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Institute
- Institut für Mathematik (1619) (remove)
Das Eigene und das Fremde
(2023)
Die vorliegende Arbeit stellt eine Untersuchung des Fremdverstehens von Lehrkräften im Mathematikunterricht dar. Mit ‚Fremdverstehen‘ soll dabei – in Anlehnung an den Soziologen Alfred Schütz – der Prozess bezeichnet werden, in welchem eine Lehrkraft versucht, das Verhalten einer Schülerin oder eines Schülers zu verstehen, indem sie dieses Verhalten auf ein Erleben zurückführt, das ihm zugrunde gelegen haben könnte. Als ein wesentliches Merkmal des Prozesses stellt Schütz in seiner Theorie des Fremdverstehens heraus, dass das Fremdverstehen eines Menschen immer auch auf seinen eigenen Erlebnissen basiert. Aus diesem Grund wird in der Arbeit ein methodischer Zweischritt vorgenommen: Es werden zunächst die mathematikbezogenen Erlebnisse zweier Lehrkräfte nachgezeichnet, bevor dann ihr Fremdverstehen in konkreten Situationen im Mathematikunterricht rekonstruiert wird. In der ersten Teiluntersuchung (= der Rekonstruktion eigener Erlebnisse der untersuchten Lehrkräfte) erfolgt die Datenerhebung mit Hilfe biographisch-narrativer Interviews, in denen die untersuchten Lehrkräfte angeregt werden, ihre mathematikbezogene Lebensgeschichte zu erzählen. Die Analyse dieser Interviews wird im Sinne der rekonstruktiven Fallanalyse vorgenommen. Insgesamt führt die erste Teiluntersuchung zu textlichen Darstellungen der rekonstruierten mathematikbezogenen Lebensgeschichte der untersuchten Mathematiklehrkräfte. In der zweiten Teiluntersuchung (= der Rekonstruktion des Fremdverstehens der untersuchten Lehrkräfte) werden dann narrative Interviews geführt, in denen die untersuchten Lehrkräfte von ihrem Fremdverstehen in konkreten Situationen im Mathematikunterricht erzählen. Die Analyse dieser Interviews erfolgt mit Hilfe eines dreischrittigen Analyseverfahrens, welches die Autorin eigens zum Zweck der Rekonstruktion von Fremdverstehen entwickelte. Am Ende dieser zweiten Teiluntersuchung werden sowohl das rekonstruierte Fremdverstehen der Lehrkräfte in verschiedenen Unterrichtssituationen dargestellt als auch Strukturen, die sich in ihrem Fremdverstehen abzeichnen. Mit Hilfe einer theoretischen Verallgemeinerung werden schließlich – auf Basis der Ergebnisse der zweiten Teiluntersuchung – Aussagen über fünf Merkmale des Fremdverstehens von Lehrkräften im Mathematikunterricht im Allgemeinen gewonnen. Mit diesen Aussagen vermag die Arbeit eine erste Beschreibung davon hervorzubringen, wie sich das Phänomen des Fremdverstehens von Lehrkräften im Mathematikunterricht ausgestalten kann.
According to Radzikowski’s celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF−G˜F+G˜A−G˜R) in the sense of Duistermaat and Hörmander [Acta Math. 128, 183–269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by Bär, Ginoux, and Pfäffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Zürich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF−G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.
Extreme value statistics is a popular and frequently used tool to model the occurrence of large earthquakes. The problem of poor statistics arising from rare events is addressed by taking advantage of the validity of general statistical properties in asymptotic regimes. In this note, I argue that the use of extreme value statistics for the purpose of practically modeling the tail of the frequency-magnitude distribution of earthquakes can produce biased and thus misleading results because it is unknown to what degree the tail of the true distribution is sampled by data. Using synthetic data allows to quantify this bias in detail. The implicit assumption that the true M-max is close to the maximum observed magnitude M-max,M-observed restricts the class of the potential models a priori to those with M-max = M-max,M-observed + Delta M with an increment Delta M approximate to 0.5... 1.2. This corresponds to the simple heuristic method suggested by Wheeler (2009) and labeled :M-max equals M-obs plus an increment." The incomplete consideration of the entire model family for the frequency-magnitude distribution neglects, however, the scenario of a large so far unobserved earthquake.
In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.
An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions
(2022)
We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature beta. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.
Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.
Conventional embeddings of the edge-graphs of Platonic polyhedra, {f,z}, where f,z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schonflies symbols I-h, O-h, and T-d). Tangled Platonic {f,z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the "theta(z)" polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.
Model-informed precision dosing (MIPD) is a quantitative dosing framework that combines prior knowledge on the drug-disease-patient system with patient data from therapeutic drug/ biomarker monitoring (TDM) to support individualized dosing in ongoing treatment. Structural models and prior parameter distributions used in MIPD approaches typically build on prior clinical trials that involve only a limited number of patients selected according to some exclusion/inclusion criteria. Compared to the prior clinical trial population, the patient population in clinical practice can be expected to also include altered behavior and/or increased interindividual variability, the extent of which, however, is typically unknown. Here, we address the question of how to adapt and refine models on the level of the model parameters to better reflect this real-world diversity. We propose an approach for continued learning across patients during MIPD using a sequential hierarchical Bayesian framework. The approach builds on two stages to separate the update of the individual patient parameters from updating the population parameters. Consequently, it enables continued learning across hospitals or study centers, because only summary patient data (on the level of model parameters) need to be shared, but no individual TDM data. We illustrate this continued learning approach with neutrophil-guided dosing of paclitaxel. The present study constitutes an important step toward building confidence in MIPD and eventually establishing MIPD increasingly in everyday therapeutic use.
We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.