Brian W. Kernighan: Millions billions zillions – defending yourself in a world of too many numbers. Princeton University Press 2018 - Hilgert, Joachim (Mathematische Semesterberichte)

Joseph Honerkamp: Denken in Strukturen und seine Geschichte – Von der Kraft des mathematischen Beweises (Springer 2018) - Hilgert, Joachim (Mathematische Semesterberichte)

Øystein Linnebo: Philosophy of Mathematics (Princeton University Press 2017) - Hilgert, Joachim (Mathematische Semesterberichte)

Jost-Hinrich Eschenburg: Sternstunden der Mathematik. Springer 2017 - Hilgert, Joachim (Mathematische Semesterberichte)

A convergent evolving finite element algorithm for mean curvature flow of closed surfaces - Kovács, Balázs; Li, Buyang; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Mathilde Vaerting. Deutschlands erste Mathematikdidaktikderin - Werth, Gerda (Beiträge zum Mathematikunterricht)

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces - Küster, Benjamin; Weich, Tobias (International Mathematics Research Notices)
Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{τ }$ over the unit cosphere bundle $S^{* }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{τ }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{* }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,σ )$ on compatible associated vector bundles $\mathbf W_{σ }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $τ$ and $σ$ defining the bundles $\mathbf V_{τ }$ and $\mathbf W_{σ }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{σ }$. Our methods of proof are based on representation theory and Lie theory.
Keywords: General Mathematics

„...weil man Fermi-Aufgaben so rechnet“. Modelltexte als sprachliche Ressource für das Erklären von Lösungswegen bei Fermi-Aufgaben - Rezat, Sara; Rezat, Sebastian (In: Mathematik differenziert 3/2019)

An Introduction to Grounded Theory with a Special Focus on Axial Coding and the Coding Paradigm - Vollstedt, Maike; Rezat, Sebastian (Compendium for Early Career Researchers in Mathematics Education)

Vocabulary in learning processes towards conceptual understanding of equivalent fractions—specifying students’ language demands on the basis of lexical trace analyses - Wessel, Lena (Mathematics Education Research Journal)
Abstract Although the teaching of vocabulary in mathematics lessons is requested in content- and language-integrated lesson designs, the clarification of the specific lexical language demands is still an open question for many mathematical topics. In a content- and language-integrated lesson design towards understanding the concept of equivalent fractions, the vocabulary (words and phrases) used by 17 students has been analyzed with qualitative means of data analysis. The qualitative in-depth analyses underline the importance of meaning-related vocabulary for making structural relations between the fractions in view explicit. Quantitative analyses of inventoried vocabulary for the four categories “self-initiated by students,” “triggered by teaching material,” “triggered by teacher,” or “triggered by peers” show the relations of collective and autonomous vocabularies from which the students retrieve their lexical means in oral and written language production.
Keywords: Education; General Mathematics

3.5 Kompetenzorientierte Didaktik der Analysis durch Orientierung an real-life tasks–Ein Beispiel für ein Lehrdesign nach dem 4C/ID-Modell - Leuders, Timo; Wessel, Lena (Professionsorientierung in der Lehrerbildung: Kompetenzorientiertes Lehren nach dem 4-Component-Instructional-Design-Modell)

Hanse-Kolloquium zur Hochschuldidaktik der Mathematik 2018
„Vielfalt, die verbindet“ ist ein Leitmotiv, welches das Hanse-Kolloquium zur Hochschuldidaktik der Mathematik 2018 in Essen gut beschreibt. Zu diesem kamen Akteure von Fachhochschulen und Universitäten mit fachmathematischer wie fachdidaktischer Perspektive vom 9. bis 10. November zusammen, um die Problematik des Übergangs von Schule zu Hochschule im Kontext mathematischer Studiengänge zu diskutieren. Der vorliegende Band bündelt die vielfältigen Projekte und Forschungsaktivitäten rund um den Übergang Schule–Hochschule und zeigt Innovationen innerhalb der mathematischen Hochschullehre gleichermaßen praxisorientiert wie theoretisch fundiert auf. Neben drei Hauptbeiträgen von Bärbel Barzel, Frode Rønning sowie Nimet Sarikaya und Peter Furlan umfasst der Band weitere 13 Sektionsbeiträge, welche u. a. die nebenstehenden Schwerpunkte fokussieren. • Heranführen von Studierenden an hochschulmathematische Denk- und Arbeitsweisen • Anpassung von Strukturen und Aufgaben für einen konstruktiven Übergang von Schule zu Hochschule • Bewährte Unterstützungsmaßnahmen für ein erfolgreiches Selbstlernen, z. B. in Form von Peer Instruction. • Etablierte Flipped-Classroom- und Blended-Learning-Formate • Messung vielfältiger Fähigkeitsprofile von Studierenden beim Eintritt in die Hochschule • Chancen der Digitalisierung nutzen: Lehren und Lernen mit digitalen Medien, z. B. mithilfe von Lernvideo oder durch dynamische Visualisierungen • Umgang mit zunehmender Heterogenität und unterschiedlichem Vorwissen auf Seite der Studierenden • Konzepte problembasierten Lernens in die Hochschullehre integrieren • Steigerung der Motivation von Studierenden • Umgang mit der doppelten Diskontinuität mathematischer Lehramtsstudiengänge

How theories of language-responsive mathematics can inform teaching designs for vocational mathematics - Wessel, Lena (Eleventh Congress of the European Society for Research in Mathematics Education (CERME11))
Keywords: Vocational education; language; percentages; scaffolding; design research

Language diversity and mathematics education: new developments - Barwell, Richard; Wessel, Lena; Parra, Aldo (Research in Mathematics Education)
Keywords: General Mathematics; Education

Mathematics education in a multilingual and multicultural environment. Post-Publikation der Topic Study Group 31 Beiträge der ICME-Konferenz 2016 in Hamburg.

Features of innovative lectures that distinguish them from traditional lectures and their evaluation by attending students - Kuklinski, Christiane; Liebendörfer, Michael; Hochmuth, Reinhard; Biehler, Rolf; Schaper, Niclas; Lankeit, Elisa; Leis, Elena; Schürmann, Mirko (Proceedings of CERME 11)

Bifurcation of solutions to Hamiltonian boundary value problems - McLachlan, Robert I; Offen, Christian (Nonlinearity)
A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.

Symplectic integration of boundary value problems - McLachlan, Robert I; Offen, Christian (Numerical Algorithms)
Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to {\em long-time} behaviour. They are directly connected to the dynamical behaviour of symplectic maps φ:M→M' on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map φ:M→M' which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not.

Koopman Operator-Based Finite-Control-Set Model Predictive Control for Electrical Drives - Hanke, Sören; Peitz, Sebastian; Wallscheid, Oliver; Klus, Stefan; Böcker, Joachim; Dellnitz, Michael (arXiv:1804.00854)
Predictive control of power electronic systems always requires a suitable model of the plant. Using typical physics-based white box models, a trade-off between model complexity (i.e. accuracy) and computational burden has to be made. This is a challenging task with a lot of constraints, since the model order is directly linked to the number of system states. Even though white-box models show suitable performance in most cases, parasitic real-world effects often cannot be modeled satisfactorily with an expedient computational load. Hence, a Koopman operator-based model reduction technique is presented which directly links the control action to the system's outputs in a black-box fashion. The Koopman operator is a linear but infinite-dimensional operator describing the dynamics of observables of nonlinear autonomous dynamical systems which can be nicely applied to the switching principle of power electronic devices. Following this data-driven approach, the model order and the number of system states are decoupled which allows us to consider more complex systems. Extensive experimental tests with an automotive-type permanent magnet synchronous motor fed by an IGBT 2-level inverter prove the feasibility of the proposed modeling technique in a finite-set model predictive control application.

Rapid Calculation of Molecular Kinetics Using Compressed Sensing - Litzinger, Florian; Boninsegna, Lorenzo; Wu, Hao; Nüske, Feliks; Patel, Raajen; Baraniuk, Richard; Noé, Frank; Clementi, Cecilia (Journal of Chemical Theory and Computation)

Data-Driven Model Reduction and Transfer Operator Approximation - Klus, Stefan; Nüske, Feliks; Koltai, Péter; Wu, Hao; Kevrekidis, Ioannis; Schütte, Christof; Noé, Frank (Journal of Nonlinear Science)

Sparse learning of stochastic dynamical equations - Boninsegna, Lorenzo; Nüske, Feliks; Clementi, Cecilia (The Journal of Chemical Physics)

Quantitative comparison of adaptive sampling methods for protein dynamics - Hruska, Eugen; Abella, Jayvee R.; Nüske, Feliks; Kavraki, Lydia E.; Clementi, Cecilia (The Journal of Chemical Physics)

Semiotische Analyse von Sinnkonstruktionsprozessen bei einem innermathematischen Zugang zum Erlernen negativer Zahlen - Schumacher, Jan (Beiträge zum Mathematikunterricht 2018)

Psychologische Grundbedürfnisse im frühen Mathematikstudium - Liebendörfer, Michael (Beiträge zum Mathematikunterricht 2018)

Auswirkungen von innovativen Vorlesungen für Lehramtsstudierende in der Studieneingangsphase - Liebendörfer, Michael; Kuklinski, Christiane; Hochmuth, Reinhard (Beiträge zum Mathematikunterricht 2018)

Motivationsentwicklung im Mathematikstudium - Liebendörfer, Michael

A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems - Gebken, Bennet; Peitz, Sebastian; Dellnitz, Michael (Numerical and Evolutionary Optimization – NEO 2017)
In this article we propose a descent method for equality and inequality constrained multiobjective optimization problems (MOPs) which generalizes the steepest descent method for unconstrained MOPs by Fliege and Svaiter to constrained problems by using two active set strategies. Under some regularity assumptions on the problem, we show that accumulation points of our descent method satisfy a necessary condition for local Pareto optimality. Finally, we show the typical behavior of our method in a numerical example.

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction - Peitz, Sebastian; Dellnitz, Michael (Mathematical and Computational Applications)
Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.

Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition - Beermann, Dennis; Dellnitz, Michael; Peitz, Sebastian; Volkwein, Stefan (Reduced-Order Modeling (ROM) for Simulation and Optimization)
In this chapter, we combine a global, derivative-free subdivision algorithm for multiobjective optimization problems with a posteriori error estimates for reduced-order models based on Proper Orthogonal Decomposition in order to efficiently solve multiobjective optimization problems governed by partial differential equations. An error bound for a semilinear heat equation is developed in such a way that the errors in the conflicting objectives can be estimated individually. The resulting algorithm constructs a library of locally valid reduced-order models online using a Greedy (worst-first) search. Using this approach, the number of evaluations of the full-order model can be reduced by a factor of more than 1000.

Tensor-based dynamic mode decomposition - Klus, Stefan; Gelß, Patrick; Peitz, Sebastian; Schütte, Christof (Nonlinearity)
Dynamic mode decomposition (DMD) is a recently developed tool for the analysis of the behavior of complex dynamical systems. In this paper, we will propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues. The goal is to reduce the computational complexity and also the amount of memory required to store the data in order to mitigate the curse of dimensionality. The efficiency of these tensor-based methods will be illustrated with the aid of several different fluid dynamics problems such as the von Kármán vortex street and the simulation of two merging vortices.

POD-based multiobjective optimal control of PDEs with non-smooth objectives - Beermann, Dennis; Dellnitz, Michael; Peitz, Sebastian; Volkwein, Stefan (PAMM)
A framework for set‐oriented multiobjective optimal control of partial differential equations using reduced order modeling has recently been developed [1]. Following concepts from localized reduced bases methods, error estimators for the reduced cost functionals are utilized to construct a library of locally valid reduced order models. This way, a superset of the Pareto set can efficiently be computed while maintaining a prescribed error bound. In this article, this algorithm is applied to a problem with non‐smooth objective functionals. Using an academic example, we show that the extension to non‐smooth problems can be realized in a straightforward manner. We then discuss the implications on the numerical results.

Vollständige Charakterisierung von piezoelektrischen Scheiben mit Ringelektroden - Feldmann, Nadine; Jurgelucks, Benjamin; Claes, Leander; Henning, Bernd

Improved Neural Control of Movements Manifests in Expertise-Related Differences in Force Output and Brain Network Dynamics - Gölz, Christian; Voelcker-Rehage, Claudia; Mora, Karin; Reuter, Eva-Maria; Godde, Ben; Dellnitz, Michael; Reinsberger, Claus; Vieluf, Solveig (Frontiers in Physiology)

Age- and Expertise-Related Differences of Sensorimotor Network Dynamics during Force Control - Vieluf, Solveig; Mora, Karin; Gölz, Christian; Reuter, Eva-Maria; Godde, Ben; Dellnitz, Michael; Reinsberger, Claus; Voelcker-Rehage, Claudia (Neuroscience)

Transition Manifolds of Complex Metastable Systems - Bittracher, Andreas; Koltai, Péter; Klus, Stefan; Banisch, Ralf; Dellnitz, Michael; Schütte, Christof (Journal of Nonlinear Science)

Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data - Peitz, Sebastian (arXiv:1801.06419)
In a recent article, we presented a framework to control nonlinear partial differential equations (PDEs) by means of Koopman operator based reduced models and concepts from switched systems. The main idea was to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. These individual systems can be approximated very efficiently by reduced order models obtained from data, and one can guarantee equality of the full and the reduced objective function under certain assumptions. In this article, we extend these results to continuous control inputs using convex combinations of multiple Koopman operators corresponding to constant controls, which results in a bilinear control system. Although equality of the objectives can be carried over when the PDE depends linearly on the control, we show that this approach is also valid in other scenarios using several flow control examples of varying complexity.

Analyzing high-dimensional time-series data using kernel transfer operator eigenfunctions - Klus, Stefan; Peitz, Sebastian; Schuster, Ingmar (arXiv:1805.10118)
Kernel transfer operators, which can be regarded as approximations of transfer operators such as the Perron-Frobenius or Koopman operator in reproducing kernel Hilbert spaces, are defined in terms of covariance and cross-covariance operators and have been shown to be closely related to the conditional mean embedding framework developed by the machine learning community. The goal of this paper is to show how the dominant eigenfunctions of these operators in combination with gradient-based optimization techniques can be used to detect long-lived coherent patterns in high-dimensional time-series data. The results will be illustrated using video data and a fluid flow example.

Measures of Edge-Uncolorability of Cubic Graphs - Fiol, M. A.; Mazzuoccolo, Guiseppe; Steffen, Eckhard (The Electronic Journal of Combinatorics)
There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.
Keywords: Cubic graph; Tait coloring; Snark; Boole coloring; Berge's conjecture; Tutte's 5-flow conjecture

Cores, joins and the Fano-flow conjectures - Jin, Ligang; Mazzuoccolo, Giuseppe; Steffen, Eckhard (Discussiones Mathematicae Graph Theory)

Approximating Vizing’s independence number conjecture - Steffen, Eckhard (Australasian Journal of Combinatorics)

Inklusiver Mathematikunterricht - vernetzt zwischen Mathematikdidaktik und Sonderpädagogik - Nührenbörger, M.; Bönig, D.; Häsel-Weide, Uta; Korff, N.; Scherer, P. (Beiträge zum Mathematikunterricht 2018. Vorträge zur Mathematikdidaktik und zur Schnittstelle Mathematik/Mathematikdidaktik auf der gemeinsamen Jahrestagung der GDM und DMV)

Fostering the Computation Competence of Low Achievers through Cooperative Learning in Inclusive Classrooms: A Longitudinal Study - Moser-Opitz, E.; Grob, U.; Wittich, C.; Häsel-Weide, Uta; Nührenbörger, M. (Learning Disabilities: A Contemporary Journal)

Förderkommentar Lernen zum Zahlenbuch 3 - Häsel-Weide, Uta; Nührenbörger, M.; Reinold, M.

Das Zahlenbuch 3. Förderheft. - Häsel-Weide, Uta; Nührenbörger, M.; Reinold, M.

Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion - Black, Tobias (SIAM Journal on Mathematical Analysis)
Keywords: Applied Mathematics; Computational Mathematics; Analysis

Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D - Black, Tobias (Journal of Differential Equations)
Keywords: Analysis; Applied Mathematics

Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions - Black, Tobias (Nonlinear Analysis)
Keywords: Applied Mathematics; Analysis

Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling - Peitz, Sebastian; Ober-Blöbaum, Sina; Dellnitz, Michael (Acta Applicandae Mathematicae)
In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient-based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the flow around a cylinder and a backward-facing-step channel flow.

Necessary optimality conditions for optimally controlled dissipative mechanical systems modelled through fractional derivatives - Jiménez, F.; Ober-Blöbaum, Sina (6th European Conference on Computational Mechanics)