Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik
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Real Closed * Rings
(2010)
In this dissertation I examine a definition of real closure of commutative unitary reduced rings. I also give a characterization of rings that are real closed in this context and how one is able to arrive to such a real closure. There are sufficient examples to help the reader get a feel for real closed * rings and the real closure * of commutative unitary rings.
Due to the immense advance of widely accessible information systems in industrial applications, science, education and every day use, it becomes more and more difficult for users of those information systems to keep track with new and updated information. An approach to cope with this problem is to go beyond traditional search facilities and instead use the users' profiles to monitor data changes and to actively inform them about these updates - an aspect that has to be explicitly developed and integrated into a variety of information systems. This is traditionally done in an individual way, depending on the application and its platform. In this dissertation, we present a novel approach to model the semantic interrelations that specify which users to inform about which updates, based on the underlying model of the respective information system. For the first time, a meta-model that allows information system designers to tag an arbitrary data model and thus specify the event-handling semantics is presented. A formal specification of how to interpret meta-models to determine the receivers of the events completes the presented concept. For the practical realization of this new concept, model driven architecture (MDA) shows to be an ideal technical means. Using our newly developed UML profile based on data-modelling standards, an implementation of the event-handling specification can automatically be generated for a variety of different target platforms, like e.g. relational databases, using triggers. This meta-approach makes the proposed solution ideal with respect to maintainability and genericity. Our solution significantly reduces the overall development efforts for an event-handling facility. In addition, the enhanced model of the information system can be used to generate an implementation that also fulfils non-functional requirements like high performance and extensibility. The overall framework, consisting of the domain specific language (i.e. the meta-model), formal and technical transformations of how to interpret the enhanced information system model and a cost-based optimizing strategy, constitutes an integrated approach, offering several advantages over traditional implementation techniques: our framework can be applied to new information systems as well as to legacy applications without having to modify existing systems; it offers an extensible, easy-to-use, generic and thus re-usable solution and it can be tailored to and optimized for many use cases, as the practical evaluation presented in this dissertation verifies.
The Sugiyama framework proposed in the seminal paper of 1981 is one of the most important algorithms in graph drawing and is widely used for visualizing directed graphs. In its common version, it draws graphs hierarchically and, hence, maps the topological direction to a geometric direction. However, such a hierarchical layout is not possible if the graph contains cycles, which have to be destroyed in a preceding step. In certain application and problem settings, e.g., bio sciences or periodic scheduling problems, it is important that the cyclic structure of the input graph is preserved and clearly visible in drawings. Sugiyama et al. also suggested apart from the nowadays standard horizontal algorithm a cyclic version they called recurrent hierarchies. However, this cyclic drawing style has not received much attention since. In this thesis we consider such cyclic drawings and investigate the Sugiyama framework for this new scenario. As our goal is to visualize cycles directly, the first phase of the Sugiyama framework, which is concerned with removing such cycles, can be neglected. The cyclic structure of the graph leads to new problems in the remaining phases, however, for which solutions are proposed in this thesis. The aim is a complete adaption of the Sugiyama framework for cyclic drawings. To complement our adaption of the Sugiyama framework, we also treat the problem of cyclic level planarity and present a linear time cyclic level planarity testing and embedding algorithm for strongly connected graphs.