Regularity of Lie groups - Hanusch, Maximilian (Communications in Analysis and Geometry)
Keywords: regularity of Lie groups

Analysis 1 und 2 Skript/Buch - Hanusch, Maximilian

Uniqueness of the Inverse First Passage Time Problem and the Shape of the Shiryaev boundary - Kolb, Martin; Klump, Alexander (Theory of Probability and its Applications)

On non-extinction in a Fleming-Viot-type particle model with Bessel drift - Kolb, Martin; Liesenfeld, Matthias (Electronic Journal of Probability)
Motivated by the work [6] of Mariusz Bieniek, Krzysztof Burdzy and Soumik Pal we study a Fleming-Viot-type particle system consisting of independently moving particles each driven by generalized Bessel processes on the positive real line. Upon hitting the boundary {0} this particle is killed and an uniformly chosen different one branches into two particles. Using the symmetry of the model and the self similarity property of Bessel processes, we obtain a criterion to decide whether the particles converge to the origin at a finite time. This addresses open problem 1.4 in [6]. Specifically, inspired by [6, Open Problem 1.5], we investigate the case of three moving particles and refine the general result of [6, Theorem 1.1(ii)] extending the regime of drift parameters, where convergence does not occur – even to values, where it does occur when considering the case of only two particles.

Persistence of autoregressive sequences with logarithmic tails - Denisov, Denis; Hinrichs, Günter; Kolb, Martin; Wachtel, Vitali (Electronic Journal of Probability)
We consider autoregressive sequences Xn = aXn−1 + ξn and Mn = max{aMn−1 , ξn} with a constant a ∈ (0, 1) and with positive, in- dependent and identically distributed innovations {ξk }. It is known that if P(ξ1 > x) ∼ d log x with some d ∈ (0, − log a) then the chains Xn and Mn are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index −1 − d/ log a. We also prove limit theorems for Xn and Mn conditioned to stay over a fixed level x0. Furthermore, we study tail asymptotics for recurrence times of Xn and Mn in the case when these chains are positive recurrent and the tail of log ξ1 is subexponential.

Schnittstellenaufgaben in der Analysis I zur Verknüpfung von Schul- und Hochschulmathematik - Aufgabenbeispiele und Ergebnisse einer Evaluationsstudie. - Hoffmann, Max; Biehler, Rolf (Bedarfsgerechte fachmathematische Lehramtsausbildung. Analyse, Zielsetzungen und Konzepte unter heterogenen Voraussetzungen)

Fachwissen als Grundlage fachdidaktischer Urteilskompetenz – Beispiele für die Herstellung konzeptueller Bezüge zwischen fachwissenschaftlicher und fachdidaktischer Lehre im gymnasialen Lehramtsstudium - Biehler, Rolf; Hoffmann, Max (Professionsorientierte Fachwissenschaft. Kohärenzstiftende Lerngelegenheiten für das Lehramtsstudium Mathematik)

Eye-Tracking Studie zu Textaufgaben in Klasse 5: Bemerken und Interpretieren syntaktischer Strukturen - Dröse, Jennifer; Neugebauer, P.; Delucchi Danhier, R.; Mertins, B. (Eye-Tracking in der Mathematik- und Naturwissenschaftsdidaktik. Forschung und Praxis)

Verification of safety critical control policies using kernel methods - Vertovec, Nikolaus; Ober-Blöbaum, Sina; Margellos, Kostas
Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example.

Prospective Teachers‘ Competence of Fostering Students’ Understanding in Script Writing Task - Dröse, Jennifer; Wessel, Lena (Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education. PME)

Semiclassical formulae For Wigner distributions - Barkhofen, Sonja; Schütte, Philipp; Weich, Tobias (Journal of Physics A: Mathematical and Theoretical)
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.

Von der Axiomatik bis zur Schnittstellenaufgabe: Entwicklung und Erforschung eines ganzheitlichen Lehrkonzepts für eine Veranstaltung Geometrie für Lehramtsstudierende - Hoffmann, Max
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.

Poisson transforms for trees of bounded degree - Bux, Kai-Uwe; Hilgert, Joachim; Weich, Tobias (Journal of Spectral Theory)
Keywords: Geometry and Topology; Mathematical Physics; Statistical and Nonlinear Physics

Ethan D. Bolker und Maura B. Mast: Common Sense Mathematics, Second Edition. AMS/MAA Press 2021 - Hilgert, Joachim (Mathematische Semesterberichte)

Viscoelastic Cahn–Hilliard models for tumor growth - Garcke, Harald; Kovács, Balázs; Trautwein, Dennis (Mathematical Models and Methods in Applied Sciences)
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].
Keywords: Applied Mathematics; Modeling and Simulation

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces - Elliott, Charles M.; Garcke, Harald; Kovács, Balázs (Numerische Mathematik)
Abstract An 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 $$H^1$$ H1 norm 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.
Keywords: Applied Mathematics; Computational Mathematics

Time-dependent electromagnetic scattering from thin layers - Nick, Jörg; Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Abstract The 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.
Keywords: Applied Mathematics; Computational Mathematics

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems - Kovács, Balázs; Li, Buyang (IMA Journal of Numerical Analysis)
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 $L^∞ (0,T;W^{1,∞ })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^∞ (0,T;W^{1,∞ })$ norm is obtained by combining the stability analysis with the consistency estimates.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions - Altmann, Robert; Kovács, Balázs; Zimmer, Christoph (IMA Journal of Numerical Analysis)
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 $τ \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error estimates for a splitting integrator for abstract semilinear boundary coupled systems - Csomós, Petra; Farkas, Bálint; Kovács, Balázs (IMA Journal of Numerical Analysis)
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.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics