TY - THES A1 - Günther, Claudia-Susanne T1 - Das Eigene und das Fremde BT - eine Untersuchung zum Fremdverstehen von Lehrkräften im Mathematikunterricht N2 - 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. KW - Fremdverstehen KW - Alfred Schütz KW - Mathematikunterricht KW - rekonstruktive Fallanalyse Y1 - 2023 ER - TY - THES A1 - Dahl, Dorothee Sophie T1 - Zahlen in den Fingern T1 - Numbers and fingers BT - eine Analyse des Lernspiels Fingu in Bezug auf den frühkindlichen Zahlerwerb im Rahmen der Artifact-Centric Activity Theory BT - an analysis of the learning game Fingu in relation to early numeracy acquisition within the framework of the artifact-centric activity theory N2 - Die Debatte über den Einsatz von digitalen Werkzeugen in der mathematischen Frühförderung ist hoch aktuell. Lernspiele werden konstruiert, mit dem Ziel, mathematisches, informelles Wissen aufzubauen und so einen besseren Schulstart zu ermöglichen. Doch allein die digitale und spielerische Aufarbeitung führt nicht zwingend zu einem Lernerfolg. Daher ist es umso wichtiger, die konkrete Implementation der theoretischen Konstrukte und Interaktionsmöglichkeiten mit den Werkzeugen zu analysieren und passend aufzubereiten. In dieser Masterarbeit wird dazu exemplarisch ein mathematisches Lernspiel namens „Fingu“ für den Einsatz im vorschulischen Bereich theoretisch und empirisch im Rahmen der Artifact-Centric Activity Theory (ACAT) untersucht. Dazu werden zunächst die theoretischen Hintergründe zum Zahlensinn, Zahlbegriffserwerb, Teil-Ganze-Verständnis, der Anzahlwahrnehmung und -bestimmung, den Anzahlvergleichen und der Anzahldarstellung mithilfe von Fingern gemäß der Embodied Cognition sowie der Verwendung von digitalen Werkzeugen und Multi-Touch-Geräten umfassend beschrieben. Anschließend wird die App Fingu erklärt und dann theoretisch entlang des ACAT-Review-Guides analysiert. Zuletzt wird die selbstständig durchgeführte Studie mit zehn Vorschulkindern erläutert und darauf aufbauend Verbesserungs- und Entwicklungsmöglichkeiten der App auf wissenschaftlicher Grundlage beigetragen. Für Fingu lässt sich abschließend festhalten, dass viele Prozesse wie die (Quasi-)Simultanerfassung oder das Zählen gefördert werden können, für andere wie das Teil-Ganze-Verständnis aber noch Anpassungen und/oder die Begleitung durch Erwachsene nötig ist. N2 - The current debate about the use of digital tools in early mathematical education has a lot of relevance these days. Educational games are designed with the aim of building mathematical informal knowledge and thus enabling a better start to school. But digital and playful implementation alone does not necessarily lead to learning. Therefore, it is important to analyze the media in detail and with regard to the theoretical constructs. In this master's thesis, a mathematical learning game called “Fingu” for preschool children is analyzed theoretically and empirically within the framework of the Artifact-Centric Activity Theory (ACAT). First, the theoretical background is described, that is the number sense, number concept acquisition, part-whole understanding, number perception and determination, number comparisons and number representation using fingers according to embodied cognition as well as the use of digital tools and multi-touch. The app itself is explained and then analyzed theoretically using the ACAT review guide. Finally, the conducted study with ten preschool children is presented. Based on those results and the scientific basis, possible improvements and development of the app are explained. For Fingu, it can be concluded that many processes such as perceptual or conceptual subitizing or counting can be improved, but for others such as part-whole understanding, adjustments and/or adult support are still necessary. KW - Zahlerwerb KW - Frühförderung KW - Lernspiele KW - Videostudie KW - ACAT KW - number KW - part-whole concept Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-607629 ER - TY - THES A1 - Gehring, Penelope T1 - Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary T1 - Nicht-lokale Randbedingungen für den spinorialen Dirac-Operator auf Raumzeiten mit zeitartigen Rand N2 - 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, Bär 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 Bär 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. N2 - Über nicht-lokale Randbedingungen – zum Beispiel dieAtiyah–Patodi–Singer (APS)-Bedingungen – für Dirac Operatoren auf Riemannschen Mannigfaltigkeiten ist recht viel bekannt, während für die hyperbolischen Dirac Operatoren auf Lorentz-Mannigfaltigkeiten dies noch nicht der Fall ist. Kürzlich haben Bär und Strohmaier [15] und Drago, Große und Murro [27] APS-ähnliche Bedingungen für den Spin Dirac Operator auf Lorentz-Mannigfaltigkeiten mit raumartigen bzw. zeitartigen Rand eingeführt. Während Bär und Strohmaier [15] zeigten, dass der Dirac Operator mit diesen Randbedingungen Fredholm ist, bewiesen Drago, Große und Murro [27] die Wohlgestelltheit des entsprechenden Anfangsrandwertproblems unter bestimmten geometrischen Annahmen. In dieser Arbeit werden wir in die Fußstapfen der letztgenannten Autoren treten und diskutieren, ob die APS-ähnlichen Bedingungen für Dirac Operatoren auf Lorentz-Mannigfaltigkeiten mit zeitartigen Rand durch allgemeinere Bedingungen ersetzt werden können, sodass die zugehörigen Anfangsrandwertprobleme immer noch wohlgestellt sind. Wir betrachten Randbedingungen, die in der Zeit lokal und in den Raumrichtungen nicht-lokal sind. Genauer gesagt verwenden wir die Raumzeitblätterung, die sich aus der Cauchy Zeitfunktion ergibt, und spalten den Dirac Operator entlang dieser Foliation auf. Daraus ergibt sich eine Familie elliptischer Operatoren, die jeweils auf Spinoren des Spinbündels über den entsprechenden Zeitschnitt wirken. Die Theorie der elliptischen Operatoren stellt dann sicher, dass wir Familien von nichtlokalen Randbedingungen bezüglich dieser Familie von Operatoren finden können. Im weiteren Verlauf verwenden wir solche Familien von Randbedingungen, um eine Lorentzsche Randbedingung auf dem gesamten zeitartigen Rand zu definieren. Durch das Analysieren der Lorentzschen Randbedingungen finden wir dann hinreichende Bedingungen für die Familie der nicht-lokalen Randbedingungen, die zur Wohlgestelltheit der entsprechenden Cauchy-Probleme führen. Die Wohlgestelltheit selbst wird dann mit Hilfe klassischer Methoden bewiesen, einschließlich Energieabschätzungen und Annäherung durch Lösungen der regularisierten Probleme. Außerdem verwenden wir diese Theorie, um explizite Randbedingungen für den Lorentzschen Dirac Operator zu konstruieren. Genauer gesagt werden wir zwei Beispiele für Randbedingungen diskutieren - das Analogon der Atiyah-Patodi-Singer- bzw. Chiralitäts-Bedingungen für unseren Fall. Dazu werden wir uns die Theorie der nicht-lokalen Randbedingungen für elliptische Operatoren genauer ansehen und die Anforderungen an die Familie der nicht-lokalen Randbedingungen für diese Beispiele analysieren. KW - Dirac operator KW - Diracoperator KW - spacetimes with timelike boundary KW - Raumzeiten mit zeitartigen Rand KW - boundary conditions KW - Randbedingungen KW - initial boundary value problem KW - Anfangsrandwertproblem Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-577755 ER - TY - THES A1 - Lopez Valencia, Diego Andres T1 - The Milnor-Moore and Poincaré-Birkhoff-Witt theorems in the locality set up and the polar structure of Shintani zeta functions T1 - Die Milnor-Moore und Poincaré-Birkhoff-Witt Theoreme in der Lokalität und die polare Struktur der Shintani-Zeta-Abbildungen N2 - 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 Poincaré-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 Poincaré-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 Poincaré-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. N2 - Diese Arbeit schlägt eine Brücke zwischen zwei Bereichen der Mathematik, einerseits der Algebra mit dem Milnor-Moore-Theorem (auch Cartier-Quillen-Milnor-Moore-Theorem genannt) sowie dem Poincaré-Birkhoff-Witt-Theorem und andererseits der Analysis mit den Shintani-Zetafunktionen, die eine Verallgemeinerung der Mehrfach-Zetafunktionen darstellen. Der erste Teil ist einer algebraischen Formulierung des Lokalitätsprinzips in der Physik und Verallgemeinerungen von Klassifikationstheoremen wie dem Milnor-Moore- und dem Poincaré-Birkhoff-Witt-Theorem auf den Lokalitätsrahmen gewidmet. Das Lokalitätsprinzip besagt grob, dass Ereignisse, die in der Raumzeit weit voneinander entfernt stattfinden, sich nicht gegenseitig beeinflussen. Die hier erörterte algebraische Formulierung dieses Prinzips ist nützlich bei der Analyse von Singularitäten, die aus weit voneinander entfernten Ereignissen im Raum entstehen, um sie zu renormalisieren und dabei die Tatsache im Gedächtnis zu behalten, dass sie sich nicht gegenseitig beeinflussen. Wir beginnen damit, dass wir einen Vektorraum mit einer symmetrischen Relation, der so genannten Lokalitätsrelation, ausstatten, die die "lokal unabhängigen" Elemente festhält. Das Paar aus einem Vektorraum und einer solchen Relation wird als Vorlokalitäts-Vektorraum bezeichnet. Dieses Konzept wird auf Tensorprodukte erweitert, die nur Tensoren aus lokal unabhängigen Elementen zulassen. Wir erweitern dieses Konzept auf die Lokalitäts-Tensor-Algebra und die symmetrische Lokalitäts-Algebra eines Vorlokalitäts-Vektorraums und beweisen die universellen Eigenschaften jeder dieser Strukturen. Wir führen auch die Vorlokalitäts-Lie-Algebren zusammen mit den zugehörigen universellen Hüllalgebren der Lokalität ein und beweisen ihre universelle Eigenschaft. Später übertragen wir alle diese Strukturen und Ergebnisse aus dem Kontext der Vorlokalität in den Kontext der Lokalität, wobei die Lokalitätsbeziehung mit der linearen Struktur des Vektorraums kompatibel sein muss. Auf diese Weise können wir Lokalitäts-Kohlengebren, Lokalitäts-Bialgebren und Lokalitäts-Hopf-Algebren definieren. Schließlich werden alle vorherigen Ergebnisse verwendet, um die Lokalitätsversionen des Milnor-Moore- und des Poincaré-Birkhoff-Witt-Theorems zu beweisen. Es ist erwähnenswert, dass die vorgestellten Beweise nicht nur die Ergebnisse im üblichen (Nichtlokalitäts-) Aufbau verallgemeinern, sondern auch oft weniger Hilfsmittel verwenden als ihre Gegenstücke in ihren Nichtlokalitäts-Gegenstücken. Der zweite Teil ist der Untersuchung der polaren Struktur der Shintani-Zeta-Funktionen gewidmet. Diese Funktionen, die u.a. die Riemman-Zetafunktion, die multiplen Zetafunktionen und die Mordell-Tornheim-Zetafunktionen verallgemeinern, werden durch Matrizen mit reellen, nicht-negativen Argumenten parametrisiert. Es ist bekannt, dass Shintani-Zetafunktionen sich zu meromorphen Funktionen mit Polen auf affinen Hyperebenen erweitern. Wir verfeinern dieses Ergebnis, indem wir zeigen, dass die Pole auf Hyperebenen liegen, die parallel zu den Facetten bestimmter konvexer Polyeder verlaufen, die mit der Definitionsmatrix für die Shintani-Zeta-Funktion assoziiert sind. Letztere sind explizit die Newton-Polytope der Polynome, die durch die Spalten der zugrunde liegenden Matrix induziert werden. Wir beweisen dann, dass die Koeffizienten der Gleichung, die die Hyperebenen in der kanonischen Basis beschreibt, entweder Null oder Eins sind, ähnlich wie die Pole, die bei der Renormierung generischer Feynman-Amplituden entstehen. Zu diesem Zweck führen wir einen Algorithmus ein, um die Gewichte über einen Graphen so zu verteilen, dass das Gewicht an jedem Knoten eine gegebene untere Schranke erfüllt. KW - locality principle KW - multizeta functions KW - meromorphic continuation KW - Milnor Moore theorem KW - Poincaré Birkhoff Witt theorem KW - Newton polytopes KW - Satz von Milnor Moore KW - Newton Polytope KW - Satz von Poincaré Birkhoff Witt KW - Lokalitätsprinzip KW - meromorphe Fortsetzung KW - Multizeta-Abbildungen Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-594213 ER - TY - THES A1 - Schindler, Daniel T1 - Mathematical modeling and simulation of protrusion-driven cell dynamics T1 - Mathematische Modellierung und Simulation von amöboiden Zelldynamiken N2 - 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. N2 - Amöboide Zellmotilität findet bei einer Vielzahl biomedizinischer Prozesse wie Krebsmetastasierung, embryonaler Morphogenese und Wundheilung statt. Im Gegensatz zu anderen Formen der Zellmotilität wird sie hauptsächlich durch erhebliche Formveränderungen der Zelle angetrieben. Sie beruht auf dem Zusammenspiel von explorativen Membranausstülpungen an der Vorderseite und einem langsamer wirkenden Membraneinzug an der Rückseite. Die Komplexität amöboider Zellmotilität machen neue Berechnungsmodelle erforderlich, um die Dynamik der Zellform mathematisch fundiert zu analysieren und zu simulieren. Ziel dieser Arbeit ist die Entwicklung (i) eines mathematischen Frameworks zur Beschreibung der Konturendynamik in Zeit und Raum, (ii) eines Computermodells, um Eigenschaften der Membranveränderungen von einzelnen Zellen zu inferieren und gleichzeitig realistische Konturdynamiken zu simulieren, (iii) und eines ergänzenden Open-Science-Ansatzes, um die oben genannten Methoden vollständig zugänglich und leicht anwendbar zu machen. Auf der Grundlage von aufeinander folgenden Mikroskopiebildern vom Modellorganismus Dictyostelium discoideum, wenden wir einen Bayesschen Ansatz an, um glatte Darstellungen der Zellmembran, sogenannte Zellkonturen, zu erhalten. Wir führen eine einparametrige Familie von regularisierten Konturflüssen ein, um Referenzpunkte auf der Kontur (virtuelle Marker) in Zeit und Raum zu verfolgen. Auf diese Weise definieren wir ein Koordinatensystem zur Visualisierung lokaler geometrischer und dynamischer Größen der individuellen Konturdynamiken in sogenannten Kymographen-Plots. Insbesondere führen wir die lokale Marker-Dispersion ein, mit der signifikante Membranveränderungen identifiziert werden können. Dieses mathematische Framework bildet die Grundlage für unser neues Modell zur Beschreibung von Konturendynamiken. Es besteht aus drei biophysiologisch motivierten Komponenten: einem stochastischen Term, der die Membranausstülpungen steuert, und zwei deterministischen Termen, die das Membraneinziehen, unter Berücksichtigung der Konturform und -fläche, steuern. Unser Modell bietet einen vollautomatisierten Ansatz zur Inferrenz der Charakteristiken von Membranveränderungen für experimentelle Zelldaten. Außerdem ermöglicht es die Simulation von realistischen und qualitativ unterschiedlichen Konturendynamiken. Mit dem ergänzenden Open-Science-Ansatz setzen wir einen hohen Standard hinsichtlich der Nutzbarkeit unserer Methoden und der Reproduzierbarkeit unserer Forschung. In diesem Kontext stellen wir die Softwarepublikation AmoePy vor, ein Open-Source-Pythonpaket zur Segmentierung, Analyse und Simulation von amöboider Zellmotilität. Darüber hinaus beschreiben wir Maßnahmen zur Verbesserung der Benutzerfreundlichkeit und Erweiterbarkeit, z. B. durch detaillierte Ausführanweisungen und eine automatisch generierte Quellcodedokumentation, und zur Gewährleistung der Funktionalität und Stabilität, z. B. durch automatische Softwaretests, Datenvalidierung und eine hierarchische Paketstruktur. Die mathematischen Methoden dieser Arbeit stellen wesentliche Verbesserungen in der Modellierung und Analyse der amöboiden Zellmotilität dar. Wir sind der Ansicht, dass die oben genannten Methoden aufgrund ihrer Verallgemeinerbarkeit von größerem Wert für andere wissenschaftliche Anwendungen sind und potentiell einsetzbar in verschiedenen Wissenschaftsfeldern sind, u. a. Mathematik, Biophysik und Medizin. Durch die Einhaltung von Open-Science-Standards ist diese Arbeit von größerem Wert und ein potenzielles Vorbild für andere Open-Science-Beiträge. KW - amöboide Bewegung KW - Zellmotilität KW - mathematische Modellierung KW - offene Wissenschaft KW - amoeboid motion KW - cell motility KW - mathematical modeling KW - open science Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-613275 ER - TY - THES A1 - Malem-Shinitski, Noa T1 - Bayesian inference and modeling for point processes with applications from neuronal activity to scene viewing T1 - Bayessche Inferenz und Modellierung für Punktprozesse mit Anwendungen von neuronaler Aktivität bis Szenenbetrachtung N2 - 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. N2 - Punktprozesse sind eine gängige Methode zur Modellierung von Ereignismengen. Von Erdbeben bis zu Social-Media-Posts, von den neuronalen Spikes bis zum Zeitpunkt von Verbrechen, von Aktienkursen bis zur Ausbreitung von Krankheiten - diese Phänomene lassen sich auf das Auftreten von Ereignissen reduzieren, die in Punkten konzentriert sind. Häufig treten diese Ereignisse nacheinander auf und bilden eine Zeitreihe. Modelle von Punktprozessen können verwendet werden, um unser Verständnis solcher Ereignisse für Klassifizierung und Vorhersage zu vertiefen. Solche Modelle umfassen einen zugrunde liegenden Zufallsprozess, der die Ereignisse erzeugt. In dieser Arbeit wird die Bayes'sche Methodik verwendet, um den zugrunde liegenden generativen Prozess aus den beobachteten Daten abzuleiten. Wir leisten einen doppelten Beitrag: Wir entwickeln neue Modelle und neue Inferenzmethoden für diese Prozesse. Wir schlagen ein Modell vor, das die Familie der Punktprozesse erweitert, bei denen das Auftreten eines Ereignisses von den vorherigen Ereignissen abhängt. Diese Familie ist als Hawkes-Prozesse bekannt. Während in den meisten bestehenden Modellen solcher Prozesse davon ausgegangen wird, dass vergangene Ereignisse nur eine exzitatorische Wirkung auf zukünftige Ereignisse haben, konzentrieren wir uns auf den neu entwickelten nichtlinearen Hawkes-Prozess, bei dem vergangene Ereignisse exzitatorische und hemmende Wirkungen haben können. Nach der Definition des Modells stellen wir seine Inferenzmethode vor und wenden sie auf Daten aus verschiedenen Bereichen an, unter anderem auf die neuronale Aktivität. Das zweite Modell, das in dieser Arbeit beschrieben wird, betrifft einen speziellen Fall von Punktprozessen - den Entscheidungsprozess, der der menschlichen Blicksteuerung zugrunde liegt. Dieser Prozess führt zu einer Reihe von fixierten Positionen in einem Bild. Wir haben ein neues Modell entwickelt, um diesen Prozess zu beschreiben, motiviert durch das bekannte Exploration-Exploitation-Dilemma. Neben dem Modell stellen wir einen Bayes'schen Inferenzalgorithmus vor, um die Modellparameter abzuleiten. Wir bleiben auf dem Gebiet der menschlichen Szenenbetrachtung und stellen fest, dass es in diesem Bereich keine bewährten Verfahren für die Bayes'sche Inferenz gibt. Wir geben einen Überblick über vier gängige Algorithmen und vergleichen ihre Leistungen bei der Ableitung von Parametern für zwei Scanpfadmodelle. Die in dieser Dissertation vorgestellten neuen Modelle und Inferenzalgorithmen bereichern das Verständnis von Punktprozessdaten und ermöglichen es uns, sinnvolle Erkenntnisse zu gewinnen. KW - Bayesian inference KW - point process KW - statistical machine learning KW - sampling KW - modeling KW - Bayessche Inferenz KW - Modellierung KW - Punktprozess KW - Stichprobenentnahme aus einem statistischen Modell KW - statistisches maschinelles Lernen Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-614952 ER - TY - GEN A1 - Ehlen, Tobias A1 - Flöge, Annie A1 - Göbel, Franziska A1 - Keller, Peter A1 - Rœlly, Sylvie ED - Keller, Peter ED - Rœlly, Sylvie T1 - Übungsbuch zur Stochastik BT - Aufgaben und Lösungen ; Grundlegende Konzepte und Anwendungen N2 - Dieses Buch stellt Übungen zu den Grundbegriffen und Grundsätzen der Stochastik und ihre Lösungen zur Verfügung. So wie man Tonleitern in der Musik trainiert, so berechnet man Übungsaufgaben in der Mathematik. In diesem Sinne soll dieses Übungsbuch vor allem als Vorlage dienen für das eigenständige, eigenverantwortliche Lernen und Üben. Die Schönheit und Einzigartigkeit der Wahrscheinlichkeitstheorie besteht darin, dass sie eine Vielzahl von realen Phänomenen modellieren kann. Daher findet man hier Aufgaben mit Verbindungen zur Geometrie, zu Glücksspielen, zur Versicherungsmathematik, zur Demographie und vielen anderen Themen. N2 - This book provides exercises on the basic concepts and principles of stochastics and their solutions. Just as one trains scales in music, one calculates exercises in mathematics. In this sense, this exercise book is primarily intended to serve as a template for independent learning and practice. The beauty and uniqueness of probability theory is that it can model a variety of real phenomena. Therefore, one can find exercises with connections to geometry, gambling, actuarial mathematics, demography and many other topics. KW - Aufgabensammlung KW - Wahrscheinlichkeitstheorie KW - Stochastik KW - Wahrscheinlichkeitsverteilung KW - Zufallsvariable KW - Grenzwertsatz KW - Konfidenzintervall KW - exercise collection KW - probability theory KW - stochastics KW - probability distribution KW - random variable KW - limit theorem KW - confidence interval Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-595939 SN - 978-3-86956-563-7 PB - Universitätsverlag Potsdam CY - Potsdam ER - 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 - THES A1 - Hain, Tobias Martin T1 - Structure formation and identification in geometrically driven soft matter systems T1 - Strukturbildung und Identifikation in geometrisch getriebenen weiche Materie-Systemen N2 - Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces. Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system's constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness. In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system "forced" to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate. We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase. The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy. N2 - Die Unterteilung eines Raums durch Grenzflächen führt zu Raumaufteilungen, die für die Selbstorganisation weicher Materie relevant sind. Bekannte Beispiele sind zelluläre Medien, wie z.B. Seifenschaum, der aus Luftblasen besteht, getrennt durch Wände aus Wasser und Seife, und komplexere Partitionen, wie sie z.B. durch bikontinuierliche Minimalflächen erzeugt werden. In dieser Arbeit werden mit Hilfe von Computersimulationen Systeme weicher Materie in Bezug auf den Zusammenhang zwischen dem im System vorherrschenden, physikalischen Kräften und der Struktur der resultierenden Grenzflächen oder Partitionen untersucht. Der Schwerpunkt liegt hierbei auf zwei Systemen, eine Copolymerschmelze und das sogenannte Quantizer Problem, bei denen der treibende Faktor der Strukturbildung, nämlich die Minimierung der freien Energie, aus einem Zusammenspiel der Minimierung der Oberfläche der Grenzflächen und der gleichzeitigen Minimierung der Elastizitätsenergie besteht. Unter diesen Gegebenheiten bevorzugen solche Systeme Zellen gleichmäßiger Größe. Im ersten Teil der Arbeit befassen wir uns mit der Bildung von scharfen Grenzflächen in Systemen von Copolymeren. Wir analysieren die zylindrische Phase eines Copolymersystems, das gezwungen wird, sich auf einer kugelförmigen Oberfläche zu organisieren. Die Topologie dieser Oberfläche erlaubt es der optimalen Konfiguration, dem Sechseckgitter, nicht, sich zu bilden. Für dreiarmige Copolymere wird die entstehende Struktur durch Lösungen des sogenannten Thomson Problems beschrieben. Letzteres sucht nach der Konfigurationen von abstoßenden Ladungen auf einer Kugeloberfläche mit minimaler Energie. Auf einem Substrat haben wir eine Kombination aus drei ineinandergreifende Lösungen des Thomson Problems gefunden, wobei der Typ der Lösungen statistisch von dem Radius des Substrates abhängt. Anschließend untersuchen wir die Bildung von amorphen und kristallinen Strukturen im Quantizersystem, einem teilchenbasierenden Modell, dessen Energiefunktional keine Oberflächenspannung enthält und möglichst kugelförmige Zellen gleicher Größe begünstigt. Wird das System quasistatisch im thermodynamischen Gleichgewicht abgekühlt, kristallisiert das Quantizersystem in den geordneten BCC Grundzustand. Wird das System allerdings zu schnell abgekühlt, sodass es sich nicht mehr im thermodynamischen Gleichgewicht befindet, bildet sich eine amorphe, annähernd hyperuniforme Struktur aus. Wir konnten zeigen, dass diese Struktur bemerkenswert unabhängig von den Ausganszuständen, sowie der Art der Energieminimierung zu sein scheint. Im Ausblick erweitern wir das Quantizersystem, indem wir Oberflächenspannung einführen. Unsere Ergebnisse deutet darauf hin, dass dieses so erweiterte Modell Mizellenphasen in Polymersystem modellieren kann. Wir beobachten einen Phasenübergang von einer ungeordneten, flüssigen Phase hin zu einer festen Frank-Caspar-Phase. Der zweite Teil der Arbeit behandelt bikontinuierliche Grenzflächen, die den Raum in zwei netzwerkartige Domänen aufteilen. Wir führen eine Open-Source Software ein, das die Identifizierung von Strukturen anhand derer Mikroskopaufnahmen ermöglicht. Hierzu erweitern und verbessern wir eine Methode, die durch den Abgleich experimentell zugänglicher Projektionen in Mikroskopaufnahmen mit berechneten Projektionen potenzieller Strukturen basiert. Dieses Verfahren wurde erstmal von Deng und Mieczkowski (1998) eingeführt. Die simulierten Strukturen basieren auf einer Nodalflächenmodellierung von dreifach-periodischen Flächen konstanter mittlerer Krümmung. Wir führen eine Fallstudie an Zellmembranen von Etioplasten, den Vorläufern von Chloroplasten, durch. Wir konnten die Struktur dieser Etioplasten als die Diamond-Struktur identifizieren. Als Ausblick automatisieren wir den Identifizierungsproyess mit Hilfe von Deep-Learning-Methoden. Erste Ergebnisse zeigen, dass mit diesem Ansatz die Identifizierung von Strukturen mit ausgezeichneter Genauigkeit gelingt. KW - soft matter KW - geometry KW - self-assembly KW - structure formation KW - quantizer KW - polymer KW - Geometrie KW - Polymere KW - Quantizer KW - Selbstassemblierung KW - weiche Materie KW - Strukturbildung Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-558808 N1 - Parts of this publication are reproduced with permission from the Royal Society of Chemistry and AIP Publishing. 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 - THES A1 - Mauerberger, Stefan T1 - Correlation based Bayesian modeling T1 - Korrelationsbasierte Bayesianische Modellierung BT - with applications in travel time tomography, seismic source inversion and magnetic field modeling BT - mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung N2 - The motivation for this work was the question of reliability and robustness of seismic tomography. The problem is that many earth models exist which can describe the underlying ground motion records equally well. Most algorithms for reconstructing earth models provide a solution, but rarely quantify their variability. If there is no way to verify the imaged structures, an interpretation is hardly reliable. The initial idea was to explore the space of equivalent earth models using Bayesian inference. However, it quickly became apparent that the rigorous quantification of tomographic uncertainties could not be accomplished within the scope of a dissertation. In order to maintain the fundamental concept of statistical inference, less complex problems from the geosciences are treated instead. This dissertation aims to anchor Bayesian inference more deeply in the geosciences and to transfer knowledge from applied mathematics. The underlying idea is to use well-known methods and techniques from statistics to quantify the uncertainties of inverse problems in the geosciences. This work is divided into three parts: Part I introduces the necessary mathematics and should be understood as a kind of toolbox. With a physical application in mind, this section provides a compact summary of all methods and techniques used. The introduction of Bayesian inference makes the beginning. Then, as a special case, the focus is on regression with Gaussian processes under linear transformations. The chapters on the derivation of covariance functions and the approximation of non-linearities are discussed in more detail. Part II presents two proof of concept studies in the field of seismology. The aim is to present the conceptual application of the introduced methods and techniques with moderate complexity. The example about traveltime tomography applies the approximation of non-linear relationships. The derivation of a covariance function using the wave equation is shown in the example of a damped vibrating string. With these two synthetic applications, a consistent concept for the quantification of modeling uncertainties has been developed. Part III presents the reconstruction of the Earth's archeomagnetic field. This application uses the whole toolbox presented in Part I and is correspondingly complex. The modeling of the past 1000 years is based on real data and reliably quantifies the spatial modeling uncertainties. The statistical model presented is widely used and is under active development. The three applications mentioned are intentionally kept flexible to allow transferability to similar problems. The entire work focuses on the non-uniqueness of inverse problems in the geosciences. It is intended to be of relevance to those interested in the concepts of Bayesian inference. N2 - Die Motivation für diese Arbeit war die Frage nach Verlässlichkeit und Belastbarkeit der seismischen Tomographie. Das Problem besteht darin, dass sehr viele Erdmodelle existieren welche die zugrundeliegenden seismischen Aufzeichnungen gleich gut beschreiben können. Die meisten Algorithmen zur Rekonstruktion von Erdmodellen liefern zwar eine Lösung, quantifizierten jedoch kaum deren Variabilität. Wenn es keine Möglichkeit gibt die abgebildeten Strukturen zu verifizieren, so ist eine Interpretation kaum verlässlich. Der ursprüngliche Gedanke war den Raum äquivalenter Erdmodelle mithilfe Bayesianische Inferenz zu erkunden. Es stellte sich jedoch schnell heraus, dass die vollständige Quantifizierung tomographischer Unsicherheiten im Rahmen einer Promotion nicht zu bewältigen ist. Um das wesentliche Konzept der statistischen Inferenz beizubehalten werden stattdessen weniger komplexe Problemstellungen aus den Geowissenschaften behandelt. Diese Dissertation hat das Ziel die Bayesianische Inferenz tiefer in den Geowissenschaften zu verankern und Wissen aus der angewandten Mathematik zu transferieren. Die zugrundeliegende Idee besteht darin auf bekannte Methoden und Techniken der Statistik zurückzugreifen um die Unsicherheiten inverser Probleme in den Geowissenschaften zu quantifizieren. Diese Arbeit gliedert sich in drei Teile: Teil I führt die notwendige Mathematik ein und soll als eine Art Werkzeugkasten verstanden werden. In Hinblick auf eine physikalische Anwendung bietet dieser Abschnitt eine kompakte Zusammenfassung aller eingesetzter Methoden und Techniken. Den Anfang macht die Einführung der Bayesianische Inferenz. Danach steht als Spezialfall die Regression mit Gauß-Prozessen unter linearen Transformationen im Vordergrund. Die Kapitel zur Herleitung von Kovarianzfunktionen und die Approximation von Nichtlinearitäten gehen etwas weiter in die Tiefe. Teil II präsentiert zwei Konzeptstudien aus dem Bereich der Seismologie. Ziel ist es bei moderater Komplexität die prinzipielle Anwendung der eingeführten Methoden und Techniken zu präsentieren. Das Beispiel zur Laufzeittomographie wendet die Näherungs\-methoden für nichtlineare Zusammenhänge an. Die Herleitung einer Kovarianzfunktion mithilfe der Wellengleichung ist am Beispiel der gedämpften Saitenschwingung gezeigt. Mit diesen beiden synthetischen Anwendungen wurde ein konsistentes Konzept zur Quantifizierung von Modellierungsunsicherheiten erarbeitet. Teil III präsentiert die Rekonstruktion des archeomagnetischen Feldes unserer Erde. Diese Anwendung nutzt den gesamten Werkzeugkasten aus Teil I und ist entsprechend umfangreich. Die Modellierung der vergangenen 1000 Jahre basiert auf echten Daten und quantifiziert zuverlässig die räumlichen Modellierungsunsicherheiten. Das präsentierte statistische Modell findet breite Anwendung und wird aktiv weiter entwickelt. Die drei genannten Anwendungen sind bewusst flexibel gehalten um die Übertragbarkeit auf ähnliche Problemstellungen zu ermöglichen. Die gesamte Arbeit legt den Fokus auf die nicht-Eindeutigkeit inverser Probleme in den Geowissenschaften. Sie will für all Jene von Relevanz sein, die sich für die Konzepte der Bayesianischen Inferenz interessieren. KW - statistical inference KW - Bayesian inversion KW - travel time tomography KW - seismic source inversion KW - magnetic field modeling KW - mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung KW - Magnetfeldmodellierung KW - seismische Quellinversion KW - statistische Inferenz KW - Laufzeittomographie Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-537827 ER - TY - THES A1 - Fischer, Jens Walter T1 - Random dynamics in collective behavior - consensus, clustering & extinction of populations T1 - Stochastische Dynamiken in kollektivem Verhalten: Konsens, Gruppenbildung, Aussterben von Populationen N2 - The echo chamber model describes the development of groups in heterogeneous social networks. By heterogeneous social network we mean a set of individuals, each of whom represents exactly one opinion. The existing relationships between individuals can then be represented by a graph. The echo chamber model is a time-discrete model which, like a board game, is played in rounds. In each round, an existing relationship is randomly and uniformly selected from the network and the two connected individuals interact. If the opinions of the individuals involved are sufficiently similar, they continue to move closer together in their opinions, whereas in the case of opinions that are too far apart, they break off their relationship and one of the individuals seeks a new relationship. In this paper we examine the building blocks of this model. We start from the observation that changes in the structure of relationships in the network can be described by a system of interacting particles in a more abstract space. These reflections lead to the definition of a new abstract graph that encompasses all possible relational configurations of the social network. This provides us with the geometric understanding necessary to analyse the dynamic components of the echo chamber model in Part III. As a first step, in Part 7, we leave aside the opinions of the inidividuals and assume that the position of the edges changes with each move as described above, in order to obtain a basic understanding of the underlying dynamics. Using Markov chain theory, we find upper bounds on the speed of convergence of an associated Markov chain to its unique stationary distribution and show that there are mutually identifiable networks that are not apparent in the dynamics under analysis, in the sense that the stationary distribution of the associated Markov chain gives equal weight to these networks. In the reversible cases, we focus in particular on the explicit form of the stationary distribution as well as on the lower bounds of the Cheeger constant to describe the convergence speed. The final result of Section 8, based on absorbing Markov chains, shows that in a reduced version of the echo chamber model, a hierarchical structure of the number of conflicting relations can be identified. We can use this structure to determine an upper bound on the expected absorption time, using a quasi-stationary distribution. This hierarchy of structure also provides a bridge to classical theories of pure death processes. We conclude by showing how future research can exploit this link and by discussing the importance of the results as building blocks for a full theoretical understanding of the echo chamber model. Finally, Part IV presents a published paper on the birth-death process with partial catastrophe. The paper is based on the explicit calculation of the first moment of a catastrophe. This first part is entirely based on an analytical approach to second degree recurrences with linear coefficients. The convergence to 0 of the resulting sequence as well as the speed of convergence are proved. On the other hand, the determination of the upper bounds of the expected value of the population size as well as its variance and the difference between the determined upper bound and the actual value of the expected value. For these results we use almost exclusively the theory of ordinary nonlinear differential equations. N2 - Beziehungen und damit Interaktion sowie Diskussion, aber auch Konflikt und Opposition bilden die Grundbausteine einer jeden Gesellschaft. Häufig wird Kommunikation als der übergreigende Begriff zur Beschreibung interner Strukturen einer Gesellschaft identifiziert. Dabei muss es sich aber nicht um eine Gesellschaft im Sinne von Nationen handeln, sondern kann auch schlicht eine Gruppe von Menschen umfassen, die miteinander strukturiert interagieren, beispielsweise, eine Gruppe von Angestellten, die an einem gemeinsamen Projekt arbeiten, oder die Mitglieder eines sozialen Netzwerks. In dieser Arbeit befassen wir uns mit der mathematischen Beschreibung solcher Prozesse innerhalb von Gruppen und Gesellschaften und legen dabei unseren Fokus auf die Bildung eines Konsens durch Interaktion aber auch die Konsequenzen von Konflikt und das potentielle Aussterben einer Population. Dabei werden zwei Modelle im Fokus des Interesses stehen: Das Echokammer Model sowie eine Erweiterung des Geburts-Todes Prozesses, die die Möglichkeit eines radikalen Abfalls der Populationsgr öße miteinschließt. Wir beginnen mit einer Einführung in Part I und teilen die verbleibende Arbeit in drei Teile auf, wobei sich die ersten beiden technischen Abschnitte, Part II und III, mit einer ausführlichen Analyse der Bausteine des Echokammer Models befassen und im dritten Abschnitt, in Part IV, der erweiterte Geburts- Todes Prozess untersucht wird. Dieser wird im Folgenden als Geburts-Todes Prozess mit teilweiser Katastrophe bezeichnet werden. Das Echokammer Model beschreibt die Entwicklung von Gruppen in zunächst heterogenen sozialen Netzwerken. Unter einem heterogenen sozialen Netzwerk verstehen wir dabei eine Menge von Individuen, von denen jedes exakt eine Meinungen vertritt. Meinungen werden vereinfacht durch Werte in [0, 1] modelliert. Bestehende Beziehungen unter den Individuen können dann durch einen Graphen dargestellt werden. Es handelt sich bei dem Echokammer Modell um ein zeit-diskretes Modell, das entsprechend, ähnlich einem Brettspiel, in Zügen abläuft. In jedem Zug wird zufällig gleichverteilt eine bestehende Beziehung aus dem Netzwerk ausgewählt und die beiden verbundenen Individuen interagieren. Dabei kann es zu zwei verschiedenen Interaktionen kommen. Sind die Meinungen der betroffenen Individuen hinreichend ähnlich, so nähern sie sich weiter in ihren Meinungen an, während sie im Fall von Meinungen, die zu weit von einander liegen, ihre Beziehung auflösen und sich eines der Individuen eine neue Beziehung sucht. 8 In dieser Arbeit untersuchen wir theoretisch die Bausteine dieses Modells. Dabei legen wir die Beobachtung zu Grunde, dass die Veränderungen der Beziehungsstruktur im Netzwerk durch einen System von interagierenden Partikeln auf einem abstrakteren Raum beschrieben werden kann. Dies erlaubt es insbesondere graphentheoretische überlegungen in die Analyse einfließen zu lassen. Diese überlegungen werden ausührlich in Part II diskutiert und führen zur Definition eines neuen, abstrahierten Graphens, der alle möglichen Beziehungskonfigurationen des sozialen Netzwerks umfasst. Dies erlaubt es uns einen ähnlichkeitsbegriff für Beziehungskonfigurationen auf Basis der benachbarten Knoten in besagtem Graphen zu definieren. Dies liefert uns das notwendige geometrische Verständnis um in Part III die dynamischen Komponenten des Echokammer models zu analysieren. Insbesondere fokusieren wir uns dabei auf die Dynamik der Kanten, für die bisher in der Literatur noch keine Ergebnisse existieren. Wir lassen zunächst in Abschnitt 7 die Meinungen der Individuen beiseite und nehmen an, dass die Position der Kanten sich in jedem Zug wie zuvor beschrieben ändert, um eine grundlegendes Verständnis der unterliegenden Dynamik zu erhalten. Unter der Verwendung der Theorie von Markovketten finden wir obere Schranken an die Konvergenzgeschwindigkeit einer assoziierten Markovkette gegen ihre eindeutige stationäre Verteilung und zeigen, dass es Netzwerke gibt, die miteinander identifizierbar und unter der analysierten Dynamik daheingehend ununterscheinbar sind, dass die stationäre Verteilung der assozierten Markovkette diesen Netzwerken dasselbe Gewicht zuordnet. Anschließend beweisen wir eine Reihe von quantitativen Resultaten, die sich insbesondere in Fällen, in denen die assozierte Markovkette reversibel ist, als berechenbar herausstellen. Insbesondere die explizite Form der stationären Verteilung sowie untere Schranken an die Cheeger Konstante zur Beschreibung der Konvergenzgeschwindigkeit stehen dabei im Fokus und werden ausführlich diskutiert. Nach dieser vertieften Analyse des reduzierten Modells, fügen wir die Meinungen unserer Betrachtung wieder hinzu. Das abschließende Result in Abschnitt 8, basierend auf absorbierenden Markovketten, liefert dann, dass in einer reduzierte Version des Echokammer Modells, in dem sich Individuen ähnlicher Meinung nicht annähern, eine hierarchische Struktur der Anzahl der konfliktreichen Beziehung identifiziert werden kann. Dies können wir ausnutzen, um eine obere Schranke an die erwartete Absorptionszeit, unter Zuhilfenahme einer quasi-stationären Verteilung, zu bestimmen. Diese hierarchische Struktur bildet außerdem eine Brücke zu klassischen Theorien von Geburts-Todes und, insbesondere, reinen Todes-Prozessen, für die eine reiche Literatur existiert. Wir zeigen abschließend auf, wie künftige Forschung diese Verbindung ausnutzen kann und diskutieren die Wichtigkeit der Ergbenisse als Bausteine eines vollständigen theoretischen Verständnisses des Echokammer Modells. Part IV stellt abschließend einen veröffentlichten Artikel vor, der sich dem Geburts- Todes Prozess mit teilweiser Katastrophe widmet. Besagter Artikel steht dabei auf zwei Säulen. Zum Einen der expliziten Berechnung des ersten Zeitpunkts einer Katastrophe, wenn die Population zu Beginn der Beobachtung von instabiler Größe ist. KW - Markov chains KW - graph theory KW - complex systems KW - interacting particle systems KW - Markovketten KW - komplexe Systeme KW - Graphentheorie KW - Systeme interagierender Partikel Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-553725 ER - TY - THES A1 - Schanner, Maximilian Arthus T1 - Correlation based modeling of the archeomagnetic field T1 - Korrelationsbasierte Modellierung des archäomagnetischen Feldes N2 - The geomagnetic main field is vital for live on Earth, as it shields our habitat against the solar wind and cosmic rays. It is generated by the geodynamo in the Earth’s outer core and has a rich dynamic on various timescales. Global models of the field are used to study the interaction of the field and incoming charged particles, but also to infer core dynamics and to feed numerical simulations of the geodynamo. Modern satellite missions, such as the SWARM or the CHAMP mission, support high resolution reconstructions of the global field. From the 19 th century on, a global network of magnetic observatories has been established. It is growing ever since and global models can be constructed from the data it provides. Geomagnetic field models that extend further back in time rely on indirect observations of the field, i.e. thermoremanent records such as burnt clay or volcanic rocks and sediment records from lakes and seas. These indirect records come with (partially very large) uncertainties, introduced by the complex measurement methods and the dating procedure. Focusing on thermoremanent records only, the aim of this thesis is the development of a new modeling strategy for the global geomagnetic field during the Holocene, which takes the uncertainties into account and produces realistic estimates of the reliability of the model. This aim is approached by first considering snapshot models, in order to address the irregular spatial distribution of the records and the non-linear relation of the indirect observations to the field itself. In a Bayesian setting, a modeling algorithm based on Gaussian process regression is developed and applied to binned data. The modeling algorithm is then extended to the temporal domain and expanded to incorporate dating uncertainties. Finally, the algorithm is sequentialized to deal with numerical challenges arising from the size of the Holocene dataset. The central result of this thesis, including all of the aspects mentioned, is a new global geomagnetic field model. It covers the whole Holocene, back until 12000 BCE, and we call it ArchKalmag14k. When considering the uncertainties that are produced together with the model, it is evident that before 6000 BCE the thermoremanent database is not sufficient to support global models. For times more recent, ArchKalmag14k can be used to analyze features of the field under consideration of posterior uncertainties. The algorithm for generating ArchKalmag14k can be applied to different datasets and is provided to the community as an open source python package. N2 - Das geomagnetische Hauptfeld ist essenziell für das Leben auf der Erde, da es unseren Lebensraum gegen den Sonnenwind und kosmische Strahlung abschirmt. Es wird vom Geodynamo im Erdkern erzeugt und zeigt eine komplexe Dynamik auf unterschiedlichen Zeitskalen. Globale Modelle des Magnetfelds werden zur Studie der Wechselwirkung von einströmenden geladenen Teilchen genutzt, aber auch um Kerndynamiken zu untersuchen und um sie in numerische Simulationen des Geodynamos einzuspeisen. Moderne Satellitenmissionen, wie SWARM und CHAMP, stützen hochauflösende Rekonstruktionen des globalen Felds. Seit dem 19. Jahrhundert wird ein globales Netzwerk von magnetischen Observatorien aufgebaut. Es wächst stetig und globale Modelle können aus den Daten, die es liefert, konstruiert werden. Geomagnetische Feldmodelle, die weiter in der Zeit zurückreichen, basieren auf indirekten Beobachtungen des Felds, d.h. auf thermoremanenten Daten, wie gebrannten Tonen oder vulkanischen Gesteinen, und auf Sedimentdaten aus Seen und Meeren. Diese indirekten Beobachtungen werden mit (teilweise sehr hohen) Unsicherheiten geliefert, die aus den komplexen Datierungs- und Messmethoden resultieren. Ziel dieser Arbeit ist die Entwicklung einer neuen Modellierungsmethode für das globale geomagnetische Feld während des Holozäns, welche die Unsicherheiten berücksichtigt und realistische Schätzungen für die Verlässlichkeit des Modells liefert. Dabei werden lediglich thermoremanente Daten betrachtet. Diesem Ziel wird sich zunächst genähert, indem ein Schnappschuss-Modell konstruiert wird, um die unregelmäßige räumliche Verteilung der Daten und die nichtlineare Beziehung zwischen Daten und Magnetfeld zu untersuchen. In einem Bayesianischen Rahmen wird ein auf Gaussprozessen basierender Algorithmus entwickelt und zunächst auf diskretisierte Daten angewendet. Dieser Algorithmus wird dann um eine zeitabhängige Komponente ergänzt und erweitert, um Datierungsfehler zu berücksichtigen. Zuletzt wird der Algorithmus sequenzialisiert, um mit numerischen Herausforderungen umzugehen, die aufgrund der Größe des Holozän-Datensatzes bestehen. Das zentrale Ergebnis dieser Arbeit, welches alle genannten Aspekte beinhaltet, ist ein neues globales geomagnetisches Feldmodell. Es deckt das gesamte Holozän ab, bis ins Jahr 12000 BCE, und wir nennen es ArchKalmag14k. Bei Betrachtung der Unsicherheiten, die gemeinsam mit dem Modell ermittelt werden, wird deutlich, dass die thermoremanente Datenbasis nicht ausreicht, um globale Modelle vor dem Jahr 6000 BCE zu stützen. Für jüngere Zeiträume kann ArchKalmag14k genutzt werden, um Merkmale des Erdmagnetfelds unter Berücksichtigung der a posteriori Unsicherheiten zu analysieren. Der Algorithmus, mit dem ArchKalmag14k erzeugt wurde, kann auf weitere Datensätze angewendet werden und wird als quelloffenes python-Paket zur Verfügung gestellt. KW - geomagnetism KW - applied mathematics KW - Gaussian processes KW - Kalman filter KW - Gauß-Prozesse KW - Kalman Filter KW - angewandte Mathematik KW - Geomagnetismus Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-555875 ER - TY - GEN A1 - Evans, Myfanwy E. A1 - Hyde, Stephen T. T1 - Symmetric Tangling of Honeycomb Networks T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1282 KW - tangles KW - knots KW - networks KW - periodic entanglement KW - molecular weaving KW - graphs Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-570842 SN - 1866-8372 IS - 1282 ER - TY - THES A1 - Hannes, Sebastian T1 - Boundary Value Problems for the Lorentzian Dirac Operator N2 - The index theorem for elliptic operators on a closed Riemannian manifold by Atiyah and Singer has many applications in analysis, geometry and topology, but it is not suitable for a generalization to a Lorentzian setting. In the case where a boundary is present Atiyah, Patodi and Singer provide an index theorem for compact Riemannian manifolds by introducing non-local boundary conditions obtained via the spectral decomposition of an induced boundary operator, so called APS boundary conditions. Bär and Strohmaier prove a Lorentzian version of this index theorem for the Dirac operator on a manifold with boundary by utilizing results from APS and the characterization of the spectral flow by Phillips. In their case the Lorentzian manifold is assumed to be globally hyperbolic and spatially compact, and the induced boundary operator is given by the Riemannian Dirac operator on a spacelike Cauchy hypersurface. Their results show that imposing APS boundary conditions for these boundary operator will yield a Fredholm operator with a smooth kernel and its index can be calculated by a formula similar to the Riemannian case. Back in the Riemannian setting, Bär and Ballmann provide an analysis of the most general kind of boundary conditions that can be imposed on a first order elliptic differential operator that will still yield regularity for solutions as well as Fredholm property for the resulting operator. These boundary conditions can be thought of as deformations to the graph of a suitable operator mapping APS boundary conditions to their orthogonal complement. This thesis aims at applying the boundary conditions found by Bär and Ballmann to a Lorentzian setting to understand more general types of boundary conditions for the Dirac operator, conserving Fredholm property as well as providing regularity results and relative index formulas for the resulting operators. As it turns out, there are some differences in applying these graph-type boundary conditions to the Lorentzian Dirac operator when compared to the Riemannian setting. It will be shown that in contrast to the Riemannian case, going from a Fredholm boundary condition to its orthogonal complement works out fine in the Lorentzian setting. On the other hand, in order to deduce Fredholm property and regularity of solutions for graph-type boundary conditions, additional assumptions for the deformation maps need to be made. The thesis is organized as follows. In chapter 1 basic facts about Lorentzian and Riemannian spin manifolds, their spinor bundles and the Dirac operator are listed. These will serve as a foundation to define the setting and prove the results of later chapters. Chapter 2 defines the general notion of boundary conditions for the Dirac operator used in this thesis and introduces the APS boundary conditions as well as their graph type deformations. Also the role of the wave evolution operator in finding Fredholm boundary conditions is analyzed and these boundary conditions are connected to notion of Fredholm pairs in a given Hilbert space. Chapter 3 focuses on the principal symbol calculation of the wave evolution operator and the results are used to proof Fredholm property as well as regularity of solutions for suitable graph-type boundary conditions. Also sufficient conditions are derived for (pseudo-)local boundary conditions imposed on the Dirac operator to yield a Fredholm operator with a smooth solution space. In the last chapter 4, a few examples of boundary conditions are calculated applying the results of previous chapters. Restricting to special geometries and/or boundary conditions, results can be obtained that are not covered by the more general statements, and it is shown that so-called transmission conditions behave very differently than in the Riemannian setting. N2 - Der Indexsatz für elliptische Operatoren auf geschlossenen Riemannschen Mannigfaltigkeiten von Atiyah und Singer hat zahlreiche Anwendungen in Analysis, Geometrie und Topologie, ist aber ungeeignet für eine Verallgemeinerung auf Lorentz-Mannigfaltigkeiten. Durch die Einführung nicht-lokaler Randbedingungen, gewonnen aus der Spektralzerlegung eines induzierten Randoperators, beweisen Atiyah, Patodi und Singer (APS) einen Indexsatz für den Fall kompakter Riemannscher Mannigfaltigkeiten mit Rand. Aufbauend auf diesem Resultat und mit Hilfe der Charakterisierung des Spektralflusses durch Philipps gelangen Bär und Strohmaier zu einem Indexsatz für den Dirac-Operator auf global hyperbolischen Lorentz-Mannigfaltigkeiten mit kompakten und raumartigen Cauchy-Hyperflächen. Ihr Ergebnis zeigt unter anderem, dass der Dirac Operator auf solchen Mannigfaltigkeiten und unter APS Randbedingungen ein Fredholm-Operator mit glattem Kern ist und das sein Index sich aus einer zum Riemannschen Fall analogen Formel berechnen lässt. Zurück im Riemannschen Setup zeigen Bär und Ballmann eine allgemeine Charakterisierung von Randbedingungen für elliptische Differentialoperatoren erster Ordnung die sowohl die Regularität von Lösungen, als auch Fredholm-Eigenschaft des resultierenden Operators garantieren. Die dort entwickelten Randbedingungen können als Deformation auf den Graphen einer geeigneten Abbildung der APS-Randbedingung auf ihr orthogonales Komplement verstanden werden. Die vorliegende Arbeit hat das Ziel die von Bär und Ballmann beschriebenen Randbedingungen auf den Dirac-Operator von global hyperbolischen Lorentz-Mannigfaltigkeiten zu übertragen um eine allgemeinere Klasse von Randbedingungen zu finden unter denen der resultierende Dirac-Operator Fredholm ist und einen glatten Lösungsraum hat. Weiterhin wird analysiert wie sich derartige Deformation von APS-Randbedingungen auf den Index solcher Operatoren auswirken und wie dieser aus den bekannten Resultaten für den APS-Index berechnet werden kann. Es wird unter anderem gezeigt, dass im Gegensatz zum Riemannschen Fall beim Übergang von Randbedingungen zu ihrem orthogonalen Komplement die Fredholm-Eigenschaft des Operators erhalten bleibt. Andererseits sind zusätzliche Annahme nötig um die Regularität von Lösungen, sowie die Fredholm-Eigenschaft für Graph-Deformationen im Fall von Lorentz-Mannigfaltigkeiten zu erhalten. Die Arbeit ist dabei wie folgt aufgebaut. In Kapitel 1 werden grundlegende Fakten zu Lorentzschen und Riemannschen Spin-Mannigfaltigkeiten, ihren Spinor-Bündeln und Dirac-Operatoren zusammengetragen. Diese Informationen dienen als Ausgangspunkt zur Definition und Analyse von Randbedingungen in späteren Kapiteln der Arbeit. Kapitel 2 definiert allgemein den Begriff der Randbedingung wie er in dieser Arbeit verwendet wird und führt zudem den sogenannten ''wave-evolution-Operator'' ein, der eine wichtige Rolle im Finden und Analysieren von Fredholm-Randbedingungen für den Dirac-Operator spielen wird. Zuletzt wird der Zusammenhang zwischen Fredholm-Paaren eines Hilbert-Raumes und Fredholm-Randbedingungen für den Dirac-Operator erklärt. Kapitel 3 beschäftigt sich mit der Berechnung des Hauptsymbols des wave-evolution-Operators und die dort erzielten Resultate werden verwendet um Fredholm-Eigenschaft, sowie Regularität von Lösungen für geeignete Deformationen von APS-Randbedingungen zu beweisen. Weiterhin werden hinreichende Bedingungen für (pseudo-)lokale Randbedingungen abgeleitet, die Fredholm-Eigenschaft und Regularität für den resultierenden Dirac-Operator garantieren. Kapitel 4 zeigt, aufbauend auf den Ergebnissen der Kapitel 1-3, einige Beispiele von lokalen und nicht-lokalen Randbedingungen für den Dirac-Operator. Unter gewissen Einschränkungen an die Geometrie der zugrunde liegenden Mannigfaltigkeit bzw. den gestellten Randbedingungen können Ergebnisse erzielt werden die in den allgemeineren Resultaten der vorangehenden Kapitel nicht enthalten sind. Zuletzt werden sogenannte Transmission-Bedingungen analysiert und die Unterschiede dieser Randbedingungen zum Riemannschen Fall aufgezeigt. T2 - Randwertprobleme für den Lorentschen Diracoperator KW - Dirac Operator KW - Boundary Value Problems KW - Lorentzian Geometry KW - Randwertprobleme KW - Diracoperator KW - Lorentzgeometrie Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-548391 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 - TY - JOUR A1 - Engbert, Ralf A1 - Rabe, Maximilian Michael A1 - Schwetlick, Lisa A1 - Seelig, Stefan A. A1 - Reich, Sebastian A1 - Vasishth, Shravan T1 - Data assimilation in dynamical cognitive science JF - Trends in cognitive sciences N2 - Dynamical models make specific assumptions about cognitive processes that generate human behavior. In data assimilation, these models are tested against timeordered data. Recent progress on Bayesian data assimilation demonstrates that this approach combines the strengths of statistical modeling of individual differences with the those of dynamical cognitive models. Y1 - 2022 U6 - https://doi.org/10.1016/j.tics.2021.11.006 SN - 1364-6613 SN - 1879-307X VL - 26 IS - 2 SP - 99 EP - 102 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Hinz, Michael A1 - Schwarz, Michael T1 - A note on Neumann problems on graphs JF - Positivity N2 - We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations. KW - Graphs KW - Discrete Dirichlet forms KW - Neumann problem KW - Royden boundary Y1 - 2022 U6 - https://doi.org/10.1007/s11117-022-00930-0 SN - 1385-1292 SN - 1572-9281 VL - 26 IS - 4 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Bär, Christian A1 - Hanke, Bernhard T1 - Local flexibility for open partial differential relations JF - Communications on pure and applied mathematics / issued by the Courant Institute of Mathematical Sciences, New York Univ. N2 - We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry. The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K is an element of Double-struck capital R every manifold of dimension at least 2 carries a complete C-1,C- 1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C-2-metrics in general. Y1 - 2021 U6 - https://doi.org/10.1002/cpa.21982 SN - 0010-3640 SN - 1097-0312 VL - 75 IS - 6 SP - 1377 EP - 1415 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Kolbe, Benedikt Maximilian A1 - Evans, Myfanwy E. T1 - Enumerating isotopy classes of tilings guided by the symmetry of triply JF - Siam journal on applied algebra and geometry N2 - We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems. KW - isotopic tiling theory KW - mapping class group KW - orbifolds KW - group KW - presentations KW - representations of groups as automorphism groups of KW - algebraic systems KW - triply periodic minimal surface KW - Delaney--Dress KW - tiling theory KW - hyperbolic tilings KW - two-dimensional topology Y1 - 2022 U6 - https://doi.org/10.1137/20M1358943 SN - 2470-6566 VL - 6 IS - 1 SP - 1 EP - 40 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Hanisch, Florian A1 - Ludewig, Matthias T1 - A rigorous construction of the supersymmetric path integral associated to a compact spin manifold JF - Communications in mathematical physics N2 - We give a rigorous construction of the path integral in N = 1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Guneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaume, Atiyah, Bismut and Witten. Y1 - 2022 U6 - https://doi.org/10.1007/s00220-022-04336-7 SN - 0010-3616 SN - 1432-0916 VL - 391 IS - 3 SP - 1209 EP - 1239 PB - Springer CY - Berlin ; Heidelberg ER - TY - JOUR A1 - Nassar, Yomna M. A1 - Hohmann, Nicolas A1 - Michelet, Robin A1 - Gottwalt, Katharina A1 - Meid, Andreas D. A1 - Burhenne, Jürgen A1 - Huisinga, Wilhelm A1 - Haefeli, Walter E. A1 - Mikus, Gerd A1 - Kloft, Charlotte T1 - Quantification of the Time Course of CYP3A Inhibition, Activation, and Induction Using a Population Pharmacokinetic Model of Microdosed Midazolam Continuous Infusion JF - Clinical Pharmacokinetics N2 - Background Cytochrome P450 (CYP) 3A contributes to the metabolism of many approved drugs. CYP3A perpetrator drugs can profoundly alter the exposure of CYP3A substrates. However, effects of such drug-drug interactions are usually reported as maximum effects rather than studied as time-dependent processes. Identification of the time course of CYP3A modulation can provide insight into when significant changes to CYP3A activity occurs, help better design drug-drug interaction studies, and manage drug-drug interactions in clinical practice. Objective We aimed to quantify the time course and extent of the in vivo modulation of different CYP3A perpetrator drugs on hepatic CYP3A activity and distinguish different modulatory mechanisms by their time of onset, using pharmacologically inactive intravenous microgram doses of the CYP3A-specific substrate midazolam, as a marker of CYP3A activity. Methods Twenty-four healthy individuals received an intravenous midazolam bolus followed by a continuous infusion for 10 or 36 h. Individuals were randomized into four arms: within each arm, two individuals served as a placebo control and, 2 h after start of the midazolam infusion, four individuals received the CYP3A perpetrator drug: voriconazole (inhibitor, orally or intravenously), rifampicin (inducer, orally), or efavirenz (activator, orally). After midazolam bolus administration, blood samples were taken every hour (rifampicin arm) or every 15 min (remaining study arms) until the end of midazolam infusion. A total of 1858 concentrations were equally divided between midazolam and its metabolite, 1'-hydroxymidazolam. A nonlinear mixed-effects population pharmacokinetic model of both compounds was developed using NONMEM (R). CYP3A activity modulation was quantified over time, as the relative change of midazolam clearance encountered by the perpetrator drug, compared to the corresponding clearance value in the placebo arm. Results Time course of CYP3A modulation and magnitude of maximum effect were identified for each perpetrator drug. While efavirenz CYP3A activation was relatively fast and short, reaching a maximum after approximately 2-3 h, the induction effect of rifampicin could only be observed after 22 h, with a maximum after approximately 28-30 h followed by a steep drop to almost baseline within 1-2 h. In contrast, the inhibitory impact of both oral and intravenous voriconazole was prolonged with a steady inhibition of CYP3A activity followed by a gradual increase in the inhibitory effect until the end of sampling at 8 h. Relative maximum clearance changes were +59.1%, +46.7%, -70.6%, and -61.1% for efavirenz, rifampicin, oral voriconazole, and intravenous voriconazole, respectively. Conclusions We could distinguish between different mechanisms of CYP3A modulation by the time of onset. Identification of the time at which clearance significantly changes, per perpetrator drug, can guide the design of an optimal sampling schedule for future drug-drug interaction studies. The impact of a short-term combination of different perpetrator drugs on the paradigm CYP3A substrate midazolam was characterized and can define combination intervals in which no relevant interaction is to be expected. Y1 - 2022 U6 - https://doi.org/10.1007/s40262-022-01175-6 SN - 0312-5963 SN - 1179-1926 VL - 61 IS - 11 SP - 1595 EP - 1607 PB - Springer CY - Northcote ER - TY - JOUR A1 - Démaris, Alix A1 - Widigson, Ella S. K. A1 - Ilvemark, Johan F. K. F. A1 - Steenholdt, Casper A1 - Seidelin, Jakob B. A1 - Huisinga, Wilhelm A1 - Michelet, Robin A1 - Aulin, Linda B. S. A1 - Kloft, Charlotte T1 - Ulcerative colitis and acute severe ulcerative colitis patients are overlooked in infliximab population pharmacokinetic models BT - results from a comprehensive review JF - Pharmaceutics / Molecular Diversity Preservation International N2 - Ulcerative colitis (UC) is part of the inflammatory bowels diseases, and moderate to severe UC patients can be treated with anti-tumour necrosis alpha monoclonal antibodies, including infliximab (IFX). Even though treatment of UC patients by IFX has been in place for over a decade, many gaps in modelling of IFX PK in this population remain. This is even more true for acute severe UC (ASUC) patients for which early prediction of IFX pharmacokinetic (PK) could highly improve treatment outcome. Thus, this review aims to compile and analyse published population PK models of IFX in UC and ASUC patients, and to assess the current knowledge on disease activity impact on IFX PK. For this, a semi-systematic literature search was conducted, from which 26 publications including a population PK model analysis of UC patients receiving IFX therapy were selected. Amongst those, only four developed a model specifically for UC patients, and only three populations included severe UC patients. Investigations of disease activity impact on PK were reported in only 4 of the 14 models selected. In addition, the lack of reported model codes and assessment of predictive performance make the use of published models in a clinical setting challenging. Thus, more comprehensive investigation of PK in UC and ASUC is needed as well as more adequate reports on developed models and their evaluation in order to apply them in a clinical setting. KW - infliximab KW - inflammatory bowel disease KW - ulcerative colitis KW - acute severe KW - disease activity KW - pharmacokinetic KW - pharmacometrics Y1 - 2022 U6 - https://doi.org/10.3390/pharmaceutics14102095 SN - 1999-4923 VL - 14 IS - 10 PB - MDPI CY - Basel ER - TY - JOUR A1 - Schanner, Maximilian A1 - Korte, Monika A1 - Holschneider, Matthias T1 - ArchKalmag14k: A kalman-filter based global geomagnetic model for the holocene JF - Journal of geophysical research : Solid earth N2 - We propose a global geomagnetic field model for the last 14 thousand years, based on thermoremanent records. We call the model ArchKalmag14k. ArchKalmag14k is constructed by modifying recently proposed algorithms, based on space-time correlations. Due to the amount of data and complexity of the model, the full Bayesian posterior is numerically intractable. To tackle this, we sequentialize the inversion by implementing a Kalman-filter with a fixed time step. Every step consists of a prediction, based on a degree dependent temporal covariance, and a correction via Gaussian process regression. Dating errors are treated via a noisy input formulation. Cross correlations are reintroduced by a smoothing algorithm and model parameters are inferred from the data. Due to the specific statistical nature of the proposed algorithms, the model comes with space and time-dependent uncertainty estimates. The new model ArchKalmag14k shows less variation in the large-scale degrees than comparable models. Local predictions represent the underlying data and agree with comparable models, if the location is sampled well. Uncertainties are bigger for earlier times and in regions of sparse data coverage. We also use ArchKalmag14k to analyze the appearance and evolution of the South Atlantic anomaly together with reverse flux patches at the core-mantle boundary, considering the model uncertainties. While we find good agreement with earlier models for recent times, our model suggests a different evolution of intensity minima prior to 1650 CE. In general, our results suggest that prior to 6000 BCE the data is not sufficient to support global models. Y1 - 2022 U6 - https://doi.org/10.1029/2021JB023166 SN - 2169-9313 SN - 2169-9356 VL - 127 IS - 2 PB - American Geophysical Union CY - Washington ER - TY - JOUR A1 - Dimitrova, Ilinka A1 - Koppitz, Jörg T1 - On relative ranks of the semigroup of orientation-preserving transformations on infinite chain with restricted range JF - Communications in algebra N2 - Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X,Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X,Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size. KW - Order-preserving transformations KW - orientation-preserving KW - transformations KW - relative rank KW - restricted range KW - transformation KW - semigroups on infinite chain Y1 - 2022 U6 - https://doi.org/10.1080/00927872.2021.2000998 SN - 0092-7872 SN - 1532-4125 VL - 50 IS - 5 SP - 2157 EP - 2168 PB - Taylor & Francis Group CY - Philadelphia ER - TY - JOUR A1 - Jia, Weihan A1 - Anslan, Sten A1 - Chen, Fahu A1 - Cao, Xianyong A1 - Dong, Hailiang A1 - Dulias, Katharina A1 - Gu, Zhengquan A1 - Heinecke, Liv A1 - Jiang, Hongchen A1 - Kruse, Stefan A1 - Kang, Wengang A1 - Li, Kai A1 - Liu, Sisi A1 - Liu, Xingqi A1 - Liu, Ying A1 - Ni, Jian A1 - Schwalb, Antje A1 - Stoof-Leichsenring, Kathleen R. A1 - Shen, Wei A1 - Tian, Fang A1 - Wang, Jing A1 - Wang, Yongbo A1 - Wang, Yucheng A1 - Xu, Hai A1 - Yang, Xiaoyan A1 - Zhang, Dongju A1 - Herzschuh, Ulrike T1 - Sedimentary ancient DNA reveals past ecosystem and biodiversity changes on the Tibetan Plateau: overview and prospects JF - Quaternary science reviews : the international multidisciplinary research and review journal N2 - Alpine ecosystems on the Tibetan Plateau are being threatened by ongoing climate warming and intensified human activities. Ecological time-series obtained from sedimentary ancient DNA (sedaDNA) are essential for understanding past ecosystem and biodiversity dynamics on the Tibetan Plateau and their responses to climate change at a high taxonomic resolution. Hitherto only few but promising studies have been published on this topic. The potential and limitations of using sedaDNA on the Tibetan Plateau are not fully understood. Here, we (i) provide updated knowledge of and a brief introduction to the suitable archives, region-specific taphonomy, state-of-the-art methodologies, and research questions of sedaDNA on the Tibetan Plateau; (ii) review published and ongoing sedaDNA studies from the Tibetan Plateau; and (iii) give some recommendations for future sedaDNA study designs. Based on the current knowledge of taphonomy, we infer that deep glacial lakes with freshwater and high clay sediment input, such as those from the southern and southeastern Tibetan Plateau, may have a high potential for sedaDNA studies. Metabarcoding (for microorganisms and plants), metagenomics (for ecosystems), and hybridization capture (for prehistoric humans) are three primary sedaDNA approaches which have been successfully applied on the Tibetan Plateau, but their power is still limited by several technical issues, such as PCR bias and incompleteness of taxonomic reference databases. Setting up high-quality and open-access regional taxonomic reference databases for the Tibetan Plateau should be given priority in the future. To conclude, the archival, taphonomic, and methodological conditions of the Tibetan Plateau are favorable for performing sedaDNA studies. More research should be encouraged to address questions about long-term ecological dynamics at ecosystem scale and to bring the paleoecology of the Tibetan Plateau into a new era. KW - Sedimentary ancient DNA (sedaDNA) KW - Tibetan Plateau KW - Environmental DNA KW - Taphonomy KW - Ecosystem KW - Biodiversity KW - Paleoecology KW - Paleogeography Y1 - 2022 U6 - https://doi.org/10.1016/j.quascirev.2022.107703 SN - 0277-3791 SN - 1873-457X VL - 293 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Bellingeri, Carlo A1 - Friz, Peter A1 - Paycha, Sylvie A1 - Preiß, Rosa Lili Dora T1 - Smooth rough paths, their geometry and algebraic renormalization JF - Vietnam journal of mathematics N2 - We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting. KW - Signatures KW - Rough paths KW - Cartan's development KW - Renormalization Y1 - 2022 U6 - https://doi.org/10.1007/s10013-022-00570-7 SN - 2305-221X SN - 2305-2228 VL - 50 IS - 3 SP - 719 EP - 761 PB - Springer CY - Singapore ER - TY - JOUR A1 - Lie, Han Cheng A1 - Stahn, Martin A1 - Sullivan, Tim J. T1 - Randomised one-step time integration methods for deterministic operator differential equations JF - Calcolo N2 - Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings. KW - Time integration KW - Operator differential equations KW - Randomisation KW - Uncertainty quantification Y1 - 2022 U6 - https://doi.org/10.1007/s10092-022-00457-6 SN - 0008-0624 SN - 1126-5434 VL - 59 IS - 1 PB - Springer CY - Milano ER - TY - JOUR A1 - Malem-Shinitski, Noa A1 - Ojeda, Cesar A1 - Opper, Manfred T1 - Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects JF - Entropy N2 - Traditionally, Hawkes processes are used to model time-continuous point processes with history dependence. Here, we propose an extended model where the self-effects are of both excitatory and inhibitory types and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and derive an inference algorithm by performing inference on an aggregated sum of Gaussian Processes. Approximate Bayesian inference is achieved via data augmentation, and we describe a mean-field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from different domains and compare it to previously reported results. KW - Bayesian inference KW - point process KW - Gaussian process Y1 - 2022 U6 - https://doi.org/10.3390/e24030356 SN - 1099-4300 VL - 24 IS - 3 PB - MDPI CY - Basel ER - TY - JOUR A1 - Pathiraja, Sahani Darschika A1 - Leeuwen, Peter Jan van T1 - Multiplicative Non-Gaussian model error estimation in data assimilation JF - Journal of advances in modeling earth systems : JAMES N2 - Model uncertainty quantification is an essential component of effective data assimilation. Model errors associated with sub-grid scale processes are often represented through stochastic parameterizations of the unresolved process. Many existing Stochastic Parameterization schemes are only applicable when knowledge of the true sub-grid scale process or full observations of the coarse scale process are available, which is typically not the case in real applications. We present a methodology for estimating the statistics of sub-grid scale processes for the more realistic case that only partial observations of the coarse scale process are available. Model error realizations are estimated over a training period by minimizing their conditional sum of squared deviations given some informative covariates (e.g., state of the system), constrained by available observations and assuming that the observation errors are smaller than the model errors. From these realizations a conditional probability distribution of additive model errors given these covariates is obtained, allowing for complex non-Gaussian error structures. Random draws from this density are then used in actual ensemble data assimilation experiments. We demonstrate the efficacy of the approach through numerical experiments with the multi-scale Lorenz 96 system using both small and large time scale separations between slow (coarse scale) and fast (fine scale) variables. The resulting error estimates and forecasts obtained with this new method are superior to those from two existing methods. KW - model uncertainty KW - non-Gaussian KW - data-driven KW - uncertainty KW - quantification KW - Lorenz 96 KW - sub-grid scale Y1 - 2022 U6 - https://doi.org/10.1029/2021MS002564 SN - 1942-2466 VL - 14 IS - 4 PB - American Geophysical Union CY - Washington ER - TY - JOUR A1 - Pohle, Jennifer A1 - Adam, Timo A1 - Beumer, Larissa T1 - Flexible estimation of the state dwell-time distribution in hidden semi-Markov models JF - Computational statistics & data analysis N2 - Hidden semi-Markov models generalise hidden Markov models by explicitly modelling the time spent in a given state, the so-called dwell time, using some distribution defined on the natural numbers. While the (shifted) Poisson and negative binomial distribution provide natural choices for such distributions, in practice, parametric distributions can lack the flexibility to adequately model the dwell times. To overcome this problem, a penalised maximum likelihood approach is proposed that allows for a flexible and data-driven estimation of the dwell-time distributions without the need to make any distributional assumption. This approach is suitable for direct modelling purposes or as an exploratory tool to investigate the latent state dynamics. The feasibility and potential of the suggested approach is illustrated in a simulation study and by modelling muskox movements in northeast Greenland using GPS tracking data. The proposed method is implemented in the R-package PHSMM which is available on CRAN. KW - Penalized likelihood KW - Smoothing KW - Time series KW - Animal movement modeling Y1 - 2022 U6 - https://doi.org/10.1016/j.csda.2022.107479 SN - 0167-9473 SN - 1872-7352 VL - 172 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Yen, Ming-Hsuan A1 - von Specht, Sebastian A1 - Lin, Yen-Yu A1 - Cotton, Fabrice A1 - Ma, Kuo-Fong T1 - Within- and between-event variabilities of strong-velocity pulses of moderate earthquakes within dense seismic arrays JF - Bulletin of the Seismological Society of America N2 - Ground motion with strong-velocity pulses can cause significant damage to buildings and structures at certain periods; hence, knowing the period and velocity amplitude of such pulses is critical for earthquake structural engineering. However, the physical factors relating the scaling of pulse periods with magnitude are poorly understood. In this study, we investigate moderate but damaging earthquakes (M-w 6-7) and characterize ground- motion pulses using the method of Shahi and Baker (2014) while considering the potential static-offset effects. We confirm that the within-event variability of the pulses is large. The identified pulses in this study are mostly from strike-slip-like earthquakes. We further perform simulations using the freq uency-wavenumber algorithm to investigate the causes of the variability of the pulse periods within and between events for moderate strike-slip earthquakes. We test the effect of fault dips, and the impact of the asperity locations and sizes. The simulations reveal that the asperity properties have a high impact on the pulse periods and amplitudes at nearby stations. Our results emphasize the importance of asperity characteristics, in addition to earthquake magnitudes for the occurrence and properties of pulses produced by the forward directivity effect. We finally quantify and discuss within- and between-event variabilities of pulse properties at short distances. Y1 - 2021 U6 - https://doi.org/10.1785/0120200376 SN - 0037-1106 SN - 1943-3573 VL - 112 IS - 1 SP - 361 EP - 380 PB - Seismological Society of America CY - El Cerito, Calif. ER - TY - JOUR A1 - Dube, Jonas A1 - Böckmann, Christine A1 - Ritter, Christoph T1 - Lidar-Derived Aerosol Properties from Ny-Ålesund, Svalbard during the MOSAiC Spring 2020 JF - Remote sensing / Molecular Diversity Preservation International (MDPI) N2 - In this work, we present Raman lidar data (from a Nd:YAG operating at 355 nm, 532 nm and 1064 nm) from the international research village Ny-Alesund for the time period of January to April 2020 during the Arctic haze season of the MOSAiC winter. We present values of the aerosol backscatter, the lidar ratio and the backscatter Angstrom exponent, though the latter depends on wavelength. The aerosol polarization was generally below 2%, indicating mostly spherical particles. We observed that events with high backscatter and high lidar ratio did not coincide. In fact, the highest lidar ratios (LR > 75 sr at 532 nm) were already found by January and may have been caused by hygroscopic growth, rather than by advection of more continental aerosol. Further, we performed an inversion of the lidar data to retrieve a refractive index and a size distribution of the aerosol. Our results suggest that in the free troposphere (above approximate to 2500 m) the aerosol size distribution is quite constant in time, with dominance of small particles with a modal radius well below 100 nm. On the contrary, below approximate to 2000 m in altitude, we frequently found gradients in aerosol backscatter and even size distribution, sometimes in accordance with gradients of wind speed, humidity or elevated temperature inversions, as if the aerosol was strongly modified by vertical displacement in what we call the "mechanical boundary layer". Finally, we present an indication that additional meteorological soundings during MOSAiC campaign did not necessarily improve the fidelity of air backtrajectories. KW - aerosol KW - Arctic haze KW - lidar KW - microphysical properties KW - backtrajectories; KW - Ny-Alesund KW - Svalbard KW - MOSAiC KW - aerosol-boundary layer interactions Y1 - 2022 U6 - https://doi.org/10.3390/rs14112578 SN - 2072-4292 VL - 14 IS - 11 PB - MDPI CY - Basel ER - TY - JOUR A1 - Lilienkamp, Henning A1 - von Specht, Sebastian A1 - Weatherill, Graeme A1 - Caire, Giuseppe A1 - Cotton, Fabrice T1 - Ground-Motion modeling as an image processing task BT - introducing a neural network based, fully data-driven, and nonergodic JF - Bulletin of the Seismological Society of America N2 - We construct and examine the prototype of a deep learning-based ground-motion model (GMM) that is both fully data driven and nonergodic. We formulate ground-motion modeling as an image processing task, in which a specific type of neural network, the U-Net, relates continuous, horizontal maps of earthquake predictive parameters to sparse observations of a ground-motion intensity measure (IM). The processing of map-shaped data allows the natural incorporation of absolute earthquake source and observation site coordinates, and is, therefore, well suited to include site-, source-, and path-specific amplification effects in a nonergodic GMM. Data-driven interpolation of the IM between observation points is an inherent feature of the U-Net and requires no a priori assumptions. We evaluate our model using both a synthetic dataset and a subset of observations from the KiK-net strong motion network in the Kanto basin in Japan. We find that the U-Net model is capable of learning the magnitude???distance scaling, as well as site-, source-, and path-specific amplification effects from a strong motion dataset. The interpolation scheme is evaluated using a fivefold cross validation and is found to provide on average unbiased predictions. The magnitude???distance scaling as well as the site amplification of response spectral acceleration at a period of 1 s obtained for the Kanto basin are comparable to previous regional studies. Y1 - 2022 U6 - https://doi.org/10.1785/0120220008 SN - 0037-1106 SN - 1943-3573 VL - 112 IS - 3 SP - 1565 EP - 1582 PB - Seismological Society of America CY - Albany ER - TY - JOUR A1 - Fischer, Florian A1 - Keller, Matthias T1 - Riesz decompositions for Schrödinger operators on graphs JF - Journal of mathematical analysis and applications N2 - We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem. KW - Potential theory KW - Green's function KW - Schrödinger operator KW - Weighted KW - graph KW - Subcritical KW - Greatest harmonic minorant Y1 - 2021 U6 - https://doi.org/10.1016/j.jmaa.2020.124674 SN - 0022-247X SN - 1096-0813 VL - 495 IS - 1 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Bär, Christian T1 - The Faddeev-LeVerrier algorithm and the Pfaffian JF - Linear algebra and its applications N2 - We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra. KW - Characteristic polynomial KW - Determinant KW - Pfaffian KW - Gauss-Bonnet-Chern KW - theorem Y1 - 2021 U6 - https://doi.org/10.1016/j.laa.2021.07.023 SN - 0024-3795 SN - 1873-1856 VL - 630 SP - 39 EP - 55 PB - Elsevier CY - New York ER - TY - JOUR A1 - Bandara, Lashi T1 - Functional calculus and harmonic analysis in geometry JF - São Paulo journal of mathematical sciences / Instituto de Matemática e Estatística da Universidade de São Paulo N2 - In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This is a succinct survey that hopes to inspire geometers and analysts alike to study these methods so that they can be further developed to be potentially applied to a broader range of questions. KW - Functional calculus KW - Real-variable harmonic analysis KW - Elliptic boundary KW - value problems KW - Kato square root problem KW - Spectral flow KW - Riesz topology KW - Gigli-Mantegazza flow KW - Bisectorial operator Y1 - 2021 U6 - https://doi.org/10.1007/s40863-019-00149-0 SN - 1982-6907 SN - 2316-9028 VL - 15 IS - 1 SP - 20 EP - 53 PB - Springer CY - Cham ER - TY - JOUR A1 - Klein, Markus A1 - Rosenberger, Elke T1 - The tunneling effect for Schrödinger operators on a vector bundle JF - Analysis and mathematical physics N2 - In the semiclassical limit (h) over bar -> 0, we analyze a class of self-adjoint Schrodinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V center dot id(E) acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m(1),... m(r) is an element of M, called potential wells. Using quasimodes of WKB-type near m(j) for eigenfunctions associated with the low lying eigenvalues of H-(h) over bar, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension l + 1. This dimension l determines the polynomial prefactor for exponentially small eigenvalue splitting. KW - Laplace-type operator KW - Vector bundle KW - WKB-expansion KW - Quasimodes KW - Tunneling KW - Spectral gap KW - Complete asymptotics Y1 - 2021 U6 - https://doi.org/10.1007/s13324-021-00485-5 SN - 1664-2368 SN - 1664-235X VL - 11 IS - 2 PB - Springer International Publishing AG CY - Cham (ZG) ER - TY - THES A1 - Etzold, Heiko T1 - Neue Zugänge zum Winkelbegriff T1 - New Ways to the Angle Concept BT - Fachdidaktische Entwicklungsforschung zur Ausbildung des Winkelfeldbegriffs bei Schülerinnen und Schülern der vierten Klassenstufe N2 - Die Vielfältigkeit des Winkelbegriffs ist gleichermaßen spannend wie herausfordernd in Hinblick auf seine Zugänge im Mathematikunterricht der Schule. Ausgehend von verschiedenen Vorstellungen zum Winkelbegriff wird in dieser Arbeit ein Lehrgang zur Vermittlung des Winkelbegriffs entwickelt und letztlich in konkrete Umsetzungen für den Schulunterricht überführt. Dabei erfolgt zunächst eine stoffdidaktische Auseinandersetzung mit dem Winkelbegriff, die von einer informationstheoretischen Winkeldefinition begleitet wird. In dieser wird eine Definition für den Winkelbegriff unter der Fragestellung entwickelt, welche Informationen man über einen Winkel benötigt, um ihn beschreiben zu können. So können die in der fachdidaktischen Literatur auftretenden Winkelvorstellungen aus fachmathematischer Perspektive erneut abgeleitet und validiert werden. Parallel dazu wird ein Verfahren beschrieben, wie Winkel – auch unter dynamischen Aspekten – informationstechnisch verarbeitet werden können, so dass Schlussfolgerungen aus der informationstheoretischen Winkeldefinition beispielsweise in dynamischen Geometriesystemen zur Verfügung stehen. Unter dem Gesichtspunkt, wie eine Abstraktion des Winkelbegriffs im Mathematikunterricht vonstatten gehen kann, werden die Grundvorstellungsidee sowie die Lehrstrategie des Aufsteigens vom Abstrakten zum Konkreten miteinander in Beziehung gesetzt. Aus der Verknüpfung der beiden Theorien wird ein grundsätzlicher Weg abgeleitet, wie im Rahmen der Lehrstrategie eine Ausgangsabstraktion zu einzelnen Winkelaspekten aufgebaut werden kann, was die Generierung von Grundvorstellungen zu den Bestandteilen des jeweiligen Winkelaspekts und zum Operieren mit diesen Begriffsbestandteilen ermöglichen soll. Hierfür wird die Lehrstrategie angepasst, um insbesondere den Übergang von Winkelsituationen zu Winkelkontexten zu realisieren. Explizit für den Aspekt des Winkelfeldes werden, anhand der Untersuchung der Sichtfelder von Tieren, Lernhandlungen und Forderungen an ein Lernmodell beschrieben, die Schülerinnen und Schüler bei der Begriffsaneignung unterstützen. Die Tätigkeitstheorie, der die genannte Lehrstrategie zuzuordnen ist, zieht sich als roter Faden durch die weitere Arbeit, wenn nun theoriebasiert Designprinzipien generiert werden, die in die Entwicklung einer interaktiven Lernumgebung münden. Hierzu wird u. a. das Modell der Artifact-Centric Activity Theory genutzt, das das Beziehungsgefüge aus Schülerinnen und Schülern, dem mathematischen Gegenstand und einer zu entwickelnden App als vermittelndes Medium beschreibt, wobei der Einsatz der App im Unterrichtskontext sowie deren regelgeleitete Entwicklung Bestandteil des Modells sind. Gemäß dem Ansatz der Fachdidaktischen Entwicklungsforschung wird die Lernumgebung anschließend in mehreren Zyklen erprobt, evaluiert und überarbeitet. Dabei wird ein qualitatives Setting angewandt, das sich der Semiotischen Vermittlung bedient und untersucht, inwiefern sich die Qualität der von den Schülerinnen und Schülern gezeigten Lernhandlungen durch die Designprinzipien und deren Umsetzung erklären lässt. Am Ende der Arbeit stehen eine finale Version der Designprinzipien und eine sich daraus ergebende Lernumgebung zur Einführung des Winkelfeldbegriffs in der vierten Klassenstufe. N2 - The diversity of the concept »angle« can be both exciting and challenging when looking at how to access it in mathematics education in schools. In this thesis, based on different ideas of the angle concept, a training course for conveying the concept will be developed and translated into concrete implementations for school teaching. First, there will be a didactical subject matter discussion of the angle concept, which will be accompanied by an angle definition from information theory. Through the didactical subject matter discussion, a definition for the angle concept will be developed which is guided by the question of what kind of information about an angle is needed in order to describe it. This way, the diverse ideas of the angle concept discussed in mathematics didactics literature can be once again derived and validated from a mathematical point of view. In parallel, a method will be described of how an angle - even one with dynamic aspects - can be handled in terms of information technology, so that conclusions can be drawn from a definition from information theory for dynamic geometry environments for instance. Considering how abstraction of the angle concept can take place in mathematics education, the Idea of Grundvorstellungen will then be connected to the structural principle of the Ascent From the Abstract to the Concrete. Based on the connection of these two theories, a training course will be developed that aims to construct an initial abstract of certain aspects of the angle concept which, in turn, aims at enabling the generating of Grundvorstellungen towards components of the angle concept and at operating with it. For this, the structural principle will be adapted – specifically to realize the transition from angle situations to angle contexts. For one aspect, the angular field, there will be a description of learning actions and demands on a learning model that supports students’ concept acquisition. The angular field, in this step, will be represented by vision fields of animals. Activity theory, on which the structural principle is based, depicts the recurring theme throughout this thesis when generating design principles that lead towards the development of an interactive learning environment. For this, the Artifact-Centric Activity Theory model will be used in order to describe connections between students, the mathematical topic and the to-be-created app. The use of the app in classroom situations, as well as its rule-governed development, are components of the model. Following a Design-Based Research approach, this learning environment will then go through several cycles of test, evaluation and revision. For this purpose, a qualitative setting will be applied using Semiotic Mediation. It will be used to investigate how far design principles, as well as their implementation, impacts on the quality of student’s learning actions. As an outcome of this thesis, a final version of the design principles and an ensuing learning environment that introduces the concept of »angular field« in grade four teaching will be created. KW - Winkel KW - Tätigkeitstheorie KW - Digitale Werkzeuge KW - Digital Tools KW - Activity Theory KW - Angle Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-504187 ER - TY - THES A1 - Zass, Alexander T1 - A multifaceted study of marked Gibbs point processes T1 - Facetten von markierten Gibbsschen Punktprozessen N2 - This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation. We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight. We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution. Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation. Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry. N2 - Diese Arbeit konzentriert sich auf die Untersuchung von markierten Gibbs-Punkt-Prozessen und stellt insbesondere einige Ergebnisse zu deren Existenz und Eindeutigkeit vor. Dabei werden Ideen und Techniken aus verschiedenen Bereichen der statistischen Mechanik verwendet: die Entropie-Methode aus der Theorie der großen Abweichungen, die Cluster-Expansion und die Kirkwood-Salsburg-Gleichungen, das Dobrushin-Kontraktionsprinzip und die Disagreement-Perkolation. Wir präsentieren zunächst ein Existenzergebnis für unendlich-volumige markierte Gibbs-Punkt-Prozesse. Genauer gesagt verwenden wir die sogenannte Entropie-Methode (und Werkzeuge der großen Abweichung), um markierte Gibbs-Punkt-Prozesse in R^d unter möglichst allgemeinen Annahmen zu konstruieren. Insbesondere gehören die zufälligen Markierungen zu einem allgemeinen normierten Raum und sind nicht beschränkt. Außerdem lassen wir Interaktionsfunktionale zu, die unbeschränkt sein können und deren Reichweite endlich, aber zufällig ist. Die Entropie-Methode beruht darauf, zu zeigen, dass eine Familie von endlich-volumigen Gibbs-Punkt-Prozessen zu sequentiell kompakten Entropie-Niveau-Mengen gehört, und daher dicht ist. Wir stellen dann unendlich-dimensionale Langevin-Diffusionen vor, die wir über eine Gibbssche Beschreibung in Wechselwirkung setzen. In dieser Umgebung sind wir in der Lage, das vorangehend vorgestellte allgemeine Ergebnis anzupassen, um die Existenz des zugehörigen unendlich-dimensionalen Maßes zu zeigen. Wir untersuchen auch seine Korrelationsfunktionen über Cluster-Expansions Techniken und erhalten die Eindeutigkeit des Gibbs-Prozesses für alle inversen Temperaturen β und Aktivitäten z unterhalb einer bestimmten Schwelle. Diese Methode beruht darauf, zunächst zu zeigen, dass die Korrelationsfunktionen des Prozesses eine so genannte Ruelle-Schranke erfüllen, um diese dann zur Lösung eines Fixpunktproblems in einem geeigneten Banach-Raum zu verwenden. Der Eindeutigkeitsbereich, den wir erhalten, wird dann aus den Modellparametern z und β definiert, für die ein solches Problem genau eine Lösung hat. Schließlich untersuchen wir die Frage nach der Eindeutigkeit von unendlich-volumigen Gibbs-Punkt-Prozessen auf R^d im unmarkierten Fall weiter. Im Zusammenhang mit repulsiven Wechselwirkungen basierend auf einer Hartkernkomponente stellen wir einen neuen Ansatz zur Eindeutigkeit vor, indem wir das diskrete Dobrushin-Kriterium im kontinuierlichen Rahmen anwenden. Wir legen zunächst einen Diskretisierungsparameter a>0 fest und untersuchen dann das Verhalten des Bereichs der Eindeutigkeit, wenn a gegen 0 geht. Mit dieser Technik sind wir in der Lage, explizite Schwellenwerte für die Parameter z und β zu erhalten, die wir dann mit bestehenden Ergebnissen aus den verschiedenen Methoden der Cluster-Expansion und der Disagreement-Perkolation vergleichen. In dieser Arbeit illustrieren wir unsere theoretischen Ergebnisse mit verschiedenen Beispielen sowohl aus der klassischen statistischen Mechanik als auch aus der stochastischen Geometrie. KW - marked Gibbs point processes KW - Langevin diffusions KW - Dobrushin criterion KW - Entropy method KW - Cluster expansion KW - Kirkwood--Salsburg equations KW - DLR equations KW - Markierte Gibbs-Punkt-Prozesse KW - Entropiemethode KW - Cluster-Expansion KW - DLR-Gleichungen KW - Dobrushin-Kriterium KW - Kirkwood-Salsburg-Gleichungen KW - Langevin-Diffusions Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-512775 ER - TY - THES A1 - Oancea, Marius-Adrian T1 - Spin Hall effects in general relativity T1 - Spin Hall Effekte in der Allgemeinen Relativitätstheorie N2 - The propagation of test fields, such as electromagnetic, Dirac or linearized gravity, on a fixed spacetime manifold is often studied by using the geometrical optics approximation. In the limit of infinitely high frequencies, the geometrical optics approximation provides a conceptual transition between the test field and an effective point-particle description. The corresponding point-particles, or wave rays, coincide with the geodesics of the underlying spacetime. For most astrophysical applications of interest, such as the observation of celestial bodies, gravitational lensing, or the observation of cosmic rays, the geometrical optics approximation and the effective point-particle description represent a satisfactory theoretical model. However, the geometrical optics approximation gradually breaks down as test fields of finite frequency are considered. In this thesis, we consider the propagation of test fields on spacetime, beyond the leading-order geometrical optics approximation. By performing a covariant Wentzel-Kramers-Brillouin analysis for test fields, we show how higher-order corrections to the geometrical optics approximation can be considered. The higher-order corrections are related to the dynamics of the spin internal degree of freedom of the considered test field. We obtain an effective point-particle description, which contains spin-dependent corrections to the geodesic motion obtained using geometrical optics. This represents a covariant generalization of the well-known spin Hall effect, usually encountered in condensed matter physics and in optics. Our analysis is applied to electromagnetic and massive Dirac test fields, but it can easily be extended to other fields, such as linearized gravity. In the electromagnetic case, we present several examples where the gravitational spin Hall effect of light plays an important role. These include the propagation of polarized light rays on black hole spacetimes and cosmological spacetimes, as well as polarization-dependent effects on the shape of black hole shadows. Furthermore, we show that our effective point-particle equations for polarized light rays reproduce well-known results, such as the spin Hall effect of light in an inhomogeneous medium, and the relativistic Hall effect of polarized electromagnetic wave packets encountered in Minkowski spacetime. N2 - Unser grundlegendes Verständnis des Universums basiert auf Einsteins allgemeiner Relativitätstheorie, die eine Beschreibung in Form einer vierdimensional gekrümmten Raumzeit liefert, in der die Anziehungskraft der Gravitation in der Krümmung der Raumzeit kodiert ist. Die überwiegende Mehrheit der experimentellen Tests, die Einsteins allgemeine Relativitätstheorie bestätigt haben, basiert auf der Beobachtung elektromagnetischer Strahlung, die von entfernten astrophysikalischen Quellen wie Sternen oder Galaxien stammt. Daher ist ein tiefgreifendes Verständnis der Dynamik der sich in der Raumzeit ausbreitenden elektromagnetischen Strahlung von entscheidender Bedeutung. Elektromagnetische Phänomene werden durch Maxwell-Gleichungen beschrieben. Die Ausbreitung elektromagnetischer Strahlung in der Raumzeit ist jedoch sehr komplexe, und es ist im Allgemeinen nützlich, Näherungen zu betrachten, welche eine vereinfachte Beschreibung liefern. Auf diese Weise können die Haupteigenschaften des Systems in einem reduzierten Gleichungssystem codiert und die Gültigkeit der Näherung quantitativ kontrolliert werden. Beispielsweise kann die Ausbreitung elektromagnetischer Strahlung in der Raumzeit durch Anwendung der geometrischen Optik auf die Maxwell-Gleichungen beschrieben werden. Diese liefert ein Modell für die Ausbreitung elektromagnetischer Strahlung in Form von Lichtstrahlen, die sich auf dem kürzesten Weg zwischen zwei Punkten ausbreiten. Im Kontext von Einsteins allgemeiner Relativitätstheorie entsprechen dise Lichtstrahlen den Nullgeodäten der zugrunde liegenden gekrümmten Raumzeit. Für die meisten astrophysikalischen Anwendungen von Interesse, wie die Beobachtung von Himmelskörpern oder Gravitationslinsen, stellen die Näherungen der geometrischen Optik und damit die Beschreibung der Ausbreitung elektromagnetischer Strahlung durch Lichtstrahlen ein zufriedenstellendes theoretisches Modell dar. In dieser Arbeit untersuchen wir mögliche Korrekturen der Ausbreitung elektromagnetischer Strahlung in der Raumzeit, welche durch die Näherung der geometrischen Optik nicht erfasst werden. Solche Korrekturen sind aus der Optik bekannt, wo beobachtet wurde, dass die Ausbreitung von Lichtstrahlen in bestimmten Materialien durch die Polarisation des Lichts beeinflusst werden kann. Diese Korrekturen sind als Spin-Hall-Effekt von Licht bekannt. In dieser Arbeit wird gezeigt, dass ein ähnlicher Effekt für elektromagnetische Strahlung auftreten kann, welche sich in gekrümmter Raumzeit in der Nähe massiver astrophysikalischer Objekte wie Schwarzer Löcher oder Sterne ausbreitet. Darüber hinaus präsentieren wir, basierend auf der Dirac-Gleichung, eine ähnliche Analyse für die Bewegung von Elektronen in gekrümmten Raumzeiten. KW - spin Hall effect KW - gravitation KW - black hole KW - Schwarzes Loch KW - Gravitation KW - Spin Hall effekte Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-502293 ER - TY - THES A1 - Möhring, Jan T1 - Stochastic inversion for core field modeling using satellite data N2 - Magnetfeldmodellierung mit Kugelflächenfunktionen basiert auf der Inversion nach hunderten bis tausenden von Parametern. Dieses hochdimensionale Problem kann grundsätzlich als ein Optimierungsproblem formuliert werden, bei dem ein globales Minimum einer gewissen Zielfunktion berechnet werden soll. Um dieses Problem zu lösen, gibt es eine Reihe bekannter Ansätze, dazu zählen etwa gradientenbasierte Verfahren oder die Methode der kleinsten Quadrate und deren Varianten. Jede dieser Methoden hat verschiedene Vor- und Nachteile, beispielsweise bezüglich der Anwendbarkeit auf nicht-differenzierbare Funktionen oder der Laufzeit zugehöriger Algorithmen. In dieser Arbeit verfolgen wir das Ziel, einen Algorithmus zu finden, der schneller als die etablierten Verfahren ist und sich auch für nichtlineare Probleme anwenden lässt. Solche nichtlinearen Probleme treten beispielsweise bei der Abschätzung von Euler-Winkeln oder bei der Verwendung der robusteren L_1-Norm auf. Dazu untersuchen wir die Anwendbarkeit stochastischer Optimierungsverfahren aus der CMAES-Familie auf die Modellierung des geomagnetischen Feldes des Erdkerns. Es werden sowohl die Grundlagen der Kernfeldmodellierung und deren Parametrisierung anhand einiger Beispiele aus der Literatur besprochen, als auch die theoretischen Hintergründe der stochastischen Verfahren gegeben. Ein CMAES-Algorithmus wurde erfolgreich angewendet, um Daten der Swarm-Satellitenmission zu invertieren und daraus das Magnetfeldmodell EvoMag abzuleiten. EvoMag zeigt gute Übereinstimmung mit etablierten Modellen, sowie mit Observatoriumsdaten aus Niemegk. Wir thematisieren einige beobachtete Schwierigkeiten und präsentieren und diskutieren die Ergebnisse unserer Modellierung. N2 - Geomagnetic field modeling using spherical harmonics requires the inversion for hundreds to thousands of parameters. This large-scale problem can always be formulated as an optimization problem, where a global minimum of a certain cost function has to be calculated. A variety of approaches is known in order to solve this inverse problem, e.g. derivative-based methods or least-squares methods and their variants. Each of these methods has its own advantages and disadvantages, which affect for example the applicability to non-differentiable functions or the runtime of the corresponding algorithm. In this work, we pursue the goal to find an algorithm which is faster than the established methods and which is applicable to non-linear problems. Such non-linear problems occur for example when estimating Euler angles or when the more robust L_1 norm is applied. Therefore, we will investigate the usability of stochastic optimization methods from the CMAES family for modeling the geomagnetic field of Earth's core. On one hand, basics of core field modeling and their parameterization are discussed using some examples from the literature. On the other hand, the theoretical background of the stochastic methods are provided. A specific CMAES algorithm was successfully applied in order to invert data of the Swarm satellite mission and to derive the core field model EvoMag. The EvoMag model agrees well with established models and observatory data from Niemegk. Finally, we present some observed difficulties and discuss the results of our model. T2 - Stochastische Inversion für Kernfeldmodellierung mit Satellitendaten KW - Geomagnetismus KW - Kernfeldmodellierung KW - Optimierung KW - Evolutionsstrategien KW - Inverse Probleme KW - Geomagnetism KW - Core Field Modeling KW - Optimization KW - Evolution Strategies KW - Inverse Problems Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-498072 ER - TY - JOUR A1 - Bär, Christian A1 - Mazzeo, Rafe T1 - Manifolds with many Rarita-Schwinger fields JF - Communications in mathematical physics N2 - The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kahler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension. Y1 - 2021 U6 - https://doi.org/10.1007/s00220-021-04030-0 SN - 0010-3616 SN - 1432-0916 VL - 384 IS - 1 SP - 533 EP - 548 PB - Springer CY - Berlin ER - TY - THES A1 - Zadorozhnyi, Oleksandr T1 - Contributions to the theoretical analysis of the algorithms with adversarial and dependent data N2 - In this work I present the concentration inequalities of Bernstein's type for the norms of Banach-valued random sums under a general functional weak-dependency assumption (the so-called $\cC-$mixing). The latter is then used to prove, in the asymptotic framework, excess risk upper bounds of the regularised Hilbert valued statistical learning rules under the τ-mixing assumption on the underlying training sample. These results (of the batch statistical setting) are then supplemented with the regret analysis over the classes of Sobolev balls of the type of kernel ridge regression algorithm in the setting of online nonparametric regression with arbitrary data sequences. Here, in particular, a question of robustness of the kernel-based forecaster is investigated. Afterwards, in the framework of sequential learning, the multi-armed bandit problem under $\cC-$mixing assumption on the arm's outputs is considered and the complete regret analysis of a version of Improved UCB algorithm is given. Lastly, probabilistic inequalities of the first part are extended to the case of deviations (both of Azuma-Hoeffding's and of Burkholder's type) to the partial sums of real-valued weakly dependent random fields (under the type of projective dependence condition). KW - Machine learning KW - nonparametric regression KW - kernel methods KW - regularisation KW - concentration inequalities KW - learning rates KW - sequential learning KW - multi-armed bandits KW - Sobolev spaces Y1 - 2021 ER - TY - JOUR A1 - Gottwald, Georg A. A1 - Reich, Sebastian T1 - Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We present a supervised learning method to learn the propagator map of a dynamical system from partial and noisy observations. In our computationally cheap and easy-to-implement framework, a neural network consisting of random feature maps is trained sequentially by incoming observations within a data assimilation procedure. By employing Takens's embedding theorem, the network is trained on delay coordinates. We show that the combination of random feature maps and data assimilation, called RAFDA, outperforms standard random feature maps for which the dynamics is learned using batch data. Y1 - 2021 U6 - https://doi.org/10.1063/5.0066080 SN - 1054-1500 SN - 1089-7682 VL - 31 IS - 10 PB - AIP CY - Melville ER - TY - THES A1 - Maier, Corinna T1 - Bayesian data assimilation and reinforcement learning for model-informed precision dosing in oncology T1 - Bayes’sche Datenassimilation und Reinforcement Learning für die modellinformierte Präzisionsdosierung in der Onkologie N2 - While patients are known to respond differently to drug therapies, current clinical practice often still follows a standardized dosage regimen for all patients. For drugs with a narrow range of both effective and safe concentrations, this approach may lead to a high incidence of adverse events or subtherapeutic dosing in the presence of high patient variability. Model-informedprecision dosing (MIPD) is a quantitative approach towards dose individualization based on mathematical modeling of dose-response relationships integrating therapeutic drug/biomarker monitoring (TDM) data. MIPD may considerably improve the efficacy and safety of many drug therapies. Current MIPD approaches, however, rely either on pre-calculated dosing tables or on simple point predictions of the therapy outcome. These approaches lack a quantification of uncertainties and the ability to account for effects that are delayed. In addition, the underlying models are not improved while applied to patient data. Therefore, current approaches are not well suited for informed clinical decision-making based on a differentiated understanding of the individually predicted therapy outcome. The objective of this thesis is to develop mathematical approaches for MIPD, which (i) provide efficient fully Bayesian forecasting of the individual therapy outcome including associated uncertainties, (ii) integrate Markov decision processes via reinforcement learning (RL) for a comprehensive decision framework for dose individualization, (iii) allow for continuous learning across patients and hospitals. Cytotoxic anticancer chemotherapy with its major dose-limiting toxicity, neutropenia, serves as a therapeutically relevant application example. For more comprehensive therapy forecasting, we apply Bayesian data assimilation (DA) approaches, integrating patient-specific TDM data into mathematical models of chemotherapy-induced neutropenia that build on prior population analyses. The value of uncertainty quantification is demonstrated as it allows reliable computation of the patient-specific probabilities of relevant clinical quantities, e.g., the neutropenia grade. In view of novel home monitoring devices that increase the amount of TDM data available, the data processing of sequential DA methods proves to be more efficient and facilitates handling of the variability between dosing events. By transferring concepts from DA and RL we develop novel approaches for MIPD. While DA-guided dosing integrates individualized uncertainties into dose selection, RL-guided dosing provides a framework to consider delayed effects of dose selections. The combined DA-RL approach takes into account both aspects simultaneously and thus represents a holistic approach towards MIPD. Additionally, we show that RL can be used to gain insights into important patient characteristics for dose selection. The novel dosing strategies substantially reduce the occurrence of both subtherapeutic and life-threatening neutropenia grades in a simulation study based on a recent clinical study (CEPAC-TDM trial) compared to currently used MIPD approaches. If MIPD is to be implemented in routine clinical practice, a certain model bias with respect to the underlying model is inevitable, as the models are typically based on data from comparably small clinical trials that reflect only to a limited extent the diversity in real-world patient populations. We propose a sequential hierarchical Bayesian inference framework that enables continuous cross-patient learning to learn the underlying model parameters of the target patient population. It is important to note that the approach only requires summary information of the individual patient data to update the model. This separation of the individual inference from population inference enables implementation across different centers of care. The proposed approaches substantially improve current MIPD approaches, taking into account new trends in health care and aspects of practical applicability. They enable progress towards more informed clinical decision-making, ultimately increasing patient benefits beyond the current practice. N2 - Obwohl Patienten sehr unterschiedlich auf medikamentöse Therapien ansprechen, werden in der klinischen Praxis häufig noch standardisierte Dosierungsschemata angewendet. Bei Arzneimitteln mit engen therapeutischen Fenstern zwischen minimal wirksamen und toxischen Konzentrationen kann dieser Ansatz bei hoher interindividueller Variabilität zu häufigem Auftreten von Toxizitäten oder subtherapeutischen Konzentrationen führen. Die modellinformierte Präzisionsdosierung (MIPD) ist ein quantitativer Ansatz zur Dosisindividualisierung, der auf der mathematischen Modellierung von Dosis-Wirkungs-Beziehungen beruht und Daten aus dem therapeutischen Drug/Biomarker-Monitoring (TDM) einbezieht. Die derzeitigen MIPD-Ansätze verwenden entweder Dosierungstabellen oder einfache Punkt-Vorhersagen des Therapieverlaufs. Diesen Ansätzen fehlt eine Quantifizierung der Unsicherheiten, verzögerte Effekte werden nicht berücksichtigt und die zugrunde liegenden Modelle werden im Laufe der Anwendung nicht verbessert. Daher sind die derzeitigen Ansätze nicht ideal für eine fundierte klinische Entscheidungsfindung auf Grundlage eines differenzierten Verständnisses des individuell vorhergesagten Therapieverlaufs. Das Ziel dieser Arbeit ist es, mathematische Ansätze für das MIPD zu entwickeln, die (i) eine effiziente, vollständig Bayes’sche Vorhersage des individuellen Therapieverlaufs einschließlich der damit verbundenen Unsicherheiten ermöglichen, (ii) Markov-Entscheidungsprozesse mittels Reinforcement Learning (RL) in einen umfassenden Entscheidungsrahmen zur Dosisindividualisierung integrieren, und (iii) ein kontinuierliches Lernen zwischen Patienten erlauben. Die antineoplastische Chemotherapie mit ihrer wichtigen dosislimitierenden Toxizität, der Neutropenie, dient als therapeutisch relevantes Anwendungsbeispiel. Für eine umfassendere Therapievorhersage wenden wir Bayes’sche Datenassimilationsansätze (DA) an, um TDM-Daten in mathematische Modelle der Chemotherapie-induzierten Neutropenie zu integrieren. Wir zeigen, dass die Quantifizierung von Unsicherheiten einen großen Mehrwert bietet, da sie eine zuverlässige Berechnung der Wahrscheinlichkeiten relevanter klinischer Größen, z.B. des Neutropeniegrades, ermöglicht. Im Hinblick auf neue Home-Monitoring-Geräte, die die Anzahl der verfügbaren TDM-Daten erhöhen, erweisen sich sequenzielle DA-Methoden als effizienter und erleichtern den Umgang mit der Unsicherheit zwischen Dosierungsereignissen. Basierend auf Konzepten aus DA und RL, entwickeln wir neue Ansätze für MIPD. Während die DA-geleitete Dosierung individualisierte Unsicherheiten in die Dosisauswahl integriert, berücksichtigt die RL-geleitete Dosierung verzögerte Effekte der Dosisauswahl. Der kombinierte DA-RL-Ansatz vereint beide Aspekte und stellt somit einen ganzheitlichen Ansatz für MIPD dar. Zusätzlich zeigen wir, dass RL Informationen über die für die Dosisauswahl relevanten Patientencharakteristika liefert. Der Vergleich zu derzeit verwendeten MIPD Ansätzen in einer auf einer klinischen Studie (CEPAC-TDM-Studie) basierenden Simulationsstudie zeigt, dass die entwickelten Dosierungsstrategien das Auftreten subtherapeutischer Konzentrationen sowie lebensbedrohlicher Neutropenien drastisch reduzieren. Wird MIPD in der klinischen Routine eingesetzt, ist eine gewisse Modellverzerrung unvermeidlich. Die Modelle basieren in der Regel auf Daten aus vergleichsweise kleinen klinischen Studien, die die Heterogenität realer Patientenpopulationen nur begrenzt widerspiegeln. Wir schlagen einen sequenziellen hierarchischen Bayes’schen Inferenzrahmen vor, der ein kontinuierliches patientenübergreifendes Lernen ermöglicht, um die zugrunde liegenden Modellparameter der Ziel-Patientenpopulation zu erlernen. Zur Aktualisierung des Modells erfordert dieser Ansatz lediglich zusammenfassende Informationen der individuellen Patientendaten, was eine Umsetzung über verschiedene Versorgungszentren hinweg erlaubt. Die vorgeschlagenen Ansätze verbessern die derzeitigen MIPD-Ansätze erheblich, wobei neue Trends in der Gesundheitsversorgung und Aspekte der praktischen Anwendbarkeit berücksichtigt werden. Damit stellen sie einen Fortschritt in Richtung einer fundierteren klinischen Entscheidungsfindung dar. KW - data assimilation KW - Datenassimilation KW - reinforcement learning KW - model-informed precision dosing KW - pharmacometrics KW - oncology KW - modellinformierte Präzisionsdosierung KW - Onkologie KW - Pharmakometrie KW - Reinforcement Learning Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-515870 ER - TY - JOUR A1 - Clavier, Pierre J. T1 - Borel-Écalle resummation of a two-point function JF - Annales Henri Poincaré : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincaré and by the Swiss Physical Society N2 - We provide an overview of the tools and techniques of resurgence theory used in the Borel-ecalle resummation method, which we then apply to the massless Wess-Zumino model. Starting from already known results on the anomalous dimension of the Wess-Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and that its Borel transform is resurgent. The Schwinger-Dyson equation of the model is then used to prove an asymptotic exponential bound for the Borel transformed two-point function on a star-shaped domain of a suitable ramified complex plane. This proves that the two-point function of the Wess-Zumino model is Borel-ecalle summable. Y1 - 2021 U6 - https://doi.org/10.1007/s00023-021-01057-w SN - 1424-0637 SN - 1424-0661 VL - 22 IS - 6 SP - 2103 EP - 2136 PB - Springer CY - Cham ER - TY - THES A1 - Dahl, Dorothee Sophie T1 - Let's have FUN! Gamification im Mathematikunterricht N2 - Spiele und spieltypische Elemente wie das Sammeln von Treuepunkten sind aus dem Alltag kaum wegzudenken. Zudem werden sie zunehmend in Unternehmen oder in Lernumgebungen eingesetzt. Allerdings ist die Methode Gamification bisher für den pädagogischen Kontext wenig klassifiziert und für Lehrende kaum zugänglich gemacht worden. Daher zielt diese Bachelorarbeit darauf ab, eine systematische Strukturierung und Aufarbeitung von Gamification sowie innovative Ansätze für die Verwendung spieltypischer Elemente im Unterricht, konkret dem Mathematikunterricht, zu präsentieren. Dies kann eine Grundlage für andere Fachgebiete, aber auch andere Lehrformen bieten und so die Umsetzbarkeit von Gamification in eigenen Lehrveranstaltungen aufzeigen. In der Arbeit wird begründet, weshalb und mithilfe welcher Elemente Gamification die Motivation und Leistungsbereitschaft der Lernenden langfristig erhöhen, die Sozial- und Personalkompetenzen fördern sowie die Lernenden zu mehr Aktivität anregen kann. Zudem wird Gamification explizit mit grundlegenden mathematikdidaktischen Prinzipien in Verbindung gesetzt und somit die Relevanz für den Mathematikunterricht hervorgehoben. Anschließend werden die einzelnen Elemente von Gamification wie Punkte, Level, Abzeichen, Charaktere und Rahmengeschichte entlang einer eigens für den pädagogischen Kontext entwickelten Klassifikation „FUN“ (Feedback – User specific elements – Neutral elements) schematisch beschrieben, ihre Funktionen und Wirkung dargestellt sowie Einsatzmöglichkeiten im Unterricht aufgezeigt. Dies beinhaltet Ideen zu lernförderlichem Feedback, Differenzierungsmöglichkeiten und Unterrichtsrahmengestaltung, die in Lehrveranstaltungen aller Art umsetzbar sein können. Die Bachelorarbeit umfasst zudem ein spezifisches Beispiel, einen Unterrichtsentwurf einer gamifizierten Mathematikstunde inklusive des zugehörigen Arbeitsmaterials, anhand dessen die Verwendung von Gamification deutlich wird. Gamification offeriert oftmals Vorteile gegenüber dem traditionellen Unterricht, muss jedoch wie jede Methode an den Inhalt und die Zielgruppe angepasst werden. Weiterführende Forschung könnte sich mit konkreten motivationalen Strukturen, personenspezifischen Unterschieden sowie mit mathematischen Inhalten wie dem Problemlösen oder dem Wechsel zwischen verschiedenen Darstellungen hinsichtlich gamifizierter Lehrformen beschäftigen. N2 - Games and game-typical elements such as collecting points are an indispensable part of everyday life. In addition, they are used increasingly in companies or in learning environments. However, the method of gamification has been little classified for the pedagogical context and it has hardly been made accessible to teachers so far. Therefore, this bachelor’s thesis aims to present a systematic structure and reconditioning of gamification as well as innovative approaches for the implementation of game-typical elements in educational contexts, specifically in teaching mathematics. This thesis can provide a basis for other subject areas, but also for other forms of teaching and thus demonstrate the feasibility of gamification in own courses. The paper explains why and with which elements gamification can increase learners' motivation and willingness to perform in the long term, promote social and personal competences and encourage learners to become more active. Moreover, gamification is explicitly linked to basic mathematics didactic principles and thus emphasizes its relevance for mathematics teaching. Afterwards the individual elements of gamification such as points, levels, badges, characters and frame story are described schematically according to the classification “FUN” (Feedback – User specific elements – Neutral elements), developed especially for the educational context in the thesis. This includes ideas for learn-enhancing feedback, opportunities for differentiation and the design of teaching frameworks that can be implemented in courses of all kinds. The bachelor’s thesis also includes a specific example, a lesson plan for a gamified mathematics lesson including the associated working material, which illustrates the use of gamification. Gamification often offers advantages over traditional teaching, but like any method, it must be adapted to the content and the target group. Further research could focus on specific motivational structures, individual differences of students, and mathematical contents such as problem solving or changing representations regarding gamified teaching. KW - Gamification KW - Spiel KW - Motivation KW - Methode KW - Unterrichtsmethode KW - Feedback KW - Innovation KW - Lernen KW - Mathematikdidaktik KW - Mathematikunterricht KW - gamification KW - game KW - game-based KW - motivation KW - learning KW - feedback KW - method KW - teaching KW - teaching methods KW - didactics of mathematics Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-515937 ER - TY - THES A1 - Engelhardt, Max Angel Ronan T1 - Zwischen Simulation und Beweis - eine mathematische Analyse des Bienaymé-Galton-Watson-Prozesses und sein Einsatz innerhalb des Mathematikunterrichts T1 - Between simulation and proof - a mathematical analysis of the Bienaymé-Galton-Watson-process and its application in mathematics lessons N2 - Die Bienaymé-Galton-Watson Prozesse können für die Untersuchung von speziellen und sich entwickelnden Populationen verwendet werden. Die Populationen umfassen Individuen, welche sich identisch, zufällig, selbstständig und unabhängig voneinander fortpflanzen und die jeweils nur eine Generation existieren. Die n-te Generation ergibt sich als zufällige Summe der Individuen der (n-1)-ten Generation. Die Relevanz dieser Prozesse begründet sich innerhalb der Historie und der inner- und außermathematischen Bedeutung. Die Geschichte der Bienaymé-Galton-Watson-Prozesse wird anhand der Entwicklung des Konzeptes bis heute dargestellt. Dabei werden die Wissenschaftler:innen verschiedener Disziplinen angeführt, die Erkenntnisse zu dem Themengebiet beigetragen und das Konzept in ihren Fachbereichen angeführt haben. Somit ergibt sich die außermathematische Signifikanz. Des Weiteren erhält man die innermathematische Bedeutsamkeit mittels des Konzeptes der Verzweigungsprozesse, welches auf die Bienaymé-Galton-Watson Prozesse zurückzuführen ist. Die Verzweigungsprozesse stellen eines der aussagekräftigsten Modelle für die Beschreibung des Populationswachstums dar. Darüber hinaus besteht die derzeitige Wichtigkeit durch die Anwendungsmöglichkeit der Verzweigungsprozesse und der Bienaymé-Galton-Watson Prozesse innerhalb der Epidemiologie. Es werden die Ebola- und die Corona-Pandemie als Anwendungsfelder angeführt. Die Prozesse dienen als Entscheidungsstütze für die Politik und ermöglichen Aussagen über die Auswirkungen von Maßnahmen bezüglich der Pandemien. Neben den Prozessen werden ebenfalls der bedingte Erwartungswert bezüglich diskreter Zufallsvariablen, die wahrscheinlichkeitserzeugende Funktion und die zufällige Summe eingeführt. Die Konzepte vereinfachen die Beschreibung der Prozesse und bilden somit die Grundlage der Betrachtungen. Außerdem werden die benötigten und weiterführenden Eigenschaften der grundlegenden Themengebiete und der Prozesse aufgeführt und bewiesen. Das Kapitel erreicht seinen Höhepunkt bei dem Beweis des Kritikalitätstheorems, wodurch eine Aussage über das Aussterben des Prozesses in verschiedenen Fällen und somit über die Aussterbewahrscheinlichkeit getätigt werden kann. Die Fälle werden anhand der zu erwartenden Anzahl an Nachkommen eines Individuums unterschieden. Es zeigt sich, dass ein Prozess bei einer zu erwartenden Anzahl kleiner gleich Eins mit Sicherheit ausstirbt und bei einer Anzahl größer als Eins, die Population nicht in jedem Fall aussterben muss. Danach werden einzelne Beispiele, wie der linear fractional case, die Population von Fibroblasten (Bindegewebszellen) von Mäusen und die Entstehungsfragestellung der Prozesse, angeführt. Diese werden mithilfe der erlangten Ergebnisse untersucht und einige ausgewählte zufällige Dynamiken werden im nachfolgenden Kapitel simuliert. Die Simulationen erfolgen durch ein in Python erstelltes Programm und werden mithilfe der Inversionsmethode realisiert. Die Simulationen stellen beispielhaft die Entwicklungen in den verschiedenen Kritikalitätsfällen der Prozesse dar. Zudem werden die Häufigkeiten der einzelnen Populationsgrößen in Form von Histogrammen angebracht. Dabei lässt sich der Unterschied zwischen den einzelnen Fällen bestätigen und es wird die Anwendungsmöglichkeit der Bienaymé-Galton-Watson Prozesse bei komplexeren Problemen deutlich. Histogramme bekräftigen, dass die einzelnen Populationsgrößen nur endlich oft vorkommen. Diese Aussage wurde von Galton aufgeworfen und in der Extinktions-Explosions-Dichotomie verwendet. Die dargestellten Erkenntnisse über das Themengebiet und die Betrachtung des Konzeptes werden mit einer didaktischen Analyse abgeschlossen. Die Untersuchung beinhaltet die Berücksichtigung der Fundamentalen Ideen, der Fundamentalen Ideen der Stochastik und der Leitidee „Daten und Zufall“. Dabei ergibt sich, dass in Abhängigkeit der gewählten Perspektive die Anwendung der Bienaymé-Galton-Watson Prozesse innerhalb der Schule plausibel ist und von Vorteil für die Schüler:innen sein kann. Für die Behandlung wird exemplarisch der Rahmenlehrplan für Berlin und Brandenburg analysiert und mit dem Kernlehrplan Nordrhein-Westfalens verglichen. Die Konzeption des Lehrplans aus Berlin und Brandenburg lässt nicht den Schluss zu, dass die Bienaymé-Galton-Watson Prozesse angewendet werden sollten. Es lässt sich feststellen, dass die zugrunde liegende Leitidee nicht vollumfänglich mit manchen Fundamentalen Ideen der Stochastik vereinbar ist. Somit würde eine Modifikation hinsichtlich einer stärkeren Orientierung des Lehrplans an den Fundamentalen Ideen die Anwendung der Prozesse ermöglichen. Die Aussage wird durch die Betrachtung und Übertragung eines nordrhein-westfälischen Unterrichtsentwurfes für stochastische Prozesse auf die Bienaymé-Galton-Watson Prozesse unterstützt. Darüber hinaus werden eine Concept Map und ein Vernetzungspentagraph nach von der Bank konzipiert um diesen Aspekt hervorzuheben. N2 - The Bienaymé-Galton-Watson processes can be used to study special and developing populations. These populations include individuals that reproduce identically, randomly, separately, independently of each other, and which exist only for one generation. The n-th generation is the random sum of the individuals of the (n-1)-th generation. The relevance of these processes is based on their history and their significance in mathematical and extra-mathematical contexts. The history of the Bienaymé-Galton-Watson processes is illustrated by the development of the concept to the present day. Various scientists from different disciplines who have contributed to the topic in their respective fields are listed. This illustrates moreover the significance in extra-mathematical contexts. Furthermore, the inner- mathematical magnitude is obtained by means of the superordinate concept of branching processes, which can be traced back to the Bienaymé-Galton-Watson processes. These branching processes are one of the most significant models for describing population growth. In addition, the current importance arises from the applicability of branching processes and the Bienaymé-Galton-Watson processes within epidemiology. The Ebola and Corona pandemics are mentioned as fields of application. The processes serve as a basis for political decision-making and enable statements made on the impact of pandemic measures. In addition to the processes, the conditional expectation value for discrete random variables, the probability generating function and the random sum are also introduced. These concepts simplify the description of the processes and thus form the basis of the considerations. Also, the required and further properties of the basic topics and processes are listed and demonstrated. The chapter reaches its climax with the proof of the criticality theorem, whereby a statement can be made about the extinction of the process in different cases and thus about the extinction probability. These cases are distinguished based on the expected number of offspring from the individuals. It turns out that a process with an expected number of less than one certainly becomes extinct. On the contrary, a process with a number greater than one does not necessarily has to die out. Individual examples are then given, such as the linear fractional case, the population of fibroblasts (connective tissue cells) of mice and the question of origin. These are investigated using the results obtained and some selected random dynamics are simulated in the following chapter. The simulations are carried out by a Python self-written program and are realized using the inversion method. These simulations exemplify the developments in the different criticality cases of the processes. Besides, the frequencies of the individual population sizes are displayed in the form of histograms. The difference between the individual cases can be confirmed and the analysis of the fibroblasts reveals the applicability of the Bienaymé-Galton-Watson processes to more complex problems. Histograms confirm that the individual population sizes occur only finitely often. This statement was raised by Galton and is used in the extinction-explosion dichotomy. The presented findings about the topic and the consideration of the concept are concluded with an analysis of didactic-background. This involves the fundamental ideas, the fundamental ideas of stochastics and the guiding idea of data and chance. Depending on the chosen perspective, the use of the Bienaymé-Galton-Watson processes within the school is plausible and may be beneficial for the students. For the treatment, the Rahmenlehrplan for Berlin and Brandenburg is analysed and compared with the core curriculum of Nord Rhine-Westphalia as an example. The design of the curriculum of Berlin and Brandenburg does not allow the conclusion of applying the Bienaymé-Galton-Watson processes. It can be seen that the underlying guiding idea is not fully compatible with some fundamental ideas of stochastics. Thus, a modification to the curriculum more oriented towards these fundamental ideas would allow the application of the processes. This statement is supported by the observation and transfer of a North Rhine-Westphalian teaching design for stochastic processes to the Bienaymé-Galton-Watson processes by means of chain letters. In addition, a concept map and a Vernetzungspentagraph by von der Bank are designed to highlight this aspect. KW - Bienaymé-Galton-Watson Prozess KW - Kritikalitätstheorem KW - Verzweigungsprozess KW - Populationen KW - linear fractional case KW - bedingter Erwartungswert KW - zufällige Summe KW - Simulation KW - wahrscheinlichkeitserzeugende Funktion KW - Historie der Verzweigungsprozesse KW - Instabilität des Prozesses KW - Aussterbewahrscheinlichkeit KW - Geometrische Reproduktionsverteilung KW - Fibroblasten KW - Entstehungsfragestellung KW - Fundamentale Ideen KW - Leitidee „Daten und Zufall“ KW - Rahmenlehrplan KW - Markov-Ketten KW - Corona KW - Bienaymé-Galton-Watson process KW - criticality theorem KW - branching process KW - populations KW - linear fractional case KW - conditional expectation value KW - random sum KW - simulation KW - probability generating function KW - history of branching processes KW - instability of the process KW - extinction probability KW - geometric reproduction distribution KW - fibroblasts KW - question of origin KW - fundamental ideas KW - guiding idea “Daten und Zufall” KW - Rahmenlehrplan KW - Markov chains KW - Corona Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-524474 ER - TY - THES A1 - Perera, Upeksha T1 - Solutions of direct and inverse Sturm–Liouville problems T1 - Lösungen von direkten und inversen Sturm-Liouville-Problemen N2 - 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 up to order eight), with a mix of boundary conditions (including non-separable and finite singular endpoints), 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. Next, a concrete implementation to the inverse Sturm–Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm–Liouville problems of order n=2,4 is verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides methods that can be adapted successfully for solving a direct (regular/singular) or inverse SLP of an arbitrary order with arbitrary boundary conditions. N2 - Die Lie-Gruppen-Methode in Kombination mit der Magnus-Expansion wird verwendet, um eine universelle Methode zu entwickeln, die zur Lösung eines Sturm-Liouville-Problems (SLP) beliebiger Ordnung mit beliebigen Randbedingungen anwendbar ist. Es wird gezeigt, dass die Methode in der Lage ist, direkte reguläre und einige singuläre SLPs gerader Ordnung (getestet bis zur 8. Ordnung) mit einer Mischung von Randbedingungen (einschließlich nicht trennbarer und endlicher singulärer Endpunkte) genau und effizient zu lösen. Die vorliegende Technik wird erfolgreich angewendet, um die Schwierigkeiten beim Finden geeigneter Sätze von Eigenwerten zu überwinden, so dass das inverse SLP-Problem effektiv gelöst werden kann. Als nächstes wird eine konkrete Implementierung des von Barcilon (1974) vorgeschlagenen inversen Sturm-Liouville-Problemalgorithmus bereitgestellt. Weiterhin wird die rechnerische Durchführbarkeit und Anwendbarkeit dieses Algorithmus zur Lösung inverser Sturm-Liouville-Probleme der Ordnung n=2,4 erfolgreich verifiziert. Es wird beobachtet, dass das Verfahren selbst bei Vorhandensein von signifikantem Rauschen erfolgreich ist, vorausgesetzt, dass die Annahmen des Algorithmus erfüllt sind. Zusammenfassend stellt diese Arbeit Methoden zur Verfügung, die erfolgreich zur Lösung eines direkten (regulär/singulären) oder inversen SLP beliebiger Ordnung mit beliebigen Randbedingungen angepasst werden können. KW - Sturm-Liouville problem KW - Inverse Sturm-Liouville problem KW - Higher-order Sturm-Liouville problem KW - Sturm-Liouville-Problem höherer Ordnung KW - Inverses Sturm-Liouville-Problem KW - Sturm-Liouville-Problem Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-530064 ER - TY - JOUR A1 - Roos, Saskia A1 - Otoba, Nobuhiko T1 - Scalar curvature and the multiconformal class of a direct product Riemannian manifold JF - Geometriae dedicata N2 - For a closed, connected direct product Riemannian manifold (M, g) = (M-1, g(1)) x ... x (M-l, g(l)), we define its multiconformal class [[g]] as the totality {integral(2)(1)g(1) circle plus center dot center dot center dot integral(2)(l)g(l)} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a positive function fi on the total space M. A multiconformal class [[ g]] contains not only all warped product type deformations of g but also the whole conformal class [(g) over tilde] of every (g) over tilde is an element of[[ g]]. In this article, we prove that [[g]] contains a metric of positive scalar curvature if and only if the conformal class of some factor (Mi, gi) does, under the technical assumption dim M-i = 2. We also show that, even in the case where every factor (M-i, g(i)) has positive scalar curvature, [[g]] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided l = 2 and dim M = 3. KW - Positive scalar curvature KW - Constant scalar curvature KW - The Yamabe KW - problem KW - Warped product KW - Umbilic product KW - Twisted product Y1 - 2021 U6 - https://doi.org/10.1007/s10711-021-00636-9 SN - 0046-5755 SN - 1572-9168 VL - 214 IS - 1 SP - 801 EP - 829 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Gottwald, Georg A. A1 - Reich, Sebastian T1 - Supervised learning from noisy observations BT - Combining machine-learning techniques with data assimilation JF - Physica : D, Nonlinear phenomena N2 - Data-driven prediction and physics-agnostic machine-learning methods have attracted increased interest in recent years achieving forecast horizons going well beyond those to be expected for chaotic dynamical systems. In a separate strand of research data-assimilation has been successfully used to optimally combine forecast models and their inherent uncertainty with incoming noisy observations. The key idea in our work here is to achieve increased forecast capabilities by judiciously combining machine-learning algorithms and data assimilation. We combine the physics-agnostic data -driven approach of random feature maps as a forecast model within an ensemble Kalman filter data assimilation procedure. The machine-learning model is learned sequentially by incorporating incoming noisy observations. We show that the obtained forecast model has remarkably good forecast skill while being computationally cheap once trained. Going beyond the task of forecasting, we show that our method can be used to generate reliable ensembles for probabilistic forecasting as well as to learn effective model closure in multi-scale systems. (C) 2021 Elsevier B.V. All rights reserved. KW - Data-driven modelling KW - Random feature maps KW - Data assimilation Y1 - 2021 U6 - https://doi.org/10.1016/j.physd.2021.132911 SN - 0167-2789 SN - 1872-8022 VL - 423 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Redmann, Martin A1 - Freitag, Melina A. T1 - Optimization based model order reduction for stochastic systems JF - Applied mathematics and computation N2 - In this paper, we bring together the worlds of model order reduction for stochastic linear systems and H-2-optimal model order reduction for deterministic systems. In particular, we supplement and complete the theory of error bounds for model order reduction of stochastic differential equations. With these error bounds, we establish a link between the output error for stochastic systems (with additive and multiplicative noise) and modified versions of the H-2-norm for both linear and bilinear deterministic systems. When deriving the respective optimality conditions for minimizing the error bounds, we see that model order reduction techniques related to iterative rational Krylov algorithms (IRKA) are very natural and effective methods for reducing the dimension of large-scale stochastic systems with additive and/or multiplicative noise. We apply modified versions of (linear and bilinear) IRKA to stochastic linear systems and show their efficiency in numerical experiments. KW - Model order reduction KW - Stochastic systems KW - Optimality conditions KW - Sylvester equations KW - Levy process Y1 - 2021 U6 - https://doi.org/10.1016/j.amc.2020.125783 SN - 0096-3003 SN - 1873-5649 VL - 398 PB - Elsevier CY - New York ER - TY - JOUR A1 - Ruchi, Sangeetika A1 - Dubinkina, Svetlana A1 - Wiljes, Jana de T1 - Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation JF - Nonlinear processes in geophysics / European Geosciences Union ; American Geophysical Union N2 - Identification of unknown parameters on the basis of partial and noisy data is a challenging task, in particular in high dimensional and non-linear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust and computationally cheap and often produce astonishingly accurate estimations despite the simplifying underlying assumptions. Yet there is a lot of room for improvement, specifically regarding a correct approximation of a non-Gaussian posterior distribution. The tempered ensemble transform particle filter is an adaptive Sequential Monte Carlo (SMC) method, whereby resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion, it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity, and the method is not as robust as ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy-inspired regularisation factor to the underlying optimal transport problem that allows the high computational cost to be considerably reduced via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as a hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov chain Monte Carlo methods results are computed as a benchmark. Y1 - 2021 U6 - https://doi.org/10.5194/npg-28-23-2021 SN - 1023-5809 SN - 1607-7946 VL - 28 IS - 1 SP - 23 EP - 41 PB - Copernicus CY - Göttingen ER - TY - JOUR A1 - Shlapunov, Alexander A1 - Tarchanov, Nikolaj Nikolaevič T1 - An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n JF - Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija N2 - We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations. KW - Navier-Stokes equations KW - de Rham complex KW - open mapping theorem Y1 - 2021 U6 - https://doi.org/10.33048/semi.2021.18.108 SN - 1813-3304 VL - 18 IS - 2 SP - 1433 EP - 1466 PB - Institut Matematiki Imeni S. L. Soboleva CY - Novosibirsk ER - TY - JOUR A1 - Beckus, Siegfried A1 - Eliaz, Latif T1 - Eigenfunctions growth of R-limits on graphs JF - Journal of spectral theory / European Mathematical Society N2 - A characterization of the essential spectrum of Schrodinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on N and Z(d) as "right-limits," captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in sigma(ss)(H) corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds. KW - Essential spectrum KW - Schrodinger operators KW - graphs KW - right limits KW - generalized eigenfunctions Y1 - 2021 U6 - https://doi.org/10.4171/JST/389 SN - 1664-039X SN - 1664-0403 VL - 11 IS - 4 SP - 1895 EP - 1933 PB - EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut für Mathematik, Technische Universität CY - Berlin ER - TY - JOUR A1 - Engbert, Ralf A1 - Rabe, Maximilian Michael A1 - Kliegl, Reinhold A1 - Reich, Sebastian T1 - Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics JF - Bulletin of mathematical biology : official journal of the Society for Mathematical Biology N2 - Newly emerging pandemics like COVID-19 call for predictive models to implement precisely tuned responses to limit their deep impact on society. Standard epidemic models provide a theoretically well-founded dynamical description of disease incidence. For COVID-19 with infectiousness peaking before and at symptom onset, the SEIR model explains the hidden build-up of exposed individuals which creates challenges for containment strategies. However, spatial heterogeneity raises questions about the adequacy of modeling epidemic outbreaks on the level of a whole country. Here, we show that by applying sequential data assimilation to the stochastic SEIR epidemic model, we can capture the dynamic behavior of outbreaks on a regional level. Regional modeling, with relatively low numbers of infected and demographic noise, accounts for both spatial heterogeneity and stochasticity. Based on adapted models, short-term predictions can be achieved. Thus, with the help of these sequential data assimilation methods, more realistic epidemic models are within reach. KW - Stochastic epidemic model KW - Sequential data assimilation KW - Ensemble Kalman KW - filter KW - COVID-19 Y1 - 2020 U6 - https://doi.org/10.1007/s11538-020-00834-8 SN - 0092-8240 SN - 1522-9602 VL - 83 IS - 1 PB - Springer CY - New York ER - TY - JOUR A1 - Schanner, Maximilian Arthus A1 - Mauerberger, Stefan A1 - Korte, Monika A1 - Holschneider, Matthias T1 - Correlation based time evolution of the archeomagnetic field JF - Journal of geophysical research : JGR ; an international quarterly. B, Solid earth N2 - In a previous study, a new snapshot modeling concept for the archeomagnetic field was introduced (Mauerberger et al., 2020, ). By assuming a Gaussian process for the geomagnetic potential, a correlation-based algorithm was presented, which incorporates a closed-form spatial correlation function. This work extends the suggested modeling strategy to the temporal domain. A space-time correlation kernel is constructed from the tensor product of the closed-form spatial correlation kernel with a squared exponential kernel in time. Dating uncertainties are incorporated into the modeling concept using a noisy input Gaussian process. All but one modeling hyperparameters are marginalized, to reduce their influence on the outcome and to translate their variability to the posterior variance. The resulting distribution incorporates uncertainties related to dating, measurement and modeling process. Results from application to archeomagnetic data show less variation in the dipole than comparable models, but are in general agreement with previous findings. Y1 - 2021 U6 - https://doi.org/10.1029/2020JB021548 SN - 2169-9313 SN - 2169-9356 VL - 126 IS - 7 PB - American Geophysical Union CY - Washington ER - TY - JOUR A1 - Reich, Sebastian A1 - Weissmann, Simon T1 - Fokker-Planck particle systems for Bayesian inference: computational approaches JF - SIAM ASA journal on uncertainty quantification N2 - Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies. KW - Bayesian inverse problems KW - Fokker-Planck equation KW - gradient flow KW - affine KW - invariance KW - gradient-free sampling methods KW - localization Y1 - 2021 U6 - https://doi.org/10.1137/19M1303162 SN - 2166-2525 VL - 9 IS - 2 SP - 446 EP - 482 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Cozzoni, Barbara A1 - Maibaum, Michael A1 - Hamm, Maximilian T1 - Thermal analysis and constraints for the MASCOT landing site selection on the asteroid Ryugu JF - Planetary and space science N2 - In June 2018, after 4 years of cruise, the Japanese space probe Hayabusa2 [1-Watanabe S. et al.: Hayabusa2 Mission Overview. (2017)] reached the Near-Earth Asteroid (162173) Ryugu. Hayabusa2 carried a small Lander named MASCOT (Mobile Asteroid Surface Scout) [2-Ho T. M. et al.: MASCOT-The Mobile Asteroid Surface Scout onboard the Hayabusa2 mission. (2017)], jointly developed by the German Aerospace Center (DLR) and the French Space Agency (CNES), to investigate Ryugu's surface structure, composition and physical properties including its thermal behaviour and magnetization in-situ. The Microgravity User Support Centre (DLR-MUSC) in Cologne was in charge of providing all thermal conditions and constraints necessary for the selection of the final landing site and for the final operations of the Lander MASCOT on the surface of the asteroid Ryugu. This article provides a comprehensive assessment of these thermal conditions and constraints, based on predictions performed with the Thermal Mathematical Model (TMM) of MASCOT using different asteroid surface thermal models, ephemeris data for approach as well as descent and hopping trajectories, the related operation sequences and scenarios and the possible environmental conditions driven by the Hayabusa2 spacecraft. A comparison with the real telemetry data confirms the analysis and provides further information about the asteroid characteristics. KW - MASCOT KW - Thermal mathematical model KW - Landing site selection KW - Small KW - spacecraft operations Y1 - 2021 U6 - https://doi.org/10.1016/j.pss.2021.105286 SN - 0032-0633 SN - 1873-5088 VL - 205 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Hastermann, Gottfried A1 - Reinhardt, Maria A1 - Klein, Rupert A1 - Reich, Sebastian T1 - Balanced data assimilation for highly oscillatory mechanical systems JF - Communications in applied mathematics and computational science : CAMCoS N2 - Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, this filter also has limitations due to its inherent assumptions of Gaussianity and linearity, which can manifest themselves in the form of dynamically inconsistent state estimates. This issue is investigated here for balanced, slowly evolving solutions to highly oscillatory Hamiltonian systems which are prototypical for applications in numerical weather prediction. It is demonstrated that the standard ensemble Kalman filter can lead to state estimates that do not satisfy the pertinent balance relations and ultimately lead to filter divergence. Two remedies are proposed, one in terms of blended asymptotically consistent time-stepping schemes, and one in terms of minimization-based postprocessing methods. The effects of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for balanced motions of two prototypical Hamiltonian reference systems. KW - data assimilation KW - ensemble Kalman filter KW - balanced dynamics KW - highly KW - oscillatory systems KW - Hamiltonian dynamics KW - geophysics Y1 - 2021 U6 - https://doi.org/10.2140/camcos.2021.16.119 SN - 1559-3940 SN - 2157-5452 VL - 16 IS - 1 SP - 119 EP - 154 PB - Mathematical Sciences Publishers CY - Berkeley ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Hounnon, Hippolyte T1 - Partial Menger algebras of terms JF - Asian-European journal of mathematics N2 - The superposition operation S-n,S-A, n >= 1, n is an element of N, maps to each (n + 1)-tuple of n-ary operations on a set A an n-ary operation on A and satisfies the so-called superassociative law, a generalization of the associative law. The corresponding algebraic structures are Menger algebras of rank n. A partial algebra of type (n + 1) which satisfies the superassociative law as weak identity is said to be a partial Menger algebra of rank n. As a generalization of linear terms we define r-terms as terms where each variable occurs at most r-times. It will be proved that n-ary r-terms form partial Menger algebras of rank n. In this paper, some algebraic properties of partial Menger algebras such as generating systems, homomorphic images and freeness are investigated. As generalization of hypersubstitutions and linear hypersubstitutions we consider r-hypersubstitutions.U KW - n-ary operation KW - n-ary term KW - superposition of n-ary operations and n-ary KW - terms KW - linear term KW - r-term KW - Menger algebra of rank n KW - partial Menger KW - algebra of rank n KW - r-hypersubstitution Y1 - 2021 U6 - https://doi.org/10.1142/S1793557121500923 SN - 1793-5571 SN - 1793-7183 VL - 14 IS - 06 PB - World Scientific CY - Singapore ER - TY - THES A1 - Hübner, Andrea T1 - Ein multityper Verzweigungsprozess als Modell zur Untersuchung der Ausbreitung von Covid-19 T1 - Modeling the spread of Covid-19 using a multitype branching process N2 - Im Zuge der Covid-19 Pandemie werden zwei Werte täglich diskutiert: Die zuletzt gemeldete Zahl der neu Infizierten und die sogenannte Reproduktionsrate. Sie gibt wieder, wie viele weitere Menschen ein an Corona erkranktes Individuum im Durchschnitt ansteckt. Für die Schätzung dieses Wertes gibt es viele Möglichkeiten - auch das Robert Koch-Institut gibt in seinem täglichen Situationsbericht stets zwei R-Werte an: Einen 4-Tage-R-Wert und einen weniger schwankenden 7-Tage-R-Wert. Diese Arbeit soll eine weitere Möglichkeit vorstellen, einige Aspekte der Pandemie zu modellieren und die Reproduktionsrate zu schätzen. In der ersten Hälfte der Arbeit werden die mathematischen Grundlagen vorgestellt, die man für die Modellierung benötigt. Hierbei wird davon ausgegangen, dass der Leser bereits ein Basisverständnis von stochastischen Prozessen hat. Im Abschnitt Grundlagen werden Verzweigungsprozesse mit einigen Beispielen eingeführt und die Ergebnisse aus diesem Themengebiet, die für diese Arbeit wichtig sind, präsentiert. Dabei gehen wir zuerst auf einfache Verzweigungsprozesse ein und erweitern diese dann auf Verzweigungsprozesse mit mehreren Typen. Um die Notation zu erleichtern, beschränken wir uns auf zwei Typen. Das Prinzip lässt sich aber auf eine beliebige Anzahl von Typen erweitern. Vor allem soll die Wichtigkeit des Parameters λ herausgestellt werden. Dieser Wert kann als durchschnittliche Zahl von Nachfahren eines Individuums interpretiert werden und bestimmt die Dynamik des Prozesses über einen längeren Zeitraum. In der Anwendung auf die Pandemie hat der Parameter λ die gleiche Rolle wie die Reproduktionsrate R. In der zweiten Hälfte dieser Arbeit stellen wir eine Anwendung der Theorie über Multitype Verzweigungsprozesse vor. Professor Yanev und seine Mitarbeiter modellieren in ihrer Veröffentlichung Branching stochastic processes as models of Covid-19 epidemic development die Ausbreitung des Corona Virus' über einen Verzweigungsprozess mit zwei Typen. Wir werden dieses Modell diskutieren und Schätzer daraus ableiten: Ziel ist es, die Reproduktionsrate zu ermitteln. Außerdem analysieren wir die Möglichkeiten, die Dunkelziffer (die Zahl nicht gemeldeter Krankheitsfälle) zu schätzen. Wir wenden die Schätzer auf die Zahlen von Deutschland an und werten diese schließlich aus. N2 - During the Covid-19 pandemic, the discussion about the situation has been dominated by two numbers: the number of daily new infected individuals and the reproduction rate. The latter is the average number of people, one infected individual will infect with the disease. Because the number of registered infected individuals is generally not equal to the actual number of people who carry the Corona virus, many facts about the pandemic have to be estimated and can not be known for certain. Since the reproduction rate is an important parameter to signify the course of the Pandemic, many ways to estimate it have been developed. The Institute of Robert Koch in Germany uses two reproduction rates R in their daily reports: The 4-days-R-value and the less fluctuating 7-days-Rvalue. This master thesis will develop another model to estimate the R-value and other interesting aspects of the pandemic. The first part of this thesis is dedicated to the mathematical foundations needed to understand the model. The reader is expected to already have basic understanding of stochastic processes. In the section Grundlagen we will discuss branching processes and present the results of their theory that are important for our work. We start by introducing simple branching processes and expand the results to multitype branching processes. In service of a simpler notation we will only consider twotype branching processes, but the results can be used for any number of types. The importance of the parameter λ shall be stressed. It can be seen as the average number of descendants of one individual and dictates the dynamic of the process over a long period of time. Applied to the modeling of the pandemic, λ plays the same role as the reproduction rate R. In the second part of this thesis will present an application of the previously developed theory about multitype branching processes. Prof. Yanev and his colleagues modeled in their publication Branching stochastic processes as models of Covid-19 epidemic development the spreading of the Corona virus by using a branching process with two types. We will discuss this model and deduce estimators from it. We want to estimate the reproduction rate and find a way to determine the number of not registered infected individuals. The estimators will be applied to the data from Germany and we will discuss the results. KW - Covid-19 KW - Corona KW - Reproduktionsrate KW - Verzweigungsprozess KW - Modellierung KW - Covid-19 KW - corona virus KW - reproduction rate KW - branching process KW - modeling Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-509225 ER - TY - JOUR A1 - De Oliveira Gomes, André A1 - Högele, Michael Anton T1 - The Kramers problem for SDEs driven by small, accelerated Lévy noise with exponentially light jumps JF - Stochastics and dynamics N2 - We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state 0, say, subject to a perturbation by a stochastic integral which is driven by an epsilon-small and (1/epsilon)-accelerated Levy process with exponentially light jumps. For this purpose, we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel, we solve the associated asymptotic first escape problem from the bounded neighborhood of 0 in the limit as epsilon -> 0 which is also known as the Kramers problem in the literature. KW - Freidlin-Wentzell theory KW - large deviations principle KW - accelerated small KW - noise Levy diffusions KW - first passage times KW - first exit location KW - strongly tempered stable Levy measure Y1 - 2021 U6 - https://doi.org/10.1142/S0219493721500192 SN - 0219-4937 SN - 1793-6799 VL - 21 IS - 04 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Garmendia, Alfonso A1 - Zambon, Marco T1 - Quotients of singular foliations and Lie 2-group actions JF - Journal of noncommutative geometry N2 - Androulidakis-Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold (M, FM ) is the quotient of a foliated manifold (P, FP ) along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid. KW - Lie groupoid KW - singular foliation KW - fibration Y1 - 2021 U6 - https://doi.org/10.4171/JNCG/434 SN - 1661-6952 SN - 1661-6960 VL - 15 IS - 4 SP - 1251 EP - 1283 PB - EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut für Mathematik, Technische Universität Berlin CY - Berlin ER - TY - JOUR A1 - Schick, Thomas A1 - Seyedhosseini, Mehran T1 - On an index theorem of Chang, Weinberger and Yu JF - Münster journal of mathematics N2 - In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that (a slight variation of) the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger and Yu (rather: a slight variation). To take the fundamental groups of the manifold and its boundary into account requires working with maximal C*-completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions. On the other hand, we introduce and propose a more conceptual and more geometric completion, which still has all the required functoriality. Y1 - 2021 U6 - https://doi.org/10.17879/59019522628 SN - 1867-5778 SN - 1867-5786 VL - 14 IS - 1 SP - 123 EP - 154 PB - WWU, Fachbereich Mathematik und Informatik CY - Münster ER - TY - JOUR A1 - Hartung, Niklas A1 - Wahl, Martin A1 - Rastogi, Abhishake A1 - Huisinga, Wilhelm T1 - Nonparametric goodness-of-fit testing for parametric covariate models in pharmacometric analyses JF - CPT: pharmacometrics & systems pharmacology N2 - The characterization of covariate effects on model parameters is a crucial step during pharmacokinetic/pharmacodynamic analyses. Although covariate selection criteria have been studied extensively, the choice of the functional relationship between covariates and parameters, however, has received much less attention. Often, a simple particular class of covariate-to-parameter relationships (linear, exponential, etc.) is chosen ad hoc or based on domain knowledge, and a statistical evaluation is limited to the comparison of a small number of such classes. Goodness-of-fit testing against a nonparametric alternative provides a more rigorous approach to covariate model evaluation, but no such test has been proposed so far. In this manuscript, we derive and evaluate nonparametric goodness-of-fit tests for parametric covariate models, the null hypothesis, against a kernelized Tikhonov regularized alternative, transferring concepts from statistical learning to the pharmacological setting. The approach is evaluated in a simulation study on the estimation of the age-dependent maturation effect on the clearance of a monoclonal antibody. Scenarios of varying data sparsity and residual error are considered. The goodness-of-fit test correctly identified misspecified parametric models with high power for relevant scenarios. The case study provides proof-of-concept of the feasibility of the proposed approach, which is envisioned to be beneficial for applications that lack well-founded covariate models. Y1 - 2021 U6 - https://doi.org/10.1002/psp4.12614 SN - 2163-8306 VL - 10 IS - 6 SP - 564 EP - 576 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - Matzka, Jürgen A1 - Stolle, Claudia A1 - Yamazaki, Yosuke A1 - Bronkalla, Oliver A1 - Morschhauser, Achim T1 - The geomagnetic Kp index and derived indices of geomagnetic activity JF - Space weather : the international journal of research and applications N2 - The geomagnetic Kp index is one of the most extensively used indices of geomagnetic activity, both for scientific and operational purposes. This article reviews the properties of the Kp index and provides a reference for users of the Kp index and associated data products as derived and distributed by the GFZ German Research Centre for Geosciences. The near real-time production of the nowcast Kp index is of particular interest for space weather services and here we describe and evaluate its current setup. Y1 - 2021 U6 - https://doi.org/10.1029/2020SW002641 SN - 1542-7390 VL - 19 IS - 5 PB - Wiley CY - New York ER - TY - JOUR A1 - Schindler, Daniel A1 - Moldenhawer, Ted A1 - Stange, Maike A1 - Lepro, Valentino A1 - Beta, Carsten A1 - Holschneider, Matthias A1 - Huisinga, Wilhelm T1 - Analysis of protrusion dynamics in amoeboid cell motility by means of regularized contour flows JF - PLoS Computational Biology : a new community journal N2 - Amoeboid cell motility is essential for a wide range of biological processes including wound healing, embryonic morphogenesis, and cancer metastasis. It relies on complex dynamical patterns of cell shape changes that pose long-standing challenges to mathematical modeling and raise a need for automated and reproducible approaches to extract quantitative morphological features from image sequences. Here, we introduce a theoretical framework and a computational method for obtaining smooth representations of the spatiotemporal contour dynamics from stacks of segmented microscopy images. Based on a Gaussian process regression we propose a one-parameter family of regularized contour flows that allows us to continuously track reference points (virtual markers) between successive cell contours. We use this approach to define a coordinate system on the moving cell boundary and to represent different local geometric quantities in this frame of reference. In particular, we introduce the local marker dispersion as a measure to identify localized membrane expansions and provide a fully automated way to extract the properties of such expansions, including their area and growth time. The methods are available as an open-source software package called AmoePy, a Python-based toolbox for analyzing amoeboid cell motility (based on time-lapse microscopy data), including a graphical user interface and detailed documentation. Due to the mathematical rigor of our framework, we envision it to be of use for the development of novel cell motility models. We mainly use experimental data of the social amoeba Dictyostelium discoideum to illustrate and validate our approach.
Author summary Amoeboid motion is a crawling-like cell migration that plays an important key role in multiple biological processes such as wound healing and cancer metastasis. This type of cell motility results from expanding and simultaneously contracting parts of the cell membrane. From fluorescence images, we obtain a sequence of points, representing the cell membrane, for each time step. By using regression analysis on these sequences, we derive smooth representations, so-called contours, of the membrane. Since the number of measurements is discrete and often limited, the question is raised of how to link consecutive contours with each other. In this work, we present a novel mathematical framework in which these links are described by regularized flows allowing a certain degree of concentration or stretching of neighboring reference points on the same contour. This stretching rate, the so-called local dispersion, is used to identify expansions and contractions of the cell membrane providing a fully automated way of extracting properties of these cell shape changes. We applied our methods to time-lapse microscopy data of the social amoeba Dictyostelium discoideum. Y1 - 2021 U6 - https://doi.org/10.1371/journal.pcbi.1009268 SN - 1553-734X SN - 1553-7358 VL - 17 IS - 8 PB - PLoS CY - San Fransisco ER - TY - JOUR A1 - Kempton, Mark A1 - Münch, Florentin A1 - Yau, Shing-Tung T1 - A homology vanishing theorem for graphs with positive curvature JF - Communications in analysis and geometry N2 - We prove a homology vanishing theorem for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Bochner on manifolds [3]. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau [11]. We moreover prove that the fundamental group is finite for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Myers on manifolds [22]. The proofs draw on several separate areas of graph theory, including graph coverings, gain graphs, and cycle spaces, in addition to the Bakry-Emery curvature, path homology, and graph homotopy. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature cannot have a non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau [12], and obtain an alternative proof of their result that the abelianization of the fundamental group of a graph is isomorphic to the first path homology over the integers. Y1 - 2021 UR - https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2021/0029/0006/CAG-2021-0029-0006-a005.pdf U6 - https://doi.org/10.4310/CAG.2021.v29.n6.a5 SN - 1019-8385 SN - 1944-9992 VL - 29 IS - 6 SP - 1449 EP - 1473 PB - International Press of Boston CY - Somerville ER - TY - JOUR A1 - Pathiraja, Sahani Darschika A1 - Reich, Sebastian A1 - Stannat, Wilhelm T1 - McKean-Vlasov SDEs in nonlinear filtering JF - SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics N2 - Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters. KW - data assimilation KW - feedback particle filter KW - Poincare inequality KW - well-posedness KW - nonlinear filtering KW - McKean-Vlasov KW - mean-field equations Y1 - 2022 U6 - https://doi.org/10.1137/20M1355197 SN - 0363-0129 SN - 1095-7138 VL - 59 IS - 6 SP - 4188 EP - 4215 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Leung, Tsz Yan A1 - Leutbecher, Martin A1 - Reich, Sebastian A1 - Shepherd, Theodore G. T1 - Forecast verification BT - relating deterministic and probabilistic metrics JF - Quarterly journal of the Royal Meteorological Society N2 - The philosophy of forecast verification is rather different between deterministic and probabilistic verification metrics: generally speaking, deterministic metrics measure differences, whereas probabilistic metrics assess reliability and sharpness of predictive distributions. This article considers the root-mean-square error (RMSE), which can be seen as a deterministic metric, and the probabilistic metric Continuous Ranked Probability Score (CRPS), and demonstrates that under certain conditions, the CRPS can be mathematically expressed in terms of the RMSE when these metrics are aggregated. One of the required conditions is the normality of distributions. The other condition is that, while the forecast ensemble need not be calibrated, any bias or over/underdispersion cannot depend on the forecast distribution itself. Under these conditions, the CRPS is a fraction of the RMSE, and this fraction depends only on the heteroscedasticity of the ensemble spread and the measures of calibration. The derived CRPS-RMSE relationship for the case of perfect ensemble reliability is tested on simulations of idealised two-dimensional barotropic turbulence. Results suggest that the relationship holds approximately despite the normality condition not being met. KW - CRPS KW - ensembles KW - idealised turbulence KW - NWP KW - RMSE KW - verification Y1 - 2021 U6 - https://doi.org/10.1002/qj.4120 SN - 0035-9009 SN - 1477-870X VL - 147 IS - 739 SP - 3124 EP - 3134 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Li, Han Cheng A1 - Sullivan, Tim J. T1 - Gamma-convergence of Onsager-Machlup functionals BT - I. With applications to maximum a posteriori estimation in Bayesian inverse problems JF - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data N2 - The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Gamma-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions. KW - Bayesian inverse problems KW - Gamma-convergence KW - maximum a posteriori KW - estimation KW - Onsager-Machlup functional KW - small ball probabilities; KW - transition path theory Y1 - 2021 U6 - https://doi.org/10.1088/1361-6420/ac3f81 SN - 0266-5611 SN - 1361-6420 VL - 38 IS - 2 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Lie, Han Cheng A1 - Sullivan, Tim J. T1 - Gamma-convergence of Onsager-Machlup functionals BT - II. Infinite product measures on Banach spaces JF - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data N2 - We derive Onsager-Machlup functionals for countable product measures on weighted l(p) subspaces of the sequence space R-N. Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Gamma-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 <= p <= 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory. KW - Bayesian inverse problems KW - Gamma-convergence KW - maximum a posteriori KW - estimation KW - Onsager-Machlup functional KW - small ball probabilities KW - transition path theory Y1 - 2021 U6 - https://doi.org/10.1088/1361-6420/ac3f82 SN - 0266-5611 SN - 1361-6420 VL - 38 IS - 2 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Eshghi, Nasim A1 - Mach, Thomas A1 - Reichel, Lothar T1 - New matrix function approximations and quadrature rules based on the Arnoldi process JF - Journal of computational and applied mathematics N2 - The Arnoldi process can be applied to inexpensively approximate matrix functions of the form f (A)v and matrix functionals of the form v*(f (A))*g(A)v, where A is a large square non-Hermitian matrix, v is a vector, and the superscript * denotes transposition and complex conjugation. Here f and g are analytic functions that are defined in suitable regions in the complex plane. This paper reviews available approximation methods and describes new ones that provide higher accuracy for essentially the same computational effort by exploiting available, but generally not used, moment information. Numerical experiments show that in some cases the modifications of the Arnoldi decompositions proposed can improve the accuracy of v*(f (A))*g(A)v about as much as performing an additional step of the Arnoldi process. KW - Arnoldi process KW - Matrix function approximation KW - Quadrature rule Y1 - 2021 U6 - https://doi.org/10.1016/j.cam.2021.113442 SN - 0377-0427 SN - 1879-1778 VL - 391 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Hethey, Christoph Philipp A1 - Hartung, Niklas A1 - Wangorsch, Gaby A1 - Weisser, Karin A1 - Huisinga, Wilhelm T1 - Physiology-based toxicokinetic modelling of aluminium in rat and man JF - Archives of toxicology : official journal of EUROTOX N2 - A sufficient quantitative understanding of aluminium (Al) toxicokinetics (TK) in man is still lacking, although highly desirable for risk assessment of Al exposure. Baseline exposure and the risk of contamination severely limit the feasibility of TK studies administering the naturally occurring isotope Al-27, both in animals and man. These limitations are absent in studies with Al-26 as a tracer, but tissue data are limited to animal studies. A TK model capable of inter-species translation to make valid predictions of Al levels in humans-especially in toxicological relevant tissues like bone and brain-is urgently needed. Here, we present: (i) a curated dataset which comprises all eligible studies with single doses of Al-26 tracer administered as citrate or chloride salts orally and/or intravenously to rats and humans, including ultra-long-term kinetic profiles for plasma, blood, liver, spleen, muscle, bone, brain, kidney, and urine up to 150 weeks; and (ii) the development of a physiology-based (PB) model for Al TK after intravenous and oral administration of aqueous Al citrate and Al chloride solutions in rats and humans. Based on the comprehensive curated Al-26 dataset, we estimated substance-dependent parameters within a non-linear mixed-effect modelling context. The model fitted the heterogeneous Al-26 data very well and was successfully validated against datasets in rats and humans. The presented PBTK model for Al, based on the most extensive and diverse dataset of Al exposure to date, constitutes a major advancement in the field, thereby paving the way towards a more quantitative risk assessment in humans. KW - PBTK KW - Toxicokinetics KW - Al-26 KW - Aluminium Y1 - 2021 U6 - https://doi.org/10.1007/s00204-021-03107-y SN - 0340-5761 SN - 1432-0738 VL - 95 IS - 9 SP - 2977 EP - 3000 PB - Springer CY - Berlin ; Heidelberg ER - TY - JOUR A1 - Cvetković, Nada A1 - Conrad, Tim A1 - Lie, Han Cheng T1 - A convergent discretization method for transition path theory for diffusion processes JF - Multiscale modeling & simulation : a SIAM interdisciplinary journal N2 - Transition path theory (TPT) for diffusion processes is a framework for analyzing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretization of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretization. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential. KW - ergodic diffusion processes KW - transition paths KW - rare events KW - Monte Carlo KW - methods Y1 - 2021 U6 - https://doi.org/10.1137/20M1329354 SN - 1540-3459 SN - 1540-3467 VL - 19 IS - 1 SP - 242 EP - 266 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Rodríguez Zuluaga, Juan A1 - Stolle, Claudia A1 - Yamazaki, Yosuke A1 - Xiong, Chao A1 - England, Scott L. T1 - A synoptic-scale wavelike structure in the nighttime equatorial ionization anomaly JF - Earth and Space Science : ESS N2 - Both ground- and satellite-based airglow imaging have significantly contributed to understanding the low-latitude ionosphere, especially the morphology and dynamics of the equatorial ionization anomaly (EIA). The NASA Global-scale Observations of the Limb and Disk (GOLD) mission focuses on far-ultraviolet airglow images from a geostationary orbit at 47.5 degrees W. This region is of particular interest at low magnetic latitudes because of the high magnetic declination (i.e., about -20 degrees) and proximity of the South Atlantic magnetic anomaly. In this study, we characterize an exciting feature of the nighttime EIA using GOLD observations from October 5, 2018 to June 30, 2020. It consists of a wavelike structure of a few thousand kilometers seen as poleward and equatorward displacements of the EIA-crests. Initial analyses show that the synoptic-scale structure is symmetric about the dip equator and appears nearly stationary with time over the night. In quasi-dipole coordinates, maxima poleward displacements of the EIA-crests are seen at about +/- 12 degrees latitude and around 20 and 60 degrees longitude (i.e., in geographic longitude at the dip equator, about 53 degrees W and 14 degrees W). The wavelike structure presents typical zonal wavelengths of about 6.7 x 10(3) km and 3.3 x 10(3) km. The structure's occurrence and wavelength are highly variable on a day-to-day basis with no apparent dependence on geomagnetic activity. In addition, a cluster or quasi-periodic wave train of equatorial plasma depletions (EPDs) is often detected within the synoptic-scale structure. We further outline the difference in observing these EPDs from FUV images and in situ measurements during a GOLD and Swarm mission conjunction. KW - equatorial ionization anomaly KW - equatorial ionosphere KW - equatorial plasma bubbles KW - wave structure KW - forcing from below Y1 - 2021 U6 - https://doi.org/10.1029/2020EA001529 SN - 2333-5084 VL - 8 IS - 2 PB - American Geophysical Union CY - Malden, Mass. ER - TY - JOUR A1 - Chang, Der-Chen A1 - Khalil, Sara A1 - Schulze, Bert-Wolfgang T1 - Analysis on regular corner spaces JF - The journal of geometric analysis N2 - We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind. KW - Boutet de Monvel's calculus KW - Pseudo-differential operators KW - Singular cones KW - Mellin symbols with values in the edge calculus KW - Parametrices of elliptic operators KW - Kegel space Y1 - 2021 U6 - https://doi.org/10.1007/s12220-021-00614-3 SN - 1050-6926 SN - 1559-002X VL - 31 IS - 9 SP - 9199 EP - 9240 PB - Springer CY - New York ER - TY - JOUR A1 - Wormell, Caroline L. A1 - Reich, Sebastian T1 - Spectral convergence of diffusion maps BT - Improved error bounds and an alternative normalization JF - SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics N2 - Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localized compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretization. We also introduce an alternative normalization for diffusion maps based on Sinkhorn weights. This normalization approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalization on flat domains, and we present a highly efficient rigorous algorithm to compute the Sinkhorn weights. KW - diffusion maps KW - graph Laplacian KW - Sinkhorn problem KW - kernel methods Y1 - 2021 U6 - https://doi.org/10.1137/20M1344093 SN - 0036-1429 SN - 1095-7170 VL - 59 IS - 3 SP - 1687 EP - 1734 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Saynisch-Wagner, Jan A1 - Bärenzung, Julien A1 - Hornschild, Aaron A1 - Irrgang, Christopher A1 - Thomas, Maik T1 - Tide-induced magnetic signals and their errors derived from CHAMP and Swarm satellite magnetometer observations JF - Earth, planets and space : EPS N2 - Satellite-measured tidal magnetic signals are of growing importance. These fields are mainly used to infer Earth's mantle conductivity, but also to derive changes in the oceanic heat content. We present a new Kalman filter-based method to derive tidal magnetic fields from satellite magnetometers: KALMAG. The method's advantage is that it allows to study a precisely estimated posterior error covariance matrix. We present the results of a simultaneous estimation of the magnetic signals of 8 major tides from 17 years of Swarm and CHAMP data. For the first time, robustly derived posterior error distributions are reported along with the reported tidal magnetic fields. The results are compared to other estimates that are either based on numerical forward models or on satellite inversions of the same data. For all comparisons, maximal differences and the corresponding globally averaged RMSE are reported. We found that the inter-product differences are comparable with the KALMAG-based errors only in a global mean sense. Here, all approaches give values of the same order, e.g., 0.09 nT-0.14 nT for M2. Locally, the KALMAG posterior errors are up to one order smaller than the inter-product differences, e.g., 0.12 nT vs. 0.96 nT for M2. KW - Tides KW - Electromagnetic induction KW - Error covariance KW - Satellite magnetometer observations Y1 - 2021 U6 - https://doi.org/10.1186/s40623-021-01557-3 SN - 1880-5981 VL - 73 IS - 1 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Junek, Heinz T1 - Zyklizität in Raum, zeit und geist : über Pflasterungen, Rollkurven, Dezimalbrüche, Schwingungen, Wellen, Iteration und Neuronale Netze JF - Zyklizität & Rhythmik: eine multidisziplinäre Vorlesungsreihe Y1 - 2020 SN - 978-3-86464-169-5 SP - 85 EP - 103 PB - trafo CY - Berlin ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Partial clones JF - Asian-European journal of mathematics : AEJM N2 - A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n-ary operations defined on set A for all natural numbers n >= 1 and the operations are the so-called superposition operations S-m(n) for natural numbers m, n >= 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties. KW - Operation KW - term KW - formula KW - superposition of operations KW - terms and KW - formulas KW - linear term KW - linear formula KW - linear tree language KW - clone KW - partial clone KW - linear hypersubstitution KW - dht-symmetric category KW - partial KW - theory Y1 - 2020 U6 - https://doi.org/10.1142/S1793557120501612 SN - 1793-5571 SN - 1793-7183 VL - 13 IS - 8 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Somogyvári, Márk A1 - Reich, Sebastian T1 - Convergence tests for transdimensional Markov chains in geoscience imaging JF - Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences N2 - Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization. KW - transdimensional inversion KW - MCMC modelling KW - convergence assessment Y1 - 2019 U6 - https://doi.org/10.1007/s11004-019-09811-x SN - 1874-8961 SN - 1874-8953 VL - 52 IS - 5 SP - 651 EP - 668 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Ly, Ibrahim T1 - A Cauchy problem for the Cauchy-Riemann operator JF - Afrika Matematika N2 - We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution. KW - nonlinear PDI KW - Cauchy problem KW - Zaremba problem Y1 - 2020 U6 - https://doi.org/10.1007/s13370-020-00810-4 SN - 1012-9405 SN - 2190-7668 VL - 32 IS - 1-2 SP - 69 EP - 76 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolaj Nikolaevič T1 - A perturbation of the de Rham complex T1 - Возмущение комплекса де Рама JF - Journal of Siberian Federal University : Mathematics & Physics JF - Žurnal Sibirskogo Federalʹnogo Universiteta : Matematika i fizika N2 - We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings. N2 - Рассмотрим возмущение комплекса де Рама на компактном многообразии с краем. Это возмущение выходит за рамки комплексов, и поэтому когомологии к нему не относятся. С другой стороны, его кривизна "мала", поэтому существует естественный способ ввести характеристику Эйлера и разработать теорию Лефшеца для возмущения. Данная работа предназначена для попытки разработать теорию когомологий для произвольных последовательностей линейных отображений. KW - de Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem KW - комплекс де Рама KW - когомологии KW - теория Ходжа KW - проблема Неймана Y1 - 2020 U6 - https://doi.org/10.17516/1997-1397-2020-13-5-519-532 SN - 1997-1397 SN - 2313-6022 VL - 13 IS - 5 SP - 519 EP - 532 PB - Siberian Federal University CY - Krasnojarsk ER - TY - JOUR A1 - Sharma, Shubham A1 - Hainzl, Sebastian A1 - Zöller, Gert A1 - Holschneider, Matthias T1 - Is Coulomb stress the best choice for aftershock forecasting? JF - Journal of geophysical research : Solid earth N2 - The Coulomb failure stress (CFS) criterion is the most commonly used method for predicting spatial distributions of aftershocks following large earthquakes. However, large uncertainties are always associated with the calculation of Coulomb stress change. The uncertainties mainly arise due to nonunique slip inversions and unknown receiver faults; especially for the latter, results are highly dependent on the choice of the assumed receiver mechanism. Based on binary tests (aftershocks yes/no), recent studies suggest that alternative stress quantities, a distance-slip probabilistic model as well as deep neural network (DNN) approaches, all are superior to CFS with predefined receiver mechanism. To challenge this conclusion, which might have large implications, we use 289 slip inversions from SRCMOD database to calculate more realistic CFS values for a layered half-space and variable receiver mechanisms. We also analyze the effect of the magnitude cutoff, grid size variation, and aftershock duration to verify the use of receiver operating characteristic (ROC) analysis for the ranking of stress metrics. The observations suggest that introducing a layered half-space does not improve the stress maps and ROC curves. However, results significantly improve for larger aftershocks and shorter time periods but without changing the ranking. We also go beyond binary testing and apply alternative statistics to test the ability to estimate aftershock numbers, which confirm that simple stress metrics perform better than the classic Coulomb failure stress calculations and are also better than the distance-slip probabilistic model. Y1 - 2020 U6 - https://doi.org/10.1029/2020JB019553 SN - 2169-9313 SN - 2169-9356 VL - 125 IS - 9 PB - American Geophysical Union CY - Washington ER - TY - JOUR A1 - Al-Saedy, Ammar Jaffar Muhesin A1 - Tarchanov, Nikolaj Nikolaevič T1 - A degree theory for Lagrangian boundary value problems JF - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University; mathematics & physics N2 - We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems. N2 - Мы изучаем те нелинейные уравнения с частными производными, которые возникают как уравнения Эйлера-Лагранжа вариационных задач. Определяя слабые граничные значения решений таких уравнений, мы инициируем теорию лагранжевых краевых задач в функциональных пространствах подходящей гладкости. Мы также анализируем, применяется ли современная концепция степени отображения к лагранжевым проблемам. KW - nonlinear equations KW - Lagrangian system KW - weak boundary values KW - quasilinear Fredholm operators KW - mapping degree Y1 - 2020 U6 - https://doi.org/10.17516/1997-1397-2020-13-1-5-25 SN - 1997-1397 SN - 2313-6022 VL - 13 IS - 1 SP - 5 EP - 25 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - CHAP A1 - Clavier, Pierre J. A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - Renormalisation and locality BT - branched zeta values T2 - Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2 Y1 - 2020 SN - 978-3-03719-205-4 print SN - 978-3-03719-705-9 online U6 - https://doi.org/10.4171/205 SP - 85 EP - 132 PB - European Mathematical Society Publishing House CY - Zürich ER - TY - JOUR A1 - Chelkh, W. A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolai T1 - A remark on the Laplace transform JF - Siberian Mathematical Journal N2 - The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function cos root z. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of cos root z using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces. KW - Fourier-Laplace transform KW - distributions with one-sided support KW - holomorphic function KW - analytic functional Y1 - 2020 U6 - https://doi.org/10.1134/S0037446620040151 SN - 0037-4466 SN - 1573-9260 VL - 61 IS - 4 SP - 755 EP - 762 PB - Consultants Bureau, Springer CY - New York ER - TY - JOUR A1 - Keller, Matthias A1 - Schwarz, Michael T1 - Courant’s nodal domain theorem for positivity preserving forms JF - Journal of spectral theory N2 - We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods. KW - Nodal domain KW - eigenfunction KW - Dirichlet form KW - compact resolvent Y1 - 2020 U6 - https://doi.org/10.4171/JST/292 SN - 1664-039X SN - 1664-0403 VL - 10 IS - 1 SP - 271 EP - 309 PB - EMS Publishing House CY - Zürich ER - TY - JOUR A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolaj Nikolaevič T1 - Asymptotic expansions at nonsymmetric cuspidal points JF - Mathematical notes N2 - We study the asymptotics of solutions to the Dirichlet problem in a domain X subset of R3 whose boundary contains a singular point O. In a small neighborhood of this point, the domain has the form {z > root x(2) + y(4)}, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point. KW - Dirichlet problem KW - singular points KW - asymptotic expansions Y1 - 2020 U6 - https://doi.org/10.1134/S0001434620070238 SN - 0001-4346 SN - 1573-8876 VL - 108 IS - 1-2 SP - 219 EP - 228 PB - Springer Science CY - New York ER - TY - JOUR A1 - Clavier, Pierre J. T1 - Double shuffle relations for arborified zeta values JF - Journal of algebra N2 - Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees. KW - Rooted trees KW - Multiple zeta values KW - Shuffle products KW - Rota-Baxter KW - algebras Y1 - 2020 U6 - https://doi.org/10.1016/j.jalgebra.2019.10.015 SN - 0021-8693 SN - 1090-266X VL - 543 SP - 111 EP - 155 PB - Elsevier CY - San Diego ER -