@misc{KolbeEvans2020, author = {Kolbe, Benedikt Maximilian and Evans, Myfanwy E.}, title = {Isotopic tiling theory for hyperbolic surfaces}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1}, issn = {1866-8372}, doi = {10.25932/publishup-54428}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-544285}, pages = {30}, year = {2020}, abstract = {In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.}, language = {en} } @misc{KellerPinchoverPogorzelski2020, author = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix}, title = {From hardy to rellich inequalities on graphs}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {3}, issn = {1866-8372}, doi = {10.25932/publishup-54214}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-542140}, pages = {22}, year = {2020}, abstract = {We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.}, language = {en} } @misc{MazzonettoSalimova2020, author = {Mazzonetto, Sara and Salimova, Diyora}, title = {Existence, uniqueness, and numerical approximations for stochastic burgers equations}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {4}, issn = {1866-8372}, doi = {10.25932/publishup-51579}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-515796}, pages = {26}, year = {2020}, abstract = {In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.}, language = {en} } @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} } @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} } @phdthesis{Schindler2023, author = {Schindler, Daniel}, title = {Mathematical modeling and simulation of protrusion-driven cell dynamics}, doi = {10.25932/publishup-61327}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-613275}, school = {Universit{\"a}t Potsdam}, pages = {VI, 161}, year = {2023}, abstract = {Amoeboid cell motility takes place in a variety of biomedical processes such as cancer metastasis, embryonic morphogenesis, and wound healing. In contrast to other forms of cell motility, it is mainly driven by substantial cell shape changes. Based on the interplay of explorative membrane protrusions at the front and a slower-acting membrane retraction at the rear, the cell moves in a crawling kind of way. Underlying these protrusions and retractions are multiple physiological processes resulting in changes of the cytoskeleton, a meshwork of different multi-functional proteins. The complexity and versatility of amoeboid cell motility raise the need for novel computational models based on a profound theoretical framework to analyze and simulate the dynamics of the cell shape. The objective of this thesis is the development of (i) a mathematical framework to describe contour dynamics in time and space, (ii) a computational model to infer expansion and retraction characteristics of individual cell tracks and to produce realistic contour dynamics, (iii) and a complementing Open Science approach to make the above methods fully accessible and easy to use. In this work, we mainly used single-cell recordings of the model organism Dictyostelium discoideum. Based on stacks of segmented microscopy images, we apply a Bayesian approach to obtain smooth representations of the cell membrane, so-called cell contours. We introduce a one-parameter family of regularized contour flows to track reference points on the contour (virtual markers) in time and space. This way, we define a coordinate system to visualize local geometric and dynamic quantities of individual contour dynamics in so-called kymograph plots. In particular, we introduce the local marker dispersion as a measure to identify membrane protrusions and retractions in a fully automated way. This mathematical framework is the basis of a novel contour dynamics model, which consists of three biophysiologically motivated components: one stochastic term, accounting for membrane protrusions, and two deterministic terms to control the shape and area of the contour, which account for membrane retractions. Our model provides a fully automated approach to infer protrusion and retraction characteristics from experimental cell tracks while being also capable of simulating realistic and qualitatively different contour dynamics. Furthermore, the model is used to classify two different locomotion types: the amoeboid and a so-called fan-shaped type. With the complementing Open Science approach, we ensure a high standard regarding the usability of our methods and the reproducibility of our research. In this context, we introduce our software publication named AmoePy, an open-source Python package to segment, analyze, and simulate amoeboid cell motility. Furthermore, we describe measures to improve its usability and extensibility, e.g., by detailed run instructions and an automatically generated source code documentation, and to ensure its functionality and stability, e.g., by automatic software tests, data validation, and a hierarchical package structure. The mathematical approaches of this work provide substantial improvements regarding the modeling and analysis of amoeboid cell motility. We deem the above methods, due to their generalized nature, to be of greater value for other scientific applications, e.g., varying organisms and experimental setups or the transition from unicellular to multicellular movement. Furthermore, we enable other researchers from different fields, i.e., mathematics, biophysics, and medicine, to apply our mathematical methods. By following Open Science standards, this work is of greater value for the cell migration community and a potential role model for other Open Science contributions.}, 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} } @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} } @misc{PereraBoeckmann2019, author = {Perera, Upeksha and B{\"o}ckmann, Christine}, title = {Solutions of direct and inverse even-order Sturm-Liouville problems using Magnus expansion}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1336}, issn = {1866-8372}, doi = {10.25932/publishup-47341}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-473414}, pages = {24}, year = {2019}, abstract = {In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm-Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.}, language = {en} } @misc{PornsawadSapsakulBoeckmann2019, author = {Pornsawad, Pornsarp and Sapsakul, Nantawan and B{\"o}ckmann, Christine}, title = {A modified asymptotical regularization of nonlinear ill-posed problems}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1335}, issn = {1866-8372}, doi = {10.25932/publishup-47343}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-473433}, pages = {19}, year = {2019}, abstract = {In this paper, we investigate the continuous version of modified iterative Runge-Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))-𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.}, language = {en} }