Institut für Mathematik
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The article deals with the pedagogical content knowledge of mathematical modelling as part of the professional competence of pre-service teachers. With the help of a test developed for this purpose from a conceptual model, we examine whether this pedagogical content knowledge can be promoted in its different facets—especially knowledge about modelling tasks and about interventions—by suitable university seminars. For this purpose, the test was administered to three groups in a seminar for the teaching of mathematical modelling: (1) to those respondents who created their own modelling tasks for use with students, (2) to those trained to intervene in mathematical modelling processes, and (3) participating students who are not required to address mathematical modelling. The findings of the study—based on variance analysis—indicate that certain facets (knowledge of modelling tasks, modelling processes, and interventions) have increased significantly in both experimental groups but to varying degrees. By contrast, pre-service teachers in the control group demonstrated no significant change to their level of pedagogical content knowledge.
Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper, we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.
Let (ϕ\(_t\))\(_{t≥0}\) be a semigroup of holomorphic functions in the unit disk \(\mathbb {D}\) and K a compact subset of \(\mathbb {D}\). We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk.
We give a collection of 16 examples which show that compositions \(g\) \(\circ\) \(f\) of well-behaved functions \(f\) and \(g\) can be badly behaved. Remarkably, in 10 of the 16 examples it suffices to take as outer function \(g\) simply a power-type or characteristic function. Such a collection of examples may serve as a source of exercises for a calculus course.
Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis.
Nowadays, science, technology, engineering, and mathematics (STEM) play a critical role in a nation’s global competitiveness and prosperity. Thus, there is a need to educate students in these subjects to meet the current and future demands of personal life and society. While applications, especially in science, engineering, and technology, are directly obvious, mathematics underpins the other STEM disciplines. It is recognized that mathematics is the foundation for all other STEM disciplines; the role of mathematics in classrooms is not clear yet. Therefore, the question arises: What is the current role of mathematics in secondary STEM classrooms? To answer this question, we conducted a systematic literature review based on three publication databases (Web of Science, ERIC, and EBSCO Teacher Referral Center). This literature review paper is intended to contribute to the current state of the role of mathematics in STEM education in secondary classrooms. Through the search, starting with 1910 documents, only 14 eligible documents were found. In these, mathematics is often seen as a minor matter and a means to an end in the eyes of science educators. From this, we conclude that the role of mathematics in the STEM classroom should be further strengthened. Overall, the paper highlights a major research gap, and proposes possible initial solutions to close it.
Ó. Blasco and S. Pott showed that the supremum of operator norms over L\(^{2}\) of all bicommutators (with the same symbol) of one-parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1 < p < ∞. The main tool is a new characterization in terms of paraproducts and two-weight John–Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multiparameter setting, with the appropriate iterated commutator in each case.
Bivariate copula monitoring
(2022)
The assumption of multivariate normality underlying the Hotelling T\(^{2}\) chart is often violated for process data. The multivariate dependency structure can be separated from marginals with the help of copula theory, which permits to model association structures beyond the covariance matrix. Copula‐based estimation and testing routines have reached maturity regarding a variety of practical applications. We have constructed a rich design matrix for the comparison of the Hotelling T\(^{2}\) chart with the copula test by Verdier and the copula test by Vuong, which allows for weighting the observations adaptively. Based on the design matrix, we have conducted a large and computationally intensive simulation study. The results show that the copula test by Verdier performs better than Hotelling T\(^{2}\) in a large variety of out‐of‐control cases, whereas the weighted Vuong scheme often fails to provide an improvement.
In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.
Our starting point is the Jacobsthal function \(j(m)\), defined for each positive integer \(m\) as the smallest number such that every \(j(m)\) consecutive integers contain at least one integer relatively prime to \(m\). It has turned out that improving on upper bounds for \(j(m)\) would also lead to advances in understanding the distribution of prime numbers among arithmetic progressions. If \(P_r\) denotes the product of the first \(r\) prime numbers, then a conjecture of Montgomery states that \(j(P_r)\) can be bounded from above by \(r (\log r)^2\) up to some constant factor. However, the until now very promising sieve methods seem to have reached a limit here, and the main goal of this work is to develop other combinatorial methods in hope of coming a bit closer to prove the conjecture of Montgomery. Alongside, we solve a problem of Recamán about the maximum possible length among arithmetic progressions in the least (positive) reduced residue system modulo \(m\). Lastly, we turn towards three additive representation functions as introduced by Erdős, Sárközy and Sós who studied their surprising different monotonicity behavior. By an alternative approach, we answer a question of Sárközy and demostrate that another conjecture does not hold.
Die vorliegende Arbeit beschäftigt sich explorativ mit Metakognition beim Umgang mit Mathematik. Aufbauend auf der vorgestellten Forschungsliteratur wird der Einsatz von Metakognition im Rahmen einer qualitativen Studie bei Studienanfänger_innen aus verschiedenen Mathematik-(Lehramts-)Studiengängen dokumentiert. Unter Verwendung der Qualitativen Inhaltsanalyse nach Mayring erfolgt die Etablierung eines Kategoriensystems für den Begriff Metakognition im Hinblick auf den Einsatz in der Mathematik, das bisherige Systematisierungen erweitert. Schließlich wird der Einsatz der entsprechenden metakognitiven Aspekte am Beispiel verschiedener Begriffe und Verfahren aus dem Analysis-Unterricht exemplarisch aufgezeigt.
Optimization problems with composite functions deal with the minimization of the sum
of a smooth function and a convex nonsmooth function. In this thesis several numerical
methods for solving such problems in finite-dimensional spaces are discussed, which are
based on proximity operators.
After some basic results from convex and nonsmooth analysis are summarized, a first-order
method, the proximal gradient method, is presented and its convergence properties are
discussed in detail. Known results from the literature are summarized and supplemented by
additional ones. Subsequently, the main part of the thesis is the derivation of two methods
which, in addition, make use of second-order information and are based on proximal Newton
and proximal quasi-Newton methods, respectively. The difference between the two methods
is that the first one uses a classical line search, while the second one uses a regularization
parameter instead. Both techniques lead to the advantage that, in contrast to many similar
methods, in the respective detailed convergence analysis global convergence to stationary
points can be proved without any restricting precondition. Furthermore, comprehensive
results show the local convergence properties as well as convergence rates of these algorithms,
which are based on rather weak assumptions. Also a method for the solution of the arising
proximal subproblems is investigated.
In addition, the thesis contains an extensive collection of application examples and a detailed
discussion of the related numerical results.
A sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed scheme, and results of numerical experiments are reported that successfully validate its computational performance.
In dieser Arbeit wird mathematisches Papierfalten und speziell 1-fach-Origami im universitären Kontext untersucht. Die Arbeit besteht aus drei Teilen.
Der erste Teil ist im Wesentlichen der Sachanalyse des 1-fach-Origami gewidmet. Im ersten Kapitel gehen wir auf die geschichtliche Einordnung des 1-fach-Origami, betrachten axiomatische Grundlagen und diskutieren, wie das Axiomatisieren von 1-fach-Origami zum Verständnis des Axiomenbegriffs beitragen könnte. Im zweiten Kapitel schildern wir das Design der zugehörigen explorativen Studie, beschreiben unsere Forschungsziele und -fragen. Im dritten Kapitel wird 1-fach-Origami mathematisiert, definiert und eingehend untersucht.
Der zweite Teil beschäftigt sich mit den von uns gestalteten und durchgeführten Kursen »Axiomatisieren lernen mit Papierfalten«. Im vierten Kapitel beschreiben wir die Lehrmethodik und die Gestaltung der Kurse, das fünfte Kapitel enthält ein Exzerpt der Kurse.
Im dritten Teil werden die zugehörigen Tests beschrieben. Im sechsten Kapitel erläutern wir das Design der Tests sowie die Testmethodik. Im siebten Kapitel findet die Auswertung ebendieser Tests statt.
Global Existence and Uniqueness Results for Nematic Liquid Crystal and Magnetoviscoelastic Flows
(2022)
Liquid crystals and polymeric fluids are found in many technical applications with liquid crystal displays probably being the most prominent one. Ferromagnetic materials are well established in industrial and everyday use, e.g. as magnets in generators, transformers and hard drive disks. Among ferromagnetic materials, we find a subclass which undergoes deformations if an external magnetic field is applied. This effect is exploited in actuators, magnetoelastic sensors, and new fluid materials have been produced which retain their induced magnetization during the flow.
A central issue consists of a proper modelling for those materials. Several models exist regarding liquid crystals and liquid crystal flows, but up to now, none of them has provided a full insight into all observed effects. On materials encompassing magnetic, elastic and perhaps even fluid dynamic effects, the mathematical literature seems sparse in terms of models. To some extent, one can unify the modeling of nematic liquid crystals and magnetoviscoelastic materials employing a so-called energetic variational approach.
Using the least action principle from theoretical physics, the actual task reduces to finding appropriate energies describing the observed behavior. The procedure leads to systems of evolutionary partial differential equations, which are analyzed in this work.
From the mathematical point of view, fundamental questions on existence, uniqueness and stability of solutions remain unsolved. Concerning the Ericksen-Leslie system modelling nematic liquid crystal flows, an approximation to this model is given by the so-called Ginzburg-Landau approximation. Solutions to the latter are intended to approximately represent solutions to the Ericksen-Leslie system. Indeed, we verify this presumption in two spatial dimensions. More precisely, it is shown that weak solutions of the Ginzburg-Landau approximation converge to solutions of the Ericksen-Leslie system in the energy space for all positive times of evolution. In order to do so, theory for the Euler equations invented by DiPerna and Majda on weak compactness and concentration measures is used.
The second part of the work deals with a system of partial differential equations modelling magnetoviscoelastic fluids. We provide a well-posedness result in two spatial dimensions for large energies and large times. Along the verification of that conclusion, existing theory on the Ericksen-Leslie system and the harmonic map flow is deployed and suitably extended.
We present a technique for computing multi-branch-point covers with prescribed ramification and demonstrate the applicability of our method in relatively large degrees by computing several families of polynomials with symplectic and linear Galois groups.
As a first application, we present polynomials over \(\mathbb{Q}(\alpha,t)\) for the primitive rank-3 groups \(PSp_4(3)\) and \(PSp_4(3).C_2\) of degree 27 and for the 2-transitive group \(PSp_6(2)\) in its actions on 28 and 36 points, respectively. Moreover, the degree-28 polynomial for \(PSp_6(2)\) admits infinitely many totally real specializations.
Next, we present the first (to the best of our knowledge) explicit polynomials for the 2-transitive linear groups \(PSL_4(3)\) and \(PGL_4(3)\) of degree 40, and the imprimitive group \(Aut(PGL_4(3))\) of degree 80.
Additionally, we negatively answer a question by König whether there exists a degree-63 rational function with rational coefficients and monodromy group \(PSL_6(2)\) ramified over at least four points. This is achieved due to the explicit computation of the corresponding hyperelliptic genus-3 Hurwitz curve parameterizing this family, followed by a search for rational points on it. As a byproduct of our calculations we obtain the first explicit \(Aut(PSL_6(2))\)-realizations over \(\mathbb{Q}(t)\).
At last, we present a technique by Elkies for bounding the transitivity degree of Galois groups. This provides an alternative way to verify the Galois groups from the previous chapters and also yields a proof that the monodromy group of a degree-276 cover computed by Monien is isomorphic to the sporadic 2-transitive Conway group \(Co_3\).
We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil (Int. Math. Res. Not. 2019: 3301–3312, 2019) on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.
We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace–Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem.
In this work, we consider impulsive dynamical systems evolving on an infinite-dimensional space and subjected to external perturbations. We look for stability conditions that guarantee the input-to-state stability for such systems. Our new dwell-time conditions allow the situation, where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov like methods are developed for this purpose. Illustrative finite and infinite dimensional examples are provided to demonstrate the application of the main results. These examples cannot be treated by any other published approach and demonstrate the effectiveness of our results.
Die Auseinandersetzung mit Simulations- und Modellierungsaufgaben, die mit digitalen Werkzeugen zu bearbeiten sind, stellt veränderte Anforderungen an Mathematiklehrkräfte in der Unterrichtsplanung und -durchführung. Werden digitale Werkzeuge sinnvoll eingesetzt, so unterstützen sie Simulations- und Modellierungsprozesse und ermöglichen realitätsnähere Sachkontexte im Mathematikunterricht. Für die empirische Untersuchung professioneller Kompetenzen zum Lehren des Simulierens und mathematischen Modellierens mit digitalen Werkzeugen ist es notwendig, Aspekte globaler Lehrkompetenzen von (angehenden) Mathematiklehrkräften bereichsspezifisch auszudeuten.
Daher haben wir ein Testinstrument entwickelt, das die Überzeugungen, die Selbstwirksamkeitserwartungen und das fachdidaktische Wissen zum Lehren des Simulierens und mathematischen Modellierens mit digitalen Werkzeugen erfasst. Ergänzt wird das Testinstrument durch selbstberichtete Vorerfahrungen zum eigenen Gebrauch digitaler Werkzeuge sowie zur Verwendung digitaler Werkzeuge in Unterrichtsplanung und -durchführung.
Das Testinstrument ist geeignet, um mittels Analysen von Veranstaltungsgruppen im Prä-Post-Design den Zuwachs der oben beschriebenen Kompetenz von (angehenden) Mathematiklehrkräften zu messen. Somit können in Zukunft anhand der Ergebnisse die Wirksamkeit von Lehrveranstaltungen, die diese Kompetenz fördern (sollen), untersucht und evaluiert werden.
Der Beitrag gliedert sich in zwei Teile: Zunächst werden in der Testbeschreibung das zugrundeliegende Konstrukt und der Anwendungsbereich des Testinstruments sowie dessen Aufbau und Hinweise zur Durchführung beschrieben. Zudem wird die Testgüte anhand der Pilotierungsergebnisse überprüft. Im zweiten Teil befindet sich das vollständige Testinstrument.
In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. https://doi.org/10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.
The dissertation investigates the wide class of Epstein zeta-functions in terms of uniform distribution modulo one of the ordinates of their nontrivial zeros. Main results are a proof of a Landau type theorem for all Epstein zeta-functions as well as uniform distribution modulo one for the zero ordinates of all Epstein zeta-functions asscoiated with binary quadratic forms.