510 Mathematik
The formulation of the decoding problem for linear block codes as an integer program (IP) with a rather tight linear programming (LP) relaxation has made a central part of channel coding accessible for the theory and methods of mathematical optimization, especially integer programming, polyhedral combinatorics and also algorithmic graph theory, since the important class of turbo codes exhibits an inherent graphical structure. We present several novel models, algorithms and theoretical results for error-correction decoding based on mathematical optimization. Our contribution includes a partly combinatorial LP decoder for turbo codes, a fast branch-and-cut algorithm for maximum-likelihood (ML) decoding of arbitrary binary linear codes, a theoretical analysis of the LP decoder's performance for 3-dimensional turbo codes, compact IP models for various heuristic algorithms as well as ML decoding in combination with higher-order modulation, and, finally, first steps towards an implementation of the LP decoder in specialized hardware. The scientific contributions are presented in the form of seven revised reprints of papers that appeared in peer-reviewed international journals or conference proceedings. They are accompanied by an extensive introductory part that reviews the basics of mathematical optimization, coding theory, and the previous results on LP decoding that we rely on afterwards.
New media are continually gaining importance in society. This process also has an increasing influence on developments in the field of education. Due to the use of computers as an integral part of schooling, new possibilities with regard to the organisation of learning processes arise. In this context, it is of great significance that appropriate computer applications for the respective learning group be prepared, so that justifiable use of computers in lessons can take place. Furthermore, efficient integration of computers requires changes in spatial organisation, in teaching methodology and in the role of the teacher. Such reflection and re-orientation are the essential basis for meaningful usage of new media in teaching and learning processes. An initial aim of this thesis is an empirical analysis of the situation regarding the usage of computers in geometry lessons in primary schools, based on a regional survey. The evaluation gives information as to how intensively the computer is used in the learning process and shows us which factors determine the use of computers in geometry lessons.
The results are an empirical foundation for the development of a computer-based learning environment called "Geolizi" (the second aim of my study). Within this learning environment, the pupils should work independently on the topics "mirror-imaged figures" and "the construction of rectangles and squares", with the help of the computer. During this process, hands-on media, traditional drawing instruments and interactive worksheets are available to the pupils. The computer (with its appropriate applications) takes over different functions in this learning process. Testing of this learning environment ("Geolizi") took place in several primary school classes, within the scope of formative and summative evaluation. With the help of questionnaires filled in by the pupils, the usability of the individual elements was tested. Based on a pre-post-investigation design, an attempt has been made to discover possible changes in the attitude of teachers regarding the usage of computers in the teaching of elementary geometry.
The results of this test phase, together with the evaluation of the questionnaires, lead to the founded presumption that usage of the multimedia-based learning environment " Geolizi " could result in greater use of computers in geometry lessons. All in all, the developed learning environment demonstrates an interesting possibility of how to use computers in the teaching of geometry at primary schools, thus making an important contribution to an independent, individualised learning process.
