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)

Mathematik (Primarstufe) in der Lehrerinnen- und Lehrerbildung. Qualifizierung für das Lehren von Mathematik in der Grundschule. - Häsel-Weide, Uta (Handbuch Lehrerinnen- und Lehrerbildung)

Linear Algebra driven by Data Science - Schramm, Thomas; Gasser, Ingenuin; Schwenker, Sören; Seiler, Ruedi; Lohse, Alexander; Zobel, Kay
Dieses Lernangebot widmet sich der linearen Algebra als dem Teil der Mathematik, der neben der Optimierung und der Stochastik die Grundlage für praktisch alle Entwicklungen im Bereich Künstliche Intelligenz (KI) darstellt. Das Fach ist jedoch für Anfänger meist ungewohnt abstrakt und wird daher oft als besonders schwierig und unanschaulich empfunden. In diesem Kurs wird das Erlernen mathematischer Kenntnisse in linearer Algebra verknüpft mit dem aktuellen und faszinierenden Anwendungsfeld der künstlichen neuronalen Netze (KNN). Daraus ergeben sich in natürlicher Weise Anwendungsbeispiele, an denen die wesentlichen Konzepte der linearen Algebra erklärt werden können. Behandelte Themen sind: Der Vektorraum der reellen Zahlentupel, reelle Vektorräume allgemein Lineare Abbildungen Matrizen Koordinaten und darstellende Matrizen Lineare Gleichungssysteme, Gaußalgorithmus Determinante Ein Ausblick auf nichtlineare Techniken, die für neuronale Netzwerke relevant sind.

A Characterization of the Finiteness of Perpetual Integrals of Levy Processes - Kolb, Martin; Savov, Mladen (Bernoulli)
We derive a criterium for the almost sure finiteness of perpetual integrals of L ́evy processes for a class of real functions including all continuous functions and for general one- dimensional L ́evy processes that drifts to plus infinity. This generalizes previous work of D ̈oring and Kyprianou, who considered L ́evy processes having a local time, leaving the general case as an open problem. It turns out, that the criterium in the general situation simplifies significantly in the situation, where the process has a local time, but we also demonstrate that in general our cri- terium can not be reduced. This answers an open problem posed in D ̈oring, L. and Kyprianou, A. (2015).
Keywords: L ́evy processes; Perpetual integrals; Potential measures

Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model - Haddenhorst, Björn; Hüllermeier, Eyke; Kolb, Martin (International Journal of Approximate Reasoning)
Reciprocal relations are binary relations Q with entries Q(i,j)∈[0,1], and such that Q(i,j)+Q(j,i)=1. Relations of this kind occur quite naturally in various domains, such as preference modeling and preference learning. For example, Q(i,j) could be the fraction of voters in a population who prefer candidate i to candidate j. In the literature, various attempts have been made at generalizing the notion of transitivity to reciprocal relations. In this paper, we compare three important frameworks of generalized transitivity: g-stochastic transitivity, T-transitivity, and cycle-transitivity. To this end, we introduce E-transitivity as an even more general notion. We also use this framework to extend an existing hierarchy of different types of transitivity. As an illustration, we study transitivity properties of probabilities of pairwise preferences, which are induced as marginals of an underlying probability distribution on rankings (strict total orders) of a set of alternatives. In particular, we analyze the interesting case of the so-called Babington Smith model, a parametric family of distributions of that kind.

The regularity problem for Lie groups with asymptotic estimate Lie algebras - Hanusch, Maximilian (Indagationes Mathematicae)
Keywords: regularity of Lie groups

The Strong Trotter Property for Locally μ-convex Lie Groups - Hanusch, Maximilian (Journal of Lie Theory)
Keywords: Lie theory; strong Trotter property

Erklärung von Mathematikleistung im Ingenieursstudium - Kuklinski, Christiane; Leis, Elena; Liebendörfer, Michael; Hochmuth, Reinhard (Beiträge zum Mathematikunterricht 2019 53. Jahrestagung der Gesellschaft für Didaktik der Mathematik.)

Design und Evaluation einer mathematischen Unterstützungsmaßnahme für Ingenieure - Khellaf, Sarah; Kuklinski, Christiane; Liebendörfer, Michael; Hochmuth, Reinhard (Beiträge zum Mathematikunterricht 2020)

Entstehung und Bedeutung von Lerngruppen in der Studieneingangsphase - Haak, Inka; Gildehaus, Lara; Liebendörfer, Michael (Beiträge zum Mathematikunterricht 2020)

Geschlechterunterschiede beim Einsatz von Lernstrategien in Mathematikveranstaltungen - Liebendörfer, Michael; Gildehaus, Lara; Göller, Robin (Beiträge zum Mathematikunterricht 2020)

studiVEMINTvideos – Mathematische Lernvideos zur Studienvorbereitung und Unterstützung im ersten Studienjahr - Biehler, Rolf; Liebendörfer, Michael; Schmitz, Angela; Fleischmann, Yael; Krämer, Sandra; Ostsieker, Laura; Schlüter, Sarah (Beiträge zum Mathematikunterricht 2020)

Single-point blow-up in the Cauchy problem for the higher-dimensional Keller-Segel system - Winkler, Michael (Nonlinearity)

Boundedness in a two-dimensional Keller-Segel-Navier-Stokes system involving a rapidly diffusing repulsive signal. - Winkler, Michael (Zeitschrift für angewandte Mathematik und Physik)

Can simultaneous density-determined enhacement of diffusion and cross-diffusion foster boundedness in Keller-Segel type systems involving signal-dependent motilities? - Winkler, Michael (Nonlinearity)

The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^1$. - Winkler, Michael (Advances in Nonlinear Analysis)

Attractiveness of constant states in logistic-type Keller-Segel systems involving subquadratic growth restrictions. - Winkler, Michael (Advanced Nonlinear Studies)

Small-mass solutions in the two-dimensional Keller-Segel system coupled to the Navier-Stokes equations. - Winkler, Michael (SIAM Journal on Mathematical Analysis)

Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains? - Winkler, Michael (Acta Applicandae Mathematicae)

Blow-up profiles and liefe beyond blow-up in the fully parabolic Keller-Segel system. - Winkler, Michael (Journal d'Analyse Mathématique)

Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation. - Rodriguez, Nancy; Winkler, Michael (Mathematical Models & Methods in Applied Sciences)

Larg time behavior in a predator-prey system with indirect pursuit-evasion interaction. - Li, Genglin; Tao, Youshan; Winkler, Michael (Discrete and Continuous Dynamical Systems --B)

Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy. - Tao, Youshan; Winkler, Michael (Journal of Differential Equations)

A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy. - Tao, Youshan; Winkler, Michael (Nonlinear Analysis)

Refined regularity and stabilization properties in a degenerate haptotaxis system - Stinner, Christian; Winkler, Michael (Discrete and Continuous Dynamical Systems --A)

Riesz distributions and Laplace transform in the Dunkl setting of type A - Rösler, Margit (Journal of Functional Analysis)
Keywords: Analysis

Multiple Points of Operator Semistable Lévy Processes - Luks, Tomasz; Xiao, Yimin (Journal of Theoretical Probability)

Mathematical Analysis of Two Competing Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow - Steiner, Evi; Winkler, Michael (Journal of Nonlinear Science)

Imaginary quadratic number fields with class groups of small exponent - Elsenhans, Andreas-Stephan; Klüners, Jürgen; Nicolae, Florin (Acta Arithmetica)
Let D<0 be a fundamental discriminant and denote by E(D) the exponent of the ideal class group Cl(D) of K=ℚ(√D). Under the assumption that no Siegel zeros exist we compute all such D with E(D) dividing 8. We compute all D with |D| ≤ 3.1⋅10²⁰ such that E(D) ≤ 8.
Keywords: Algebra and Number Theory

The conductor density of local function fields with abelian Galois group - Klüners, Jürgen; Müller, Raphael (Journal of Number Theory)
We give an exact formula for the number of G-extensions of local function fields Fq((t)) for finite abelian groups G up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
Keywords: Algebra and Number Theory

Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems - Burban, Igor; Zheglov, A. (Proceedings of the London Mathematical Society)
In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen–Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen–Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero–Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, providing explicit computations of some ‘isospectral deformations’ of the planar rational Calogero–Moser system in the case of the split rational potential.

Lesen lernen von Aufgabentexten - Dröse, Jennifer; Prediger, Susanne (Sprachbildender Mathematikunterricht in der Sekundarstufe - ein forschungsbasiertes Praxisbuch)

Frühe alltagsintegrierte mathematische Bildung. Eine Einleitung - Bruns, Julia; Vogtländer, A. (Frühe alltagsintegrierte mathematische Bildung. Handreichung für Lehrende)

Situative Beobachtung und Wahrnehmung angehender frühpädagogischer Fachpersonen im Bereich Mathematik – Entwicklung und Validierung eines Testinstruments - Bruns, Julia; Strahl, Carolin; Gasteiger, Hedwig (Unterrichtswissenschaft)
Zusammenfassung Frühe mathematische Bildung findet in der Kindertagesstätte insbesondere in spontanen Alltagssituationen statt. Damit diese Alltagssituationen zu natürlichen Lernsituationen für die Kinder werden, müssen (angehende) frühpädagogische Fachkräfte zunächst das mathematische Potenzial in diesen Situationen erkennen. Diese Fähigkeit zur situativen Beobachtung und Wahrnehmung ist bisher unzureichend erforscht, was unter anderem auf begrenzte Methoden zur standardisierten Erfassung dieses Konstrukts zurückzuführen ist. Um diesem Bedarf zu begegnen, wurde das videobasierte Testinstrument Vimas_num zur Erfassung der Fähigkeit zur situativen Beobachtung und Wahrnehmung angehender frühpädagogischer Fachpersonen im Bereich Mengen und Zahlen entwickelt. Auf der Grundlage von drei aufeinander aufbauenden Studien wird die Validität verschiedener Aspekte der Testwertinterpretation untersucht. Die Ergebnisse geben Hinweise auf die Validität der Testwertinterpretation in Bezug auf den Testinhalt, die interne Struktur des Tests und die Zusammenhänge mit theoretisch verbundenen Konstrukten.
Keywords: Education

Interaktionen in mathematischen Spielsituationen in Kindertagesstätte und Familie - Meyer, Lena; Tabeling, Laura; Gasteiger, Hedwig; Bruns, Julia (Beiträge zum Mathematikunterricht 2020)

Adaptionshandlungen von Multiplikatorinnen und Multiplikatoren zur frühen mathematischen Bildung – Beschreibung und Bewertung aus fachbezogener Perspektive - Bruns, Julia; Schopferer, Theresa; Gasteiger, Hedwig (Journal für Mathematik-Didaktik)
Zusammenfassung Die Implementation früher mathematischer Bildung in Kindertagesstätten kann durch fachdidaktische Fortbildungen für frühpädagogische Fachkräfte unterstützt werden. Um Fachkräfte in der Breite zu erreichen, erscheint ein scaling up Ansatz geeignet, bei dem aufbauend auf das Fortbildungskonzept EmMa – Erzieherinnen und Erzieher machen Mathematik Multiplikatorinnen und Multiplikatoren qualifiziert werden. Dabei ergibt sich die Herausforderung, dass Fortbildende im Elementarbereich kaum mathematisches und mathematikdidaktisches Wissen mitbringen. Der Kontext dieser Studie ist der scaling up Prozess der Fortbildung EmMa unter diesen Umständen. Das Erkenntnisinteresse liegt in der Beschreibung und Bewertung der Adaptionshandlungen der Multiplikatorinnen und Multiplikatoren. Für die Bewertung der Adaptionen werden zwei Kriterien herangezogen: (1) das Kriterium der Präzision, das die Vollständigkeit und Klarheit der Inhalte umfasst und (2) das Kriterium der fachlichen Korrektheit, das durch das Ausbleiben fachbezogener Fehler beschrieben wird. Analysiert werden dabei die theoretischen, fachbezogenen Phasen der von Multiplikatorinnen und Multiplikatoren durchgeführten Fortbildung zur frühen mathematischen Bildung, da davon ausgegangen wird, dass insbesondere diese theoretischen Inhalte für diese Gruppe von Multiplikatorinnen und Multiplikatoren eine Herausforderung darstellen. Zur Untersuchung wurde eine videogestützte, qualitative Feldstudie mit N = 8 Fällen durchgeführt. Mit Hilfe der qualitativen Inhaltsanalyse wurden die Adaptionshandlungen der Multiplikatorinnen und Multiplikatoren charakterisiert und hinsichtlich ihrer Präzision und der fachlichen Fehler in diesen Adaptionen analysiert. Dabei zeigte sich, dass der eigenständige Umgang mit den mathematischen und mathematikdidaktischen Inhalten des Fortbildungskonzepts EmMa trotz intensiver Unterstützung anspruchsvoll und bei vielen Multiplikatorinnen und Multiplikatoren mit fachbezogenen Fehlern verbunden ist. Implikationen dieser Ergebnisse für die weitere Forschung werden diskutiert.

Situational Perception in Mathematics (SPiM)—Results of a Cross-Country Study in Austria and Norway - Bruns, Julia; Carlsen, Martin; Eichen, Lars; Erfjord, Ingvald; Hundeland, Per Sigurd (Mathematics Education in the Early Years)

Mathematisches Fachwissen von frühpädagogischen Fachpersonen im Spannungsfeld zwischen Disziplin- und Praxis - Bruns, Julia (Beiträge zum Mathematikunterricht 2020)

“Why don’t you make a drawing?" Motivation and strategy use in mathematical modelling - Schönherr, Johanna; Blomberg, J.; Schukajlow, S.; Rellensmann, J. (Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education: PME)

Measuring and investigating strategic knowledge about drawing to solve geometry modelling problems - Schönherr, Johanna; Schukajlow, S.; Leopold, C. (ZDM – Mathematics Education)

Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties - Black, Tobias (Mathematical Models and Methods in Applied Sciences)
Keywords: Applied Mathematics; Modeling and Simulation

Zirkel und Lineal ohne Parallelenaxiom: Ein konstruktiver Zugang zur hyperbolischen Geometrie. - Hoffmann, Max (Der Mathematikunterricht)

Enhancing Fifth Graders’ Awareness of Syntactic Features in Mathematical Word Problems: A Design Research Study on the Variation Principle - Dröse, Jennifer; Prediger, Susanne (Journal für Mathematik-Didaktik, 41(2))

Symmetry in optimal control: A multiobjective model predictive control approach - Flaßkamp, K.; Ober-Blöbaum, Sina; Peitz, S. (Advances in Dynamics, Optimization and Computation)

Expertise von Lehrpersonen für inklusiven Mathematikunterricht der Sekundarstufe - Ausgangspunkte zur Professionalisierungsforschung. - Seitz, Susanne; Häsel-Weide, Uta; Wilke, Yannik; Wallner, Melina (K:ON Kölner Online-Journal für Lehrer*innenbildung)

Verstehensgrundlagen diagnostizieren - Welche Wissenselemente fokussieren Lehrkräfte? - Dröse, Jennifer (Beiträge zum Mathematikunterricht 2020)

Textaufgaben lesen lernen – eine digital gestützte Einheit mit App - Dröse, Jennifer; Eisen, V.; Prediger, Susanne; Altieri, M.; Schellenbach, M.; Menning, R. (Mathematik lehren 223)

The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems - Gerlach, Raphael; Ziessler, Adrian (Advances in Dynamics, Optimization and Computation)
In this work we review the novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems developed in [6] and [36]. By utilizing results on embedding techniques for infinite dimensional systems we extend a classical subdivision scheme [8] as well as a continuation algorithm [7] for the computation of attractors and invariant manifolds of finite dimensional systems to the infinite dimensional case. We show how to implement this approach for the analysis of delay differential equations and partial differential equations and illustrate the feasibility of our implementation by computing the attractor of the Mackey-Glass equation and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation.

On the equivariance properties of self-adjoint matrices - Dellnitz, Michael; Gebken, Bennet; Gerlach, Raphael; Klus, Stefan (Dynamical Systems)
We investigate self-adjoint matrices A∈Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A)⊂O(n) which is isomorphic to ⊗nk=1Z2. If the self-adjoint matrix possesses multiple eigenvalues – this may, for instance, be induced by symmetry properties of an underlying dynamical system – then A is even equivariant with respect to the action of a group Γ(A)≃∏ki=1O(mi) where m1,…,mk are the multiplicities of the eigenvalues λ1,…,λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.

Spectral Properties of Convex Bodies - Hilgert, Joachim; Barnum, H. (J. of Lie Theory)

Mathematik studieren -- Ein Ratgeber für Erstsemester und solche, die es vielleicht werden wollen

Algebraically Independent Generators for the Algebra of Invariant Differential Operators on SLn(ℝ)/SOn(ℝ) - Hilgert, Joachim; Brennecken, D.; Ciardo, L.

Titu Andreescu und Vlad Crisan: Mathematical Induction – A powerful and elegant method of proof. XYZ Press 2017 und Florian André Dalwigk: Vollständige Induktion – Beispiele und Aufgaben bis zum Umfallen. Springer Spektrum 2019 - Hilgert, Joachim (Mathematische Semesterberichte)

Fabio Toscano: The Secret Formula – How a Mathematical Duel Inflamed Renaissance Italy and Uncovered the Cubic Equation. Princeton University Press 2020 - Hilgert, Joachim (Mathematische Semesterberichte)

Robert Bosch: OPT ART – From Mathematical Optimization to Visual Design. Princeton University Press 2019 - Hilgert, Joachim (Mathematische Semesterberichte)

David M. Bressoud: Calculus Reordered -- A History of the Big Ideas. Princeton University Press 2019 - Hilgert, Joachim (Mathematische Semesterberichte)

Daniel Grieser: Mathematisches Problemlösen und Beweisen – Eine Entdeckungsreise in die Mathematik. 2. Auflage (Springer 2017) - Hilgert, Joachim (Mathematische Semesterberichte)

Claas Lattmann: Mathematische Modellierung bai Platon zwischen Thales und Euklid (De Gruyter 2019) - Hilgert, Joachim (Mathematische Semesterberichte)

Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates - Hipp, David; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates - Hipp, David; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Higher-order linearly implicit full discretization of the Landau–Lifshitz–Gilbert equation - Akrivis, Georgios; Feischl, Michael; Kovács, Balázs; Lubich, Christian (Mathematics of Computation)
Keywords: Applied Mathematics; Computational Mathematics; Algebra and Number Theory

A convergent algorithm for forced mean curvature flow driven by diffusion on the surface - Kovács, Balázs; Li, Buyang; Lubich, Christian (Interfaces and Free Boundaries)
Keywords: Applied Mathematics

RBF-based penalized least-squares approximation of noisy scattered data on the sphere - Hesse, Kerstin (Multivariate Algorithms and Information-Based Complexity)

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

Schulbücher. Werkzeuge zum Üben in den Fächern Deutsch und Mathematik - Rezat, Sara; Rezat, Sebastian (Die Grundschulzeitschrift 320)

Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer Zahlen - Schumacher, Jan; Rezat, Sebastian (Zeichen und Sprache im Mathematikunterricht: Semiotik in Theorie und Praxis)

Mathematiklernen mit digitalen Schulbüchern im Spannungsfeld zwischen Individualisierung und Kooperation - Rezat, Sebastian (Mobile Medien im Schulkontext)

Differenziertes Üben gestalten. Zwischen Umsetzung in der Praxis und Fundierung in der Forschung. - Leuders, Timo; Wessel, Lena (Pädagogik 2/2020)

Sprachbildender Vorstellungsaufbau für verschiedene mathematische Konzepte: Brüche in Klasse 6 - Wessel, Lena; Kuzu, T.; Prediger, Susanne (Sprachbildender Mathematikunterricht in der Sekundarstufe - ein forschungsbasiertes Praxisbuch)

Fach- und sprachintegrierte Ansätze für Neuzugewanderte - Sprütten, F.; Wessel, Lena; Zentgraf, K.; Prediger, S. (Sprachbildender Mathematikunterricht in der Sekundarstufe. Ein forschungsbasiertes Praxisbuch)

Language for Mathematics in Vocational Contexts. Handbook for teachers and facilitators - Wessel, Lena; Prediger, S.; Stein, A.; Wijers, M.; Jonker, V.