Digitalisierung als Chance für alle Prinzipien guten Unterrichts - Barzel, Bärbel; Greefrath, Gilbert; Nagel, Mareike; Hoffmann, Max (mathematik lehren)
athematische Kompetenzen digital zu fördern und digitale Kompetenzen mathematisch zu fördern – dies ist eine Forderung der neuen Bildungsstandards mit Blick auf eine Bildung in der digitalen Welt. Gerade das Potenzial digitaler Medien für das fachliche Lernen wurde in vielen Studien bestätigt. Eine sinnvoll gestaltete Einbettung digitaler Medien bietet die Chance, allen fünf Prinzipien eines guten Unterrichts gerecht zu werden: Verstehensorientierung, Durchgängigkeit, kognitive Aktivierung, Lernendenorientierung & Adaptivität und Kommunikationsförderung. Die flächendeckende Nutzung digitaler Medien etabliert sich bislang nur zögerlich. Aber wie können wir Lehrkräfte stärken, digitale Medien sinnvoll einzusetzen? Wir möchten hier die Bandbreite der Möglichkeiten an Beispielen verdeutlichen, ihren Einsatz motivieren und Wege für einen guten Unterricht aufzeigen.

31 Sprachbildung im berufsbezogenen Mathematikunterricht. - Prediger, Susanne; Wessel, Lena (Berufs-und Fachsprache Deutsch in Wissenschaft und Praxis)

Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification - Offen, Christian
The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero is provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.

Temperedness of locally symmetric spaces: The product case - Weich, Tobias; Wolf, Lasse Lennart (Geom Dedicata)
Let $X=X_1× X_2$ be a product of two rank one symmetric spaces of non-compact type and $Γ$ a torsion-free discrete subgroup in $G_1× G_2$. We show that the spectrum of $Γ \backslash X$ is related to the asymptotic growth of $Γ$ in the two direction defined by the two factors. We obtain that $L^2(Γ \backslash G)$ is tempered for large class of $Γ$.

Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces - Lutsko, Christopher; Weich, Tobias; Wolf, Lasse Lennart (arXiv:2402.02530)
Given a real semisimple connected Lie group $G$ and a discrete torsion-free subgroup $Γ < G$ we prove a precise connection between growth rates of the group $Γ$, polyhedral bounds on the joint spectrum of the ring of invariant differential operators, and the decay of matrix coefficients. In particular, this allows us to completely characterize temperedness of $L^2(Γ\backslash G)$ in this general setting.

Fachbezogene Professionalisierung von fachfremd Mathematik unterrichtenden Lehrkräften: Retrospektive Selbsteinschätzungen zur Expertise im Umgang mit Schwierigkeiten beim Mathematiklernen im Anfangsunterricht der Grundschule - Häsel-Weide, Uta; Graf, Lara Marie; Höveler, K.; Nührenbörger, M. (HLZ – Herausforderung Lehrer*innenbildung)

The meaning landscape of the concept of the total derivative in multivariable real analysis textbooks: an analysis based on a new model of meaning - Lankeit, Elisa; Biehler, Rolf (ZDM – Mathematics Education)
Abstract In this paper, we propose a novel conceptual framework tailored for modeling the meaning of mathematical concepts in university-level mathematics, addressing their rigorous nature and their relationships with related concepts as well as interpretations in various contexts. Within this framework, we present a model of meaning for the concepts of total differentiability and total derivative that provides a variety of possible interpretations and aspects. We then use the proposed model of meaning as a tool for analyzing three different textbooks for mathematics majors on the topic of multidimensional differentiability. ﻿Our paper is an example of a subject matter analysis of a topic in university mathematics carried out in a structured way. The model of meaning for total differentiability presented in this paper could inspire course design and analysis including the design of assignments and assessments for students. Moreover, it could serve as a valuable research tool for further analyses. For example, it could be used as a framework for analyzing courses taught or as a basis for developing a test instrument to assess students’ understanding. With our textbook analysis, we begin to examine the landscape of textbooks regarding differentiability concepts in the multidimensional case, shedding light on the diversity of meaning facets that are covered in the textbooks. These results could be useful for guiding instructors and learners in selecting and using textbooks for teaching and learning based on their respective needs.

On the Universal Transformation of Data-Driven Models to Control Systems - Peitz, Sebastian; Bieker, Katharina (Automatica)
As in almost every other branch of science, the major advances in data science and machine learning have also resulted in significant improvements regarding the modeling and simulation of nonlinear dynamical systems. It is nowadays possible to make accurate medium to long-term predictions of highly complex systems such as the weather, the dynamics within a nuclear fusion reactor, of disease models or the stock market in a very efficient manner. In many cases, predictive methods are advertised to ultimately be useful for control, as the control of high-dimensional nonlinear systems is an engineering grand challenge with huge potential in areas such as clean and efficient energy production, or the development of advanced medical devices. However, the question of how to use a predictive model for control is often left unanswered due to the associated challenges, namely a significantly higher system complexity, the requirement of much larger data sets and an increased and often problem-specific modeling effort. To solve these issues, we present a universal framework (which we call QuaSiModO: Quantization-Simulation-Modeling-Optimization) to transform arbitrary predictive models into control systems and use them for feedback control. The advantages of our approach are a linear increase in data requirements with respect to the control dimension, performance guarantees that rely exclusively on the accuracy of the predictive model, and only little prior knowledge requirements in control theory to solve complex control problems. In particular the latter point is of key importance to enable a large number of researchers and practitioners to exploit the ever increasing capabilities of predictive models for control in a straight-forward and systematic fashion.

Practice-Oriented Research in Tertiary Mathematics Education

A $C^k$-seeley-extension-theorem for Bastiani’s differential calculus - Hanusch, Maximilian (Canadian Journal of Mathematics)
Keywords: extension of differentiable maps

On the structure of regularization paths for piecewise differentiable regularization terms - Gebken, Bennet; Bieker, Katharina; Peitz, Sebastian (Journal of Global Optimization)
Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression and machine learning. Since the choice of the regularization parameter is crucial but often difficult, path-following methods are used to approximate the entire regularization path, i.e., the set of all possible solutions for all regularization parameters. Due to their nature, the development of these methods requires structural results about the regularization path. The goal of this article is to derive these results for the case of a smooth objective function which is penalized by a piecewise differentiable regularization term. We do this by treating regularization as a multiobjective optimization problem. Our results suggest that even in this general case, the regularization path is piecewise smooth. Moreover, our theory allows for a classification of the nonsmooth features that occur in between smooth parts. This is demonstrated in two applications, namely support-vector machines and exact penalty methods.

Student Teachers ’ Knowledge of Congruence before a University Course on Geometry - Hoffmann, Max; Biehler, Rolf (Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022))
Keywords: Teaching and learning of specific topics in university mathematics; Transition to; across and from university mathematics; Student Teachers; Geometry; Congruence; Double Discontinuity.

Innovations in university teaching based on mathematic education research - Florensa, Ignasio; Hoffmann, Max; Romo Vázquez, Avenilde; Zandieh, Michelle; Martínez-Planell, Rafael (Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022))

Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source - Black, Tobias; Fuest, Mario; Lankeit, Johannes; Mizukami, Masaaki (Nonlinear Analysis: Real World Applications)
Keywords: Applied Mathematics; Computational Mathematics; General Economics; Econometrics and Finance; General Engineering; General Medicine; Analysis

Reziprokes Peer-Tutoring zur Förderung von Schüler*innen mit Schwierigkeiten beim Mathematiklernen. - Vitt, Vivian; Häsel-Weide, Uta (Mathematica Didactica, 46)

Inklusive Praktiken unterrichtsintegrierter Förderung im Mathematikunterricht. - Häsel-Weide, Uta; Nührenbörger, M. (Mathematica Didactica, 46)

The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups - Hanusch, Maximilian (Annals of Global Analysis and Geometry)
Keywords: Lax equation; generalized Baker-Campbell-Dynkin-Hausdorff formula; regularity of Lie groups

Critically 3-frustrated signed graphs - Cappello, Chiara; Naserasr, Reza; Steffen, Eckhard; Wang, Zhouningxin (arXiv:2304.10243)
Extending the notion of maxcut, the study of the frustration index of signed graphs is one of the basic questions in the theory of signed graphs. Recently two of the authors initiated the study of critically frustrated signed graphs. That is a signed graph whose frustration index decreases with the removal of any edge. The main focus of this study is on critical signed graphs which are not edge-disjoint unions of critically frustrated signed graphs (namely non-decomposable signed graphs) and which are not built from other critically frustrated signed graphs by subdivision. We conjecture that for any given k there are only finitely many critically k-frustrated signed graphs of this kind. Providing support for this conjecture we show that there are only two of such critically 3-frustrated signed graphs where there is no pair of edge-disjoint negative cycles. Similarly, we show that there are exactly ten critically 3-frustrated signed planar graphs that are neither decomposable nor subdivisions of other critically frustrated signed graphs. We present a method for building non-decomposable critically frustrated signed graphs based on two given such signed graphs. We also show that the condition of being non-decomposable is necessary for our conjecture.

Algorithms for the genome median under a restricted measure of rearrangement - Silva, Helmuth O.M.; Rubert, Diego P.; Araujo, Eloi; Steffen, Eckhard; Doerr, Daniel; Martinez, Fábio V. (RAIRO - Operations Research)
Ancestral reconstruction is a classic task in comparative genomics. Here, we study the genome median problem, a related computational problem which, given a set of three or more genomes, asks to find a new genome that minimizes the sum of pairwise distances between it and the given genomes. The distance stands for the amount of evolution observed at the genome level, for which we determine the minimum number of rearrangement operations necessary to transform one genome into the other. For almost all rearrangement operations the median problem is NP-hard, with the exception of the breakpoint median that can be constructed efficiently for multichromosomal circular and mixed genomes. In this work, we study the median problem under a restricted rearrangement measure called c4-distance, which is closely related to the breakpoint and the DCJ distance. We identify tight bounds and decomposers of the c4-median and develop algorithms for its construction, one exact ILP-based and three combinatorial heuristics. Subsequently, we perform experiments on simulated data sets. Our results suggest that the c4-distance is useful for the study the genome median problem, from theoretical and practical perspectives.
Keywords: Management Science and Operations Research; Computer Science Applications; Theoretical Computer Science

Sets of r-graphs that color all r-graphs - Ma, Yulai; Mattiolo, Davide; Steffen, Eckhard; Wolf, Isaak Hieronymus (arXiv:2305.08619)