Hamiltonian Neural Networks with Automatic Symmetry Detection - Dierkes, Eva; Offen, Christian; Ober-Blöbaum, Sina; Flaßkamp, Kathrin (Chaos)
Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered.

Finite-data error bounds for Koopman-based prediction and control - Nüske, Feliks; Peitz, Sebastian; Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl (Journal of Nonlinear Science)
The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d. samples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.

Efficient time stepping for numerical integration using reinforcement learning - Dellnitz, Michael; Hüllermeier, Eyke; Lücke, Marvin; Ober-Blöbaum, Sina; Offen, Christian; Peitz, Sebastian; Pfannschmidt, Karlson (SIAM Journal on Scientific Computing)
Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.

Second grader´s understanding of doubling and halfing in various representations - Schwerin, Imke; Häsel-Weide, Uta (International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education)

Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern - Schmidt, Rebekka; Tenberge, Claudia; Häsel-Weide, Uta (Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre)

Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course - Hoffmann, Max; Biehler, Rolf (ZDM – Mathematics Education)
Abstract Intending to counteract Klein’s second discontinuity in teacher education, we explored and applied the innovation of “ interface ePortfolio ” in the context of a geometry course for preservice teachers (PSTs). The tool offers the possibility of implementing the design principle of profession orientation . In the article, we theoretically clarify what we understand by this principle and locate our innovative concept against this theoretical background. We empirically investigate the extent to which counteraction against the second discontinuity is successful by analyzing reflection texts created in the interface ePortfolio, focusing on PSTs’ perspectives. Our qualitative content analysis shows that most of them perceive the innovation as helpful in the intended sense and indicates that the course concept, in general, and the interface ePortfolio, in particular, have helped establish relevant links between the course content and their later work as teachers.

Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements - Dröse, Jennifer; Prediger, Susanne (Journal für Mathematik-Didaktik)
Abstract Teachers’ 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.
Keywords: Education; General Mathematics

Teacher expertise for fostering at-risk students’ understanding of basic concepts: conceptual model and evidence for growth - Prediger, S.; Dröse, Jennifer; Stahnke, R.; Ademmer, C. (Journal of Mathematics Teacher Education)

Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects? - Bruns, Julia; Hagena, Maike; Gasteiger, Hedwig (Teaching and Teacher Education)
Keywords: Education

Introducing mathematics undergraduate students to the theorem prover lean - Garnelo Abellanas, Irene; Liebendörfer, Michael (Thirteenth Congress of the European Society for Research in Mathematics (CERME13))

Feynman–Kac Formula and Asymptotic Behavior of the Minimal Energy for the Relativistic Nelson Model in Two Spatial Dimensions - Hinrichs, Benjamin; Matte, Oliver (Annales Henri Poincaré)

Super-Gaussian decay of exponentials: A sufficient condition - Hinrichs, Benjamin; Janssen, Daan W.; Ziebell, Jobst (Journal of Mathematical Analysis and Applications)
Keywords: Applied Mathematics; Analysis

Higher rank quantum-classical correspondence - Hilgert, Joachim; Weich, Tobias; Wolf, Lasse Lennart (Analysis & PDE)
For a compact Riemannian locally symmetric space $Γ\backslash G/K$ of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle-Taylor resonances and establish a spectral gap which is uniform in $Γ$ if $G/K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G/K$.

Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems - Schütte, Philipp; Weich, Tobias; Barkhofen, Sonja (Communications in Mathematical Physics)
In this article we prove meromorphic continuation of weighted zeta functions in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (2016). We obtain a residue formula proving equality between residues of weighted zetas and invariant Ruelle distributions. We combine this equality with results of Guillarmou, Hilgert and Weich (2021) in order to relate the residues to Patterson-Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems.

Einblicke in ein Referenzmodell zur Analyse der Einführung von Vektoren in Schulbüchern - Mai, Tobias; Biehler, Rolf (Beiträge zum Mathematikunterricht 2022. 56. Jahrestagung der Gesellschaft für Didaktik der Mathematik)

Fachdidaktisches Design von Begründungsvideos im Projekt studiVEMINTvideos - Biehler, Rolf; Guntermann, Dominik; Liebendörfer, Michael; Krämer, Sandra; Schlüter, Sarah (Beiträge zum Mathematikunterricht 2022. 56. Jahrestagung der Gesellschaft für Didaktik der Mathematik)

Das totale Differential und die Richtungsableitung – Eine Analyse mit Blick in ausgewählte Lehrbücher - Lankeit, Elisa; Biehler, Rolf (Beiträge zum Mathematikunterricht 2022. 56. Jahrestagung der Gesellschaft für Didaktik der Mathematik)

Spectral correspondences for rank one locally symmetric spaces - The case of exceptional parameters - Hilgert, Joachim; Arends, C. (J. de l'École polytechnique — Mathématiques)

Aspects of control theory on infinite-dimensional Lie groups and G-manifolds - Hilgert, Joachim; Glöckner, H. (J. Diff. Equations)

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds - Papageorgiou, Efthymia (Potential Analysis)
Abstract This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with $$L^1$$ L 1 initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.
Keywords: Analysis