TY - BOOK ED - Biehler, Rolf ED - Liebendörfer, Michael ED - Gueudet, Ghislaine ED - Rasmussen, Chris ED - Winsløw, Carl ID - 37469 SN - 1869-4918 TI - Practice-Oriented Research in Tertiary Mathematics Education ER - TY - JOUR AU - Brennecken, Dominik AU - Rösler, Margit ID - 36294 JF - Transaction of the American Mathematical Society TI - The Dunkl-Laplace transform and Macdonald’s hypergeometric series ER - TY - JOUR AU - Hanusch, Maximilian ID - 34814 IS - 1 JF - Canadian Journal of Mathematics KW - extension of differentiable maps SN - 0008-414X TI - A $C^k$-seeley-extension-theorem for Bastiani’s differential calculus VL - 75 ER - TY - JOUR AB - Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression and machine learning. Since the choice of the regularization parameter is crucial but often difficult, path-following methods are used to approximate the entire regularization path, i.e., the set of all possible solutions for all regularization parameters. Due to their nature, the development of these methods requires structural results about the regularization path. The goal of this article is to derive these results for the case of a smooth objective function which is penalized by a piecewise differentiable regularization term. We do this by treating regularization as a multiobjective optimization problem. Our results suggest that even in this general case, the regularization path is piecewise smooth. Moreover, our theory allows for a classification of the nonsmooth features that occur in between smooth parts. This is demonstrated in two applications, namely support-vector machines and exact penalty methods. AU - Gebken, Bennet AU - Bieker, Katharina AU - Peitz, Sebastian ID - 27426 IS - 3 JF - Journal of Global Optimization TI - On the structure of regularization paths for piecewise differentiable regularization terms VL - 85 ER - TY - CONF AU - Hoffmann, Max AU - Biehler, Rolf ED - Trigueros, Marı́a ED - Barquero, Berta ED - Hochmuth, Reinhard ED - Peters, Jana ID - 31849 KW - Teaching and learning of specific topics in university mathematics KW - Transition to KW - across and from university mathematics KW - Student Teachers KW - Geometry KW - Congruence KW - Double Discontinuity. T2 - Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022) TI - Student Teachers ’ Knowledge of Congruence before a University Course on Geometry ER - TY - CONF AU - Florensa, Ignasio AU - Hoffmann, Max AU - Romo Vázquez, Avenilde AU - Zandieh, Michelle AU - Martínez-Planell, Rafael ED - Trigueros, Marı́a ED - Barquero, Berta ED - Hochmuth, Reinhard ED - Peters, Jana ID - 43097 T2 - Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022) TI - Innovations in university teaching based on mathematic education research ER - TY - JOUR AU - Biehler, Rolf AU - Liebendörfer, Michael AU - Schmitz, A. ID - 43504 JF - Mitteilungen der Gesellschaft für Didaktik der Mathematik TI - Lernvideos und ihre Erstellung - Das Projekt studiVEMINTvideos VL - 114 ER - TY - JOUR AU - Black, Tobias AU - Fuest, Mario AU - Lankeit, Johannes AU - Mizukami, Masaaki ID - 43105 JF - Nonlinear Analysis: Real World Applications KW - Applied Mathematics KW - Computational Mathematics KW - General Economics KW - Econometrics and Finance KW - General Engineering KW - General Medicine KW - Analysis SN - 1468-1218 TI - Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source VL - 73 ER - TY - CHAP AU - Vitt, Vivian AU - Häsel-Weide, Uta ID - 43227 T2 - Mathematica Didactica, 46 TI - Reziprokes Peer-Tutoring zur Förderung von Schüler*innen mit Schwierigkeiten beim Mathematiklernen. ER - TY - CHAP AU - Häsel-Weide, Uta AU - Nührenbörger, M. ID - 43226 T2 - Mathematica Didactica, 46 TI - Inklusive Praktiken unterrichtsintegrierter Förderung im Mathematikunterricht. ER - TY - JOUR AU - Hanusch, Maximilian ID - 34832 IS - 21 JF - Annals of Global Analysis and Geometry KW - Lax equation KW - generalized Baker-Campbell-Dynkin-Hausdorff formula KW - regularity of Lie groups TI - The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups VL - 63 ER - TY - GEN AB - Extending the notion of maxcut, the study of the frustration index of signed graphs is one of the basic questions in the theory of signed graphs. Recently two of the authors initiated the study of critically frustrated signed graphs. That is a signed graph whose frustration index decreases with the removal of any edge. The main focus of this study is on critical signed graphs which are not edge-disjoint unions of critically frustrated signed graphs (namely non-decomposable signed graphs) and which are not built from other critically frustrated signed graphs by subdivision. We conjecture that for any given k there are only finitely many critically k-frustrated signed graphs of this kind. Providing support for this conjecture we show that there are only two of such critically 3-frustrated signed graphs where there is no pair of edge-disjoint negative cycles. Similarly, we show that there are exactly ten critically 3-frustrated signed planar graphs that are neither decomposable nor subdivisions of other critically frustrated signed graphs. We present a method for building non-decomposable critically frustrated signed graphs based on two given such signed graphs. We also show that the condition of being non-decomposable is necessary for our conjecture. AU - Cappello, Chiara AU - Naserasr, Reza AU - Steffen, Eckhard AU - Wang, Zhouningxin ID - 44501 T2 - arXiv:2304.10243 TI - Critically 3-frustrated signed graphs ER - TY - JOUR AB - Ancestral reconstruction is a classic task in comparative genomics. Here, we study the genome median problem, a related computational problem which, given a set of three or more genomes, asks to find a new genome that minimizes the sum of pairwise distances between it and the given genomes. The distance stands for the amount of evolution observed at the genome level, for which we determine the minimum number of rearrangement operations necessary to transform one genome into the other. For almost all rearrangement operations the median problem is NP-hard, with the exception of the breakpoint median that can be constructed efficiently for multichromosomal circular and mixed genomes. In this work, we study the median problem under a restricted rearrangement measure called c4-distance, which is closely related to the breakpoint and the DCJ distance. We identify tight bounds and decomposers of the c4-median and develop algorithms for its construction, one exact ILP-based and three combinatorial heuristics. Subsequently, we perform experiments on simulated data sets. Our results suggest that the c4-distance is useful for the study the genome median problem, from theoretical and practical perspectives. AU - Silva, Helmuth O.M. AU - Rubert, Diego P. AU - Araujo, Eloi AU - Steffen, Eckhard AU - Doerr, Daniel AU - Martinez, Fábio V. ID - 44857 IS - 3 JF - RAIRO - Operations Research KW - Management Science and Operations Research KW - Computer Science Applications KW - Theoretical Computer Science SN - 0399-0559 TI - Algorithms for the genome median under a restricted measure of rearrangement VL - 57 ER - TY - GEN AU - Ma, Yulai AU - Mattiolo, Davide AU - Steffen, Eckhard AU - Wolf, Isaak Hieronymus ID - 44859 T2 - arXiv:2305.08619 TI - Sets of r-graphs that color all r-graphs ER - TY - JOUR AU - Hanusch, Maximilian ID - 34833 IS - 4 JF - Indagationes Mathematicae. KW - Lie group actions and analytic 1-submanifolds TI - Decompositions of Analytic 1-Manifolds VL - 34 ER - TY - GEN AB - We present a novel method for high-order phase reduction in networks of weakly coupled oscillators and, more generally, perturbations of reducible normally hyperbolic (quasi-)periodic tori. Our method works by computing an asymptotic expansion for an embedding of the perturbed invariant torus, as well as for the reduced phase dynamics in local coordinates. Both can be determined to arbitrary degrees of accuracy, and we show that the phase dynamics may directly be obtained in normal form. We apply the method to predict remote synchronisation in a chain of coupled Stuart-Landau oscillators. AU - von der Gracht, Sören AU - Nijholt, Eddie AU - Rink, Bob ID - 45498 T2 - arXiv:2306.03320 TI - A parametrisation method for high-order phase reduction in coupled oscillator networks ER - TY - JOUR AU - Häsel-Weide, Uta ID - 45712 IS - 339 JF - Die Grundschulzeitschrift TI - Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen ER - TY - JOUR AU - Graf, Lara Marie AU - Wienhues, Inga AU - Häsel-Weide, Uta ID - 45713 IS - 339 JF - Die Grundschulzeitschrift TI - Addition und Subtraktion verstehen ER - TY - JOUR AB - 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. AU - Hoffmann, Max AU - Biehler, Rolf ID - 45786 JF - ZDM – Mathematics Education KW - General Mathematics KW - Education SN - 1863-9690 TI - 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 ER - TY - JOUR AU - Hinrichs, Benjamin AU - Janssen, Daan W. AU - Ziebell, Jobst ID - 46100 IS - 1 JF - Journal of Mathematical Analysis and Applications KW - Applied Mathematics KW - Analysis SN - 0022-247X TI - Super-Gaussian decay of exponentials: A sufficient condition VL - 528 ER - TY - GEN AB - Let $X=X_1\times X_2$ be a product of two rank one symmetric spaces of non-compact type and $\Gamma$ a torsion-free discrete subgroup in $G_1\times G_2$. We show that the spectrum of $\Gamma \backslash X$ is related to the asymptotic growth of $\Gamma$ in the two direction defined by the two factors. We obtain that $L^2(\Gamma \backslash G)$ is tempered for large class of $\Gamma$. AU - Weich, Tobias AU - Wolf, Lasse L. ID - 46117 T2 - arXiv:2304.09573 TI - Temperedness of locally symmetric spaces: The product case ER - TY - JOUR AU - Bruns, Julia AU - Hagena, Maike AU - Gasteiger, Hedwig ID - 46155 JF - Teaching and Teacher Education KW - Education SN - 0742-051X TI - Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects? VL - 132 ER - TY - BOOK ED - Biehler, Rolf ED - Liebendörfer, Michael ED - Gueudet, Ghislaine ED - Rasmussen, Chris ED - Winsløw, Carl ID - 46157 SN - 1869-4918 TI - Practice-Oriented Research in Tertiary Mathematics Education ER - TY - JOUR AU - Ma, Yulai AU - Mattiolo, Davide AU - Steffen, Eckhard AU - Wolf, Isaak Hieronymus ID - 46256 IS - 3 JF - SIAM Journal on Discrete Mathematics KW - General Mathematics SN - 0895-4801 TI - Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs VL - 37 ER - TY - CONF AB - The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE. AU - Offen, Christian AU - Ober-Blöbaum, Sina ED - Nielsen, F ED - Barbaresco, F ID - 42163 KW - System identification KW - discrete Lagrangians KW - travelling waves T2 - Geometric Science of Information TI - Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves VL - 14071 ER - TY - JOUR AB - The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis. AU - Ober-Blöbaum, Sina AU - Offen, Christian ID - 29240 JF - Journal of Computational and Applied Mathematics KW - Lagrangian learning KW - variational backward error analysis KW - modified Lagrangian KW - variational integrators KW - physics informed learning SN - 0377-0427 TI - Variational Learning of Euler–Lagrange Dynamics from Data VL - 421 ER - TY - JOUR AB - The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients. AU - McLachlan, Robert AU - Offen, Christian ID - 29236 IS - 1 JF - Journal of Geometric Mechanics KW - variational integrators KW - backward error analysis KW - Euler--Lagrange equations KW - multistep methods KW - conjugate symplectic methods TI - Backward error analysis for conjugate symplectic methods VL - 15 ER - TY - JOUR AB - 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. AU - Dierkes, Eva AU - Offen, Christian AU - Ober-Blöbaum, Sina AU - Flaßkamp, Kathrin ID - 37654 IS - 6 JF - Chaos SN - 1054-1500 TI - Hamiltonian Neural Networks with Automatic Symmetry Detection VL - 33 ER - TY - JOUR AB - 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. AU - Nüske, Feliks AU - Peitz, Sebastian AU - Philipp, Friedrich AU - Schaller, Manuel AU - Worthmann, Karl ID - 23428 JF - Journal of Nonlinear Science TI - Finite-data error bounds for Koopman-based prediction and control VL - 33 ER - TY - JOUR AB - 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. AU - Dellnitz, Michael AU - Hüllermeier, Eyke AU - Lücke, Marvin AU - Ober-Blöbaum, Sina AU - Offen, Christian AU - Peitz, Sebastian AU - Pfannschmidt, Karlson ID - 21600 IS - 2 JF - SIAM Journal on Scientific Computing TI - Efficient time stepping for numerical integration using reinforcement learning VL - 45 ER - TY - CONF AU - Schwerin, Imke AU - Häsel-Weide, Uta ED - Novotna, J. ED - Moraova, H. ID - 46757 T2 - International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education TI - Second grader´s understanding of doubling and halfing in various representations ER - TY - CHAP AU - Schmidt, Rebekka AU - Tenberge, Claudia AU - Häsel-Weide, Uta ED - Vöing, N. ED - Schmidt, R. ED - Neiske, I. ID - 46758 T2 - Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre TI - Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern ER - TY - CHAP AU - Biehler, Rolf AU - Engel, Joachim AU - Frischemeier, Daniel ED - Bruder, Regina ED - Büchter, A. ED - Gasteiger, H. ED - Schmidt-Thieme, B. ED - Weigand, HG. ID - 47416 SN - 9783662666036 T2 - Handbuch der Mathematikdidaktik TI - Stochastik: Leitidee Daten und Zufall ER - TY - GEN AB - In this proceeding we consider a translation invariant Nelson type model in two spatial dimensions modeling a scalar relativistic particle in interaction with a massive radiation field. As is well-known, the corresponding Hamiltonian can be defined with the help of an energy renormalization. First, we review a Feynman-Kac formula for the semigroup generated by this Hamiltonian proven by the authors in a recent preprint (where several matter particles and exterior potentials are treated as well). After that, we employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian. AU - Hinrichs, Benjamin AU - Matte, Oliver ID - 47534 T2 - arXiv:2309.09005 TI - Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions ER - TY - CHAP AU - Biehler, Rolf AU - Frischemeier, Daniel AU - Gould, Ronald AU - Pfannkuch, Maxine ED - Pepin, Birgit ED - Gueudet, Ghislaine ED - Choppin, Jeffrey ID - 48042 T2 - Handbook of Digital Resources in Mathematics Education TI - Impacts of Digitalization on Content and Goals of Statistics Education ER - TY - CHAP AU - Leiss, Dominik AU - Gerlach, Kerstin AU - Wessel, Lena AU - Schmidt-Thieme, Barbara ID - 48319 SN - 9783662666036 T2 - Handbuch der Mathematikdidaktik TI - Sprache und Mathematiklernen ER - TY - CHAP AU - Kortmeyer, Jörg AU - Biehler, Rolf ED - Dreyfus, T. ED - Gonzalez-Martin, A. S. ED - Nardi, E. ED - Monaghan, J. ED - Thompson, P. W. ID - 48480 T2 - The Learning and Teaching of Calculus Across Disciplines – Proceedings of the Second Calculus Conference TI - The use of integrals for accumulation and mean values in basic electrical engineering courses ER - TY - JOUR AU - Biehler, Rolf AU - Krüger, Katja ID - 49049 IS - 115 JF - GDM-Mitteilungen TI - Von Glückstal (Ukraine) nach Frankfurt: Problemlösen, Algorithmen, Stochastik – Nachruf auf Arthur Engel ER - TY - CONF AU - Wallner, Melina AU - Häsel-Weide, Uta ED - Novotna, J. ED - Moraova, H. ID - 46756 T2 - International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education TI - Conceptual understanding of third grades for axial symmetry ER - TY - JOUR AB - Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree. AU - von der Gracht, Sören AU - Nijholt, Eddie AU - Rink, Bob ID - 49326 IS - 6 JF - SIAM Journal on Applied Mathematics KW - Applied Mathematics SN - 0036-1399 TI - Hypernetworks: Cluster Synchronization Is a Higher-Order Effect VL - 83 ER - TY - GEN AB - To model dynamical systems on networks with higher order (non-pairwise) interactions, we recently introduced a new class of ODEs on hypernetworks. Here we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady state branch "reluctant" if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady state branch. AU - von der Gracht, Sören AU - Nijholt, Eddie AU - Rink, Bob ID - 49371 T2 - arXiv:2311.17186 TI - Higher order interactions lead to "reluctant" synchrony breaking ER - TY - JOUR AU - Klüners, Jürgen AU - Wang, Jiuya ID - 49372 JF - La Matematica SN - 2730-9657 TI - Idélic Approach in Enumerating Heisenberg Extensions ER - TY - JOUR AU - Seitz, Simone AU - Häsel-Weide, Uta AU - Wilke, Yannik AU - Wallner, Melina ID - 49425 JF - Teachers and Teaching TI - Expertise and professionalism for inclusive (mathematics) teaching and learning: reflections on findings from interdisciplinary professionalisation research ER - TY - CONF AB - By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conservation laws, such as conservation of energy (time invariance), momentum (translation invariance), or angular momentum (rotational invariance). To learn a system representation, one could learn the discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function Ld which defines them. Based on ideas from Lie group theory, in this work we introduce a framework to learn a discrete Lagrangian along with its symmetry group from discrete observations of motions and, therefore, identify conserved quantities. The learning process does not restrict the form of the Lagrangian, does not require velocity or momentum observations or predictions and incorporates a cost term which safeguards against unwanted solutions and against potential numerical issues in forward simulations. The learnt discrete quantities are related to their continuous analogues using variational backward error analysis and numerical results demonstrate the improvement such models can have both qualitatively and quantitatively even in the presence of noise. AU - Lishkova, Yana AU - Scherer, Paul AU - Ridderbusch, Steffen AU - Jamnik, Mateja AU - Liò, Pietro AU - Ober-Blöbaum, Sina AU - Offen, Christian ID - 34135 IS - 2 T2 - IFAC-PapersOnLine TI - Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery VL - 56 ER - TY - GEN AB - In this article we investigate the convergence behavior of gathering protocols with fixed circulant topologies using tools form dynamical systems. Given a fixed number of mobile entities moving in the Euclidean plane, we model a gathering protocol as a system of ordinary differential equations whose equilibria are exactly all possible gathering points. Then, we find necessary and sufficient conditions for the structure of the underlying interaction graph such that the protocol is stable and converging, i.e., gathering, in the distributive computing sense by using tools from dynamical systems. Moreover, these tools allow for a more fine grained analysis in terms of speed of convergence in the dynamical systems sense. In fact, we derive a decomposition of the state space into stable invariant subspaces with different convergence rates. In particular, this decomposition is identical for every (linear) circulant gathering protocol, whereas only the convergence rates depend on the weights in interaction graph itself. AU - Gerlach, Raphael AU - von der Gracht, Sören AU - Dellnitz, Michael ID - 44840 KW - Dynamical Systems KW - Coupled Systems KW - Distributed Computing KW - Robot Swarms KW - Autonomous Mobile Robots KW - Gathering T2 - arXiv:2305.06632 TI - On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies ER - TY - JOUR AB - A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $n$-spaces such that each totally isotropic $t$-space is contained in exactly one member of $Y$. Nontrivial examples are known only for $t=1$ and $t=n-1$. We give an almost complete classification of such $t$-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces. AU - Schmidt, Kai-Uwe AU - Weiß, Charlene ID - 50298 IS - 1 JF - Combinatorial Theory TI - Packings and Steiner systems in polar spaces VL - 3 ER - TY - JOUR AB - We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled. AU - Schmidt, Kai‐Uwe AU - Weiß, Charlene ID - 50297 IS - 9 JF - Journal of Combinatorial Designs TI - Existence of small ordered orthogonal arrays VL - 31 ER - TY - THES AB - Digital communications relies heavily on the usage of different types of codes. Prominent codes nowadays are rank-metric codes and subspace codes - the q-analogs of binary codes and binary codes with constant weight. All these codes can be viewed as subsets of classical association schemes. A central coding-theoretic problem is to derive upper bounds for the size of codes. This thesis investigates Delsartes powerful linear program whose optimum is precisely such a bound for codes in association schemes. The linear programs for binary codes and binary constant-weight codes have been extensively studied since the 1970s, but their optimum is still unknown. We determine in a unified way the optimum of the linear program in several ordinary q-analogs as well as in their affine counterparts. In particular, bounds and constructions for codes in polar spaces are established, where the bounds are sharp up to a constant factor in many cases. Moreover, based on these results, an almost complete classification of Steiner systems in polar spaces is provided by showing that they could only exist in some corner cases. AU - Weiß, Charlene ID - 50300 TI - Linear programming bounds in classical association schemes ER - TY - GEN AB - A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space over $\mathbb{F}_p$ equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-$(n,k,\lambda)$ design in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $k$-spaces such that each totally isotropic $t$-space is contained in exactly $\lambda$ members of $Y$. Nontrivial examples are currently only known for $t\leq 2$. We show that $t$-$(n,k,\lambda)$ designs in polar spaces exist for all $t$ and $p$ provided that $k>\frac{21}{2}t$ and $n$ is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive. AU - Weiß, Charlene ID - 50299 T2 - arXiv:2311.08288 TI - Nontrivial $t$-designs in polar spaces exist for all $t$ ER - TY - JOUR AU - Häsel-Weide, Uta AU - Schmidt, R. AU - Büker, Petra ID - 48596 IS - 1 JF - Zeitschrift für Schul- Und Professionsentwicklung. (PFLB) TI - „FInDig“: Fach – Inklusion – Digitalisierung vernetzen. Ein Planungs- und Reflexionsmodell für die Lehrkräftebildung VL - 5 ER -