@article{32097,
author = {{Weich, Tobias and Guedes Bonthonneau, Yannick and Guillarmou, Colin}},
journal = {{Journal of Differential Geometry (to appear) -- arXiv:2103.12127}},
title = {{{SRB Measures of Anosov Actions}}},
year = {{2024}},
}
@article{46469,
abstract = {{We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schrödinger equation. }},
author = {{Offen, Christian and Ober-Blöbaum, Sina}},
issn = {{1054-1500}},
journal = {{Chaos}},
number = {{1}},
publisher = {{AIP Publishing}},
title = {{{Learning of discrete models of variational PDEs from data}}},
doi = {{10.1063/5.0172287}},
volume = {{34}},
year = {{2024}},
}
@inbook{50554,
author = {{Prediger, Susanne and Wessel, Lena}},
booktitle = {{Berufs-und Fachsprache Deutsch in Wissenschaft und Praxis}},
editor = {{Efing, Christian and Kalkavan-Aydin, Zeynep}},
isbn = {{978-3-11-074544-3}},
pages = {{363--372}},
publisher = {{DE GRUYTER}},
title = {{{31 Sprachbildung im berufsbezogenen Mathematikunterricht.}}},
volume = {{Band 3}},
year = {{2024}},
}
@article{51208,
abstract = {{AbstractApproximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions.}},
author = {{Gebken, Bennet}},
issn = {{0926-6003}},
journal = {{Computational Optimization and Applications}},
keywords = {{Applied Mathematics, Computational Mathematics, Control and Optimization}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A note on the convergence of deterministic gradient sampling in nonsmooth optimization}}},
doi = {{10.1007/s10589-024-00552-0}},
year = {{2024}},
}
@unpublished{51204,
abstract = {{Given a real semisimple connected Lie group $G$ and a discrete torsion-free
subgroup $\Gamma < G$ we prove a precise connection between growth rates of the
group $\Gamma$, polyhedral bounds on the joint spectrum of the ring of
invariant differential operators, and the decay of matrix coefficients. In
particular, this allows us to completely characterize temperedness of
$L^2(\Gamma\backslash G)$ in this general setting.}},
author = {{Lutsko, Christopher and Weich, Tobias and Wolf, Lasse Lennart}},
booktitle = {{arXiv:2402.02530}},
title = {{{Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces}}},
year = {{2024}},
}
@article{51374,
author = {{Hasler, David and Hinrichs, Benjamin and Siebert, Oliver}},
issn = {{0022-1236}},
journal = {{Journal of Functional Analysis}},
keywords = {{Analysis}},
number = {{7}},
publisher = {{Elsevier BV}},
title = {{{Non-Fock ground states in the translation-invariant Nelson model revisited non-perturbatively}}},
doi = {{10.1016/j.jfa.2024.110319}},
volume = {{286}},
year = {{2024}},
}
@article{32101,
author = {{Weich, Tobias and Guedes Bonthonneau, Yannick and Guillarmou, Colin and Hilgert, Joachim}},
journal = {{J. Europ. Math. Soc.}},
pages = {{1--36}},
title = {{{Ruelle-Taylor resonaces of Anosov actions}}},
year = {{2024}},
}
@unpublished{51501,
author = {{Hilgert, Joachim}},
title = {{{Quantum-Classical Correspondences for Locally Symmetric Spaces}}},
year = {{2024}},
}
@article{46019,
abstract = {{We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical system in the multiobjective setting, which trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plain multiobjective steepest descent method (Algorithm 1). A further improvement in terms of efficiency is achieved by avoiding the solution of a quadratic subproblem to compute a common step direction for all objective functions, which is usually required in first-order methods. Using a different discretization of our inertial gradient-like dynamical system, we obtain an accelerated multiobjective gradient method that does not require the solution of a subproblem in each step (Algorithm 2). While this algorithm does not converge in general, it yields good results on test problems while being faster than standard steepest descent.}},
author = {{Sonntag, Konstantin and Peitz, Sebastian}},
journal = {{Journal of Optimization Theory and Applications}},
publisher = {{Springer}},
title = {{{Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-Like Systems}}},
doi = {{10.1007/s10957-024-02389-3}},
year = {{2024}},
}
@unpublished{51334,
abstract = {{The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from [1] is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.}},
author = {{Sonntag, Konstantin and Gebken, Bennet and Müller, Georg and Peitz, Sebastian and Volkwein, Stefan}},
booktitle = {{arXiv:2402.06376}},
title = {{{A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces}}},
year = {{2024}},
}
@article{51841,
abstract = {{athematische Kompetenzen digital zu fördern und digitale Kompetenzen mathematisch zu fördern – dies ist eine Forderung der neuen Bildungsstandards mit Blick auf eine Bildung in der digitalen Welt. Gerade das Potenzial digitaler Medien für das fachliche Lernen wurde in vielen Studien bestätigt. Eine sinnvoll gestaltete Einbettung digitaler Medien bietet die Chance, allen fünf Prinzipien eines guten Unterrichts gerecht zu werden: Verstehensorientierung, Durchgängigkeit, kognitive Aktivierung, Lernendenorientierung & Adaptivität und Kommunikationsförderung. Die flächendeckende Nutzung digitaler Medien etabliert sich bislang nur zögerlich. Aber wie können wir Lehrkräfte stärken, digitale Medien sinnvoll einzusetzen? Wir möchten hier die Bandbreite der Möglichkeiten an Beispielen verdeutlichen, ihren Einsatz motivieren und Wege für einen guten Unterricht aufzeigen.}},
author = {{Barzel, Bärbel and Greefrath, Gilbert and Nagel, Mareike and Hoffmann, Max}},
journal = {{mathematik lehren}},
pages = {{42 -- 47}},
title = {{{Digitalisierung als Chance für alle Prinzipien guten Unterrichts}}},
volume = {{242}},
year = {{2024}},
}
@unpublished{52342,
abstract = {{We introduce the concept of a k-token signed graph and study some of its combinatorial and algebraic properties. We prove that two switching isomorphic signed graphs have switching isomorphic token graphs. Moreover, we show that the Laplacian spectrum of a balanced signed graph is contained in the Laplacian spectra of its k-token signed graph. Besides, we introduce and study the unbalance level of a signed graph, which is a new parameter that measures how far a signed graph is from being balanced. Moreover, we study the relation between the frustration index and the unbalance level of signed graphs and their token signed graphs. }},
author = {{Dalfó, C. and Fiol, M. A. and Steffen, Eckhard}},
booktitle = {{arXiv:2403.02924}},
title = {{{On token signed graphs}}},
year = {{2024}},
}
@unpublished{52691,
abstract = {{We prove Feynman-Kac formulas for the semigroups generated by selfadjoint
operators in a class containing Fr\"ohlich Hamiltonians known from solid state
physics. The latter model multi-polarons, i.e., a fixed number of quantum
mechanical electrons moving in a polarizable crystal and interacting with the
quantized phonon field generated by the crystal's vibrational modes. Both the
electrons and phonons can be confined to suitable open subsets of Euclidean
space. We also include possibly very singular magnetic vector potentials and
electrostatic potentials. Our Feynman-Kac formulas comprise Fock space
operator-valued multiplicative functionals and can be applied to every vector
in the underlying Hilbert space. In comparison to the renormalized Nelson
model, for which analogous Feynman-Kac formulas are known, the analysis of the
creation and annihilation terms in the multiplicative functionals requires
novel ideas to overcome difficulties caused by the phonon dispersion relation
being constant. Getting these terms under control and generalizing other
construction steps so as to cover confined systems are the main achievements of
this article.}},
author = {{Hinrichs, Benjamin and Matte, Oliver}},
booktitle = {{arXiv:2403.12147}},
title = {{{Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains}}},
year = {{2024}},
}
@article{52726,
abstract = {{Heteroclinic structures organize global features of dynamical systems. We analyse whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analysed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design of heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures.}},
author = {{Bick, Christian and von der Gracht, Sören}},
issn = {{2051-1329}},
journal = {{Journal of Complex Networks}},
keywords = {{Applied Mathematics, Computational Mathematics, Control and Optimization, Management Science and Operations Research, Computer Networks and Communications}},
number = {{2}},
publisher = {{Oxford University Press (OUP)}},
title = {{{Heteroclinic dynamics in network dynamical systems with higher-order interactions}}},
doi = {{10.1093/comnet/cnae009}},
volume = {{12}},
year = {{2024}},
}
@article{49905,
abstract = {{For 0 ≤ t ≤ r let m(t, r) be the maximum number s such that every t-edge-connected r-graph has s pairwise disjoint perfect matchings. There are only a few values of m(t, r) known, for instance m(3, 3) = m(4, r) = 1, and m(t, r) ≤ r − 2 for all t = 5,
and m(t, r) ≤ r − 3 if r is even. We prove that m(2l, r) ≤ 3l − 6 for every l ≥ 3 and r ≥ 2l.}},
author = {{Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}},
issn = {{0209-9683}},
journal = {{Combinatorica}},
keywords = {{Computational Mathematics, Discrete Mathematics and Combinatorics}},
pages = {{429--440}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs}}},
doi = {{10.1007/s00493-023-00078-9}},
volume = {{44}},
year = {{2024}},
}
@article{21199,
abstract = {{As in almost every other branch of science, the major advances in data
science and machine learning have also resulted in significant improvements
regarding the modeling and simulation of nonlinear dynamical systems. It is
nowadays possible to make accurate medium to long-term predictions of highly
complex systems such as the weather, the dynamics within a nuclear fusion
reactor, of disease models or the stock market in a very efficient manner. In
many cases, predictive methods are advertised to ultimately be useful for
control, as the control of high-dimensional nonlinear systems is an engineering
grand challenge with huge potential in areas such as clean and efficient energy
production, or the development of advanced medical devices. However, the
question of how to use a predictive model for control is often left unanswered
due to the associated challenges, namely a significantly higher system
complexity, the requirement of much larger data sets and an increased and often
problem-specific modeling effort. To solve these issues, we present a universal
framework (which we call QuaSiModO:
Quantization-Simulation-Modeling-Optimization) to transform arbitrary
predictive models into control systems and use them for feedback control. The
advantages of our approach are a linear increase in data requirements with
respect to the control dimension, performance guarantees that rely exclusively
on the accuracy of the predictive model, and only little prior knowledge
requirements in control theory to solve complex control problems. In particular
the latter point is of key importance to enable a large number of researchers
and practitioners to exploit the ever increasing capabilities of predictive
models for control in a straight-forward and systematic fashion.}},
author = {{Peitz, Sebastian and Bieker, Katharina}},
journal = {{Automatica}},
publisher = {{Elsevier}},
title = {{{On the Universal Transformation of Data-Driven Models to Control Systems}}},
doi = {{10.1016/j.automatica.2022.110840}},
volume = {{149}},
year = {{2023}},
}
@inbook{35697,
author = {{Ostsieker, Laura and Biehler, Rolf}},
booktitle = {{Practice-Oriented Research in Tertiary Mathematics Education}},
isbn = {{9783031141744}},
issn = {{1869-4918}},
pages = {{181--201}},
publisher = {{Springer International Publishing}},
title = {{{Supporting Students in Developing Adequate Concept Images and Definitions at University: The Case of the Convergence of Sequences}}},
doi = {{10.1007/978-3-031-14175-1_9}},
year = {{2023}},
}
@inbook{35678,
author = {{Kortemeyer, Jörg and Biehler, Rolf}},
booktitle = {{Practice-Oriented Research in Tertiary Mathematics Education}},
isbn = {{9783031141744}},
issn = {{1869-4918}},
pages = {{669--692}},
publisher = {{Springer International Publishing}},
title = {{{Analyzing the Interface Between Mathematics and Engineering in Basic Engineering Courses}}},
doi = {{10.1007/978-3-031-14175-1_32}},
year = {{2023}},
}
@inbook{35669,
author = {{Biehler, Rolf and Liebendörfer, Michael and Gueudet, Ghislaine and Rasmussen, Chris and Winsløw, Carl}},
booktitle = {{Practice-Oriented Research in Tertiary Mathematics Education}},
isbn = {{9783031141744}},
issn = {{1869-4918}},
publisher = {{Springer International Publishing}},
title = {{{Practice-Oriented Research in Tertiary Mathematics Education – An Introduction}}},
doi = {{10.1007/978-3-031-14175-1_1}},
year = {{2023}},
}
@inbook{35681,
author = {{Liebendörfer, Michael and Büdenbender-Kuklinski, Christiane and Lankeit, Elisa and Schürmann, Mirko and Biehler, Rolf and Schaper, Niclas}},
booktitle = {{Practice-Oriented Research in Tertiary Mathematics Education}},
isbn = {{9783031141744}},
issn = {{1869-4918}},
pages = {{91--117}},
publisher = {{Springer International Publishing}},
title = {{{Framing Goals of Mathematics Support Measures}}},
doi = {{10.1007/978-3-031-14175-1_5}},
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{36294,
author = {{Brennecken, Dominik and Rösler, Margit}},
journal = {{Transaction of the American Mathematical Society}},
publisher = {{ American Mathematical Society}},
title = {{{The Dunkl-Laplace transform and Macdonald’s hypergeometric series}}},
doi = {{10.1090/tran/8860}},
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{43504,
author = {{Biehler, Rolf and Liebendörfer, Michael and Schmitz, A.}},
journal = {{Mitteilungen der Gesellschaft für Didaktik der Mathematik}},
pages = {{8--12}},
title = {{{Lernvideos und ihre Erstellung - Das Projekt studiVEMINTvideos}}},
volume = {{114}},
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}},
}
@inbook{43226,
author = {{Häsel-Weide, Uta and Nührenbörger, M.}},
booktitle = {{Mathematica Didactica, 46}},
title = {{{Inklusive Praktiken unterrichtsintegrierter Förderung im Mathematikunterricht.}}},
doi = {{https://doi.org/10.18716/ojs/md/2023.1670}},
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}},
}
@unpublished{44501,
abstract = {{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. }},
author = {{Cappello, Chiara and Naserasr, Reza and Steffen, Eckhard and Wang, Zhouningxin}},
booktitle = {{arXiv:2304.10243}},
title = {{{Critically 3-frustrated signed graphs}}},
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}},
}
@article{45786,
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.}},
author = {{Hoffmann, Max and Biehler, Rolf}},
issn = {{1863-9690}},
journal = {{ZDM – Mathematics Education}},
keywords = {{General Mathematics, Education}},
publisher = {{Springer}},
title = {{{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}}},
doi = {{10.1007/s11858-023-01505-3}},
year = {{2023}},
}
@article{46100,
author = {{Hinrichs, Benjamin and Janssen, Daan W. and Ziebell, Jobst}},
issn = {{0022-247X}},
journal = {{Journal of Mathematical Analysis and Applications}},
keywords = {{Applied Mathematics, Analysis}},
number = {{1}},
publisher = {{Elsevier BV}},
title = {{{Super-Gaussian decay of exponentials: A sufficient condition}}},
doi = {{10.1016/j.jmaa.2023.127558}},
volume = {{528}},
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{46155,
author = {{Bruns, Julia and Hagena, Maike and Gasteiger, Hedwig}},
issn = {{0742-051X}},
journal = {{Teaching and Teacher Education}},
keywords = {{Education}},
publisher = {{Elsevier BV}},
title = {{{Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects?}}},
doi = {{10.1016/j.tate.2023.104270}},
volume = {{132}},
year = {{2023}},
}
@book{46157,
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{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}},
}
@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}},
}
@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}},
}
@article{37654,
abstract = {{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.}},
author = {{Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}},
issn = {{1054-1500}},
journal = {{Chaos}},
number = {{6}},
publisher = {{AIP Publishing}},
title = {{{Hamiltonian Neural Networks with Automatic Symmetry Detection}}},
doi = {{10.1063/5.0142969}},
volume = {{33}},
year = {{2023}},
}
@article{23428,
abstract = {{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.}},
author = {{Nüske, Feliks and Peitz, Sebastian and Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl}},
journal = {{Journal of Nonlinear Science}},
title = {{{Finite-data error bounds for Koopman-based prediction and control}}},
doi = {{10.1007/s00332-022-09862-1}},
volume = {{33}},
year = {{2023}},
}
@article{21600,
abstract = {{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.}},
author = {{Dellnitz, Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen, Christian and Peitz, Sebastian and Pfannschmidt, Karlson}},
journal = {{SIAM Journal on Scientific Computing}},
number = {{2}},
pages = {{A579--A595}},
title = {{{Efficient time stepping for numerical integration using reinforcement learning}}},
doi = {{10.1137/21M1412682}},
volume = {{45}},
year = {{2023}},
}