Kaiserslautern - Fachbereich Mathematik
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This survey provides the reader with an overview of numerous results on p-permu- tation modules and the closely related classes of endo-trivial, endo-permutation and endo-p- permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of categorical equivalences between blocks of finite group alge- bras. For this reason, there has been, since the late 1990’s, much interest in classifying such modules. The aim of this manuscript is to review classical results as well as all the major recent advances in the area. The first part of this survey serves as an introduction to the topic for non-experts in modular representation theory of finite groups, outlining proof ideas of the most important results at the foundations of the theory. Simultaneously, the connections between the aforementioned classes of modules are emphasised. In this respect, results, which are dispersed in the literature, are brought together, and emphasis is put on common properties and the role played by the p-permutation modules throughout the theory. Finally, in the last part of the manuscript, lifting results from positive characteristic to characteristic zero are collected and their proofs sketched.
Das MINT-EC-Girls-Camp: Math-Talent-School richtet sich an mathematikbegeisterte Schülerinnen von MINT-EC-Schulen, die Einblicke in die Berufswelt von Mathematikerinnen und Mathematikern bekommen möchten. Die Veranstaltung veranschaulicht den Schülerinnen die steigende Relevanz angewandter mathematischer Forschungsgebiete, wie der Techno- und der Wirtschaftsmathematik. Sie soll dazu dienen, Schüler:innen die Bedeutung mathematischer Arbeitsweisen in der heutigen Berufswelt, insbesondere in Industrie und Wirtschaft, begreifbar zu machen. Die Talent-School wird organisiert von MINT-EC und dem Felix-Klein-Zentrum für Mathematik. Die fachwissenschaftliche Betreuung der Schülerinnen während dieser Talent-School wurde durch Mitarbeitende des Kompetenzzentrums für Mathematische Modellierung in MINT-Projekten in der Schule (KOMMS) der TU Kaiserslautern und des Fraunhofer ITWM umgesetzt. In diesem Report beschreiben wir die Projekte, die während der Talent-School im Oktober 2022 durchgeführt wurden.
Seit 1993 veranstaltet der Fachbereich Mathematik der TU Kaiserslautern jährlich die mathematischen Modellierungswochen. Die Veranstaltung erwuchs parallel zu der steigenden Relevanz angewandter mathematischer Forschungsgebiete, wie der Techno- und der Wirtschaftsmathematik. Sie soll dazu dienen, Schülerinnen und Schülern die Bedeutung mathematischer Arbeitsweisen in der heutigen Berufswelt, insbesondere in Industrie und Wirtschaft, begreifbar zu machen. Darüber hinaus bietet die Modellierungswoche den teilnehmenden Lehrkräften einen Einblick in die Projektarbeit mit offenen Fragestellungen im Rahmen der mathematischen Modellierung. In diesem Report beschreiben wir die Projekte, die während der Modellierungswoche im Dezember 2022 durchgeführt wurden.
Load modeling is one of the crucial tasks for improving smart grids’ energy efficiency. Among many alternatives, machine learning-based load models have become popular in applications and have shown outstanding performance in recent years. The performance of these models highly relies on data quality and quantity available for training. However, gathering a sufficient amount of high-quality data is time-consuming and extremely expensive. In the last decade, Generative Adversarial Networks (GANs) have demonstrated their potential to solve the data shortage problem by generating synthetic data by learning from recorded/empirical data. Educated synthetic datasets can reduce prediction error of electricity consumption when combined with empirical data. Further, they can be used to enhance risk management calculations. Therefore, we propose RCGAN, TimeGAN, CWGAN, and RCWGAN which take individual electricity consumption data as input to provide synthetic data in this study. Our work focuses on one dimensional times series, and numerical experiments on an empirical dataset show that GANs are indeed able to generate synthetic data with realistic appearance.
Seit 1993 veranstaltet der Fachbereich Mathematik der TU Kaiserslautern jährlich die mathematischen Modellierungswochen. Die Veranstaltung erwuchs parallel zu der steigenden Relevanz angewandter mathematischer Forschungsgebiete, wie der Technomathematik und der Wirtschaftsmathematik. Sie soll dazu dienen, Schülerinnen und Schülern die Bedeutung mathematischer Arbeitsweisen in der heutigen Berufswelt, insbesondere in Industrie und Wirtschaft, begreifbar zu machen. Darüber hinaus bietet die Modellierungswoche den teilnehmenden Lehrkräften einen Einblick in die Projektarbeit mit offenen Fragestellungen im Rahmen der mathematischen Modellierung. In diesem Report beschreiben wir die Projekte, die während der Modellierungswoche im Dezember 2021 durchgeführt wurden. Der Themenschwerpunkt der Veranstaltung lautete "Wetter und Katastrophenschutz".
In this paper, a prediction model for the tensile behaviour of ultra-high performance
fibre-reinforced concrete is proposed. It is based on integrating force contributions of all fibres
crossing the crack plane. Piecewise linear models for the force contributions depending on fibre
orientation and embedded length are fitted to force–slip curves obtained in single-fibre pull-out tests.
Fibre characteristics in the crack are analysed in a micro-computed tomography image of a concrete
sample. For more general predictions, a stochastic fibre model with a one-parametric orientation
distribution is introduced. Simple estimators for the orientation parameter are presented, which only
require fibre orientations in the crack plane. Our prediction method is calibrated to fit experimental
tensile curves.
This contribution defends two claims. The first is about why thought experiments are so relevant and powerful in mathematics. Heuristics and proof are not strictly and, therefore, the relevance of thought experiments is not contained to heuristics. The main argument is based on a semiotic analysis of how mathematics works with signs. Seen in this way, formal symbols do not eliminate thought experiments (replacing them by something rigorous), but rather provide a new stage for them. The formal world resembles the empirical world in that it calls for exploration and offers surprises. This presents a major reason why thought experiments occur both in empirical sciences and in mathematics. The second claim is about a looming aporia that signals the limitation of thought experiments. This aporia arises when mathematical arguments cease to be fully accessible, thus violating a precondition for experimenting in thought. The contribution focuses on the work of Vladimir Voevodsky (1966–2017, Fields medalist in 2002) who argued that even very pure branches of mathematics cannot avoid inaccessibility of proof. Furthermore, he suggested that computer verification is a feasible path forward, but only if proof is not modeled in terms of formal logic.
We consider the optimization problem of a large insurance company that wants to maximize the expected utility of its surplus through the optimal control of the proportional reinsurance. In addition, the insurer is exposed to the risk of default of its reinsurer at the worst possible time, a setting that is closely related to a scenario of the Swiss Solvency Test.