Heteroclinic dynamics in network dynamical systems with higher-order interactions - Bick, Christian; von der Gracht, Sören (Journal of Complex Networks)
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.
Keywords: Applied Mathematics; Computational Mathematics; Control and Optimization; Management Science and Operations Research; Computer Networks and Communications

Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs - Ma, Yulai; Mattiolo, Davide; Steffen, Eckhard; Wolf, Isaak Hieronymus (Combinatorica)
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.
Keywords: Computational Mathematics; Discrete Mathematics and Combinatorics

A new Lagrangian approach to control affine systems with a quadratic Lagrange term - Leyendecker, Sigrid; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Almagro, Rodrigo T. Sato Martín de; Szemenyei, Flóra Orsolya (Journal of Computational Dynamics)
In this work, we consider optimal control problems for mechanical systems with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, the differential geometric structure (similar to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretisation of the optimal control problem via variational integrators in a straightforward way.
Keywords: Optimal control problem; Lagrangian system; Hamiltonian system; Variations; Pontryagin's maximum principle.

Numerical analysis of an evolving bulk--surface model of tumour growth - Edelmann, Dominik; Kovács, Balázs; Lubich, Christian (arXiv)

Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing - Delarue, Benjamin; Monclair, Daniel; Sanders, Andrew (arXiv:2403.14257)
By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Gu\'eritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

Quantum resonances and scattering poles of classical rank one locally symmetric spaces - Delarue, Benjamin; Hilgert, Joachim (arXiv:2403.14426)
For negatively curved symmetric spaces it is known that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-cocompact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.

Numerical surgery for mean curvature flow of surfaces - Kovács, Balázs (SIAM Journal on Scientific Computing)

Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting - Brennecken, Dominik (Journal of Mathematical Analysis and Applications)
Keywords: Applied Mathematics; Analysis

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces - Papageorgiou, Efthymia (Journal of Evolution Equations)
Abstract This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G / K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$ L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi- K -invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi- K -invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G / K . In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$ L 1 asymptotic convergence without the assumption of bi- K -invariance.
Keywords: Mathematics (miscellaneous)

Research on curriculum resources in mathematics education: a survey of the field - Rezat, Sebastian (ZDM – Mathematics Education)

Digitalisierung als Chance für alle Prinzipien guten Unterrichts - Barzel, Bärbel; Greefrath, Gilbert; Nagel, Mareike; Hoffmann, Max (mathematik lehren)
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.

31 Sprachbildung im berufsbezogenen Mathematikunterricht. - Prediger, Susanne; Wessel, Lena (Berufs-und Fachsprache Deutsch in Wissenschaft und Praxis)

Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification - Offen, Christian
The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero is provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.

Temperedness of locally symmetric spaces: The product case - Weich, Tobias; Wolf, Lasse Lennart (Geom Dedicata)
Let $X=X_1× X_2$ be a product of two rank one symmetric spaces of non-compact type and $Γ$ a torsion-free discrete subgroup in $G_1× G_2$. We show that the spectrum of $Γ \backslash X$ is related to the asymptotic growth of $Γ$ in the two direction defined by the two factors. We obtain that $L^2(Γ \backslash G)$ is tempered for large class of $Γ$.

Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces - Lutsko, Christopher; Weich, Tobias; Wolf, Lasse Lennart (arXiv:2402.02530)
Given a real semisimple connected Lie group $G$ and a discrete torsion-free subgroup $Γ < G$ we prove a precise connection between growth rates of the group $Γ$, 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 $L^2(Γ\backslash G)$ in this general setting.

Fachbezogene Professionalisierung von fachfremd Mathematik unterrichtenden Lehrkräften: Retrospektive Selbsteinschätzungen zur Expertise im Umgang mit Schwierigkeiten beim Mathematiklernen im Anfangsunterricht der Grundschule - Häsel-Weide, Uta; Graf, Lara Marie; Höveler, K.; Nührenbörger, M. (HLZ – Herausforderung Lehrer*innenbildung)

The meaning landscape of the concept of the total derivative in multivariable real analysis textbooks: an analysis based on a new model of meaning - Lankeit, Elisa; Biehler, Rolf (ZDM – Mathematics Education)
Abstract In this paper, we propose a novel conceptual framework tailored for modeling the meaning of mathematical concepts in university-level mathematics, addressing their rigorous nature and their relationships with related concepts as well as interpretations in various contexts. Within this framework, we present a model of meaning for the concepts of total differentiability and total derivative that provides a variety of possible interpretations and aspects. We then use the proposed model of meaning as a tool for analyzing three different textbooks for mathematics majors on the topic of multidimensional differentiability. ﻿Our paper is an example of a subject matter analysis of a topic in university mathematics carried out in a structured way. The model of meaning for total differentiability presented in this paper could inspire course design and analysis including the design of assignments and assessments for students. Moreover, it could serve as a valuable research tool for further analyses. For example, it could be used as a framework for analyzing courses taught or as a basis for developing a test instrument to assess students’ understanding. With our textbook analysis, we begin to examine the landscape of textbooks regarding differentiability concepts in the multidimensional case, shedding light on the diversity of meaning facets that are covered in the textbooks. These results could be useful for guiding instructors and learners in selecting and using textbooks for teaching and learning based on their respective needs.

On the Universal Transformation of Data-Driven Models to Control Systems - Peitz, Sebastian; Bieker, Katharina (Automatica)
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.

Practice-Oriented Research in Tertiary Mathematics Education

A $C^k$-seeley-extension-theorem for Bastiani’s differential calculus - Hanusch, Maximilian (Canadian Journal of Mathematics)
Keywords: extension of differentiable maps

On the structure of regularization paths for piecewise differentiable regularization terms - Gebken, Bennet; Bieker, Katharina; Peitz, Sebastian (Journal of Global Optimization)
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.

Student Teachers ’ Knowledge of Congruence before a University Course on Geometry - Hoffmann, Max; Biehler, Rolf (Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022))
Keywords: Teaching and learning of specific topics in university mathematics; Transition to; across and from university mathematics; Student Teachers; Geometry; Congruence; Double Discontinuity.

Innovations in university teaching based on mathematic education research - Florensa, Ignasio; Hoffmann, Max; Romo Vázquez, Avenilde; Zandieh, Michelle; Martínez-Planell, Rafael (Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022))

Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source - Black, Tobias; Fuest, Mario; Lankeit, Johannes; Mizukami, Masaaki (Nonlinear Analysis: Real World Applications)
Keywords: Applied Mathematics; Computational Mathematics; General Economics; Econometrics and Finance; General Engineering; General Medicine; Analysis

Reziprokes Peer-Tutoring zur Förderung von Schüler*innen mit Schwierigkeiten beim Mathematiklernen. - Vitt, Vivian; Häsel-Weide, Uta (Mathematica Didactica, 46)

Inklusive Praktiken unterrichtsintegrierter Förderung im Mathematikunterricht. - Häsel-Weide, Uta; Nührenbörger, M. (Mathematica Didactica, 46)

The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups - Hanusch, Maximilian (Annals of Global Analysis and Geometry)
Keywords: Lax equation; generalized Baker-Campbell-Dynkin-Hausdorff formula; regularity of Lie groups

Critically 3-frustrated signed graphs - Cappello, Chiara; Naserasr, Reza; Steffen, Eckhard; Wang, Zhouningxin (arXiv:2304.10243)
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.

Algorithms for the genome median under a restricted measure of rearrangement - Silva, Helmuth O.M.; Rubert, Diego P.; Araujo, Eloi; Steffen, Eckhard; Doerr, Daniel; Martinez, Fábio V. (RAIRO - Operations Research)
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.
Keywords: Management Science and Operations Research; Computer Science Applications; Theoretical Computer Science

Sets of r-graphs that color all r-graphs - Ma, Yulai; Mattiolo, Davide; Steffen, Eckhard; Wolf, Isaak Hieronymus (arXiv:2305.08619)

Decompositions of Analytic 1-Manifolds - Hanusch, Maximilian (Indagationes Mathematicae.)
Keywords: Lie group actions and analytic 1-submanifolds

A parametrisation method for high-order phase reduction in coupled oscillator networks - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (arXiv:2306.03320)
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.

Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen - Häsel-Weide, Uta (Die Grundschulzeitschrift)

Addition und Subtraktion verstehen - Graf, Lara Marie; Wienhues, Inga; Häsel-Weide, Uta (Die Grundschulzeitschrift)

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

Temperedness of locally symmetric spaces: The product case - Weich, Tobias; Wolf, Lasse L. (arXiv:2304.09573)
Let $X=X_1× X_2$ be a product of two rank one symmetric spaces of non-compact type and $Γ$ a torsion-free discrete subgroup in $G_1× G_2$. We show that the spectrum of $Γ \backslash X$ is related to the asymptotic growth of $Γ$ in the two direction defined by the two factors. We obtain that $L^2(Γ \backslash G)$ is tempered for large class of $Γ$.

Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs - Ma, Yulai; Mattiolo, Davide; Steffen, Eckhard; Wolf, Isaak Hieronymus (SIAM Journal on Discrete Mathematics)
Keywords: General Mathematics

Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves - Offen, Christian; Ober-Blöbaum, Sina (Geometric Science of Information)
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.
Keywords: System identification; discrete Lagrangians; travelling waves

Variational Learning of Euler–Lagrange Dynamics from Data - Ober-Blöbaum, Sina; Offen, Christian (Journal of Computational and Applied Mathematics)
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.
Keywords: Lagrangian learning; variational backward error analysis; modified Lagrangian; variational integrators; physics informed learning

Backward error analysis for conjugate symplectic methods - McLachlan, Robert; Offen, Christian (Journal of Geometric Mechanics)
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.
Keywords: variational integrators; backward error analysis; Euler--Lagrange equations; multistep methods; conjugate symplectic methods

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

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

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

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

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

Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions - Hinrichs, Benjamin; Matte, Oliver (arXiv:2309.09005)
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.

Conceptual understanding of third grades for axial symmetry - Wallner, Melina; Häsel-Weide, Uta (International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education)

Hypernetworks: Cluster Synchronization Is a Higher-Order Effect - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (SIAM Journal on Applied Mathematics)
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.
Keywords: Applied Mathematics

Higher order interactions lead to "reluctant" synchrony breaking - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (arXiv:2311.17186)
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.

Idélic Approach in Enumerating Heisenberg Extensions - Klüners, Jürgen; Wang, Jiuya (La Matematica)

Expertise and professionalism for inclusive (mathematics) teaching and learning: reflections on findings from interdisciplinary professionalisation research - Seitz, Simone; Häsel-Weide, Uta; Wilke, Yannik; Wallner, Melina (Teachers and Teaching)

Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery - Lishkova, Yana; Scherer, Paul; Ridderbusch, Steffen; Jamnik, Mateja; Liò, Pietro; Ober-Blöbaum, Sina; Offen, Christian (IFAC-PapersOnLine)
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.

On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies - Gerlach, Raphael; von der Gracht, Sören; Dellnitz, Michael (arXiv:2305.06632)
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.
Keywords: Dynamical Systems; Coupled Systems; Distributed Computing; Robot Swarms; Autonomous Mobile Robots; Gathering

Packings and Steiner systems in polar spaces - Schmidt, Kai-Uwe; Weiß, Charlene (Combinatorial Theory)
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.

Existence of small ordered orthogonal arrays - Schmidt, Kai‐Uwe; Weiß, Charlene (Journal of Combinatorial Designs)
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.

Linear programming bounds in classical association schemes - Weiß, Charlene
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.

Nontrivial $t$-designs in polar spaces exist for all $t$ - Weiß, Charlene (arXiv:2311.08288)
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 $t$-$(n,k,λ)$ 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 $λ$ members of $Y$. Nontrivial examples are currently only known for $t≤ 2$. We show that $t$-$(n,k,λ)$ 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.

„FInDig“: Fach – Inklusion – Digitalisierung vernetzen. Ein Planungs- und Reflexionsmodell für die Lehrkräftebildung - Häsel-Weide, Uta; Schmidt, R.; Büker, Petra (Zeitschrift für Schul- Und Professionsentwicklung. (PFLB))

Insights into out-of-field teachers’ self-reports: Fostering the understanding of addition and subtraction as a basis for children to overcome difficulties in mathematics - Graf, Lara Marie; Häsel-Weide, Uta; Höveler, K.; Nührenbörger, M. (Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13))

Absence of principal eigenvalues for higher rank locally symmetric spaces - Weich, Tobias; Wolf, Lasse Lennart (Communications in Mathematical Physics)
Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $≥ 1/4$. In this article we prove a generalization of this result for the joint $L^2$-eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces $Γ\backslash G/K$ of higher rank. We derive dynamical assumptions on the $Γ$-action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups.

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

Invariant Ruelle Distributions on Convex-Cocompact Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions - Schütte, Philipp; Weich, Tobias (arXiv:2308.13463)
We present a numerical algorithm for the computation of invariant Ruelle distributions on convex co-compact hyperbolic surfaces. This is achieved by exploiting the connection between invariant Ruelle distributions and residues of meromorphically continued weighted zeta functions established by the authors together with Barkhofen (2021). To make this applicable for numerics we express the weighted zeta as the logarithmic derivative of a suitable parameter dependent Fredholm determinant similar to Borthwick (2014). As an additional difficulty our transfer operator has to include a contracting direction which we account for with techniques developed by Rugh (1992). We achieve a further improvement in convergence speed for our algorithm in the case of surfaces with additional symmetries by proving and applying a symmetry reduction of weighted zeta functions.

Bounds for the chromatic index of signed multigraphs - Steffen, Eckhard; Wolf, Isaak Hieronymus (Discrete Applied Mathematics)
Keywords: Applied Mathematics; Discrete Mathematics and Combinatorics

Graph-Theoretical Models for the Analysis and Design of Socio-Technical Networks - Cappello, Chiara; Steffen, Eckhard (The Digital Twin of Humans)

Rotation r-graphs - Steffen, Eckhard; Wolf, Isaak Hieronymus (Discrete Mathematics)
Keywords: Discrete Mathematics and Combinatorics; Theoretical Computer Science

On Lieb-Robinson bounds for a class of continuum fermions - Hinrichs, Benjamin; Lemm, Marius; Siebert, Oliver (arXiv:2310.17736)
We consider the quantum dynamics of a many-fermion system in $\mathbb R^d$ with an ultraviolet regularized pair interaction as previously studied in [M. Gebert, B. Nachtergaele, J. Reschke, and R. Sims, Ann. Henri Poincar\'e 21.11 (2020)]. We provide a Lieb-Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb-Robinson bound on $L^2$-overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb-Robinson bounds.

A Lower Bound on the Critical Momentum of an Impurity in a Bose-Einstein Condensate - Hinrichs, Benjamin; Lampart, Jonas (arXiv:2311.05361)
In the Bogoliubov-Fr\"ohlich model, we prove that an impurity immersed in a Bose-Einstein condensate forms a stable quasi-particle when the total momentum is less than its mass times the speed of sound. The system thus exhibits superfluid behavior, as this quasi-particle does not experience friction. We do not assume any infrared or ultraviolet regularization of the model, which contains massless excitations and point-like interactions.

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

Spectral correspondences for finite graphs without dead ends - Bux, Kai-Uwe; Hilgert, Joachim; Weich, Tobias (arXiv:2307.10876)
We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on vector spaces associated with the set of non backtracking paths in the graphs. The second kind of operators are averaging operators acting on vector spaces associated with the space of vertices of the graph. The choice of vector spaces reflects regularity properties. Our main results are correspondences between classical and quantum spectral objects as well as some automatic regularity properties for eigenfunctions of transfer operators.

Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters - Arends, Christian; Hilgert, Joachim (Journal de l’École polytechnique — Mathématiques)
In this paper we complete the program of relating the Laplace spectrum for rank one compact locally symmetric spaces with the first band Ruelle-Pollicott resonances of the geodesic flow on its sphere bundle. This program was started by Flaminio and Forni for hyperbolic surfaces, continued by Dyatlov, Faure and Guillarmou for real hyperbolic spaces and by Guillarmou, Hilgert and Weich for general rank one spaces. Except for the case of hyperbolic surfaces a countable set of exceptional spectral parameters always left untreated since the corresponding Poisson transforms are neither injective nor surjective. We use vector valued Poisson transforms to treat also the exceptional spectral parameters. For surfaces the exceptional spectral parameters lead to discrete series representations of $SL(2,\mathbb R)$. In higher dimensions the situation is more complicated, but can be described completely.
Keywords: Ruelle resonances; Poisson transforms; locally symmetric spaces; principal series representations

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

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

A pairing formula for resonant states on finite regular graphs - Hilgert, Joachim; Arends, C.; Frahm, J.

Edge Laplacians and vector valued Poisson transforms for graphs - Hilgert, Joachim; Arends, C.; Frahm, J.

Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem - Hilgert, Joachim; Baier, T.; Kaya, O.; Mourao, J.; Nunes, J.

Ruelle-Taylor resonances of Anosov actions - Hilgert, Joachim; Guedes Bonthonneau, Y.; Guillarmou, C.; Weich, Tobias

Fast Convergence of Inertial Multiobjective Gradient-like Systems with Asymptotic Vanishing Damping - Sonntag, Konstantin; Peitz, Sebastian (arXiv:2307.00975)
We present a new gradient-like dynamical system related to unconstrained convex smooth multiobjective optimization which involves inertial effects and asymptotic vanishing damping. To the best of our knowledge, this system is the first inertial gradient-like system for multiobjective optimization problems including asymptotic vanishing damping, expanding the ideas laid out in [H. Attouch and G. Garrigos, Multiobjective optimization: an inertial approach to Pareto optima, preprint, arXiv:1506.02823, 201]. We prove existence of solutions to this system in finite dimensions and further prove that its bounded solutions converge weakly to weakly Pareto optimal points. In addition, we obtain a convergence rate of order O(t−2) for the function values measured with a merit function. This approach presents a good basis for the development of fast gradient methods for multiobjective optimization.

Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Bernreuther, Marco; Dellnitz, Michael; Gebken, Bennet; Müller, Georg; Peitz, Sebastian; Sonntag, Konstantin; Volkwein, Stefan (arXiv:2308.01113)
Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

Deep neural networks on diffeomorphism groups for optimal shape reparametrization - Celledoni, Elena; Glöckner, Helge; Riseth, Jørgen; Schmeding, Alexander (BIT Numerical Mathematics)

Aspects of control theory on infinite-dimensional Lie groups and G-manifolds - Glöckner, Helge; Hilgert, Joachim (Journal of Differential Equations)
Keywords: 22E65; 28B05; 34A12; 34H05; 46E30; 46E40

Diffeomorphism groups of convex polytopes - Glöckner, Helge (Journal of Convex Analysis)
Let $E$ be a finite-dimensional real vector space and $M\subseteq E$ be a convex polytope with non-empty interior. We turn the group of all $C^∞$-diffeomorphisms of $M$ into a regular Lie group.

Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groups - Glöckner, Helge; Tárrega, Luis (Journal of Lie Theory)

The Digital Twin of Humans

Post-pandemic online mathematics and statistics support: Practitioners’ opinions in Germany and Great Britain & Ireland - Gilbert, H.; Schürmann, M.; Liebendörfer, M.; Lawson, D.; Hodds, M. (International Journal of Mathematical Education in Science and Technology)
Keywords: Applied Mathematics; Education; Mathematics (miscellaneous)

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

Erfassung und Vergleich (mathematischer) Eingangsvoraussetzungen angehender Studierender verschiedener mathematikhaltiger Studiengänge - Göller, Robin; Gildehaus, Lara; Liebendörfer, Michael; Besser, Michael (Hanse-Kolloquium zur Hochschuldidaktik der Mathematik 2021. Beiträge zum gleichnamigen Online-Symposium am 12 November 2021 aus Bochum)

Zu digital - zu viel - zu schwer? Qualitative Einsichten in das Erleben und Handeln von Erstsemester-Studierenden der Mathematik während der Corona-Pandemie - Kempen, Leander; Liebendörfer, Michael (Hanse-Kolloquium zur Hochschuldidaktik der Mathematik 2021. Beiträge zum gleichnamigen Online-Symposium am 12 November 2021 aus Bochum)

Bearbeitungsmuster von Studierenden im Umgang mit formalen Definitionen im Kontext konstanter Folgen - Schlüter, Sarah; Liebendörfer, Michael (Beiträge zum Mathematikunterricht 2022. 56. Jahrestagung der Gesellschaft für Didaktik der Mathematik)

Förderung prozeduraler Flexibilität durch Lernvideos mit interaktiven Aufgaben - Krämer, Sandra; Liebendörfer, Michael (Beiträge zum Mathematikunterricht 2022. 56. Jahrestagung der Gesellschaft für Didaktik der Mathematik)

Preservice teachers’ mathematics-related values and expectancy in the transition from school to university
Many preservice mathematics teachers lose their motivation during their first year at university. This phenomenon has been repeatedly described in recent years but is not yet fully understood. Since motivation may relate to different objects such as mathematics or teaching, we aim to qualitatively reconstruct different facets of the central motivational constructs of Situated-Expectancy-Value theory (intrinsic value, attainment value, utility value, cost, and expectancy of success) for preservice mathematics teachers. The analysis of longitudinal group interviews of 14 preservice higher-secondary mathematic teachers from a German university revealed different objects of motivation (e.g., teaching mathematics, scientific mathematics, procedural mathematics, or proof-based mathematics) in preservice teachers' values and expectancy of success. Furthermore, relations between those values and expectancy of success were identified that played a significant role in preservice teachers’ motivational development over their first semester (e.g., relations of attainment value for scientific mathematics and psychological cost). Theoretical and practical implications towards a teaching-specific conceptualization of expectancy of success and values and value interventions are being discussed.

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

Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints - Bartels, Sören; Kovács, Balázs; Wang, Zhangxian (IMA Journal of Numerical Analysis)
Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates - Contri, Alessandro; Kovács, Balázs; Massing, André (arXiv)
We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.

Funnel control of linear systems with arbitrary relative degree under output measurement losses - Berger, Thomas; Lanza, Lukas (IMA Journal of Mathematical Control and Information,)

Edge-wise funnel output synchronization of heterogeneous agents with relative degree one - Lee, J. G.; Berger, Thomas; Trenn, S.; Shim, H. (Automatica)

Neue Wege im mathematischen Unterricht: Auf den Spuren Mathilde Vaertings - Werth, Gerda

Standardkonstruktionen eigenständig entdecken. Mathilde Vaertings Idee der Selbstständigkeitsprobe - Werth, Gerda (mathematik lehren)

$L^2$-spectrum, growth indicator function and critical exponent on locally symmetric spaces - Wolf, Lasse L.; Zhang, Hong-Wei (arXiv:2311.11770)
In this short note we observe, on locally symmetric spaces of higher rank, a connection between the growth indicator function introduced by Quint and the modified critical exponent of the Poincar\'e series equipped with the polyhedral distance. As a consequence, we provide a different characterization of the bottom of the $L^2$-spectrum of the Laplace-Beltrami operator in terms of the growth indicator function. Moreover, we explore the relationship between these three objects and the temperedness.

Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models - Delarue, Benjamin; Schütte, Philipp; Weich, Tobias (Annales Henri Poincaré)
Abstract We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing, the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016) can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.
Keywords: Mathematical Physics; Nuclear and High Energy Physics; Statistical and Nonlinear Physics

Singular cohomology of symplectic quotients by circle actions and Kirwan surjectivity - Delarue, Benjamin; Ramacher, Pablo; Schmitt, Maximilian (arXiv:2312.03634)
Let $M$ be a symplectic manifold carrying a Hamiltonian $S^1$-action with momentum map $J:M → \mathbb{R}$ and consider the corresponding symplectic quotient $\mathcal{M}_0:=J^{-1}(0)/S^1$. We extend Sjamaar's complex of differential forms on $\mathcal{M}_0$, whose cohomology is isomorphic to the singular cohomology $H(\mathcal{M}_0;\mathbb{R})$ of $\mathcal{M}_0$ with real coefficients, to a complex of differential forms on $\mathcal{M}_0$ associated with a partial desingularization $\widetilde{\mathcal{M}}_0$, which we call resolution differential forms. The cohomology of that complex turns out to be isomorphic to the de Rham cohomology $H(\widetilde{ \mathcal{M}}_0)$ of $\widetilde{\mathcal{M}}_0$. Based on this, we derive a long exact sequence involving both $H(\mathcal{M}_0;\mathbb{R})$ and $H(\widetilde{ \mathcal{M}}_0)$ and give conditions for its splitting. We then define a Kirwan map $\mathcal{K}:H_{S^1}(M) → H(\widetilde{\mathcal{M}}_0)$ from the equivariant cohomology $H_{S^1}(M)$ of $M$ to $H(\widetilde{\mathcal{M}}_0)$ and show that its image contains the image of $H(\mathcal{M}_0;\mathbb{R})$ in $H(\widetilde{\mathcal{M}}_0)$ under the natural inclusion. Combining both results in the case that all fixed point components of $M$ have vanishing odd cohomology we obtain a surjection $\check κ:H^ev_{S^1}(M) → H^ev(\mathcal{M}_0;\mathbb{R})$ in even degrees, while already simple examples show that a similar surjection in odd degrees does not exist in general. As an interesting class of examples we study abelian polygon spaces.