TY - JOUR AB - In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby ‘modified’ equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation. AU - McLachlan, Robert I AU - Offen, Christian ID - 19941 IS - 3 JF - Journal of Geometric Mechanics TI - Backward error analysis for variational discretisations of partial differential equations VL - 14 ER - TY - JOUR AB - Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes. AU - Offen, Christian AU - Ober-Blöbaum, Sina ID - 23382 JF - Chaos: An Interdisciplinary Journal of Nonlinear Science TI - Symplectic integration of learned Hamiltonian systems VL - 32(1) ER - TY - CONF AU - Dröse, Jennifer ID - 40613 T2 - Beiträge zum Mathematikunterricht TI - Verstehensgrundlagen diagnostizieren - Diagnostisches Denken von drei Professionalisierungsgruppen ER - TY - CONF AU - Graf, Lara Marie AU - Häsel-Weide, Uta AU - Nührenbörger, Marcus AU - Höveler, Karina ID - 47520 T2 - Beiträge zum Mathematikunterricht 2022 - 56. Jahrestagung der Gesellschaft für Didaktik der Mathematik TI - Lernwege von fachfremd unterrichtenden Lehrkräften zur Ablösung vom zählenden Rechnen ER - TY - CONF AU - Schwerin, Imke ID - 47519 T2 - Beiträge zum Mathematikunterricht 2022 - 56. Jahrestagung der Gesellschaft für Didaktik der Mathematik TI - Verdoppeln und Halbieren im 2. Schuljahr - Vorgehensweisen und Verständnis ER - TY - CHAP AU - Dellori, Anna AU - Wessel, Lena ED - Karunakaran, S.S. ED - Higgins, A. ID - 48407 T2 - Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education TI - Pre-service Teachers' Professional Development: Relating Abstract Algebra and School Algebra ER - TY - JOUR AU - Wessel, Lena AU - Dröse, Jennifer ID - 48408 JF - mathematik lehren TI - Schreiben will gelernt sein: Schreiblerngelegenheiten adaptiv gestalten VL - 233 ER - TY - CHAP AU - Wessel, Lena AU - Leuders, Timo ED - Biehler, Rolf ED - Liebendörfer, Michael ED - Gueudet, G. ED - Rasmussen, C. ED - Winslow, C. ID - 46160 SN - 1869-4918 T2 - Practice-Oriented Research in Tertiary Mathematics Education TI - Profession-Specific Curriculum Design in Mathematics Teacher Education: Connecting Disciplinary Practice to the Learning of Group Theory ER - TY - JOUR AU - Dellori, Anna AU - Wessel, Lena ID - 48325 JF - Beiträge zum Mathematikunterricht 2022 TI - Entwicklung und Erprobung von professionsorientierten Lernumgebungen zur Wissensvernetzung in der Algebra ER - TY - CHAP AU - Dröse, Jennifer AU - Neugebauer, P. AU - Delucchi Danhier, R. AU - Mertins, B. ED - Kleine, P. ED - Graulich, N. ED - Kuhn, J. ED - Schindler, M. ID - 45373 T2 - Eye-Tracking in der Mathematik- und Naturwissenschaftsdidaktik. Forschung und Praxis TI - Eye-Tracking Studie zu Textaufgaben in Klasse 5: Bemerken und Interpretieren syntaktischer Strukturen ER - TY - JOUR AU - Prediger, S. AU - Dröse, Jennifer AU - Stahnke, R. AU - Ademmer, C. ID - 45374 JF - Journal of Mathematics Teacher Education TI - Teacher expertise for fostering at-risk students’ understanding of basic concepts: conceptual model and evidence for growth ER - TY - CONF AU - Dröse, Jennifer AU - Griese, B. AU - Wessel, Lena ID - 45379 T2 - Twelfth Congress of the European Society for Research in Mathematics Education (CERME12) TI - Prosepctive teachers‘ diagnostic judgements on students’ under- standing of conditional probabilities ER - TY - CONF AB - Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example. AU - Vertovec, Nikolaus AU - Ober-Blöbaum, Sina AU - Margellos, Kostas ID - 30733 TI - Verification of safety critical control policies using kernel methods ER - TY - CHAP AU - Dellori, Anna AU - Wessel, Lena ED - Trigueros, M. ED - Barquero, B. ED - Hochmuth, R. ED - Peters, J. ID - 48385 T2 - Proceedings of INDRUM2022 TI - Design principles for intertwining local and nonlocal mathematics - The case of relating registers and representations in abstract algebra ER - TY - JOUR AB - AbstractTeachers’ in-depth diagnostic thinking has been shown to be crucial for student-centered teaching as they need to perceive and interpret students’ understanding for well-informed decision-making on adaptive teaching practices. The paper presents a content-related approach to analyzing diagnostic thinking processes with respect to the mathematical knowledge elements that prospective teachers identify as students’ resources and obstacles. Prospective teachers’ challenge is that some relevant knowledge elements first have to be unpacked, because compact concepts (such as the place value concept) or procedures (such as for multi-digit multiplication) comprise several smaller knowledge elements (such as the positional property) that have to be made explicit for students to foster their learning processes adequately. Our study examines what knowledge elements prospective teachers perceive and interpret in a transcript vignettes on multi-digit multiplication (of decimal and natural numbers) and its underlying basic arithmetic concepts (place value understanding and meaning of multiplication) in written diagnostic judgments on students’ resources and obstacles (N = 196). A comparative design within the vignette is used to investigate how far the process of perceiving can be supported by thematic cues. The analysis reveals that those knowledge elements cued in the vignette by being already unpacked and explicitly addressed are perceived and interpreted more often (but with lower correctness) than those that are uncued and therefore have to be unpacked by the prospective teachers themselves. This confirms the need to prepare prospective teachers for unpacking mathematical concepts themselves. AU - Dröse, Jennifer AU - Prediger, Susanne ID - 40607 JF - Journal für Mathematik-Didaktik KW - Education KW - General Mathematics SN - 0173-5322 TI - Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements ER - TY - CONF AU - Dröse, Jennifer AU - Griese, Birgit AU - Wessel, Lena ID - 48389 T2 - Twelfth Congress of the European Society for Research in Mathematics Education (CERME12) TI - Prospective teachers’ diagnostic judgments on students’ understanding of conditional probabilities ER - TY - CONF AU - Dröse, Jennifer AU - Wessel, Lena ED - Fernandez, C. ED - Llinares, S. ED - Gutiérrez, A. ED - Planas, N. ID - 45378 T2 - Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education. PME TI - Prospective Teachers‘ Competence of Fostering Students’ Understanding in Script Writing Task ER - TY - JOUR AB - In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems. AU - Barkhofen, Sonja AU - Schütte, Philipp AU - Weich, Tobias ID - 31057 IS - 24 JF - Journal of Physics A: Mathematical and Theoretical TI - Semiclassical formulae For Wigner distributions VL - 55 ER - TY - THES AB - Vorgestellt wird ein Entwicklungsforschungsprojekt zur Konzeption und Durchführung einer Veranstaltung "Geometrie für Lehramtsstudierende". Die Schwerpunkte des Projekts sind zum einen die inhaltliche Gestaltung der Veranstaltung und zum anderen die Umsetzung von Professionsorientierung. Bezogen auf den inhaltlichen Aufbau wird das auf metrischen Räumen aufbauende Axiomensystem der "Saccheri-Ebene" vorgestellt und mit alternativen axiomatischen Zugängen zur ebenen Geometrie verglichen. Die Frage nach der Umsetzung von Professionsorientierung in Fachveranstaltungen ist eng mit der Problematik der zweiten Diskontinuität verbunden. In der Arbeit wird dieses Problem auf Grundlage der Synthese von theoretischen Hintergründen zur Bedeutung von mathematischem Wissen und Können für professionelle Handlungskompetenz von Mathematiklehrkräften diskutiert und darauf aufbauend werden theoriebasierte Entwurfsprinzipien für professionsorientierte Fachveranstaltungen entworfen. Zentrale Elemente der methodischen Gestaltung sind die sogenannten "Schnittstellenwochen" zu den Themen Kongruenz und Symmetrie sowie das begleitende Schnittstellen-ePortfolio. Das zentrale Ergebnis der Arbeit ist ein theoretisch fundiertes und empirisch evaluiertes ganzheitliches Veranstaltungskonzept für eine professionsorientierte Geometrie-Veranstaltung für Lehramtsstudierende, dessen Konzeption auf andere Fachveranstaltungen übertragbar ist. Darüber hinaus ergeben sich im Rahmen der durchgeführten Entwicklungsforschung verschiedene neue Beiträge zur Geometriedidaktik in Schule- und Hochschule. AU - Hoffmann, Max ID - 31363 TI - Von der Axiomatik bis zur Schnittstellenaufgabe: Entwicklung und Erforschung eines ganzheitlichen Lehrkonzepts für eine Veranstaltung Geometrie für Lehramtsstudierende ER - TY - JOUR AU - Bux, Kai-Uwe AU - Hilgert, Joachim AU - Weich, Tobias ID - 35322 IS - 2 JF - Journal of Spectral Theory KW - Geometry and Topology KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 1664-039X TI - Poisson transforms for trees of bounded degree VL - 12 ER -