@article{31057, abstract = {{In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems.}}, author = {{Barkhofen, Sonja and Schütte, Philipp and Weich, Tobias}}, journal = {{Journal of Physics A: Mathematical and Theoretical}}, number = {{24}}, publisher = {{IOP Publishing Ltd}}, title = {{{Semiclassical formulae For Wigner distributions}}}, doi = {{10.1088/1751-8121/ac6d2b}}, volume = {{55}}, year = {{2022}}, } @phdthesis{31363, abstract = {{Vorgestellt wird ein Entwicklungsforschungsprojekt zur Konzeption und Durchführung einer Veranstaltung "Geometrie für Lehramtsstudierende". Die Schwerpunkte des Projekts sind zum einen die inhaltliche Gestaltung der Veranstaltung und zum anderen die Umsetzung von Professionsorientierung. Bezogen auf den inhaltlichen Aufbau wird das auf metrischen Räumen aufbauende Axiomensystem der "Saccheri-Ebene" vorgestellt und mit alternativen axiomatischen Zugängen zur ebenen Geometrie verglichen. Die Frage nach der Umsetzung von Professionsorientierung in Fachveranstaltungen ist eng mit der Problematik der zweiten Diskontinuität verbunden. In der Arbeit wird dieses Problem auf Grundlage der Synthese von theoretischen Hintergründen zur Bedeutung von mathematischem Wissen und Können für professionelle Handlungskompetenz von Mathematiklehrkräften diskutiert und darauf aufbauend werden theoriebasierte Entwurfsprinzipien für professionsorientierte Fachveranstaltungen entworfen. Zentrale Elemente der methodischen Gestaltung sind die sogenannten "Schnittstellenwochen" zu den Themen Kongruenz und Symmetrie sowie das begleitende Schnittstellen-ePortfolio. Das zentrale Ergebnis der Arbeit ist ein theoretisch fundiertes und empirisch evaluiertes ganzheitliches Veranstaltungskonzept für eine professionsorientierte Geometrie-Veranstaltung für Lehramtsstudierende, dessen Konzeption auf andere Fachveranstaltungen übertragbar ist. Darüber hinaus ergeben sich im Rahmen der durchgeführten Entwicklungsforschung verschiedene neue Beiträge zur Geometriedidaktik in Schule- und Hochschule.}}, author = {{Hoffmann, Max}}, pages = {{410}}, title = {{{Von der Axiomatik bis zur Schnittstellenaufgabe: Entwicklung und Erforschung eines ganzheitlichen Lehrkonzepts für eine Veranstaltung Geometrie für Lehramtsstudierende}}}, doi = {{10.17619/UNIPB/1-1313}}, year = {{2022}}, } @article{35322, author = {{Bux, Kai-Uwe and Hilgert, Joachim and Weich, Tobias}}, issn = {{1664-039X}}, journal = {{Journal of Spectral Theory}}, keywords = {{Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{2}}, pages = {{659--681}}, publisher = {{European Mathematical Society - EMS - Publishing House GmbH}}, title = {{{Poisson transforms for trees of bounded degree}}}, doi = {{10.4171/jst/414}}, volume = {{12}}, year = {{2022}}, } @misc{51554, author = {{Hilgert, Joachim}}, booktitle = {{Mathematische Semesterberichte}}, pages = {{151–153}}, title = {{{Ethan D. Bolker und Maura B. Mast: Common Sense Mathematics, Second Edition. AMS/MAA Press 2021}}}, doi = {{10.1007/s00591-021-00314-7}}, volume = {{69}}, year = {{2022}}, } @article{45970, abstract = {{ We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions [Formula: see text] by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula: see text], is required. Moreover, in arbitrary dimensions [Formula: see text], we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions [Formula: see text] and in two dimensions [Formula: see text], where a CFL condition is required. Then, in two dimensions [Formula: see text], we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions [Formula: see text]. }}, author = {{Garcke, Harald and Kovács, Balázs and Trautwein, Dennis}}, issn = {{0218-2025}}, journal = {{Mathematical Models and Methods in Applied Sciences}}, keywords = {{Applied Mathematics, Modeling and Simulation}}, number = {{13}}, pages = {{2673--2758}}, publisher = {{World Scientific Pub Co Pte Ltd}}, title = {{{Viscoelastic Cahn–Hilliard models for tumor growth}}}, doi = {{10.1142/s0218202522500634}}, volume = {{32}}, year = {{2022}}, } @article{45969, abstract = {{AbstractAn evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the$$H^1$$H1norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.}}, author = {{Elliott, Charles M. and Garcke, Harald and Kovács, Balázs}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{873--925}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces}}}, doi = {{10.1007/s00211-022-01301-3}}, volume = {{151}}, year = {{2022}}, } @article{45963, abstract = {{AbstractThe scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge–Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.}}, author = {{Nick, Jörg and Kovács, Balázs and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{1123--1164}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Time-dependent electromagnetic scattering from thin layers}}}, doi = {{10.1007/s00211-022-01277-0}}, volume = {{150}}, year = {{2022}}, } @article{45964, abstract = {{Abstract Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^\infty (0,T;W^{1,\infty })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^\infty (0,T;W^{1,\infty })$ norm is obtained by combining the stability analysis with the consistency estimates.}}, author = {{Kovács, Balázs and Li, Buyang}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems}}}, doi = {{10.1093/imanum/drac033}}, year = {{2022}}, } @article{45966, abstract = {{Abstract This paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.}}, author = {{Altmann, Robert and Kovács, Balázs and Zimmer, Christoph}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{2}}, pages = {{950--975}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions}}}, doi = {{10.1093/imanum/drac002}}, volume = {{43}}, year = {{2022}}, } @article{45968, abstract = {{Abstract We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components that describe the coupling and the dynamics in the abstract bulk- and surface-spaces, and treats the nonlinear terms similarly to an exponential integrator. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups, and a careful comparison of the exact and approximate flows. This analysis also requires a deep understanding of the effects of the Dirichlet operator (the abstract version of the harmonic extension operator), which is essential for the stable coupling in our method. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.}}, author = {{Csomós, Petra and Farkas, Bálint and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Error estimates for a splitting integrator for abstract semilinear boundary coupled systems}}}, doi = {{10.1093/imanum/drac079}}, year = {{2022}}, } @article{45958, abstract = {{AbstractIn this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.}}, author = {{Beschle, Cedric Aaron and Kovács, Balázs}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{1}}, pages = {{1--48}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces}}}, doi = {{10.1007/s00211-022-01280-5}}, volume = {{151}}, year = {{2022}}, } @article{45956, abstract = {{Abstract The full Maxwell equations in the unbounded three-dimensional space coupled to the Landau–Lifshitz–Gilbert equation serve as a well-tested model for ferromagnetic materials. We propose a weak formulation of the coupled system based on the boundary integral formulation of the exterior Maxwell equations. We show existence and partial uniqueness of a weak solution and propose a new numerical algorithm based on finite elements and boundary elements as spatial discretization with backward Euler and convolution quadrature for the time domain. This is the first numerical algorithm which is able to deal with the coupled system of Landau–Lifshitz–Gilbert equation and full Maxwell’s equations without any simplifications like quasi-static approximations (e.g. eddy current model) and without restrictions on the shape of the domain (e.g. convexity). We show well-posedness and convergence of the numerical algorithm under minimal assumptions on the regularity of the solution. This is particularly important as there are few regularity results available and one generally expects the solution to be non-smooth. Numerical experiments illustrate and expand on the theoretical results.}}, author = {{Bohn, Jan and Feischl, Michael and Kovács, Balázs}}, issn = {{1609-4840}}, journal = {{Computational Methods in Applied Mathematics}}, keywords = {{Applied Mathematics, Computational Mathematics, Numerical Analysis}}, number = {{1}}, pages = {{19--48}}, publisher = {{Walter de Gruyter GmbH}}, title = {{{FEM-BEM Coupling for the Maxwell–Landau–Lifshitz–Gilbert Equations via Convolution Quadrature: Weak Form and Numerical Approximation}}}, doi = {{10.1515/cmam-2022-0145}}, volume = {{23}}, year = {{2022}}, } @inproceedings{52574, author = {{Werth, Gerda}}, booktitle = {{Beiträge zum Mathematikunterricht}}, location = {{Frankfurt am. Main}}, publisher = {{WTM}}, title = {{{Neue Wege im mathematischen Unterricht - Auf den Spuren Mathilde Vaertings}}}, doi = {{https://doi.org/10.37626/GA9783959872089.0}}, year = {{2022}}, } @inproceedings{53480, author = {{Malik, Sara Naseem and Rezat, Sebastian}}, booktitle = {{Proceedings on the Twelfth Congress on the European Society for research in Mathematics Education (CERME 12)}}, publisher = {{ERME / Free University of Bozen-Bolzano}}, title = {{{Linguistic features of word problems that cause difficulties for learners across the curriculum: A literature review}}}, year = {{2022}}, } @article{44689, abstract = {{AbstractEven in the digital age, learning mathematics at an academic level still requires much reading of mathematical text. Research has shown that reading mathematical text requires readers to engage with all the structures of the book and with its pedagogical voice, making connections, and plausible reasoning. Specific practices and strategies that support the close reading of mathematical text have been suggested; however, descriptions and empirical evaluations of materials designed to support these activities are rare. We present the design and first evaluation cycle of materials developed in a design research project that aims to scaffold close reading of mathematical text. The materials were designed and evaluated in a German university course on elementary geometry for first-year teacher education students who study mathematics to become primary teachers. The reading strategies were explained and modeled for students in reading-strategy videos. Additionally, close reading of mathematical text was scaffolded by close-reading tasks and homework tasks and problems that build on the reading strategies and were specifically designed to foster understanding of the mathematical text. Survey data were collected from 296 students to evaluate their use of and attitude toward the different materials. The quantitative results indicate that students used the materials and were generally able to learn the course content by themselves. From all provided materials, they found the close-reading tasks most helpful. A qualitative analysis of answers to open questions revealed issues with different materials, particularly with the script, and requests for additional materials. The issues with the script were categorized inductively. The categories are presented as a qualitative result of the study and discussed.}}, author = {{Rezat, Sebastian and Malik, Sara Naseem and Leifeld, Markus}}, issn = {{1571-0068}}, journal = {{International Journal of Science and Mathematics Education}}, keywords = {{General Mathematics, Education}}, number = {{S1}}, pages = {{215--236}}, publisher = {{Springer}}, title = {{{Scaffolding Close Reading of Mathematical Text in Pre-service Primary Teacher Education at the Tertiary Level: Design and Evaluation}}}, doi = {{10.1007/s10763-022-10309-y}}, volume = {{20}}, year = {{2022}}, } @inproceedings{48389, author = {{Dröse, Jennifer and Griese, Birgit and Wessel, Lena}}, booktitle = {{Twelfth Congress of the European Society for Research in Mathematics Education (CERME12)}}, location = {{Bozen-Bolzano,Italy}}, title = {{{Prospective teachers’ diagnostic judgments on students’ understanding of conditional probabilities}}}, year = {{2022}}, } @article{48325, author = {{Dellori, Anna and Wessel, Lena}}, journal = {{Beiträge zum Mathematikunterricht 2022}}, pages = {{665--668}}, publisher = {{LibreCat University}}, title = {{{Entwicklung und Erprobung von professionsorientierten Lernumgebungen zur Wissensvernetzung in der Algebra}}}, doi = {{10.17877/DE290R-23598}}, year = {{2022}}, } @inbook{48385, author = {{Dellori, Anna and Wessel, Lena}}, booktitle = {{Proceedings of INDRUM2022}}, editor = {{Trigueros, M. and Barquero, B. and Hochmuth, R. and Peters, J.}}, pages = {{572--573}}, title = {{{Design principles for intertwining local and nonlocal mathematics - The case of relating registers and representations in abstract algebra}}}, year = {{2022}}, } @inbook{48407, author = {{Dellori, Anna and Wessel, Lena}}, booktitle = {{Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education}}, editor = {{Karunakaran, S.S. and Higgins, A.}}, pages = {{1177}}, publisher = {{MA}}, title = {{{Pre-service Teachers' Professional Development: Relating Abstract Algebra and School Algebra}}}, year = {{2022}}, } @article{48408, author = {{Wessel, Lena and Dröse, Jennifer}}, journal = {{mathematik lehren}}, pages = {{33--36}}, title = {{{Schreiben will gelernt sein: Schreiblerngelegenheiten adaptiv gestalten}}}, volume = {{233}}, year = {{2022}}, }