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.