TY - JOUR A1 - Kortenkamp, Ulrich A1 - Kuzle, Ana A1 - Reitz-Koncebovski, Karen T1 - Fachdidaktisches Wissen aus dem Fachwissen generieren BT - Design Research zur Verknüpfung von Fachwissenschaft und Fachdidaktik in der Lehrkräftebildung Mathematik JF - PSI-Potsdam: Ergebnisbericht zu den Aktivitäten im Rahmen der Qualitätsoffensive Lehrerbildung (2019-2023) (Potsdamer Beiträge zur Lehrerbildung und Bildungsforschung ; 3) N2 - Das Mathematik-Teilprojekt SPIES-M zielt auf eine stärkere Professionsorientierung und die Verknüpfung von Fachwissenschaft und Fachdidaktik in der universitären Lehrkräftebildung. Zu allen großen Inhaltsgebieten der Mathematik wurden neue Lehrveranstaltungen konzipiert und in den Studienordnungen sämtlicher Lehrämter Mathematik an der Universität Potsdam implementiert. Für die Konzeption wurden theoriebasiert Gestaltungsprinzipien herausgearbeitet, die sowohl für das Design als auch für die Evaluation und Weiterentwicklung der Lehrveranstaltungen nach dem Design-Research-Ansatz genutzt werden können. Die Umsetzung der Gestaltungsprinzipien wird am Beispiel der Fundamentalen Idee der Proportionalität verdeutlicht und dabei aufgezeigt, wie Studierende dazu befähigt werden können, fachdidaktisches Wissen aus fachmathematischen Inhalten zu generieren. Die Entwicklung des Professionswissens der Studierenden wird mithilfe unterschiedlicher Instrumente untersucht, um Rückschlüsse auf die Wirksamkeit der neu konzipierten Lehrveranstaltungen zu ziehen. Für die Untersuchungen im Mixed-Methods-Design werden neben Beobachtungen in Lehrveranstaltungen eigens konzipierte Wissenstests, Gruppeninterviews, Unterrichtsentwürfe aus Praxisphasen und Lerntagebücher genutzt. Die Studierendenperspektive wird durch Befragungen zur wahrgenommenen (Berufs-)Relevanz der Lehrveranstaltungen erhoben. Weiteres wesentliches Element der Begleitforschung ist die kollegiale Supervision durch sogenannte „Spies“ (Spione), die die Veranstaltungen kriteriengeleitet beobachten und anschließend gemeinsam mit den Dozierenden reflektieren. Die bisherigen Ergebnisse werden hier präsentiert und hinsichtlich ihrer Implikationen diskutiert. Die im Projekt entwickelten Gestaltungsprinzipien als Werkzeug für Design und Evaluation sowie das Spies-Konzept der kollegialen Supervision werden für die Qualitätsentwicklung von Lehrveranstaltungen zum Transfer vorgeschlagen. N2 - The mathematics sub-project SPIES-M aims at a stronger professional orientation and the linking of subject-specific knowledge and subject-specific didactics in university teacher training. New courses were designed for all major mathematical content areas and implemented in the academic regulations of all mathematics teacher training programs at the University of Potsdam. For the course design, theory-based design principles were developed, which can be used both for the design, and for the evaluation and further development of the courses according to the design-research approach. The implementation of the design principles is exemplary illustrated for the fundamental idea of proportionality, by showing how students can be empowered to generate subject didactic knowledge from subject mathematical content. For this study, an explorative mixed-methods design was chosen, in which the development of the students’ professional knowledge was examined with the help of different instruments in order to draw conclusions about the effectiveness of the newly designed courses. In addition to course observations, specially designed knowledge tests, group interviews, lesson plans from practical phases, and learning diaries were used. The students’ perspective was examined through surveys on the perceived (professional) relevance of the courses. Another important element of the accompanying research was the collegial supervision by so-called „spies“, who observed the courses according to criteria and then reflected on them together with the course lecturers. Here, the current results are presented and discussed regarding their diverse implications. Lastly, the developed design principles as a tool for the design and evaluation of the mathematics courses as well as the spies concept of collegial supervision are proposed for transfer for the quality development of courses in general. KW - Lehrkräftebildung Mathematik KW - Professionswissen KW - Verknüpfung Fachwissenschaft und Fachdidaktik KW - Design Research KW - Gestaltungsprinzipien KW - kollegiale Supervision KW - teacher training mathematics KW - professional knowledge KW - linking of subject science and didactic KW - design research KW - design elements KW - collegial supervision Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-617602 SN - 978-3-86956-568-2 SN - 2626-3556 SN - 2626-4722 IS - 3 SP - 171 EP - 191 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Lewandowski, Max T1 - Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes JF - Journal of mathematical physics N2 - 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. Y1 - 2022 U6 - https://doi.org/10.1063/5.0055753 SN - 0022-2488 SN - 1089-7658 VL - 63 IS - 1 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Zöller, Gert T1 - A note on the estimation of the maximum possible earthquake magnitude based on extreme value theory for the Groningen Gas Field JF - The bulletin of the Seismological Society of America : BSSA N2 - 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. Y1 - 2022 U6 - https://doi.org/10.1785/0120210307 SN - 0037-1106 SN - 1943-3573 VL - 112 IS - 4 SP - 1825 EP - 1831 PB - Seismological Society of America CY - El Cerito, Calif. ER - TY - JOUR A1 - Kaya, Adem A1 - Freitag, Melina A. T1 - Conditioning analysis for discrete Helmholtz problems JF - Computers and mathematics with applications : an international journal N2 - 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. KW - Helmholtz problem KW - Condition number KW - Ill-conditioning KW - Indefinite KW - matrices Y1 - 2022 U6 - https://doi.org/10.1016/j.camwa.2022.05.016 SN - 0898-1221 SN - 1873-7668 VL - 118 SP - 171 EP - 182 PB - Elsevier Science CY - Amsterdam ER - TY - JOUR A1 - Houdebert, Pierre A1 - Zass, Alexander T1 - An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions JF - Journal of applied probability / Applied Probability Trust N2 - 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. KW - Gibbs point process KW - DLR equations KW - uniqueness KW - Dobrushin criterion; KW - cluster expansion KW - disagreement percolation Y1 - 2022 U6 - https://doi.org/10.1017/jpr.2021.70 SN - 0021-9002 SN - 1475-6072 VL - 59 IS - 2 SP - 541 EP - 555 PB - Cambridge Univ. Press CY - Cambridge ER - TY - JOUR A1 - Evans, Myfanwy E. A1 - Hyde, Stephen T. T1 - Symmetric Tangling of Honeycomb Networks JF - Symmetry N2 - 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. KW - tangles KW - knots KW - networks KW - periodic entanglement KW - molecular weaving KW - graphs Y1 - 2022 U6 - https://doi.org/10.3390/sym14091805 SN - 2073-8994 VL - 14 SP - 1 EP - 13 PB - MDPI CY - Basel, Schweiz ET - 9 ER - TY - JOUR A1 - Hyde, Stephen T. A1 - Evans, Myfanwy E. T1 - Symmetric tangled Platonic polyhedra JF - Proceedings of the National Academy of Sciences of the United States of America N2 - 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. KW - regular polyhedra KW - compound polyhedra KW - helicates KW - metal-organic KW - frameworks KW - clathrin Y1 - 2022 U6 - https://doi.org/10.1073/pnas.2110345118 SN - 0027-8424 SN - 1091-6490 VL - 119 IS - 1 PB - National Acad. of Sciences CY - Washington ER - TY - JOUR A1 - Maier, Corinna Sabrina A1 - Wiljes, Jana de A1 - Hartung, Niklas A1 - Kloft, Charlotte A1 - Huisinga, Wilhelm T1 - A continued learning approach for model-informed precision dosing BT - Updating models in clinical practice JF - CPT: pharmacometrics & systems pharmacology N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1002/psp4.12745 SN - 2163-8306 VL - 11 IS - 2 SP - 185 EP - 198 PB - London CY - Nature Publ. Group ER - TY - JOUR A1 - Bär, Christian A1 - Bandara, Lashi T1 - Boundary value problems for general first-order elliptic differential operators JF - Journal of functional analysis N2 - 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. KW - elliptic differential operators of firstorder KW - elliptic boundary KW - conditions KW - boundary regularity KW - Fredholm property KW - H-infinity-functional calculus KW - maximal regularity KW - Rarita-Schwinger KW - operator Y1 - 2022 U6 - https://doi.org/10.1016/j.jfa.2022.109445 SN - 0022-1236 SN - 1096-0783 VL - 282 IS - 12 PB - Elsevier CY - Amsterdam [u.a.] ER - TY - JOUR A1 - Mera, Azal Jaafar Musa A1 - Tarkhanov, Nikolai T1 - An elliptic equation of finite index in a domain JF - Boletin de la Sociedad Matemática Mexicana N2 - We give an example of first order elliptic equation for a complex-valued function in a plane domain which has a finite number of linearly independent solutions for any right-hand side. No boundary value conditions are thus required. KW - elliptic equation KW - Fredholm operator KW - index Y1 - 2022 U6 - https://doi.org/10.1007/s40590-022-00442-7 SN - 1405-213X SN - 2296-4495 VL - 28 IS - 2 PB - Springer International CY - New York [u.a.] ER -