@article{HanischStrohmaierWaters2022, author = {Hanisch, Florian and Strohmaier, Alexander and Waters, Alden}, title = {A relative trace formula for obstacle scattering}, series = {Duke mathematical journal}, volume = {171}, journal = {Duke mathematical journal}, number = {11}, publisher = {Duke Univ. Press}, address = {Durham, NC}, issn = {0012-7094}, doi = {10.1215/00127094-2022-0053}, pages = {2233 -- 2274}, year = {2022}, abstract = {We consider the case of scattering by several obstacles in Rd for d ≥ 2. In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator Δ0 are unitarily equivalent. For suitable functions that decay sufficiently fast, we have that the difference g(Δ) - g(Δ0) is a trace-class operator and its trace is described by the Krein spectral shift function. In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators Δ1 and Δ2 obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then g(Δ) - g(Δ1) - g(Δ2) C g(Δ0) is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman-Krein formula. In case g(x) D x 2 , 1 the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization.}, language = {en} } @phdthesis{Günther2023, author = {G{\"u}nther, Claudia-Susanne}, title = {Das Eigene und das Fremde}, school = {Universit{\"a}t Potsdam}, pages = {245}, year = {2023}, abstract = {Die vorliegende Arbeit stellt eine Untersuchung des Fremdverstehens von Lehrkr{\"a}ften im Mathematikunterricht dar. Mit ‚Fremdverstehen' soll dabei - in Anlehnung an den Soziologen Alfred Sch{\"u}tz - der Prozess bezeichnet werden, in welchem eine Lehrkraft versucht, das Verhalten einer Sch{\"u}lerin oder eines Sch{\"u}lers zu verstehen, indem sie dieses Verhalten auf ein Erleben zur{\"u}ckf{\"u}hrt, das ihm zugrunde gelegen haben k{\"o}nnte. Als ein wesentliches Merkmal des Prozesses stellt Sch{\"u}tz 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 zun{\"a}chst die mathematikbezogenen Erlebnisse zweier Lehrkr{\"a}fte nachgezeichnet, bevor dann ihr Fremdverstehen in konkreten Situationen im Mathematikunterricht rekonstruiert wird. In der ersten Teiluntersuchung (= der Rekonstruktion eigener Erlebnisse der untersuchten Lehrkr{\"a}fte) erfolgt die Datenerhebung mit Hilfe biographisch-narrativer Interviews, in denen die untersuchten Lehrkr{\"a}fte angeregt werden, ihre mathematikbezogene Lebensgeschichte zu erz{\"a}hlen. Die Analyse dieser Interviews wird im Sinne der rekonstruktiven Fallanalyse vorgenommen. Insgesamt f{\"u}hrt die erste Teiluntersuchung zu textlichen Darstellungen der rekonstruierten mathematikbezogenen Lebensgeschichte der untersuchten Mathematiklehrkr{\"a}fte. In der zweiten Teiluntersuchung (= der Rekonstruktion des Fremdverstehens der untersuchten Lehrkr{\"a}fte) werden dann narrative Interviews gef{\"u}hrt, in denen die untersuchten Lehrkr{\"a}fte von ihrem Fremdverstehen in konkreten Situationen im Mathematikunterricht erz{\"a}hlen. 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 Lehrkr{\"a}fte 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 {\"u}ber f{\"u}nf Merkmale des Fremdverstehens von Lehrkr{\"a}ften im Mathematikunterricht im Allgemeinen gewonnen. Mit diesen Aussagen vermag die Arbeit eine erste Beschreibung davon hervorzubringen, wie sich das Ph{\"a}nomen des Fremdverstehens von Lehrkr{\"a}ften im Mathematikunterricht ausgestalten kann.}, language = {de} } @misc{Dahl2023, type = {Master Thesis}, author = {Dahl, Dorothee Sophie}, title = {Zahlen in den Fingern}, doi = {10.25932/publishup-60762}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-607629}, school = {Universit{\"a}t Potsdam}, pages = {118}, year = {2023}, abstract = {Die Debatte {\"u}ber den Einsatz von digitalen Werkzeugen in der mathematischen Fr{\"u}hf{\"o}rderung ist hoch aktuell. Lernspiele werden konstruiert, mit dem Ziel, mathematisches, informelles Wissen aufzubauen und so einen besseren Schulstart zu erm{\"o}glichen. Doch allein die digitale und spielerische Aufarbeitung f{\"u}hrt nicht zwingend zu einem Lernerfolg. Daher ist es umso wichtiger, die konkrete Implementation der theoretischen Konstrukte und Interaktionsm{\"o}glichkeiten mit den Werkzeugen zu analysieren und passend aufzubereiten. In dieser Masterarbeit wird dazu exemplarisch ein mathematisches Lernspiel namens „Fingu" f{\"u}r den Einsatz im vorschulischen Bereich theoretisch und empirisch im Rahmen der Artifact-Centric Activity Theory (ACAT) untersucht. Dazu werden zun{\"a}chst die theoretischen Hintergr{\"u}nde zum Zahlensinn, Zahlbegriffserwerb, Teil-Ganze-Verst{\"a}ndnis, der Anzahlwahrnehmung und -bestimmung, den Anzahlvergleichen und der Anzahldarstellung mithilfe von Fingern gem{\"a}ß der Embodied Cognition sowie der Verwendung von digitalen Werkzeugen und Multi-Touch-Ger{\"a}ten umfassend beschrieben. Anschließend wird die App Fingu erkl{\"a}rt und dann theoretisch entlang des ACAT-Review-Guides analysiert. Zuletzt wird die selbstst{\"a}ndig durchgef{\"u}hrte Studie mit zehn Vorschulkindern erl{\"a}utert und darauf aufbauend Verbesserungs- und Entwicklungsm{\"o}glichkeiten der App auf wissenschaftlicher Grundlage beigetragen. F{\"u}r Fingu l{\"a}sst sich abschließend festhalten, dass viele Prozesse wie die (Quasi-)Simultanerfassung oder das Z{\"a}hlen gef{\"o}rdert werden k{\"o}nnen, f{\"u}r andere wie das Teil-Ganze-Verst{\"a}ndnis aber noch Anpassungen und/oder die Begleitung durch Erwachsene n{\"o}tig ist.}, language = {de} } @phdthesis{Gehring2023, author = {Gehring, Penelope}, title = {Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary}, doi = {10.25932/publishup-57775}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-577755}, school = {Universit{\"a}t Potsdam}, pages = {100}, year = {2023}, abstract = {Non-local boundary conditions - for example the Atiyah-Patodi-Singer (APS) conditions - for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, B{\"a}r and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While B{\"a}r and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions. In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed. We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems. Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions - the analogue of the Atiyah-Patodi-Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.}, language = {en} } @phdthesis{LopezValencia2023, author = {Lopez Valencia, Diego Andres}, title = {The Milnor-Moore and Poincar{\´e}-Birkhoff-Witt theorems in the locality set up and the polar structure of Shintani zeta functions}, doi = {10.25932/publishup-59421}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-594213}, school = {Universit{\"a}t Potsdam}, pages = {147}, year = {2023}, abstract = {This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincar{\´e}-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions. The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincar{\´e}-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincar{\´e}-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts. The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.}, language = {en} } @phdthesis{MalemShinitski2023, author = {Malem-Shinitski, Noa}, title = {Bayesian inference and modeling for point processes with applications from neuronal activity to scene viewing}, doi = {10.25932/publishup-61495}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-614952}, school = {Universit{\"a}t Potsdam}, pages = {vii, 129}, year = {2023}, abstract = {Point processes are a common methodology to model sets of events. From earthquakes to social media posts, from the arrival times of neuronal spikes to the timing of crimes, from stock prices to disease spreading -- these phenomena can be reduced to the occurrences of events concentrated in points. Often, these events happen one after the other defining a time--series. Models of point processes can be used to deepen our understanding of such events and for classification and prediction. Such models include an underlying random process that generates the events. This work uses Bayesian methodology to infer the underlying generative process from observed data. Our contribution is twofold -- we develop new models and new inference methods for these processes. We propose a model that extends the family of point processes where the occurrence of an event depends on the previous events. This family is known as Hawkes processes. Whereas in most existing models of such processes, past events are assumed to have only an excitatory effect on future events, we focus on the newly developed nonlinear Hawkes process, where past events could have excitatory and inhibitory effects. After defining the model, we present its inference method and apply it to data from different fields, among others, to neuronal activity. The second model described in the thesis concerns a specific instance of point processes --- the decision process underlying human gaze control. This process results in a series of fixated locations in an image. We developed a new model to describe this process, motivated by the known Exploration--Exploitation dilemma. Alongside the model, we present a Bayesian inference algorithm to infer the model parameters. Remaining in the realm of human scene viewing, we identify the lack of best practices for Bayesian inference in this field. We survey four popular algorithms and compare their performances for parameter inference in two scan path models. The novel models and inference algorithms presented in this dissertation enrich the understanding of point process data and allow us to uncover meaningful insights.}, language = {en} } @misc{EhlenFloegeGoebeletal.2023, author = {Ehlen, Tobias and Fl{\"o}ge, Annie and G{\"o}bel, Franziska and Keller, Peter and Rœlly, Sylvie}, title = {{\"U}bungsbuch zur Stochastik}, editor = {Keller, Peter and Rœlly, Sylvie}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-563-7}, doi = {10.25932/publishup-59593}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-595939}, pages = {306}, year = {2023}, abstract = {Dieses Buch stellt {\"U}bungen zu den Grundbegriffen und Grunds{\"a}tzen der Stochastik und ihre L{\"o}sungen zur Verf{\"u}gung. So wie man Tonleitern in der Musik trainiert, so berechnet man {\"U}bungsaufgaben in der Mathematik. In diesem Sinne soll dieses {\"U}bungsbuch vor allem als Vorlage dienen f{\"u}r das eigenst{\"a}ndige, eigenverantwortliche Lernen und {\"U}ben. Die Sch{\"o}nheit und Einzigartigkeit der Wahrscheinlichkeitstheorie besteht darin, dass sie eine Vielzahl von realen Ph{\"a}nomenen modellieren kann. Daher findet man hier Aufgaben mit Verbindungen zur Geometrie, zu Gl{\"u}cksspielen, zur Versicherungsmathematik, zur Demographie und vielen anderen Themen.}, language = {de} } @article{KortenkampKuzleReitzKoncebovski2023, author = {Kortenkamp, Ulrich and Kuzle, Ana and Reitz-Koncebovski, Karen}, title = {Fachdidaktisches Wissen aus dem Fachwissen generieren}, series = {PSI-Potsdam: Ergebnisbericht zu den Aktivit{\"a}ten im Rahmen der Qualit{\"a}tsoffensive Lehrerbildung (2019-2023) (Potsdamer Beitr{\"a}ge zur Lehrerbildung und Bildungsforschung ; 3)}, journal = {PSI-Potsdam: Ergebnisbericht zu den Aktivit{\"a}ten im Rahmen der Qualit{\"a}tsoffensive Lehrerbildung (2019-2023) (Potsdamer Beitr{\"a}ge zur Lehrerbildung und Bildungsforschung ; 3)}, number = {3}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-568-2}, issn = {2626-3556}, doi = {10.25932/publishup-61760}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-617602}, pages = {171 -- 191}, year = {2023}, abstract = {Das Mathematik-Teilprojekt SPIES-M zielt auf eine st{\"a}rkere Professionsorientierung und die Verkn{\"u}pfung von Fachwissenschaft und Fachdidaktik in der universit{\"a}ren Lehrkr{\"a}ftebildung. Zu allen großen Inhaltsgebieten der Mathematik wurden neue Lehrveranstaltungen konzipiert und in den Studienordnungen s{\"a}mtlicher Lehr{\"a}mter Mathematik an der Universit{\"a}t Potsdam implementiert. F{\"u}r die Konzeption wurden theoriebasiert Gestaltungsprinzipien herausgearbeitet, die sowohl f{\"u}r das Design als auch f{\"u}r die Evaluation und Weiterentwicklung der Lehrveranstaltungen nach dem Design-Research-Ansatz genutzt werden k{\"o}nnen. Die Umsetzung der Gestaltungsprinzipien wird am Beispiel der Fundamentalen Idee der Proportionalit{\"a}t verdeutlicht und dabei aufgezeigt, wie Studierende dazu bef{\"a}higt werden k{\"o}nnen, fachdidaktisches Wissen aus fachmathematischen Inhalten zu generieren. Die Entwicklung des Professionswissens der Studierenden wird mithilfe unterschiedlicher Instrumente untersucht, um R{\"u}ckschl{\"u}sse auf die Wirksamkeit der neu konzipierten Lehrveranstaltungen zu ziehen. F{\"u}r die Untersuchungen im Mixed-Methods-Design werden neben Beobachtungen in Lehrveranstaltungen eigens konzipierte Wissenstests, Gruppeninterviews, Unterrichtsentw{\"u}rfe aus Praxisphasen und Lerntageb{\"u}cher 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 pr{\"a}sentiert und hinsichtlich ihrer Implikationen diskutiert. Die im Projekt entwickelten Gestaltungsprinzipien als Werkzeug f{\"u}r Design und Evaluation sowie das Spies-Konzept der kollegialen Supervision werden f{\"u}r die Qualit{\"a}tsentwicklung von Lehrveranstaltungen zum Transfer vorgeschlagen.}, language = {de} } @article{Metzger2023, author = {Metzger, Jan}, title = {Refined position estimates for surfaces of Willmore type in Riemannian manifolds}, series = {Communications in analysis and geometry}, volume = {30}, journal = {Communications in analysis and geometry}, number = {10}, publisher = {International Press of Boston}, address = {Somerville, Mass.}, issn = {1019-8385}, doi = {10.4310/CAG.2022.v30.n10.a5}, pages = {2315 -- 2346}, year = {2023}, abstract = {In this paper we consider surfaces which are critical points of the Willmore functional subject to constrained area. In the case of small area we calculate the corrections to the intrinsic geometry induced by the ambient curvature. These estimates together with the choice of an adapted geometric center of mass lead to refined position estimates in relation to the scalar curvature of the ambient manifold.}, language = {en} } @article{FalkenhagenKnoechelKloftetal.2023, author = {Falkenhagen, Undine and Kn{\"o}chel, Jane and Kloft, Charlotte and Huisinga, Wilhelm}, title = {Deriving mechanism-based pharmacodynamic models by reducing quantitative systems pharmacology models}, series = {CPT: Pharmacometrics \& Systems Pharmacology}, volume = {12}, journal = {CPT: Pharmacometrics \& Systems Pharmacology}, number = {4}, publisher = {Wiley}, address = {Hoboken}, issn = {2163-8306}, doi = {10.1002/psp4.12903}, pages = {432 -- 443}, year = {2023}, abstract = {Quantitative systems pharmacology (QSP) models integrate comprehensive qualitative and quantitative knowledge about pharmacologically relevant processes. We previously proposed a first approach to leverage the knowledge in QSP models to derive simpler, mechanism-based pharmacodynamic (PD) models. Their complexity, however, is typically still too large to be used in the population analysis of clinical data. Here, we extend the approach beyond state reduction to also include the simplification of reaction rates, elimination of reactions, and analytic solutions. We additionally ensure that the reduced model maintains a prespecified approximation quality not only for a reference individual but also for a diverse virtual population. We illustrate the extended approach for the warfarin effect on blood coagulation. Using the model-reduction approach, we derive a novel small-scale warfarin/international normalized ratio model and demonstrate its suitability for biomarker identification. Due to the systematic nature of the approach in comparison with empirical model building, the proposed model-reduction algorithm provides an improved rationale to build PD models also from QSP models in other applications.}, language = {en} }