Einführung: Lehrinnovationen in der Hochschulmathematik – praxisrelevant – didaktisch fundiert – forschungsbasiert - Biehler, Rolf; Eichler, Andreas; Hochmuth, Reinhard; Rach, Stefanie; Schaper, Niclas (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Mathematikvorkurse als Brücke in das Studium–Einführung - Biehler, Rolf (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Introducing students to machine learning with decision trees using CODAP and Jupyter Notebooks - Biehler, Rolf; Fleischer, Yannik (Teaching Statistics)
Keywords: Education; Statistics and Probability

Lehrinnovationen in der Hochschulmathematik

3D-Geometrie–virtuell und real - Biehler, Rolf; Weigand, Hans-Georg (mathematik lehren)

Mathematics in Engineering Education: a Review of the Recent Literature with a View towards Innovative Practices - Pepin, Birgit; Biehler, Rolf; Gueudet, Ghislaine (International Journal of Research in Undergraduate Mathematics Education)
Abstract The aim of the special issue is to bring together important current international research on innovative teaching and learning practices in mathematics in engineering education, and to develop deeper understandings of the characteristics of current teaching and learning practices that can inform the design and implementation of future innovative practice. The focus of this review paper is to provide a state-of-the-art overview of this emerging field at the cross-roads between mathematics and engineering education, in addition to introducing the papers of this special issue. To guide this paper, we posed three review questions: (1) How can current (teaching/learning/study) practices of mathematics in engineering education be characterized with a view towards innovation?; (2) What are the ‘resources’ (cognitive, material, digital, social) used, and what are those that appear also well suited for innovative courses?; (3) What are promising innovative practices in mathematics in engineering education, and what are the implications for curriculum reform? Looking back across the studies we summarized in the review, we conclude that they are lagging behind the more fundamental changes that are happening in engineering education, whilst addressing selected aspects of innovative changes within the current system of engineering education. At the same time, the nine papers of this special issue contribute new perspectives for innovative practices in mathematics in engineering education, for a better understanding of current practices and for future research.
Keywords: Education; Mathematics (miscellaneous)

Design-Based Research in der Hochschullehre am Beispiel der Lehrveranstaltung „Einführung in die Kultur der Mathematik“ - Kempen, Leander; Biehler, Rolf (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Education and professional development of university mathematics teachers - Winsløw, Carl; Biehler, Rolf; Jaworski, Barbara; Rønning, Frode; Wawro, Megan (Research and Development in University Mathematics Education)

Schatten von 3D-Objekten: Modellierung mit GeoGebra 3D und Anwendungen in der Computergrafik - Höper, Lukas; Malin, Leah; Biehler, Rolf (mathematik lehren)

Faszination 3D-Film: Entwicklung einer 3D-Konstruktion - Hüsing, Sven; Weiser, Niklas; Biehler, Rolf (mathematik lehren)

Consultation Phases in Mathematics Learning and Support Centres - Schürmann, Mirko; Panse, Anja; Shaikh, Zain; Biehler, Rolf; Schaper, Niclas; Liebendörfer, Michael; Hilgert, Joachim (International Journal of Research in Undergraduate Mathematics Education)
Abstract Mathematics Learning Support Centres are becoming more and more common in higher education both internationally and in Germany. Whereas it is clear that their quality largely depends on a functioning interaction in consultations, little is known about how such consultations proceed in detail. On the basis of models from the literature and recorded support sessions (N = 36), we constructed a process model that divides consultations into four ideal–typical phases. In the individual consultations, forward or backward leaps occur, but overall the model seems to describe the data well. A high intercoder reliability shows that it can be applied consistently on real data by different researchers. An analysis of the consultations between students and tutors shows that both mainly work on past attempts or thoughts of the students to solve the exercise or problems and on concrete strategies to solve a problem within the session. In contrast, very little time is dedicated to summarizing and reflecting the solution. The data allows for a more in-depth discussion of what constitutes quality in advising processes and how it might be further explored. Practically, the model may structure support sessions and help in focussing on different goals in different phases.
Keywords: Education; Mathematics (miscellaneous)

The process of actively building a model for a randomization test – insights into learners’ modeling activities based on a case study - Podworny, Susanne; Biehler, Rolf (Mathematical Thinking and Learning)
Keywords: Developmental and Educational Psychology; Education; General Mathematics

A first introduction to data science education in secondary schools: Teaching and learning about data exploration withCODAPusing survey data - Frischemeier, Daniel; Biehler, Rolf; Podworny, Susanne; Budde, Lea (Teaching Statistics)
Keywords: Education; Statistics and Probability

Opportunities and Possibilities of a Network of Mathematical Learning and Support Centres in Germany - Schürmann, Mirko; Liebendörfer, Michael; Gildehaus, Lara; Schaper, Niclas; Hochmuth, Reinhard; Biehler, Rolf; Lankeit, Elisa; Kuklinski, Christiane; Ruge, Johanna (Sigma newsletter)

Was ist eine gute Skizze? Strategiewissen beim mathematischen Modellieren im Bereich der Geometrie fördern - Schönherr, Johanna; Schukajlow, S.; Blomberg, J. (mathematik lehren)

Does strategic knowledge matter? Effects of strategic knowledge about drawing on students’ modeling competencies in the domain of geometry - Schönherr, Johanna; Schukajlow, S.; Blomberg, J.; Leopold, C. (Mathematical Thinking and Learning)

Do emotions and prior performance facilitate the use of the learner-generated drawing strategy? Effects of enjoyment, anxiety, and intramathematical performance on the use of the drawing strategy and modelling performance - Schönherr, Johanna; Blomberg, J.; Schukajlow, S.; Leopold, C. (Contemporary Educational Psychology)

Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation - McLachlan, Robert I; Offen, Christian (Foundations of Computational Mathematics)
We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error.

Detection of high codimensional bifurcations in variational PDEs - Kreusser, Lisa Maria; McLachlan, Robert I; Offen, Christian (Nonlinearity)

Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach - Flaßkamp, Kathrin; Ober-Blöbaum, Sina; Peitz, Sebastian (Advances in Dynamics, Optimization and Computation)
Many dynamical systems possess symmetries, e.g. rotational and translational invariances of mechanical systems. These can be beneficially exploited in the design of numerical optimal control methods. We present a model predictive control scheme which is based on a library of precomputed motion primitives. The primitives are equivalence classes w.r.t. the symmetry of the optimal control problems. Trim primitives as relative equilibria w.r.t. this symmetry, play a crucial role in the algorithm. The approach is illustrated using an academic mobile robot example.

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator - Klus, Stefan; Nüske, Feliks; Hamzi, Boumediene (Entropy)
Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.

Das Kompetenzzentrum Hochschuldidaktik Mathematik (khdm) - Hochmuth, Reinhard; Liebendörfer, Michael; Biehler, Rolf; Eichler, Andreas (Neues Handbuch Hochschullehre)

Pareto Explorer: a global/local exploration tool for many-objective optimization problems - Schütze, Oliver; Cuate, Oliver; Martín, Adanay; Peitz, Sebastian; Dellnitz, Michael (Engineering Optimization)
Multi-objective optimization is an active field of research that has many applications. Owing to its success and because decision-making processes are becoming more and more complex, there is a recent trend for incorporating many objectives into such problems. The challenge with such problems, however, is that the dimensions of the solution sets—the so-called Pareto sets and fronts—grow with the number of objectives. It is thus no longer possible to compute or to approximate the entire solution set of a given problem that contains many (e.g. more than three) objectives. On the other hand, the computation of single solutions (e.g. via scalarization methods) leads to unsatisfying results in many cases, even if user preferences are incorporated. In this article, the Pareto Explorer tool is presented—a global/local exploration tool for the treatment of many-objective optimization problems (MaOPs). In the first step, a solution of the problem is computed via a global search algorithm that ideally already includes user preferences. In the second step, a local search along the Pareto set/front of the given MaOP is performed in user specified directions. For this, several continuation-like procedures are proposed that can incorporate preferences defined in decision, objective, or in weight space. The applicability and usefulness of Pareto Explorer is demonstrated on benchmark problems as well as on an application from industrial laundry design.

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control - Klus, Stefan; Nüske, Feliks; Peitz, Sebastian; Niemann, Jan-Hendrik; Clementi, Cecilia; Schütte, Christof (Physica D: Nonlinear Phenomena)
We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.

Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator - Peitz, Sebastian; Klus, Stefan (Lecture Notes in Control and Information Sciences)
In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, proper orthogonal decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model is obtained via a convex combination of these autonomous systems. Using a recent convergence result for extended dynamic mode decomposition (EDMD), convergence of the reduced objective function can be shown. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1-dimensional Burgers equation and the 2-dimensional Navier–Stokes equations as examples. Finally, an extension for online adaptivity is presented.

Deep model predictive flow control with limited sensor data and online learning - Bieker, Katharina; Peitz, Sebastian; Brunton, Steven L.; Kutz, J. Nathan; Dellnitz, Michael (Theoretical and Computational Fluid Dynamics)
The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.

Data-Driven Model Predictive Control using Interpolated Koopman Generators - Peitz, Sebastian; Otto, Samuel E.; Rowley, Clarence W. (SIAM Journal on Applied Dynamical Systems)
In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation --- realized by linear interpolation between two operators --- does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.

Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge - Schumacher, Jan; Rezat, Sebastian (Zeichen und Sprache im Mathematikunterricht)
Diagrammatisches Schließen wird im Zusammenhang mit dem Lernen von Mathmematik und ihrer Symbolsprache als wesentliche Theorie der Wissenskonstruktion diskutiert. Dabei wird häufig davon ausgegangen, dass die Wissenskonstruktion im Sinne diagrammatischen Schließens erfolgt. Deskriptive Rekonstruktionen diagrammatischen Schließens bei Lernenden stellen jedoch ein Desiderat der mathematikdidaktischen Forschung dar. Der vorliegende Beitrag befasst sich mit der Fragestellung, wie sich diagrammatisches Schließen bei Lernenden rekonstruieren lässt. Als mögliche Werkzeuge für eine solche Rekonstruktion werden Toulmins Argumentationsschema und Vergnauds Schema-Begriff exemplarisch angewandt, um das diagrammatische Schließen eines Schülerpaars beim Einstieg in die Subtraktion negativer Zahlen zu rekonstruieren. Abschließend wird die tatsächliche Eignung der beiden Ansätze zur Rekonstruktion diagrammatischen Schließens diskutiert.

Deduktion und Abduktion beim diagrammatischen Schließen – das didaktische Potential der Peirceschen Semiotik - Schumacher, Jan

A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors - Gerlach, Raphael; Ziessler, Adrian; Eckhardt, Bruno; Dellnitz, Michael (SIAM Journal on Applied Dynamical Systems)
In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method. By using previously obtained approximations of the parameter-dependent relative global attractor we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole subdivision procedure. We illustrate the feasibility of the set-oriented path following method by exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence and of large-scale atmospheric regime changes .

Lie groupoids of mappings taking values in a Lie groupoid - Amiri, Habib; Glöckner, Helge; Schmeding, Alexander (Archivum Mathematicum)
Keywords: 22A22; 22E65; 22E67; 46T10; 47H30; 58D15; 58H05

Products of regular locally compact spaces are k_R-spaces - Glöckner, Helge; Masbough, Niku (Topology Proceedings)
Keywords: 54B10; 54D45; 54D50

Smoothing operators for vector-valued functions and extension operators - Glöckner, Helge (arXiv:2006.00254)
For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M, endowed with a C^k-manifold structure turning the inclusion map from L to M into a C^k-map. Moreover, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.

Lie groups of real analytic diffeomorphisms are L^1-regular - Glöckner, Helge (arXiv:2007.15611)
Let $M$ be a compact, real analytic manifold and $G$ be the Lie group of all real-analytic diffeomorphisms of $M$, which is modelled on the (DFS)-space ${\mathfrak g}$ of real-analytic vector fields on $M$. We study flows of time-dependent real-analytic vector fields on $M$ which are integrable functions in time, and their dependence on the time-dependent vector field. Notably, we show that the Lie group $G$ is $L^1$-regular in the sense that each $[\gamma]$ in $L^1([0,1],{\mathfrak g})$ has an evolution which is an absolutely continuous $G$-valued function on $[0,1]$ and smooth in $[\gamma]$. As tools for the proof, we develop several new results concerning $L^p$-regularity of infinite-dimensional Lie groups, for $1\leq p\leq \infty$, which will be useful also for the discussion of other classes of groups. Moreover, we obtain new results concerning the continuity and complex analyticity of non-linear mappings on open subsets of locally convex direct limits.

Variational integrators for dissipative systems - Limebeer, D. J. N.; Ober-Blöbaum, Sina; Farshi, F. H. (IEEE Transactions on Automatic Control)

Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty - Hernández Castellanos, Carlos Ignacio; Ober-Blöbaum, Sina; Peitz, Sebastian (International Journal of Robust and Nonlinear Control)
In real-world problems, uncertainties (e.g., errors in the measurement, precision errors) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where optimal solutions can degrade in quality or even become infeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multi-objective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control (MPC). In multi-objective optimal control, an optimal compromise between multiple conflicting criteria has to be found. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. This approach is closely related to explicit MPC for nonlinear systems, where the potentially expensive optimization problem is solved in an offline phase in order to enable fast solutions in the online phase. In order to reduce the numerical cost of the offline phase, we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multi-objective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multi-objective framework allows for online adaptations of the desired objective. Alternatively, an automatic scalarizing procedure yields very efficient feedback controls. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.

Numerical computation of lightly multi-objective robust optimal solutions by means of generalized cell mapping - Hernández Castellanos, C. I. O.; Schütze, G.; Sun, J.-Q.; Ober-Blöbaum, Sina; Morales-Luna, G. (Mathematics)

A multirate variational approach to simulation and optimal control for flexible spacecraft - Lishkova, Y.; Ober-Blöbaum, Sina; Cannon, M.; Leyendecker, S. (Accepted for publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist Conference - Lake Tahoe)

A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systems - Faulwasser, T.; Flaßkamp, K.; Ober-Blöbaum, Sina; Worthmann, K. (24th International Symposium on Mathematical Theory of Networks and Systems)

Modelling of the convection-diffusion equation through fractional restricted calculus of variations - Cresson, J.; Jiménez, F.; Ober-Blöbaum, Sina (24th International Symposium on Mathematical Theory of Networks and Systems)

On Weyl’s type theorems and genericity of projective rigidity in sub-Riemannian geometry - Jean, Frédéric; Maslovskaya, Sofya; Zelenko, Igor (Geometriae Dedicata)
Keywords: Geometry and Topology

The turnpike property in maximization of microbial metabolite production - Maslovskaya, Sofya; Caillau, Jean-Baptiste; Djema, Walid; Giraldi, Laetitia; Jean-Luc, Jean-Luc; Pomet, Jean-Baptiste

Pollicott-Ruelle Resonant States and Betti Numbers - Küster, Benjamin; Weich, Tobias (Communications in Mathematical Physics)
Abstract Given a closed orientable hyperbolic manifold of dimension $$\ne 3$$ ≠ 3 we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numbers. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers.
Keywords: Mathematical Physics; Statistical and Nonlinear Physics

Schnittstellenaktivitäten zum Kongruenzsatz WSW - Hoffmann, Max (Beiträge zum Mathematikunterricht 2020)

Rezension: Andrew Granville und Jenniver Granville: Prime Supects: The Anatomy of Integers and Permutations - Hoffmann, Max (Mathematische Semesterberichte)

Rezension: Ehrhard Behrends: Parkettierungen der Ebene – Von Escher über Möbius zu Penrose - Hoffmann, Max (Mathematische Semesterberichte)

Der Mathematikunterricht 66 (6): Geometrie in Schule und Lehramtsausbildung – ein Nachwuchsheft

Tragfähige Grundlagen - Häsel-Weide, Uta; Nührenbörger, Marcus (Kinder lernen Zukunft – Anforderungen und tragfähige Grundlagen)

Aufgaben für den inklusiven Mathematikunterricht - aus der Sicht von Lehrkräften. - Heckmann, Lara; Häsel-Weide, Uta (Beiträge zum Mathematikunterricht 2020)

Sachrechnen - Häsel-Weide, Uta (Didaktik des Unterrichts bei Lernschwierigkeiten: Ein Handbuch für Studium und Praxis)