@unpublished{55575, author = {{Jakob, Johanna}}, title = {{{Der Whitneysche Fortsetzungssatz für vektorwertige Funktionen}}}, year = {{2023}}, } @inproceedings{42163, abstract = {{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.}}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, booktitle = {{Geometric Science of Information}}, editor = {{Nielsen, F and Barbaresco, F}}, keywords = {{System identification, discrete Lagrangians, travelling waves}}, location = {{Saint-Malo, Palais du Grand Large, France}}, pages = {{569--579}}, publisher = {{Springer, Cham.}}, title = {{{Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves}}}, doi = {{10.1007/978-3-031-38271-0_57}}, volume = {{14071}}, year = {{2023}}, } @book{37469, editor = {{Biehler, Rolf and Liebendörfer, Michael and Gueudet, Ghislaine and Rasmussen, Chris and Winsløw, Carl}}, isbn = {{9783031141744}}, issn = {{1869-4918}}, publisher = {{Springer International Publishing}}, title = {{{Practice-Oriented Research in Tertiary Mathematics Education}}}, doi = {{10.1007/978-3-031-14175-1}}, year = {{2023}}, } @article{34814, author = {{Hanusch, Maximilian}}, issn = {{0008-414X}}, journal = {{Canadian Journal of Mathematics}}, keywords = {{extension of differentiable maps}}, number = {{1}}, pages = {{170--201}}, publisher = {{Canadian Mathematical Society}}, title = {{{A $C^k$-seeley-extension-theorem for Bastiani’s differential calculus}}}, doi = {{10.4153/s0008414x21000596}}, volume = {{75}}, year = {{2023}}, } @article{27426, abstract = {{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.}}, author = {{Gebken, Bennet and Bieker, Katharina and Peitz, Sebastian}}, journal = {{Journal of Global Optimization}}, number = {{3}}, pages = {{709--741}}, title = {{{On the structure of regularization paths for piecewise differentiable regularization terms}}}, doi = {{10.1007/s10898-022-01223-2}}, volume = {{85}}, year = {{2023}}, } @inproceedings{31849, author = {{Hoffmann, Max and Biehler, Rolf}}, booktitle = {{Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022)}}, editor = {{Trigueros, Marı́a and Barquero, Berta and Hochmuth, Reinhard and Peters, Jana}}, keywords = {{Teaching and learning of specific topics in university mathematics, Transition to, across and from university mathematics, Student Teachers, Geometry, Congruence, Double Discontinuity.}}, publisher = {{University of Hannover and INDRUM.}}, title = {{{Student Teachers ’ Knowledge of Congruence before a University Course on Geometry}}}, year = {{2023}}, } @inproceedings{43097, author = {{Florensa, Ignasio and Hoffmann, Max and Romo Vázquez, Avenilde and Zandieh, Michelle and Martínez-Planell, Rafael}}, booktitle = {{Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022)}}, editor = {{Trigueros, Marı́a and Barquero, Berta and Hochmuth, Reinhard and Peters, Jana}}, title = {{{Innovations in university teaching based on mathematic education research}}}, year = {{2023}}, } @article{43105, author = {{Black, Tobias and Fuest, Mario and Lankeit, Johannes and Mizukami, Masaaki}}, issn = {{1468-1218}}, journal = {{Nonlinear Analysis: Real World Applications}}, keywords = {{Applied Mathematics, Computational Mathematics, General Economics, Econometrics and Finance, General Engineering, General Medicine, Analysis}}, publisher = {{Elsevier BV}}, title = {{{Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source}}}, doi = {{10.1016/j.nonrwa.2023.103868}}, volume = {{73}}, year = {{2023}}, } @inbook{43227, author = {{Vitt, Vivian and Häsel-Weide, Uta}}, booktitle = {{Mathematica Didactica, 46}}, title = {{{Reziprokes Peer-Tutoring zur Förderung von Schüler*innen mit Schwierigkeiten beim Mathematiklernen.}}}, doi = {{https://doi.org/10.18716/ojs/md/2023.1671}}, year = {{2023}}, } @article{34832, author = {{Hanusch, Maximilian}}, journal = {{Annals of Global Analysis and Geometry}}, keywords = {{Lax equation, generalized Baker-Campbell-Dynkin-Hausdorff formula, regularity of Lie groups}}, number = {{21}}, title = {{{The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups}}}, doi = {{10.1007/s10455-023-09888-y}}, volume = {{63}}, year = {{2023}}, } @article{44857, abstract = {{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.}}, author = {{Silva, Helmuth O.M. and Rubert, Diego P. and Araujo, Eloi and Steffen, Eckhard and Doerr, Daniel and Martinez, Fábio V.}}, issn = {{0399-0559}}, journal = {{RAIRO - Operations Research}}, keywords = {{Management Science and Operations Research, Computer Science Applications, Theoretical Computer Science}}, number = {{3}}, pages = {{1045--1058}}, publisher = {{EDP Sciences}}, title = {{{Algorithms for the genome median under a restricted measure of rearrangement}}}, doi = {{10.1051/ro/2023052}}, volume = {{57}}, year = {{2023}}, } @unpublished{44859, author = {{Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}}, booktitle = {{arXiv:2305.08619}}, title = {{{Sets of r-graphs that color all r-graphs}}}, year = {{2023}}, } @article{34833, author = {{Hanusch, Maximilian}}, journal = {{Indagationes Mathematicae.}}, keywords = {{Lie group actions and analytic 1-submanifolds}}, number = {{4}}, pages = {{752--811}}, title = {{{Decompositions of Analytic 1-Manifolds}}}, doi = {{10.1016/j.indag.2023.02.003}}, volume = {{34}}, year = {{2023}}, } @unpublished{45498, abstract = {{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.}}, author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}}, booktitle = {{arXiv:2306.03320}}, pages = {{29}}, title = {{{A parametrisation method for high-order phase reduction in coupled oscillator networks}}}, year = {{2023}}, } @article{45712, author = {{Häsel-Weide, Uta}}, journal = {{Die Grundschulzeitschrift}}, number = {{339}}, pages = {{6--11}}, publisher = {{Friedrich Verlag}}, title = {{{ Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen}}}, year = {{2023}}, } @article{45713, author = {{Graf, Lara Marie and Wienhues, Inga and Häsel-Weide, Uta}}, journal = {{Die Grundschulzeitschrift}}, number = {{339}}, pages = {{20--23}}, publisher = {{Friedrich Verlag}}, title = {{{Addition und Subtraktion verstehen}}}, year = {{2023}}, } @unpublished{46117, abstract = {{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 $\Gamma$.}}, author = {{Weich, Tobias and Wolf, Lasse L.}}, booktitle = {{arXiv:2304.09573}}, title = {{{Temperedness of locally symmetric spaces: The product case}}}, year = {{2023}}, } @article{46256, author = {{Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}}, issn = {{0895-4801}}, journal = {{SIAM Journal on Discrete Mathematics}}, keywords = {{General Mathematics}}, number = {{3}}, pages = {{1548--1565}}, publisher = {{Society for Industrial & Applied Mathematics (SIAM)}}, title = {{{Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs}}}, doi = {{10.1137/22m1500654}}, volume = {{37}}, year = {{2023}}, } @article{29240, abstract = {{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.}}, author = {{Ober-Blöbaum, Sina and Offen, Christian}}, issn = {{0377-0427}}, journal = {{Journal of Computational and Applied Mathematics}}, keywords = {{Lagrangian learning, variational backward error analysis, modified Lagrangian, variational integrators, physics informed learning}}, pages = {{114780}}, publisher = {{Elsevier}}, title = {{{Variational Learning of Euler–Lagrange Dynamics from Data}}}, doi = {{10.1016/j.cam.2022.114780}}, volume = {{421}}, year = {{2023}}, } @article{29236, abstract = {{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.}}, author = {{McLachlan, Robert and Offen, Christian}}, journal = {{Journal of Geometric Mechanics}}, keywords = {{variational integrators, backward error analysis, Euler--Lagrange equations, multistep methods, conjugate symplectic methods}}, number = {{1}}, pages = {{98--115}}, publisher = {{AIMS Press}}, title = {{{Backward error analysis for conjugate symplectic methods}}}, doi = {{10.3934/jgm.2023005}}, volume = {{15}}, year = {{2023}}, }