The model of curricular frames is discussed as an approach to specify connections in teaching and learning processes; it serves, together with the modelling of connections as graphs, as aids for the empirical investigation. The empirical study particularly focuses on the topic "sets of two equations in straight form" in middle grade classes and restricts on the investigation of some network relations according to subject systematics and a special relation according to the application of mathematical objects, the model relation. The main research questions are: 1. Which relations according to subject systematics and which model relations to the topic focused on are part of curricular frames? 2. Which network transformations result in teaching and learning processes, on the way from the frame of the intended curriculum into the frame of the implemented curriculum and further in the frame of the achieved curriculum? The results show that connections as they are presented in text books are nearly unchanged translated into the frame of the implemented curriculum: teachers follow almost exactly the textbook in their lessons. On the further transfer of the implemented network into students' minds many connections are filtered out: the mainly learned connections by students are part of the links according to subject systematics (links between problems and solving algorithms, linkages according to a subconcept-superconcept relation), model links are hardly known. Students show great difficulties, when they have to use connections in problem solving processes. The study reveals the incompleteness of the transfer of implemented networks into students' minds. Moreover, the missing relations in the achieved networks are pointed out. The results of the study provide useful information for a possible improvement of teaching and learning processes with respect to the different sorts of connections.

We consider Lorentz manifolds which carry a conformal gradient field with at least one zero. All such manifolds are necessarily locally conformally flat. If in addition the manifold is developable and the image of a development map is great enough one can describe the global conformal type by a 3-regular graph provided with two additional datas. We get this conformal invariants using certain conformal development maps into the projective standard quadric. All conformal diffeomorphisms of this projective quadric are nothing else than the projectivities of the ambient real projective space which preserve the quadric. In the proofs we often use this fact. As a by-product of a general existence theorem we give an infinite number of locally conformally flat Lorentz metrics on the differentiable manifold R^n which are in pairs not conformally equivalent.