TY - JOUR AU - Winkler, Michael ID - 35617 JF - Applied Mathematics Letters TI - Boundedness in a three-dimensional Keller-Segel-Stokes system with subcritical sensitivity. VL - 112 ER - TY - JOUR AU - Winkler, Michael ID - 35613 JF - Transactions of the American Mathematical Society TI - Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow. VL - 374 ER - TY - JOUR AU - Rösler, Margit AU - Voit, Michael ID - 37659 IS - 3 JF - Proceedings of the American Mathematical Society KW - Applied Mathematics KW - General Mathematics SN - 0002-9939 TI - Positive intertwiners for Bessel functions of type B VL - 149 ER - TY - JOUR AU - Tao, Youshan AU - Winkler, Michael ID - 35603 JF - Journal of Functional Analysis TI - A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation. VL - 281 ER - TY - JOUR AU - Tao, Youshan AU - Winkler, Michael ID - 35606 JF - Nonlinear Analysis, Theory, Methods and Applications TI - The dampening role of large repulsive convection in a chemotaxis system modeling tumor angiogenesis. VL - 208 ER - TY - JOUR AU - Wang, Yulan AU - Winkler, Michael AU - Xiang, Zhaoyin ID - 35612 JF - Science China Mathematics TI - Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption. VL - 64 ER - TY - JOUR AU - Ding, Mengyao AU - Winkler, Michael ID - 35601 JF - Nonlinear Differential Equatiions and Applicatiions TI - Small-density solutiions in Keller-Segel systems involving rapidly decaying diffusivities. VL - 28 ER - TY - JOUR AU - Tao, Youshan AU - Winkler, Michael ID - 35605 JF - Communications in Mathematical Sciences TI - Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime. VL - 19 ER - TY - JOUR AU - Tao, Youshan AU - Winkler, Michael ID - 35604 JF - SIAM Journal on Mathematical Analysis TI - Taxis-driven formation of singular hotspots in a May-Nowak type model for virus infection. VL - 53 ER - TY - JOUR AU - Wang, Yulan AU - Winkler, Michael AU - Xiang, Zhaoyin ID - 35610 JF - Communications in Partial Differential Equations TI - Local energy estimates and global solvability in a threee-dimensional chemotaxis-fluid system with prescribed signal on the boundary. VL - 46 ER - TY - JOUR AU - Wang, Yulan AU - Winkler, Michael AU - Xiang, Zhaoyin ID - 35611 JF - Advances in Nonlinear Analysis TI - Global solvability in a threee-dimensional Keller-Segel.Stokes system involving arbitrary superlinear logistic degradation VL - 10 ER - TY - JOUR AU - Tao, Youshan AU - Winkler, Michael ID - 35608 IS - --A JF - Discrete and Continuous Dynamical Systems TI - Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction VL - 41 ER - TY - JOUR AB - In this paper we obtain a complete list of imaginary n-quadratic fields with class groups of exponent 3 and 5 under ERH for every positive integer n where an n-quadratic field is a number field of degree 2ⁿ represented as the composite of n quadratic fields. AU - Klüners, Jürgen AU - Komatsu, Toru ID - 34840 IS - 329 JF - Mathematics of Computation KW - Applied Mathematics KW - Computational Mathematics KW - Algebra and Number Theory SN - 0025-5718 TI - Imaginary multiquadratic number fields with class group of exponent $3$ and $5$ VL - 90 ER - TY - JOUR AB - Let E be an ordinary elliptic curve over a finite field and g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E⁹ . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E³ and of the Igusa modular form in dimension 4. We illustrate our algorithms with examples of curves with many rational points over finite fields. AU - Kirschmer, Markus AU - Narbonne, Fabien AU - Ritzenthaler, Christophe AU - Robert, Damien ID - 34912 IS - 333 JF - Mathematics of Computation KW - Applied Mathematics KW - Computational Mathematics KW - Algebra and Number Theory SN - 0025-5718 TI - Spanning the isogeny class of a power of an elliptic curve VL - 91 ER - TY - JOUR AB - This paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang–Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang–Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples. AU - Burban, Igor AU - Abedin, R. ID - 44329 IS - 2 JF - Communications in Mathematical Physics TI - Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation VL - 387 ER - TY - JOUR AB - The role of domain-specific content knowledge is discussed controversially for the early childhood context. Therefore, this review aims at untangling the research on domain-specific content knowledge for early childhood educators by systematically reviewing the conceptual and operational definition of and results on early childhood educators' content knowledge in different domains. Using the scientific databases ERIC, PsycInfo and Web of Sciences, we identified 36 studies on early childhood educators' domain-specific content knowledge. By comparing these studies, we found that conceptualizations of early childhood educators' content knowledge move on a continuum between a scientific related perspective and a practice related perspective. The scientific related perspective defines content knowledge as the knowledge of key concepts, facts and rules of the domain integrating knowledge taught in primary, secondary or upper secondary school. The practice related perspective includes knowledge of key concepts, facts and rules of the domain limited to the knowledge explicitly relevant for teaching in early childhood education as well as selected domain-specific knowledge of children and teaching. Our review shows that the results and implications drawn by the study authors depend on how these authors conceptualize early childhood educators' content knowledge on this continuum. Further research, therefore, needs to consider carefully how early childhood educators' content knowledge is conceptualized. The paper further discusses gaps in this research field, such as validating methods for measuring early childhood educators' content knowledge or implementing more rigorous experimental designs to examine effects of early childhood educators' content knowledge. AU - Bruns, Julia AU - Gasteiger, Hedwig AU - Strahl, Carolin ID - 34822 IS - 2 JF - Review of Education KW - content knowledge KW - domain-specific learning KW - early childhood education KW - teacher knowledge SN - 2049-6613 TI - Conceptualising and measuring domain-specific content knowledge of early childhood educators: A systematic review VL - 9 ER - TY - JOUR AU - Bruns, Julia AU - Schopferer, T. ID - 45687 JF - . TPS – Theorie und Praxis der Sozialpädagogik TI - Eine mustergültige Wissenschaft VL - 11 ER - TY - CONF AU - Jensen, Solveig AU - Gasteiger, Hedwig AU - Bruns, Julia ED - Inprasitha, Maitree ED - Changsri, Narumon ED - Boonsena, Nisakorn ID - 36527 T2 - Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education TI - Place value and regrouping as seperate constructs of place value understanding VL - 3 ER - TY - JOUR AU - Bruns, Julia AU - Gasteiger, Hedwig AU - Strahl, Carolin ID - 34823 IS - 2 JF - Review of Education SN - 2049-6613 TI - Context and Implications Document for: Conceptualising and measuring domain-specific content knowledge of early childhood educators: A systematic review VL - 9 ER - TY - CONF AU - Jensen, Solveig AU - Gasteiger, Hedwig AU - Bruns, Julia ID - 36536 T2 - Beiträge zum Mathematikunterricht 2021 TI - Stellenwertverständnis: Verständnis von Stellenwertprinzip und Bündelungsprinzip als separate Konstrukte ER - TY - JOUR AU - Schönherr, Johanna AU - Blomberg, J. AU - Schukajlow, S. AU - Leopold, C. ID - 45342 JF - Contemporary Educational Psychology TI - 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 VL - 65 ER - TY - JOUR AU - Schönherr, Johanna AU - Schukajlow, S. AU - Blomberg, J. AU - Leopold, C. ID - 45333 JF - Mathematical Thinking and Learning TI - Does strategic knowledge matter? Effects of strategic knowledge about drawing on students’ modeling competencies in the domain of geometry ER - TY - JOUR AU - Schönherr, Johanna AU - Schukajlow, S. AU - Blomberg, J. AU - Leopold, C. ID - 45343 JF - Learning and Instruction TI - The role of strategy-based motivation in mathematical problem solving: The case of learner-generated drawings VL - 80 ER - TY - JOUR AU - Schönherr, Johanna AU - Schukajlow, S. AU - Blomberg, J. ID - 45335 JF - mathematik lehren TI - Was ist eine gute Skizze? Strategiewissen beim mathematischen Modellieren im Bereich der Geometrie fördern VL - 224 ER - TY - JOUR AU - Black, Tobias AU - Fuest, Mario AU - Lankeit, Johannes ID - 34673 IS - 3 JF - Zeitschrift für angewandte Mathematik und Physik KW - Applied Mathematics KW - General Physics and Astronomy KW - General Mathematics SN - 0044-2275 TI - Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems VL - 72 ER - TY - JOUR AU - Black, Tobias AU - Wu, Chunyan ID - 34675 IS - 4 JF - Zeitschrift für angewandte Mathematik und Physik KW - Applied Mathematics KW - General Physics and Astronomy KW - General Mathematics SN - 0044-2275 TI - Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation VL - 72 ER - TY - JOUR AU - Dröse, Jennifer AU - Prediger, S. AU - Neugebauer, P. AU - Danhier, R. D. AU - Mertins, B. ID - 45381 JF - International Electronic Journal of Mathematics Education, 16(1), em0625 TI - Investigating students' processes of noticing and interpreting syntactic language features in word problem solving through eye-tracking ER - TY - JOUR AU - Dröse, Jennifer AU - Prediger, S. ID - 45380 JF - Studies in Educational Evaluation, 68 (100953) TI - Identifying obstacles is not enough for everybody – Differential efficacy of an intervention fostering fifth graders’ comprehension for word problems ER - TY - JOUR AU - Häsel-Weide, Uta AU - Seitz, Simone AU - Wallner, Melina AU - Wilke, Yannik AU - Heckmann, Lara ID - 31578 IS - 1 JF - QfI - Qualifizierung für Inklusion. Online-Zeitschrift zur Forschung über Aus-, Fort- und Weiterbildung pädagogischer Fachkräfte TI - Mit Aufgaben im inklusiven Mathematikunterricht professionell umgehen - Erkenntnisse einer Interviewstudie mit Lehrpersonen der Sekundarstufe VL - 3 ER - TY - CONF AU - Hattermann, Mathias AU - Häsel-Weide, Uta AU - Wallner, Melina ED - Inprasitha, M. ED - Changsri, N. ED - Boonsena, N. ID - 31583 T2 - Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education TI - Conceptualiziation processes of 6th graders for rotational symmetry VL - 3 ER - TY - CONF AB - The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid. AU - Offen, Christian AU - Ober-Blöbaum, Sina ID - 22894 KW - optimal control KW - catastrophe theory KW - bifurcations KW - variational methods KW - symplectic integrators SN - 2405-8963 TI - Bifurcation preserving discretisations of optimal control problems VL - 54(19) ER - TY - CONF AU - Ridderbusch, Steffen AU - Offen, Christian AU - Ober-Blöbaum, Sina AU - Goulart, Paul ID - 21572 T2 - 2021 60th IEEE Conference on Decision and Control (CDC) TI - Learning ODE Models with Qualitative Structure Using Gaussian Processes ER - TY - CONF AB - We propose a reachability approach for infinite and finite horizon multi-objective optimization problems for low-thrust spacecraft trajectory design. The main advantage of the proposed method is that the Pareto front can be efficiently constructed from the zero level set of the solution to a Hamilton-Jacobi-Bellman equation. We demonstrate the proposed method by applying it to a low-thrust spacecraft trajectory design problem. By deriving the analytic expression for the Hamiltonian and the optimal control policy, we are able to efficiently compute the backward reachable set and reconstruct the optimal trajectories. Furthermore, we show that any reconstructed trajectory will be guaranteed to be weakly Pareto optimal. The proposed method can be used as a benchmark for future research of applying reachability analysis to low-thrust spacecraft trajectory design. AU - Vertovec, Nikolaus AU - Ober-Blöbaum, Sina AU - Margellos, Kostas ID - 21592 TI - Multi-objective minimum time optimal control for low-thrust trajectory design ER - TY - CONF AU - Jiménez, F. AU - Ober-Blöbaum, Sina ID - 29868 T2 - Nichtlineare Sci 31 TI - Fractional Damping Through Restricted Calculus of Variations VL - 46 ER - TY - JOUR AB - ZusammenfassungZu den ersten geometrischen Begriffen, die Kinder bereits im Elementar- und Primarbereich lernen, zählen u. a. Viereck, Rechteck und Quadrat. Studien zeigen, dass Lernende bereits früh individuelle Vorstellungen, sog. individuelle Begriffskonzepte, zu diesen Begriffen aufbauen. Zwar wird die Entwicklung von Begriffsverständnis in verschiedenen mathematikdidaktischen Stufenmodellen dargestellt, diese sind jedoch generisch und beschreiben nicht explizit die Entwicklung der ersten individuellen Begriffskonzepte von Lernenden zu Viereck, Rechteck und Quadrat. Aus empirischer Sicht liegen verschiedene Studien vor, die einzelne Aspekte der individuellen Begriffskonzepte von Lernenden unterschiedlicher Altersgruppen zu diesen Begriffen ausleuchten. Um Begriffsbildungsprozesse aus empirischer Sicht detaillierter entlang der jeweils vorherrschenden individuellen Begriffskonzepte zu beschreiben, fehlen insbesondere Studien in der Grundschule, die alle vier Klassenstufen betrachten und dabei differenzierte Erkenntnisse zu verschiedenen theoretischen Indikatoren des Begriffsverständnisses liefern. Daher geht die vorliegende Studie der Frage nach, welches Verständnis der Begriffe Viereck, Rechteck und Quadrat Schülerinnen und Schüler der Jahrgangsstufen 1, 2, 3 und 4 zeigen. Dazu wurde eine Quasi-Längsschnittstudie mit N = 456 Grundschulkindern (ca. 100 pro Jahrgangsstufe) durchgeführt. Die Ergebnisse geben detaillierte Einblicke in die individuellen Begriffskonzepte der Lernenden und zeigen, dass Lernende zunehmend Eigenschaften der Figuren berücksichtigen, jedoch individuelle Begriffskonzepte über lange Zeit auch prototypisch geprägt sind. Implikationen dieser Ergebnisse für Forschung und Praxis werden diskutiert. AU - Bruns, Julia AU - Unterhauser, Elisabeth AU - Gasteiger, Hedwig ID - 34827 IS - 2 JF - Journal für Mathematik-Didaktik KW - Education KW - General Mathematics SN - 0173-5322 TI - Geometrisches Begriffsverständnis in der Grundschule am Beispiel der Begriffe Viereck, Rechteck und Quadrat VL - 42 ER - TY - JOUR AU - Prediger, Susanne AU - Dröse, Jennifer ID - 45382 JF - Lernen und Lernstörungen, 10(2) TI - Fehlerbearbeitung bei mathematischen Textaufgaben – Sprachliche und strategische Fehlerursachen und ihre Bearbeitung ER - TY - JOUR AB - AbstractMathematics 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. AU - Schürmann, Mirko AU - Panse, Anja AU - Shaikh, Zain AU - Biehler, Rolf AU - Schaper, Niclas AU - Liebendörfer, Michael AU - Hilgert, Joachim ID - 35702 IS - 1 JF - International Journal of Research in Undergraduate Mathematics Education KW - Education KW - Mathematics (miscellaneous) SN - 2198-9745 TI - Consultation Phases in Mathematics Learning and Support Centres VL - 8 ER - TY - JOUR AU - Guillarmou, Colin AU - Hilgert, Joachim AU - Weich, Tobias ID - 31263 JF - Annales Henri Lebesgue SN - 2644-9463 TI - High frequency limits for invariant Ruelle densities VL - 4 ER - TY - JOUR AU - Brennecken, Dominik AU - Hilgert, Joachim AU - Ciardo, Lorenzo ID - 36271 IS - 2 JF - Journal of Lie Theory TI - Algebraically Independent Generators for the Algebra of Invariant Differential Operators on SLn(R)/SOn(R) VL - 31 ER - TY - BOOK ED - Hilgert, Joachim ED - Hilgert, I. ID - 51493 TI - Mathematik - Ein Reiseführer 2. Auflage ER - TY - GEN AU - Hilgert, Joachim AU - Weich, Tobias AU - Wolf, L. ID - 51503 TI - Higher-rank quantum-classical correspondence ER - TY - GEN AU - Hilgert, Joachim ID - 51556 T2 - Mathematische Semesterberichte TI - Philip Ording: 99 Variations on a Proof. Princeton University Press 2019 VL - 68 ER - TY - GEN AU - Hilgert, Joachim ID - 51555 T2 - Mathematische Semesterberichte TI - Georg Glaeser (Hrsg.): 77-mal Mathematik für zwischendurch – Unterhaltsame Kuriositäten und unorthodoxe Anwendungen. Springer Spektrum 2020 VL - 68 ER - TY - CHAP AU - Rezat, Sebastian AU - Schacht, Florian AU - Häsel-Weide, Uta ED - Clark-Wilson, A. ED - Donevska-Todorova, A. ED - Faggiano, E. ED - Trgalová , J. ED - Weigang, H.-G. ID - 34161 T2 - Mathematics Education in the Digital Age. Learning, Practice and Theory TI - Challenges of making sense of tasks and automated feedback in digital mathematics textbooks ER - TY - JOUR AB - 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. AU - McLachlan, Robert I AU - Offen, Christian ID - 19938 IS - 6 JF - Foundations of Computational Mathematics TI - Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation VL - 20 ER - TY - JOUR AU - Kreusser, Lisa Maria AU - McLachlan, Robert I AU - Offen, Christian ID - 19939 IS - 5 JF - Nonlinearity SN - 0951-7715 TI - Detection of high codimensional bifurcations in variational PDEs VL - 33 ER - TY - CHAP AB - 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. AU - Flaßkamp, Kathrin AU - Ober-Blöbaum, Sina AU - Peitz, Sebastian ED - Junge, Oliver ED - Schütze, Oliver ED - Froyland, Gary ED - Ober-Blöbaum, Sina ED - Padberg-Gehle, Kathrin ID - 17411 SN - 2198-4182 T2 - Advances in Dynamics, Optimization and Computation TI - Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach ER - TY - JOUR AB - 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. AU - Klus, Stefan AU - Nüske, Feliks AU - Hamzi, Boumediene ID - 21819 JF - Entropy SN - 1099-4300 TI - Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator ER - TY - JOUR AU - Hochmuth, Reinhard AU - Liebendörfer, Michael AU - Biehler, Rolf AU - Eichler, Andreas ID - 16964 JF - Neues Handbuch Hochschullehre TI - Das Kompetenzzentrum Hochschuldidaktik Mathematik (khdm) VL - 95 ER - TY - JOUR AB - 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. AU - Schütze, Oliver AU - Cuate, Oliver AU - Martín, Adanay AU - Peitz, Sebastian AU - Dellnitz, Michael ID - 10596 IS - 5 JF - Engineering Optimization SN - 0305-215X TI - Pareto Explorer: a global/local exploration tool for many-objective optimization problems VL - 52 ER - TY - JOUR AB - 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. AU - Klus, Stefan AU - Nüske, Feliks AU - Peitz, Sebastian AU - Niemann, Jan-Hendrik AU - Clementi, Cecilia AU - Schütte, Christof ID - 16288 JF - Physica D: Nonlinear Phenomena SN - 0167-2789 TI - Data-driven approximation of the Koopman generator: Model reduction, system identification, and control VL - 406 ER - TY - CHAP AB - 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. AU - Peitz, Sebastian AU - Klus, Stefan ID - 16289 SN - 0170-8643 T2 - Lecture Notes in Control and Information Sciences TI - Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator VL - 484 ER - TY - JOUR AB - 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. AU - Bieker, Katharina AU - Peitz, Sebastian AU - Brunton, Steven L. AU - Kutz, J. Nathan AU - Dellnitz, Michael ID - 16290 JF - Theoretical and Computational Fluid Dynamics SN - 0935-4964 TI - Deep model predictive flow control with limited sensor data and online learning VL - 34 ER - TY - JOUR AB - 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. AU - Peitz, Sebastian AU - Otto, Samuel E. AU - Rowley, Clarence W. ID - 16309 IS - 3 JF - SIAM Journal on Applied Dynamical Systems TI - Data-Driven Model Predictive Control using Interpolated Koopman Generators VL - 19 ER - TY - CHAP AB - Diagrammatisches Schlie{\ss}en wird im Zusammenhang mit dem Lernen von Mathmematik und ihrer Symbolsprache als wesentliche Theorie der Wissenskonstruktion diskutiert. Dabei wird h{\"{a}}ufig davon ausgegangen, dass die Wissenskonstruktion im Sinne diagrammatischen Schlie{\ss}ens erfolgt. Deskriptive Rekonstruktionen diagrammatischen Schlie{\ss}ens bei Lernenden stellen jedoch ein Desiderat der mathematikdidaktischen Forschung dar. Der vorliegende Beitrag befasst sich mit der Fragestellung, wie sich diagrammatisches Schlie{\ss}en bei Lernenden rekonstruieren l{\"{a}}sst. Als m{\"{o}}gliche Werkzeuge f{\"{u}}r eine solche Rekonstruktion werden Toulmins Argumentationsschema und Vergnauds Schema-Begriff exemplarisch angewandt, um das diagrammatische Schlie{\ss}en eines Sch{\"{u}}lerpaars beim Einstieg in die Subtraktion negativer Zahlen zu rekonstruieren. Abschlie{\ss}end wird die tats{\"{a}}chliche Eignung der beiden Ans{\"{a}}tze zur Rekonstruktion diagrammatischen Schlie{\ss}ens diskutiert. AU - Schumacher, Jan AU - Rezat, Sebastian ED - Kadunz, Gert ID - 13108 T2 - Zeichen und Sprache im Mathematikunterricht TI - Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge ER - TY - CONF AU - Schumacher, Jan ID - 31873 TI - Deduktion und Abduktion beim diagrammatischen Schließen – das didaktische Potential der Peirceschen Semiotik ER - TY - JOUR AB - 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 . AU - Gerlach, Raphael AU - Ziessler, Adrian AU - Eckhardt, Bruno AU - Dellnitz, Michael ID - 16710 JF - SIAM Journal on Applied Dynamical Systems SN - 1536-0040 TI - A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors ER - TY - JOUR AU - Amiri, Habib AU - Glöckner, Helge AU - Schmeding, Alexander ID - 34789 IS - 5 JF - Archivum Mathematicum KW - 22A22 KW - 22E65 KW - 22E67 KW - 46T10 KW - 47H30 KW - 58D15 KW - 58H05 SN - 0044-8753 TI - Lie groupoids of mappings taking values in a Lie groupoid VL - 56 ER - TY - JOUR AU - Glöckner, Helge AU - Masbough, Niku ID - 34787 JF - Topology Proceedings KW - 54B10 KW - 54D45 KW - 54D50 SN - 0146-4124 TI - Products of regular locally compact spaces are k_R-spaces VL - 55 ER - TY - GEN AB - 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. AU - Glöckner, Helge ID - 34808 T2 - arXiv:2006.00254 TI - Smoothing operators for vector-valued functions and extension operators ER - TY - GEN AB - 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. AU - Glöckner, Helge ID - 34807 T2 - arXiv:2007.15611 TI - Lie groups of real analytic diffeomorphisms are L^1-regular ER - TY - JOUR AU - Limebeer, D. J. N. AU - Ober-Blöbaum, Sina AU - Farshi, F. H. ID - 29399 JF - IEEE Transactions on Automatic Control TI - Variational integrators for dissipative systems VL - 65(4) ER - TY - JOUR AB - 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. AU - Hernández Castellanos, Carlos Ignacio AU - Ober-Blöbaum, Sina AU - Peitz, Sebastian ID - 16297 JF - International Journal of Robust and Nonlinear Control TI - Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty VL - 30(17) ER - TY - JOUR AU - Hernández Castellanos, C. I. O. AU - Schütze, G. AU - Sun, J.-Q. AU - Ober-Blöbaum, Sina AU - Morales-Luna, G. ID - 29398 JF - Mathematics TI - Numerical computation of lightly multi-objective robust optimal solutions by means of generalized cell mapping VL - 8(11):1959 ER - TY - CONF AU - Lishkova, Y. AU - Ober-Blöbaum, Sina AU - Cannon, M. AU - Leyendecker, S. ID - 29422 T2 - Accepted for publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist Conference - Lake Tahoe TI - A multirate variational approach to simulation and optimal control for flexible spacecraft ER - TY - CONF AU - Faulwasser, T. AU - Flaßkamp, K. AU - Ober-Blöbaum, Sina AU - Worthmann, K. ID - 29423 T2 - 24th International Symposium on Mathematical Theory of Networks and Systems TI - A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systems ER - TY - CONF AU - Cresson, J. AU - Jiménez, F. AU - Ober-Blöbaum, Sina ID - 29424 T2 - 24th International Symposium on Mathematical Theory of Networks and Systems TI - Modelling of the convection-diffusion equation through fractional restricted calculus of variations ER - TY - JOUR AU - Jean, Frédéric AU - Maslovskaya, Sofya AU - Zelenko, Igor ID - 29545 IS - 1 JF - Geometriae Dedicata KW - Geometry and Topology SN - 0046-5755 TI - On Weyl’s type theorems and genericity of projective rigidity in sub-Riemannian geometry VL - 213 ER - TY - GEN AU - Maslovskaya, Sofya AU - Caillau, Jean-Baptiste AU - Djema, Walid AU - Giraldi, Laetitia AU - Jean-Luc, Jean-Luc AU - Pomet, Jean-Baptiste ID - 29546 TI - The turnpike property in maximization of microbial metabolite production ER - TY - JOUR AB - AbstractGiven 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. AU - Küster, Benjamin AU - Weich, Tobias ID - 31264 IS - 2 JF - Communications in Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0010-3616 TI - Pollicott-Ruelle Resonant States and Betti Numbers VL - 378 ER - TY - CONF AU - Hoffmann, Max ED - Siller, Hans-Stefan ED - Weigel, Wolfgang ED - Wörler, Jan Franz ID - 31372 T2 - Beiträge zum Mathematikunterricht 2020 TI - Schnittstellenaktivitäten zum Kongruenzsatz WSW ER - TY - GEN AU - Hoffmann, Max ID - 31386 T2 - Mathematische Semesterberichte TI - Rezension: Andrew Granville und Jenniver Granville: Prime Supects: The Anatomy of Integers and Permutations VL - 67 ER - TY - GEN AU - Hoffmann, Max ID - 31384 T2 - Mathematische Semesterberichte TI - Rezension: Ehrhard Behrends: Parkettierungen der Ebene – Von Escher über Möbius zu Penrose VL - 67 ER - TY - BOOK ED - Hoffmann, Max ID - 31381 TI - Der Mathematikunterricht 66 (6): Geometrie in Schule und Lehramtsausbildung – ein Nachwuchsheft ER - TY - CHAP AU - Häsel-Weide, Uta AU - Nührenbörger, Marcus ED - Hecker, Ulrich ED - Lassek, Maresi ED - Ramseger, Jörg ID - 31551 T2 - Kinder lernen Zukunft – Anforderungen und tragfähige Grundlagen TI - Tragfähige Grundlagen VL - Band 150 ER - TY - CHAP AU - Heckmann, Lara AU - Häsel-Weide, Uta ED - Siller, Hans-Stefan ED - Weigel, Wolfgang ED - Wörler, Jan Franz ID - 31552 SN - 978-3-95987-139-6 T2 - Beiträge zum Mathematikunterricht 2020 TI - Aufgaben für den inklusiven Mathematikunterricht - aus der Sicht von Lehrkräften. ER - TY - CHAP AU - Häsel-Weide, Uta ED - Heimlich, Ulrich ED - Wember, Franz B. ID - 31549 SN - 978-3170355699 T2 - Didaktik des Unterrichts bei Lernschwierigkeiten: Ein Handbuch für Studium und Praxis TI - Sachrechnen ER - TY - CHAP AU - Häsel-Weide, Uta ED - Cramer, Colin ED - König, Johannes ED - Rothland, Martin ED - Blömeke, Sigrid ID - 31550 SN - 978-3-8252-5473-5 T2 - Handbuch Lehrerinnen- und Lehrerbildung TI - Mathematik (Primarstufe) in der Lehrerinnen- und Lehrerbildung. Qualifizierung für das Lehren von Mathematik in der Grundschule. ER - TY - GEN AB - 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. AU - Schramm, Thomas AU - Gasser, Ingenuin AU - Schwenker, Sören AU - Seiler, Ruedi AU - Lohse, Alexander AU - Zobel, Kay ID - 33273 TI - Linear Algebra driven by Data Science ER - TY - JOUR AB - 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). AU - Kolb, Martin AU - Savov, Mladen ID - 33282 IS - 2 JF - Bernoulli KW - L ́evy processes KW - Perpetual integrals KW - Potential measures TI - A Characterization of the Finiteness of Perpetual Integrals of Levy Processes VL - 26 ER - TY - JOUR AB - 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. AU - Haddenhorst, Björn AU - Hüllermeier, Eyke AU - Kolb, Martin ID - 33330 IS - 2 JF - International Journal of Approximate Reasoning TI - Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model VL - 119 ER - TY - JOUR AU - Hesse, Kerstin AU - Sloan, Ian H. AU - Womersley, Robert S. ID - 34629 JF - Journal of Computational and Applied Mathematics KW - Applied Mathematics KW - Computational Mathematics SN - 0377-0427 TI - Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data VL - 382 ER - TY - CHAP AU - Hesse, Kerstin ED - Hickernell, Fred J. ED - Kritzer, Peter ID - 34632 SN - 9783110633115 T2 - Multivariate Algorithms and Information-Based Complexity TI - RBF-based penalized least-squares approximation of noisy scattered data on the sphere ER - TY - JOUR AU - Hanusch, Maximilian ID - 34828 IS - 1 JF - Indagationes Mathematicae KW - regularity of Lie groups SN - 0019-3577 TI - The regularity problem for Lie groups with asymptotic estimate Lie algebras VL - 31 ER - TY - JOUR AU - Hanusch, Maximilian ID - 34830 IS - 1 JF - Journal of Lie Theory KW - Lie theory KW - strong Trotter property TI - The Strong Trotter Property for Locally μ-convex Lie Groups VL - 30 ER - TY - CONF AU - Biehler, Rolf AU - Fleischer, Yannik AU - Budde, Lea AU - Frischemeier, Daniel AU - Gerstenberger, Dietrich AU - Podworny, Susanne AU - Schulte, Carsten ED - Arnold, P. ID - 35814 T2 - New Skills in the Changing World of Statistics Education Proceedings of the Roundtable conference of the International Association for Statistical Education (IASE) TI - Data science education in secondary schools: Teaching and learning decision trees with CODAP and Jupyter Notebooks as an example of integrating machine learning into statistics education ER - TY - CHAP AU - Lankeit, Elisa AU - Biehler, Rolf ED - Hausberger, T. ED - Bosch, M. ED - Chelloughi, F. ID - 35912 T2 - Proceedings of the Third Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2020, 12-19 September 2020) TI - “I only know the absolute value function”–About students’ concept images and example spaces concerning continuity and differentiability ER - TY - CHAP AU - Lankeit, Elisa AU - Bauer, T. AU - Biehler, Rolf ED - Siller, H.-S. ED - Weigel, W. ED - Wörler, J. F. ID - 35911 T2 - Beiträge zum Mathematikunterricht 2020 TI - Votingfragen in den Übungen zur Analysis – Wirkung verschiedener Einsatzszenarien ER - TY - JOUR AU - Lünne, Steffen AU - Schnell, Susanne AU - Biehler, Rolf ID - 35701 IS - 5 JF - European Journal of Teacher Education KW - Education SN - 0261-9768 TI - Motivation of out-of-field teachers for participating in professional development courses in mathematics VL - 44 ER - TY - CHAP AU - Barzel, Bärbel AU - Biehler, Rolf ED - Zehetmeier, Stefan ED - Potari, Despina ED - Ribeiro, Miguel ID - 35808 T2 - Professional development and knowledge of mathematics teachers TI - Theory-Based Design of Professional Development for Upper Secondary Teachers–Focusing on the Content-Specific Use of Digital Tools ER - TY - CHAP AU - Biehler, Rolf AU - Durand-Guerrier, Viviane ED - Hausberger, T. ED - Bosch, M. ED - Chelloughi, F. ID - 35811 KW - Number Theory KW - Algebra KW - Discrete Mathematics KW - Logic KW - Research in University Mathematics Edcuation T2 - Proceedings of the Third Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2020, 12-19 September 2020) TI - University Mathematics Didactic Research on Number Theory, Algebra, Discrete Mathematics, Logic ER - TY - CHAP AU - Budde, Lea AU - Frischemeier, Daniel AU - Biehler, Rolf AU - Fleischer, Yannik AU - Gerstenberger, Dietrich AU - Podworny, Susanne AU - Schulte, Carsten ED - Arnold, P. ID - 35821 T2 - New Skills in the Changing World of Statistics Education: Proceedings of the Roundtable conference of the International Association for Statistical Education (IASE), July 2020 TI - Data Science Education in Secondary School: How to Develop Statistical Reasoning When Exploring Data Using CODAP ER - TY - CHAP AU - Biehler, Rolf AU - Kempen, Leander AU - Paravicini, Walther ED - Siller, H.-S. ED - Weigel, W. ED - Wörler, J. F. ID - 35819 T2 - Beiträge zum Mathematikunterricht 2020 TI - Hochschuldidaktik Mathematik: Fachbezogenes Design und empirische Studien ER - TY - JOUR AB - AbstractThe derivative concept plays a major role in economics. However, its use in economics is very heterogeneous, sometimes inconsistent, and contradicts students’ prior knowledge from school. This applies in particular to the common economic interpretation of the derivative as the amount of change while increasing the production by one unit. Hence, in calculus courses for economics students, learners should acquire an understanding of the derivative that is mathematically acceptable and connected to their prior knowledge, but which also takes into account its practical use in economics. In this paper we first develop a theoretical model describing such an understanding of the derivative for economics students. We then present an exploratory study investigating the extent to which economics students have such an understanding after their calculus course. The results indicate that many of them might not have acquired this kind of understanding, in particular concerning the common economic interpretation of the derivative. The study furthermore yields possible gaps in students’ understanding and possible misconceptions. AU - Feudel, Frank AU - Biehler, Rolf ID - 35822 IS - 1 JF - Journal für Mathematik-Didaktik KW - Education KW - General Mathematics SN - 0173-5322 TI - Students’ Understanding of the Derivative Concept in the Context of Mathematics for Economics VL - 42 ER - TY - CHAP AU - Fleischer, Yannik AU - Biehler, Rolf ED - Siller, H.-S. ED - Weigel, W. ED - Wörler, J. F. ID - 35823 T2 - Beiträge zum Mathematikunterricht 2020 TI - Automatisierte Entscheidungsverfahren als Thema im allgemeinbildenden Mathematikunterricht ER - TY - JOUR AU - Griese, Birgit AU - Nieszporek, Ralf AU - Biehler, Rolf ID - 35826 IS - 1 JF - Stochastik in der Schule TI - Frei verfügbare Materialien für Unterricht und Fortbildung: Stochastik verständnisorientiert unterrichten VL - 40 ER - TY - JOUR AU - Kempen, Leander AU - Biehler, Rolf ID - 35709 JF - Frontiers in Psychology KW - General Psychology SN - 1664-1078 TI - Using Figurate Numbers in Elementary Number Theory – Discussing a ‘Useful’ Heuristic From the Perspectives of Semiotics and Cognitive Psychology VL - 11 ER - TY - CHAP AU - Kempen, Leander AU - Krämer, Sandra AU - Biehler, Rolf ED - Hausberger, T. ED - Bosch, M. ED - Chelloughi, F. ID - 35829 T2 - Proceedings of the Third Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2020, 12-19 September 2020) TI - Investigating high school graduates’ personal meaning of the notion of “mathematical proof” ER - TY - CHAP AU - Kempen, Leander AU - Krämer, Sandra AU - Biehler, Rolf ED - Siller, H.-S. ED - Weigel, W. ED - Wörler, J. F. ID - 35827 T2 - Beiträge zum Mathematikunterricht 2020 TI - Was verstehen Schülerinnen und Schüler unter „Beweis“? – ausgewählte Ergebnisse einer Pilotstudie ER - TY - CHAP AU - Lünne, Steffen AU - Schnell, Susanne AU - Biehler, Rolf ED - Porsch, R. ED - Rösken-Winter, B. ID - 35914 SN - 9783658272920 T2 - Professionelles Handeln im fachfremd erteilten Mathematikunterricht TI - Ffunt@OWL – Konzept und Gestaltungsprinzipien zur Qualifizierung fachfremd Mathematik unterrichtender Lehrkräfte der Sekundarstufe I ER -