TY - THES A1 - Zhai, Xiaomin T1 - Design, Development and Evaluation of a Virtual Classroom and Teaching Contents for Bernoulli Stochastics T1 - Design, Entwicklung und Evaluierung einer virtuellen Lernumgebung und des Lehrstoffs für die Bernoulli Stochastik N2 - This thesis is devoted to Bernoulli Stochastics, which was initiated by Jakob Bernoulli more than 300 years ago by his master piece 'Ars conjectandi', which can be translated as 'Science of Prediction'. Thus, Jakob Bernoulli's Stochastics focus on prediction in contrast to the later emerging disciplines probability theory, statistics and mathematical statistics. Only recently Jakob Bernoulli's focus was taken up von Collani, who developed a unified theory of uncertainty aiming at making reliable and accurate predictions. In this thesis, teaching material as well as a virtual classroom are developed for fostering ideas and techniques initiated by Jakob Bernoulli and elaborated by Elart von Collani. The thesis is part of an extensively construed project called 'Stochastikon' aiming at introducing Bernoulli Stochastics as a unified science of prediction and measurement under uncertainty. This ambitious aim shall be reached by the development of an internet-based comprehensive system offering the science of Bernoulli Stochastics on any level of application. So far it is planned that the 'Stochastikon' system (http://www.stochastikon.com/) will consist of five subsystems. Two of them are developed and introduced in this thesis. The first one is the e-learning programme 'Stochastikon Magister' and the second one 'Stochastikon Graphics' that provides the entire Stochastikon system with graphical illustrations. E-learning is the outcome of merging education and internet techniques. E-learning is characterized by the facts that teaching and learning are independent of place and time and of the availability of specially trained teachers. Knowledge offering as well as knowledge transferring are realized by using modern information technologies. Nowadays more and more e-learning environments are based on the internet as the primary tool for communication and presentation. E-learning presentation tools are for instance text-files, pictures, graphics, audio and videos, which can be networked with each other. There could be no limit as to the access to teaching contents. Moreover, the students can adapt the speed of learning to their individual abilities. E-learning is particularly appropriate for newly arising scientific and technical disciplines, which generally cannot be presented by traditional learning methods sufficiently well, because neither trained teachers nor textbooks are available. The first part of this dissertation introduces the state of the art of e-learning in statistics, since statistics and Bernoulli Stochastics are both based on probability theory and exhibit many similar features. Since Stochastikon Magister is the first e-learning programme for Bernoulli Stochastics, the educational statistics systems is selected for the purpose of comparison and evaluation. This makes sense as both disciplines are an attempt to handle uncertainty and use methods that often can be directly compared. The second part of this dissertation is devoted to Bernoulli Stochastics. This part aims at outlining the content of two courses, which have been developed for the anticipated e-learning programme Stochastikon Magister in order to show the difficulties in teaching, understanding and applying Bernoulli Stochastics. The third part discusses the realization of the e-learning programme Stochastikon Magister, its design and implementation, which aims at offering a systematic learning of principles and techniques developed in Bernoulli Stochastics. The resulting e-learning programme differs from the commonly developed e-learning programmes as it is an attempt to provide a virtual classroom that simulates all the functions of real classroom teaching. This is in general not necessary, since most of the e-learning programmes aim at supporting existing classroom teaching. The forth part presents two empirical evaluations of Stochastikon Magister. The evaluations are performed by means of comparisons between traditional classroom learning in statistics and e-learning of Bernoulli Stochastics. The aim is to assess the usability and learnability of Stochastikon Magister. Finally, the fifth part of this dissertation is added as an appendix. It refers to Stochastikon Graphics, the fifth component of the entire Stochastikon system. Stochastikon Graphics provides the other components with graphical representations of concepts, procedures and results obtained or used in the framework of Bernoulli Stochastics. The primary aim of this thesis is the development of an appropriate software for the anticipated e-learning environment meant for Bernoulli Stochastics, while the preparation of the necessary teaching material constitutes only a secondary aim used for demonstrating the functionality of the e-learning platform and the scientific novelty of Bernoulli Stochastics. To this end, a first version of two teaching courses are developed, implemented and offered on-line in order to collect practical experiences. The two courses, which were developed as part of this projects are submitted as a supplement to this dissertation. For the time being the first experience with the e-learning programme Stochastikon Magister has been made. Students of different faculties of the University of Würzburg, as well as researchers and engineers, who are involved in the Stochastikon project have obtained access to Stochastikon Magister via internet. They have registered for Stochastikon Magister and participated in the course programme. This thesis reports on two assessments of these first experiences and the results will lead to further improvements with respect to content and organization of Stochastikon Magister. N2 - Diese Dissertation ist der Bernoulli Stochastik gewidmet, die von Jakob Bernoulli von mehr als 300 Jahren begonnen wurde und zwar mit seinem Hauptwerk 'Ars conjectandi', was mit 'Wissenschaft der Vorhersagen' übersetzt werden kann. Im Zentrum von Jakob Bernoullis Stochastik stehen Vorhersagen im Gegensatz zu den später entstandenen Disziplinen Wahrscheinlichkeitstheorie, Statistik und Mathematische Statistik. Vor Kurzem wurde Jakob Bernoullis Vorschlag von Elart von Collani aufgegriffen, der daraus eine einheitliche Theorie der Unsicherheit entwickelte mit dem Ziel zuverlässige und genaue Vorhersagen zu erlauben. In dieser Dissertation wird eine virtuelle Lernumgebung und der entsprechende Lehrstoff für die Ideen und Resultate von Jakob Bernoulli und Elart von Collani entwickelt. Die Dissertation ist Teil des groß angelegten Projekts 'Stochastikon' dessen Ziel es ist, die Bernoulli Stochastik als allgemeine Wissenschaft der Vorhersage und des Messens unter Unsicherheit zu etablieren. Dieses ehrgeizige Ziel soll mit Hilfe eines umfassenden Internet-Systems erreicht werden, das die Bernoulli Stochastik für alle Gebiete der Anwendung zur Verfügung stellt. Zur Zeit besteht das Stochastikon-System (http://www.stochastikon.com/) aus fünf Teilsystemen. Zwei dieser Systeme sind im Rahmen dieser Dissertation entstanden. Das erste ist das E-Learning Programm 'Stochastikon Magister' und das zweite heißt 'Stochastikon Graphics'. Es beliefert das gesamte Stochastikon Systems mit grafischen Darstellungen. E-Learning entsteht durch Anwendung moderner Informationstechnologien auf die Ausbildung. E-Learning zeichnet sich dadurch aus, dass Lehren und Lernen unabhängig von Ort und Zeit und vom Vorhandensein von speziell ausgebildeten Lehrern ist. Das Angebot und Übermittlung von Wissen erfolgt mit Hilfe der modernen Informationstechnologien. Heutzutage basieren immer mehr E-Learning Programme auf dem Internet, das zum wichtigstes Mittel der Wissensvermittlung und Wissensdarstellung geworden ist. Für die Darstellung des Lehrstoffes im Rahmen des E-Learning existieren kaum Grenzen und darüber hinaus können die Studenten die Lerngeschwindigkeit ihren eigenen Bedürfnissen anpassen. E-Learning ist dabei besonders geeignet für neue wissenschaftliche und technische Disziplinen, die mangels Lehrer und Lehrmaterialien nicht mit Hilfe traditioneller Lernmethoden unterrichtet werden können. Im ersten Teil der Dissertation wird der State of the Art im Bereich der E-Learning Porgramme im Fach Statistik dargestellt, da Statistik und Bernoulli Stochastik beide auf der Wahrscheinlichkeitstheorie basieren und daher viele Ähnlichkeiten aufweisen. Stochastikon Magister ist das erste E-Learning Programm im Rahmen der Bernoulli Stochastik und daher werden zum Vergleich und zur Evaluierung Lehrsysteme aus der Statistik verwendet. Dies ist schon deshalb sinnvoll, weil beide Diszipline sich mit der Unsicherheit beschäftigen und in vielen Fällen die entsprechenden Methoden direkt miteinander verglichen werden können. Der zweite Teil der Dissertation enthält eine kurze Einführung in die Bernoulli Stochastik und zwar anhand von zwei Kursen die im Rahmen der Dissertation entwickelt wurden. Dieser Teil soll die Schwierigkeiten veranschaulichen, die beim Lehren, Verstehen und Anwenden der Bernoulli Stochastik überwunden werden müssen. Im dritten Teil der Dissertation wird die Realisierung des E-Learning Programms Stochastikon Magister diskutiert. Charakteristisch für das E-Learning Programm Stochastikon Magister ist der Versuch ein vollständiges virtuelles Klassenzimmer zu verwirklichen in dem alle Funktionen des traditionellen Schulunterrichts simuliert werden. Dies ist im Allgemeinen nicht notwendig, da das Ziel der meisten E-Learning Programme nur aus der Unterstützung eines traditionellen Lehrprogrammes besteht. Der vierte Teil der Dissertation besteht aus einer empirischen Evaluation des Stochastikon Magisters. Dabei wird eine traditionelle Statistikausbildung mit der Ausbildung in Bernoulli Stochastik auf der Grundlage des E-Learning Programms Stochastikon Magister verglichen. Ziel ist es die Bedienungsfreundlichkeit des Magisters und die Erlernbarkeit der Bernoulli Stochastik nachzuweisen. Der fünfte Teil schließlich besteht aus einem Anhang, der die fünfte Komponente des Stochastik Systems (Stochasticon Graphics) beschreibt. Stochasticon Graphics sorgt dafür, dass alle übrigen Komponenten mit graphischen Abbildungen der Konzepte, Verfahren und Resultate versorgt werden. Das Hauptziel dieser Dissertation ist die Entwicklung geeigneter Software für eine E-Learning-Umgebung für das Fach Bernoulli Stochastik. Die Bereitstellung des entsprechenden Lehrstoff stellt dabei ein sekundäres, aber notwendiges Ziel dar. Notwendig um vor allem die Funktionalität der E-Learning Plattform und die wissenschaftliche Neuheit der Bernoulli Stochastik demonstrieren zu können. Dazu wurden erste Versionen von zwei Kursen entwickelt und on-line angeboten, um Erfahrungen zu sammeln. Die beiden Kurse, die im Rahmen dieser Dissertation entwickelt worden sind, sind als Anhang der Dissertation beigefügt. Zur Zeit werden die ersten Erfahrungen mit dem E-Learning Programm Stochastikon Magister gemacht. Studenten verschiedener Fakultäten der Universität Würzburg aber auch Wissenschaftler und Techniker, die im Rahmen des Stochastikon Projekts mitarbeiten, haben die Kurse belegt und absolviert. Die Ergebnisse dieser ersten Erfahrungen werden in der Dissertation dargestellt und sie werden zu weiteren Verbesserungen des E-Learning Programms führen. KW - Moment KW - Wahrscheinlichkeitsverteilung KW - E-Learning KW - Stochastik KW - Vorhersagetheorie KW - Unsicherheit KW - Zufall KW - Ignoranz KW - Bernoulli Raum KW - Vorhersageverfahren KW - Bernoulli KW - Jakob KW - Uncertainty KW - Randomness KW - Ignorance KW - Bernoulli Space KW - Prediction Procedure Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-56106 ER - TY - THES A1 - Curtef, Oana T1 - Rayleigh–quotient optimization on tensor products of Grassmannians T1 - Rayleigh–Quotient Optimierung auf Tensorprodukte von Graßmann-Mannigfaltigkeiten N2 - Applications in various research areas such as signal processing, quantum computing, and computer vision, can be described as constrained optimization tasks on certain subsets of tensor products of vector spaces. In this work, we make use of techniques from Riemannian geometry and analyze optimization tasks on subsets of so-called simple tensors which can be equipped with a differentiable structure. In particular, we introduce a generalized Rayleigh-quotient function on the tensor product of Grassmannians and on the tensor product of Lagrange- Grassmannians. Its optimization enables a unified approach to well-known tasks from different areas of numerical linear algebra, such as: best low-rank approximations of tensors (data compression), computing geometric measures of entanglement (quantum computing) and subspace clustering (image processing). We perform a thorough analysis on the critical points of the generalized Rayleigh-quotient and develop intrinsic numerical methods for its optimization. Explicitly, using the techniques from Riemannian optimization, we present two type of algorithms: a Newton-like and a conjugated gradient algorithm. Their performance is analysed and compared with established methods from the literature. N2 - Viele Fragestellungen aus den unterschiedlichen mathematischen Disziplinen, wie z.B. Signalverarbeitung, Quanten-Computing und Computer-Vision, können als Optimierungsprobleme auf Teilmengen von Tensorprodukten von Vektorräumen beschrieben werden. In dieser Arbeit verwenden wir Techniken aus der Riemannschen Geometrie, um Optimierungsprobleme für Mengen von sogenannten einfachen Tensoren, welche mit einer differenzierbaren Struktur ausgestattet werden können, zu untersuchen. Insbesondere führen wir eine verallgemeinerte Rayleigh-Quotienten-Funktion auf dem Tensorprodukt von Graßmann-Mannigfaltigkeiten bzw. Lagrange-Graßmann-Mannigfaltigkeiten ein. Dies führt zu einem einheitlichen Zugang zu bekannten Problemen aus verschiedenen Bereichen der numerischen linearen Algebra, wie z.B. die Niedrig–Rang–Approximation von Tensoren (Datenkompression), die Beschreibung geometrischer Maße für Quantenverschränkung (Quanten-Computing) und Clustering (Bildverarbeitung). Wir führen eine gründliche Analyse der kritischen Punkte des verallgemeinerten Rayleigh-Quotienten durch und entwickeln intrinsische numerische Methoden für dessen Optimierung. Wir stellen zwei Arten von Algorithmen vor, die wir mit Hilfe von Techniken aus der Riemannsche Optimierung entwickeln: eine mit Gemeinsamkeiten zum Newton-Verfahren und eine zum CG-Verfahren ähnliche. Wir analysieren die Performance der Algorithmen und vergleichen sie mit gängigen Methoden aus der Literatur. KW - Optimierung KW - Riemannsche Optimierung KW - Newtonverfahren KW - Verfahren der konjugierten Gradienten KW - Maße für Quantenverschränkung KW - Riemannian optimization KW - Grassmann Manifold KW - Newton method KW - Conjugate gradient method KW - tensor rank KW - subspace clustering KW - enatnglement measure KW - Riemannsche Geometrie KW - Grassmann-Mannigfaltigkeit KW - Konjugierte-Gradienten-Methode KW - Newton-Verfahren Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-83383 ER - TY - THES A1 - Christ, Thomas T1 - Value-distribution of the Riemann zeta-function and related functions near the critical line T1 - Werteverteilung der Riemannschen Zetafunktion und verwandter Funktionen nahe der kritischen Geraden N2 - The Riemann zeta-function forms a central object in multiplicative number theory; its value-distribution encodes deep arithmetic properties of the prime numbers. Here, a crucial role is assigned to the analytic behavior of the zeta-function on the so called critical line. In this thesis we study the value-distribution of the Riemann zeta-function near and on the critical line. Amongst others we focus on the following. PART I: A modified concept of universality, a-points near the critical line and a denseness conjecture attributed to Ramachandra. The critical line is a natural boundary of the Voronin-type universality property of the Riemann zeta-function. We modify Voronin's concept by adding a scaling factor to the vertical shifts that appear in Voronin's universality theorem and investigate whether this modified concept is appropriate to keep up a certain universality property of the Riemann zeta-function near and on the critical line. It turns out that it is mainly the functional equation of the Riemann zeta-function that restricts the set of functions which can be approximated by this modified concept around the critical line. Levinson showed that almost all a-points of the Riemann zeta-function lie in a certain funnel-shaped region around the critical line. We complement Levinson's result: Relying on arguments of the theory of normal families and the notion of filling discs, we detect a-points in this region which are very close to the critical line. According to a folklore conjecture (often attributed to Ramachandra) one expects that the values of the Riemann zeta-function on the critical line lie dense in the complex numbers. We show that there are certain curves which approach the critical line asymptotically and have the property that the values of the zeta-function on these curves are dense in the complex numbers. Many of our results in part I are independent of the Euler product representation of the Riemann zeta-function and apply for meromorphic functions that satisfy a Riemann-type functional equation in general. PART II: Discrete and continuous moments. The Lindelöf hypothesis deals with the growth behavior of the Riemann zeta-function on the critical line. Due to classical works by Hardy and Littlewood, the Lindelöf hypothesis can be reformulated in terms of power moments to the right of the critical line. Tanaka showed recently that the expected asymptotic formulas for these power moments are true in a certain measure-theoretical sense; roughly speaking he omits a set of Banach density zero from the path of integration of these moments. We provide a discrete and integrated version of Tanaka's result and extend it to a large class of Dirichlet series connected to the Riemann zeta-function. N2 - Die Riemannsche Zetafunktion ist ein zentraler Gegenstand der multiplikativen Zahlentheorie; in ihrer Werteverteilung liegen wichtige arithmetische Eigenschaften der Primzahlen kodiert. Besondere Bedeutung kommt hierbei dem analytischen Verhalten der Zetafunktion auf der sog. kritischen Geraden zu. Wir untersuchen in dieser Arbeit die Werteverteilung der Riemannschen Zetafunktion auf und nahe der kritischen Geraden. Wir fokusieren wir uns dabei u.a. auf folgende Punkte. TEIL I: Ein modifiziertes Universalitätskonzept, a-Stellen nahe der kritischen Geraden und eine Dichtheitsvermutung nach Ramachandra. Die kritische Gerade fungiert als natürliche Grenze für die Voroninsche Universalitätseigenschaft der Riemannschen Zetafunktion. Wir modifizieren Voronins Universalitätskonzept dahingehend, dass wir die vertikalen Translationen aus Voronins Universalitätssatz mit einer zusätzlichen Skalierung versehen. Wir untersuchen, ob durch dieses modifizierte Konzept eine abgeschwächte Universalitätseigenschaft der Riemannschen Zetafunktion um die kritschen Gerade aufrecht erhalten werden kann. Es stellt sich heraus, dass die Gestalt der Funktionen, die sich auf diese Weise durch die Zetafunktion approximieren lassen, stark von der Funktionalgleichung und der Wahl des skalierenden Faktors abhängt. Nach einem Resultat von Levinson liegen fast alle a-Stellen der Riemannschen Zetafunktion in einem trichterförmigen Bereich um die kritische Gerade. Gewisse Normalitätsargumenten sowie das Konzept der 'filling discs' erlauben uns Levinsons Resultat zu ergänzen und a-Stellen in diesem trichterförmigen Bereich aufzuspüren, die sehr nahe an der kritischen Geraden liegen. Man vermutet, dass die Werte der Riemannschen Zetafunktion auf der kritischen Geraden dicht in den komplexen Zahlen liegen. Wir nähern uns dieser Vermutung (die man oft Ramachandra zuschreibt), indem wir die Existenz gewisser Kurven nachweisen, die sich asymptotisch an die kritische Gerade anschmiegen und die Eigenschaft besitzen, dass die Werte der Zetafunktion auf diesen Kurven dicht in den komplexen Zahlen liegen. Viele unserer Ergebnisse in Teil I sind unabhängig von der Eulerproduktdarstellung der Zetafunktion und gelten allgemein für beliebige meromorphe Funktionen, die einer Funktionalgleichung vom Riemann-Typ genügen. TEIL II: Diskrete und kontinuierliche Momente. Die Lindelöf Vermutung trifft eine Aussage über das Wachstumsverhalten der Zetafunktion auf der kritischen Geraden. Nach klassischen Arbeiten von Hardy und Littlewood lässt sie sich mittels Potenzmomente der Zetafunktion rechts von der kritischen Geraden umformulieren. Tanaka konnte kürzlich nachweisen, dass die asymptotischen Formeln, die man für diese Potenzmomente erwartet in einem gewissen maßtheoretischem Sinne Gülitgkeit besitzen: grob gesprochen wird heibei eine Menge mit Banachdichte null vom Integrationsweg der Potenzmomente ausgespart. Wir stellen eine diskrete und eine integrierte Version von Tanakas Resultat zur Verfügung. Zudem verallgemeinern wir Tanakas Ergebnis auf eine große Klasse von Dirichletreihen. KW - Riemannsche Zetafunktion KW - Riemann zeta-function KW - universality KW - a-point distribution Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-97763 ER - TY - INPR A1 - Geiselhart, Roman A1 - Gielen, Rob H. A1 - Lazar, Mircea A1 - Wirth, Fabian R. T1 - An Alternative Converse Lyapunov Theorem for Discrete-Time Systems N2 - This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete–time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and semi–global practical Lyapunov functions for globally asymptotically stable systems. Furthermore, for specific classes of sys- tems, the developed converse theorem can be used to establish non–conservatism of a particular type of Lyapunov functions. Most notably, a proof that conewise linear Lyapunov functions are non–conservative for GES conewise linear systems is given and, as a by–product, tractable construction of polyhedral Lyapunov functions for linear systems is attained. KW - Ljapunov-Funktion KW - stability analysis KW - conewise linear systems KW - discrete-time systems KW - converse Lyapunov theorems Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-78512 ER - TY - THES A1 - Sen, Surath T1 - Character Analysis and Numerical Computations of Standard M.I. Probability Distributions T1 - Charakteranalyse und Numerische Berechnungen der Standard M.I. Wahrscheinlichkeitsverteilungen N2 - Development and character analysis of software programs, which compute minimum information probability distributions. N2 - Entwicklung und Charakteranalyse von Softwareprogrammen, welche Minimuminformations Wahrscheinlichkeitsverteilungen berechnen. KW - Newton-Verfahren KW - Charakteranalyse KW - Softwareentwicklung KW - Minimum Information Wahrscheinlichkeitsverteilung KW - Newton-Raphson Verfahren KW - Minimum Information Probability Distribution KW - Newton-Raphson Method Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-78623 ER - TY - JOUR A1 - Harrach, Bastian T1 - Recent Progress on the Factorization Method for Electrical Impedance Tomography JF - Computational and Mathematical Methods in Medicine N2 - The Factorization Method is a noniterative method to detect the shape and position of conductivity anomalies inside an object. The method was introduced by Kirsch for inverse scattering problems and extended to electrical impedance tomography (EIT) by Brühl and Hanke. Since these pioneering works, substantial progress has been made on the theoretical foundations of the method. The necessary assumptions have been weakened, and the proofs have been considerably simplified. In this work, we aim to summarize this progress and present a state-of-the-art formulation of the Factorization Method for EIT with continuous data. In particular, we formulate the method for general piecewise analytic conductivities and give short and self-contained proofs. KW - Mathematik Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-96229 ER - TY - THES A1 - Barth, Dominik T1 - Computation of multi-branch-point covers and applications in Galois theory T1 - Berechnung von Mehrpunktüberlagerungen und Anwendungen in der Galoistheorie N2 - We present a technique for computing multi-branch-point covers with prescribed ramification and demonstrate the applicability of our method in relatively large degrees by computing several families of polynomials with symplectic and linear Galois groups. As a first application, we present polynomials over \(\mathbb{Q}(\alpha,t)\) for the primitive rank-3 groups \(PSp_4(3)\) and \(PSp_4(3).C_2\) of degree 27 and for the 2-transitive group \(PSp_6(2)\) in its actions on 28 and 36 points, respectively. Moreover, the degree-28 polynomial for \(PSp_6(2)\) admits infinitely many totally real specializations. Next, we present the first (to the best of our knowledge) explicit polynomials for the 2-transitive linear groups \(PSL_4(3)\) and \(PGL_4(3)\) of degree 40, and the imprimitive group \(Aut(PGL_4(3))\) of degree 80. Additionally, we negatively answer a question by König whether there exists a degree-63 rational function with rational coefficients and monodromy group \(PSL_6(2)\) ramified over at least four points. This is achieved due to the explicit computation of the corresponding hyperelliptic genus-3 Hurwitz curve parameterizing this family, followed by a search for rational points on it. As a byproduct of our calculations we obtain the first explicit \(Aut(PSL_6(2))\)-realizations over \(\mathbb{Q}(t)\). At last, we present a technique by Elkies for bounding the transitivity degree of Galois groups. This provides an alternative way to verify the Galois groups from the previous chapters and also yields a proof that the monodromy group of a degree-276 cover computed by Monien is isomorphic to the sporadic 2-transitive Conway group \(Co_3\). N2 - Wir stellen eine Technik zur Berechnung von Mehrpunktüberlagerungen mit vorgeschriebener Verzweigung vor und demonstrieren die Anwendbarkeit unserer Methode in relativ großen Graden durch die Berechnung mehrerer Familien von Polynomen mit symplektischen und linearen Galoisgruppen. Als erste Anwendung präsentieren wir Polynome über \(\mathbb{Q}(\alpha,t)\) für die primitiven Rang-3-Gruppen \(PSp_4(3)\) und \(PSp_4(3).C_2\) vom Grad 27 und für die 2-fach transitive Gruppe \(PSp_6(2)\) in ihren Operationen auf 28 bzw. 36 Punkten. Außerdem lässt das Polynom vom Grad 28 für \(PSp_6(2)\) unendlich viele total-reelle Spezialisierungen zu. Als Nächstes präsentieren wir die (unseres Wissens nach) ersten expliziten Polynome für die 2-fach transitiven linearen Gruppen \(PSL_4(3)\) und \(PGL_4(3)\) vom Grad 40 und die imprimitive Gruppe \(Aut(PGL_4(3))\) vom Grad 80. Zusätzlich geben wir eine negative Antwort auf die Frage von König, ob es eine rationale Funktion vom Grad 63 mit rationalen Koeffizienten gibt, die über mindestens vier Punkten verzweigt ist und Monodromiegruppe \(PSL_6(2)\) besitzt. Dies wird durch die explizite Berechnung der entsprechenden hyperelliptischen Geschlecht-3 Hurwitzkurve erreicht, die diese Familie parametrisiert, gefolgt von einer Suche nach rationalen Punkten auf ihr. Als Nebenprodukt unserer Berechnungen erhalten wir die ersten expliziten \(Aut(PSL_6(2))\)-Realisierungen über \(\mathbb{Q}(t)\). Schließlich stellen wir eine Technik von Elkies zur Beschränkung des Transitivitätsgrades von Galoisgruppen vor. Diese bietet einen alternativen Weg, die Galoisgruppen aus den vorherigen Kapiteln zu verifizieren und liefert auch einen Beweis dafür, dass die Monodromiegruppe einer von Monien berechneten Grad-276 Überlagerung isomorph zur sporadischen 2-fach transitiven Conway-Gruppe \(Co_3\) ist. KW - Galois-Theorie KW - Hurwitz-Raum KW - Monodromie KW - Überlagerung KW - Belyi map Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-277025 ER - TY - THES A1 - Fuller, Timo T1 - Contributions to the Multivariate Max-Domain of Attraction T1 - Beiträge zum Multivariaten Max-Stabilen Anziehungsbereich N2 - This thesis covers a wide range of results for when a random vector is in the max-domain of attraction of max-stable random vector. It states some new theoretical results in D-norm terminology, but also gives an explaination why most approaches to multivariate extremes are equivalent to this specific approach. Then it covers new methods to deal with high-dimensional extremes, ranging from dimension reduction to exploratory methods and explaining why the Huessler-Reiss model is a powerful parametric model in multivariate extremes on par with the multivariate Gaussian distribution in multivariate regular statistics. It also gives new results for estimating and inferring the multivariate extremal dependence structure, strategies for choosing thresholds and compares the behavior of local and global threshold approaches. The methods are demonstrated in an artifical simulation study, but also on German weather data. N2 - Diese Disseration behandelt eine Reihe von Resultaten, die gelten, wenn sich ein Zufallsvektor im multivariaten Anziehungsbereich eines max-stabilen Zufallsvektors befindet. Es gibt einige neue theoretische Resulte, die in D-Norm Terminologie behandelt werden, aber es wird auch erläutert, warum dieser Ansatz äquivalent zu den meisten anderen Ansätzen in multivariater Extremwerttheorie ist. Es werden neue Methoden dargestellt, mit denen hoch-dimensionale Extremwertprobleme angegangen werden können, darunter Konzepte die Dimensionsreduzierung bis explorative Methoden, und es wird weiter erklärt, warum das parametrische Hüssler-Reiss model eine mächtige Rolle in der multivariaten Extremwerttheorie einnimmt, vergleichbar mit Gausschen Zufallsvektoren in der Standard-Statistik. Es gibt außerdem neue Methoden für Schätzung und Interferenz der extremalen Abhängigkeitsstruktur, Strategien, wie Thresholds gewählt werden können und ein Vergleich von lokalen gegenüber globalen Thresholds. Die Methoden werden in einer Simulationsstudie mit künstlichen Daten, aber auch mit deutschen Wetterdaten demonstriert. KW - Extremwertstatistik KW - Multivariate statistics KW - Dimension reduction KW - Estimation KW - Data Exploration KW - Tail-behavior Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-207422 ER - TY - THES A1 - Wenz, Andreas T1 - Computation of Belyi maps with prescribed ramification and applications in Galois theory T1 - Berechnung von Belyi-Funktionen mit vorgegebener Monodromiegruppe und Anwendungen in der Galoistheorie N2 - We compute genus-0 Belyi maps with prescribed monodromy and strictly verify the computed results. Among the computed examples are almost simple primitive groups that satisfy the rational rigidity criterion yielding polynomials with prescribed Galois groups over Q(t). We also give an explicit version of a theorem of Magaard, which lists all sporadic groups occurring as composition factors of monodromy groups of rational functions. N2 - Wir berechnen Geschlecht-0 Belyi-Funktionen mit vorgegebener Monodromiegruppe und liefern rigorose Verifikationsbeweise. Unter den berechneten Exemplaren finden sich fast-einfache primitive Gruppen, welche das sogenannte "Rationale-Starrheitskriterium" erfüllen, die zu Galois-Realisierungen über Q(t) führen. Außerdem liefern wir eine explizite Version eines Satzes von Magaard, der alle sporadischen Gruppen auflistet, die als Kompositionsfaktoren von Monodromiegruppen rationaler Funktionen auftreten. KW - Galois-Theorie KW - Überlagerung KW - Belyi map KW - Explicit Computation KW - Belyi-Funktionen KW - Explizite Berechnung Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-240838 ER - TY - JOUR A1 - Kakaroumpas, Spyridon A1 - Soler i Gibert, Odí T1 - Dyadic product BMO in the Bloom setting JF - Journal of the London Mathematical Society N2 - Ó. Blasco and S. Pott showed that the supremum of operator norms over L\(^{2}\) of all bicommutators (with the same symbol) of one-parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1 < p < ∞. The main tool is a new characterization in terms of paraproducts and two-weight John–Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multiparameter setting, with the appropriate iterated commutator in each case. KW - dyadic product BMO KW - bicommutators KW - Bloom setting Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-318602 SN - 0024-6107 VL - 106 IS - 2 SP - 899 EP - 935 ER - TY - THES A1 - Lechner, Theresa T1 - Proximal Methods for Nonconvex Composite Optimization Problems T1 - Proximal-Verfahren für nichtkonvexe zusammengesetzte Optimierungsprobleme N2 - Optimization problems with composite functions deal with the minimization of the sum of a smooth function and a convex nonsmooth function. In this thesis several numerical methods for solving such problems in finite-dimensional spaces are discussed, which are based on proximity operators. After some basic results from convex and nonsmooth analysis are summarized, a first-order method, the proximal gradient method, is presented and its convergence properties are discussed in detail. Known results from the literature are summarized and supplemented by additional ones. Subsequently, the main part of the thesis is the derivation of two methods which, in addition, make use of second-order information and are based on proximal Newton and proximal quasi-Newton methods, respectively. The difference between the two methods is that the first one uses a classical line search, while the second one uses a regularization parameter instead. Both techniques lead to the advantage that, in contrast to many similar methods, in the respective detailed convergence analysis global convergence to stationary points can be proved without any restricting precondition. Furthermore, comprehensive results show the local convergence properties as well as convergence rates of these algorithms, which are based on rather weak assumptions. Also a method for the solution of the arising proximal subproblems is investigated. In addition, the thesis contains an extensive collection of application examples and a detailed discussion of the related numerical results. N2 - In Optimierungsproblemen mit zusammengesetzten Funktionen wird die Summe aus einer glatten und einer konvexen, nicht glatten Funktion minimiert. Die vorliegende Arbeit behan- delt mehrere numerische Verfahren zur Lösung solcher Probleme in endlich-dimensionalen Räumen, welche auf Proximity Operatoren basieren. Nach der Zusammenfassung einiger grundlegender Resultate aus der konvexen und nicht- glatten Analysis wird ein Verfahren erster Ordnung, das Proximal-Gradienten-Verfahren, vorgestellt und dessen Konvergenzeigenschaften ausführlich behandelt. Bekannte Resultate aus der Literatur werden dabei zusammengefasst und durch weitere Ergebnisse ergänzt. Im Anschluss werden im Hauptteil der Arbeit zwei Verfahren hergeleitet, die zusätzlich Informationen zweiter Ordnung nutzen und auf Proximal-Newton- beziehungsweise Proximal-Quasi- Newton-Verfahren beruhen. Der Unterschied zwischen beiden Verfahren liegt darin, dass bei ersterem eine klassische Schrittweitensuche verwendet wird, während das zweite stattdessen einen Regularisierungsparameter nutzt. Beide Techniken führen dazu, dass im Gegensatz zu vielen verwandten Verfahren in der jeweils ausführlichen Konvergenzanalyse die globale Konvergenz zu stationären Punkten ohne weitere einschränkende Voraussetzungen bewiesen werden kann. Ferner zeigen umfassende Resultate die lokalen Konvergenzeigenschaften sowie Konvergenzraten der Algorithmen auf, welche auf lediglich schwachen Annahmen beruhen. Ein Verfahren zur Lösung auftretender Proximal-Teilprobleme ist ebenfalls Bestandteil dieser Arbeit. Die Dissertation beinhaltet zudem eine umfangreiche Sammlung von Anwendungsbeispielen und zugehörigen numerischen Ergebnissen. KW - Optimierung KW - composite optimization KW - proximal gradient method KW - proximal Newton method Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-289073 ER - TY - JOUR A1 - Dashkovskiy, Sergey A1 - Slynko, Vitalii T1 - Stability conditions for impulsive dynamical systems JF - Mathematics of Control, Signals, and Systems N2 - In this work, we consider impulsive dynamical systems evolving on an infinite-dimensional space and subjected to external perturbations. We look for stability conditions that guarantee the input-to-state stability for such systems. Our new dwell-time conditions allow the situation, where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov like methods are developed for this purpose. Illustrative finite and infinite dimensional examples are provided to demonstrate the application of the main results. These examples cannot be treated by any other published approach and demonstrate the effectiveness of our results. KW - lyapunov methods KW - stability KW - robustness KW - impulsive systems KW - infinite-dimensional systems KW - nonlinear systems KW - input-to-state stability Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-268390 SN - 1435-568X VL - 34 IS - 1 ER - TY - JOUR A1 - Kanzow, Christian A1 - Raharja, Andreas B. A1 - Schwartz, Alexandra T1 - An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems JF - Journal of Optimization Theory and Applications N2 - A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided. KW - quasinormality constraint qualification KW - cardinality constraints KW - augmented Lagrangian KW - global convergence KW - stationarity Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269166 SN - 1573-2878 VL - 189 IS - 3 ER - TY - THES A1 - Stumpf, Pascal T1 - On coverings and reduced residues in combinatorial number theory T1 - Über Abdeckungen und prime Restklassen in kombinatorischer Zahlentheorie N2 - Our starting point is the Jacobsthal function \(j(m)\), defined for each positive integer \(m\) as the smallest number such that every \(j(m)\) consecutive integers contain at least one integer relatively prime to \(m\). It has turned out that improving on upper bounds for \(j(m)\) would also lead to advances in understanding the distribution of prime numbers among arithmetic progressions. If \(P_r\) denotes the product of the first \(r\) prime numbers, then a conjecture of Montgomery states that \(j(P_r)\) can be bounded from above by \(r (\log r)^2\) up to some constant factor. However, the until now very promising sieve methods seem to have reached a limit here, and the main goal of this work is to develop other combinatorial methods in hope of coming a bit closer to prove the conjecture of Montgomery. Alongside, we solve a problem of Recamán about the maximum possible length among arithmetic progressions in the least (positive) reduced residue system modulo \(m\). Lastly, we turn towards three additive representation functions as introduced by Erdős, Sárközy and Sós who studied their surprising different monotonicity behavior. By an alternative approach, we answer a question of Sárközy and demostrate that another conjecture does not hold. N2 - Der Startpunkt dieser Arbeit ist die Jacobsthal-Funktion \(j(m)\), die für jede natürliche Zahl \(m\) als die kleinste Zahl definiert ist, so dass je \(j(m)\) aufeinanderfolgende ganze Zahlen mindestens eine zu \(m\) teilerfremde Zahl enthalten. Es hat sich herausgestellt, dass Verbesserungen oberer Abschätzungen für \(j(m)\) gleichzeitig zu Fortschritten im Verständnis der Verteilung der Primzahlen in arithmetischen Folgen führen. Bezeichnet \(P_r\) das Produkt der ersten \(r\) Primzahlen, dann besagt eine Vermutung von Montgomery, dass \(j(P_r)\) bis auf einen konstanten Faktor durch \(r (\log r)^2\) von oben abgeschätzt werden kann. Allerdings scheinen die hier bisher sehr vielversprechenden Siebmethoden eine Grenze erreicht zu haben, und das Hauptziel dieser Arbeit ist es andere kombinatorische Methoden zu entwickeln, in der Hoffnung einem Beweis der Vermutung von Montgomery ein wenig näher zu kommen. Auf diesem Weg lösen wir nebenbei ein Problem von Recamán über die maximal mögliche Länge unter den arithmetischen Folgen im kleinsten (positiven) primen Restklassensystem modulo \(m\). Außerdem wenden wir uns am Ende drei additiven Darstellungsfunktionen zu, wie sie von Erdős, Sárközy und Sós eingeführt wurden, die deren überraschend unterschiedliches Monotonieverhalten untersucht haben. Mit einem alternativen Ansatz beantworten wir hier eine Frage von Sárközy und zeigen auf, dass eine andere Vermutung nicht bestehen kann. KW - Kombinatorische Zahlentheorie KW - coverings KW - reduced residues KW - Jacobsthal function Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-293504 ER - TY - JOUR A1 - Weishäupl, Sebastian T1 - The weak Gram law for Hecke \(L\)-functions JF - The Ramanujan Journal N2 - We generalize a theorem by Titchmarsh about the mean value of Hardy’s \(Z\)-function at the Gram points to the Hecke \(L\)-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration. KW - Gram’s law KW - Gram points KW - Hecke eigenforms KW - Hecke L-functions Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324404 SN - 1382-4090 VL - 60 IS - 4 ER - TY - JOUR A1 - Appell, Jürgen A1 - Brito, Belén López A1 - Reinwand, Simon T1 - Counterexamples on compositions JF - Mathematische Semesterberichte N2 - We give a collection of 16 examples which show that compositions \(g\) \(\circ\) \(f\) of well-behaved functions \(f\) and \(g\) can be badly behaved. Remarkably, in 10 of the 16 examples it suffices to take as outer function \(g\) simply a power-type or characteristic function. Such a collection of examples may serve as a source of exercises for a calculus course. KW - composition of functions KW - examples and counterexamples Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324306 SN - 0720-728X VL - 70 IS - 1 ER - TY - JOUR A1 - Lu, Lu A1 - Mönius, Katja T1 - Algebraic degree of Cayley graphs over abelian groups and dihedral groups JF - Journal of Algebraic Combinatorics N2 - For a graph \(\Gamma\) , let K be the smallest field containing all eigenvalues of the adjacency matrix of \(\Gamma\) . The algebraic degree \(\deg (\Gamma )\) is the extension degree \([K:\mathbb {Q}]\). In this paper, we completely determine the algebraic degrees of Cayley graphs over abelian groups and dihedral groups. KW - Cayley graph KW - integral graph KW - algebraic degree Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324380 SN - 0925-9899 VL - 57 IS - 3 ER - TY - JOUR A1 - Kanzow, Christian T1 - Y. Cui, J.-S. Pang: “Modern Nonconvex Nondifferentiable Optimization” JF - Jahresbericht der Deutschen Mathematiker-Vereinigung N2 - No abstract available. KW - Kanzow, C. Y. Cui, J.-S. Pang: “Modern Nonconvex Nondifferentiable Optimization” KW - Rezension Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324346 SN - 0012-0456 VL - 124 IS - 2 ER - TY - JOUR A1 - Greefrath, Gilbert A1 - Oldenburg, Reinhard A1 - Siller, Hans-Stefan A1 - Ulm, Volker A1 - Weigand, Hans-Georg T1 - Mathematics students’ characteristics of basic mental models of the derivative JF - Journal für Mathematik-Didaktik N2 - The concept of derivative is characterised with reference to four basic mental models. These are described as theoretical constructs based on theoretical considerations. The four basic mental models—local rate of change, tangent slope, local linearity and amplification factor—are not only quantified empirically but are also validated. To this end, a test instrument for measuring students’ characteristics of basic mental models is presented and analysed regarding quality criteria. Mathematics students (n = 266) were tested with this instrument. The test results show that the four basic mental models of the derivative can be reconstructed among the students with different characteristics. The tangent slope has the highest agreement values across all tasks. The agreement on explanations based on the basic mental model of rate of change is not as strongly established among students as one would expect due to framework settings in the school system by means of curricula and educational standards. The basic mental model of local linearity plays a rather subordinate role. The amplification factor achieves the lowest agreement values. In addition, cluster analysis was conducted to identify different subgroups of the student population. Moreover, the test results can be attributed to characteristics of the task types as well as to the students’ previous experiences from mathematics classes by means of qualitative interpretation. These and other results of students’ basic mental models of the derivative are presented and discussed in detail. N2 - Der Begriff der Ableitung wird anhand von vier Grundvorstellungen charakterisiert. Diese werden als theoretische Konstrukte beschrieben, die auf theoretischen Überlegungen beruhen. Die vier Grundvorstellungen – lokale Änderungsrate, Tangentensteigung, lokale Linearität und Verstärkungsfaktor – werden empirisch quantifiziert und validiert. Zu diesem Zweck wird ein Testinstrument zur Messung der Charakteristika dieser Grundvorstellungen von Lernenden erstellt, bzgl. Gütekriterien ausgewertet und an Mathematikstudierenden (n = 266) getestet. Die Ergebnisse zeigen, dass die vier Grundvorstellungen der Ableitung bei den Lernenden mit unterschiedlichen Merkmalen rekonstruiert werden können. Die Tangentensteigung weist über alle Aufgaben hinweg die höchsten Übereinstimmungswerte auf. Die Übereinstimmung bei Erklärungen, die auf der Grundvorstellung der lokalen Änderungsrate beruhen, ist bei den Studierenden nicht so stark ausgeprägt, wie man es aufgrund der Rahmenbedingungen im Schulsystem durch Lehrpläne und Bildungsstandards erwarten würde. Die Grundvorstellung der lokalen Linearität spielt eine eher untergeordnete Rolle. Der Verstärkungsfaktor erzielt die geringsten Übereinstimmungswerte. Darüber hinaus wurde eine Clusteranalyse durchgeführt, um verschiedene Untergruppen der Schülerpopulation zu identifizieren. Die Testergebnisse können mittels qualitativer Interpretation auf Merkmale der Aufgabentypen sowie auf die Vorerfahrungen der Studierenden aus dem Mathematikunterricht zurückgeführt werden. Diese und weitere Ergebnisse zu den grundlegenden mentalen Modellen der Studierenden zur Ableitung werden ausführlich dargestellt und diskutiert. KW - derivative KW - basic mental models KW - structure KW - test instrument KW - Ableitung KW - Grundvorstellung KW - Struktur KW - Testinstrument Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324317 SN - 0173-5322 VL - 44 IS - 1 ER - TY - JOUR A1 - Kanzow, Christian A1 - Mehlitz, Patrick T1 - Convergence properties of monotone and nonmonotone proximal gradient methods revisited JF - Journal of Optimization Theory and Applications N2 - Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis. KW - non-Lipschitz optimization KW - nonsmooth optimization KW - proximal gradient method Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324351 SN - 0022-3239 VL - 195 IS - 2 ER - TY - JOUR A1 - Siller, Hans-Stefan A1 - Elschenbroich, Hans-Jürgen A1 - Greefrath, Gilbert A1 - Vorhölter, Katrin T1 - Mathematical modelling of exponential growth as a rich learning environment for mathematics classrooms JF - ZDM Mathematics Education N2 - Mathematical concepts are regularly used in media reports concerning the Covid-19 pandemic. These include growth models, which attempt to explain or predict the effectiveness of interventions and developments, as well as the reproductive factor. Our contribution has the aim of showing that basic mental models about exponential growth are important for understanding media reports of Covid-19. Furthermore, we highlight how the coronavirus pandemic can be used as a context in mathematics classrooms to help students understand that they can and should question media reports on their own, using their mathematical knowledge. Therefore, we first present the role of mathematical modelling in achieving these goals in general. The same relevance applies to the necessary basic mental models of exponential growth. Following this description, based on three topics, namely, investigating the type of growth, questioning given course models, and determining exponential factors at different times, we show how the presented theoretical aspects manifest themselves in teaching examples when students are given the task of reflecting critically on existing media reports. Finally, the value of the three topics regarding the intended goals is discussed and conclusions concerning the possibilities and limits of their use in schools are drawn. KW - basic mental models KW - exponential growth KW - mathematics classrooms KW - mathematical modelling KW - growth models Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324393 SN - 1863-9690 VL - 55 IS - 1 ER - TY - JOUR A1 - Steuding, Jörn A1 - Tongsomporn, Janyarak T1 - On the order of growth of Lerch zeta functions JF - Mathematics N2 - We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t\(^{13/84+ϵ}\) as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t\(^ϵ\) (which is the so-called Lindelöf hypothesis). The growth of an analytic function is closely related to the distribution of its zeros. KW - Lerch zeta function KW - Hurwitz zeta function KW - (approximate) functional equation KW - order of growth KW - exponent pairs KW - MSC 11M35 Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-303981 SN - 2227-7390 VL - 11 IS - 3 ER - TY - JOUR A1 - Heins, Michael A1 - Roth, Oliver A1 - Waldmann, Stefan T1 - Convergent star products on cotangent bundles of Lie groups JF - Mathematische Annalen N2 - For a connected real Lie group G we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of G. This star product trivially converges on polynomial functions on T\(^*\)G thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on T\(^*\)G, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter \(\hbar\). This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on G with an appropriate nuclear Fréchet algebra of functions on \({\mathfrak {g}}^*\). The passage to the Weyl-ordered star product, i.e. the Gutt star product on T\(^*\)G, is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on \(\hbar\). KW - Lie groups KW - star products Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324324 SN - 0025-5831 VL - 386 IS - 1-2 ER - TY - JOUR A1 - Jotz, M. A1 - Mehta, R. A. A1 - Papantonis, T. T1 - Modules and representations up to homotopy of Lie n-algebroids JF - Journal of Homotopy and Related Structures N2 - This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general \(n\in {\mathbb {N}}\). The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles. KW - Lie n-algebroids KW - representations up to homotopy KW - differential graded modules KW - Poisson algebras KW - adjoint and coadjoint representations Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324333 SN - 2193-8407 VL - 18 IS - 1 ER - TY - JOUR A1 - Kourou, Maria A1 - Zarvalis, Konstantinos T1 - Compact sets in petals and their backward orbits under semigroups of holomorphic functions JF - Potential Analysis N2 - Let (ϕ\(_t\))\(_{t≥0}\) be a semigroup of holomorphic functions in the unit disk \(\mathbb {D}\) and K a compact subset of \(\mathbb {D}\). We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk. KW - semigroup of holomorphic functions KW - backward orbit KW - petal KW - harmonic measure KW - condenser capacity KW - Koenigs function KW - green energy KW - hyperbolic area Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324368 SN - 0926-2601 VL - 59 IS - 4 ER - TY - THES A1 - Scherz, Jan T1 - Weak Solutions to Mathematical Models of the Interaction between Fluids, Solids and Electromagnetic Fields T1 - Schwache Lösungen für mathematische Modelle der Wechselwirkung zwischen Flüssigkeiten, Festkörpern und elektromagnetischen Feldern N2 - We analyze the mathematical models of two classes of physical phenomena. The first class of phenomena we consider is the interaction between one or more insulating rigid bodies and an electrically conducting fluid, inside of which the bodies are contained, as well as the electromagnetic fields trespassing both of the materials. We take into account both the cases of incompressible and compressible fluids. In both cases our main result yields the existence of weak solutions to the associated system of partial differential equations, respectively. The proofs of these results are built upon hybrid discrete-continuous approximation schemes: Parts of the systems are discretized with respect to time in order to deal with the solution-dependent test functions in the induction equation. The remaining parts are treated as continuous equations on the small intervals between consecutive discrete time points, allowing us to employ techniques which do not transfer to the discretized setting. Moreover, the solution-dependent test functions in the momentum equation are handled via the use of classical penalization methods. The second class of phenomena we consider is the evolution of a magnetoelastic material. Here too, our main result proves the existence of weak solutions to the corresponding system of partial differential equations. Its proof is based on De Giorgi's minimizing movements method, in which the system is discretized in time and, at each discrete time point, a minimization problem is solved, the associated Euler-Lagrange equations of which constitute a suitable approximation of the original equation of motion and magnetic force balance. The construction of such a minimization problem is made possible by the realization that, already on the continuous level, both of these equations can be written in terms of the same energy and dissipation potentials. The functional for the discrete minimization problem can then be constructed on the basis of these potentials. N2 - Wir analysieren die mathematischen Modelle von zwei Arten physikalischer Phänomene. Die erste Art von Phänomenen, die wir betrachten, ist die Wechselwirkung zwischen einem oder mehreren isolierenden starren Körpern und einem elektrisch leitenden Fluid, das die Körper umgibt, sowie den elektromagnetischen Feldern in beiden Materialien. Wir untersuchen sowohl den Fall inkompressibler als auch kompressibler Fluide. In beiden Fällen liefert unser Hauptresultat die Existenz von schwachen Lösungen für das zugehörige System partieller Differentialgleichungen. Die Beweise dieser Resultate beruhen auf hybriden diskret-kontinuierlichen Approximationsmethoden: Teile der Systeme werden in der Zeit diskretisiert, um das Problem der lösungsabhängigen Testfunktionen in der Induktionsgleichung zu bewältigen. Die verbleibenden Gleichungen werden als kontinuierliche Gleichungen auf den kleinen Intervallen zwischen aufeinanderfolgenden diskreten Zeitpunkten behandelt, sodass wir Techniken anwenden können, die sich nicht auf das diskretisierte System übertragen lassen. Darüber hinaus wird das Problem der lösungsabhängigen Testfunktionen in der Impulsgleichung durch die Verwendung klassischer Penalisierungsmethoden gelöst. Die zweite Art von Phänomenen, die wir betrachten, ist die Entwicklung eines magnetoelastischen Materials. Auch hier beweist unser Hauptresultat die Existenz schwacher Lösungen für das zugehörige System partieller Differentialgleichungen. Der Beweis basiert auf der Methode von De Giorgi, bei der das System in der Zeit diskretisiert und in jedem diskreten Zeitpunkt ein Minimierungsproblem gelöst wird, dessen zugehörige Euler-Lagrange-Gleichungen eine geeignete Approximation an die ursprüngliche Bewegungsgleichung und mikromagnetische Gleichung darstellen. Die Konstruktion eines solchen Minimierungsproblems wird durch die Erkenntnis ermöglicht, dass diese beiden Gleichungen bereits im kontinuierlichen System mithilfe derselben Energie- und Dissipationspotenziale ausgedrückt werden können. Das Funktional für das diskrete Minimierungsproblem kann dann auf Grundlage dieser Potenziale konstruiert werden. KW - Fluid-Struktur-Wechselwirkung KW - Magnetoelastizität KW - Magnetohydrodynamik KW - Navier-Stokes-Gleichung KW - Zeitdiskrete Approximation KW - Fluid-structure interaction KW - Magnetoelasticity KW - Magnetohydrodynamics KW - Minimizing movements KW - Navier-Stokes equations KW - Rothe method Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-349205 ER - TY - JOUR A1 - Dashkovskiy, Sergey A1 - Kapustyan, Oleksiy A1 - Schmid, Jochen T1 - A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations JF - Mathematics of Control, Signals, and Systems N2 - We establish the local input-to-state stability of a large class of disturbed nonlinear reaction–diffusion equations w.r.t. the global attractor of the respective undisturbed system. KW - local input-to-state stability KW - global attractor KW - lonlinear reaction-diffusion equations Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-281099 VL - 32 IS - 3 ER - TY - JOUR A1 - Gerber, Sebastian A1 - Quarder, Jascha A1 - Greefrath, Gilbert A1 - Siller, Hans-Stefan T1 - Promoting adaptive intervention competence for teaching simulations and mathematical modelling with digital tools BT - theoretical background and empirical analysis of a university course in teacher education JF - Frontiers in Education N2 - Providing adaptive, independence-preserving and theory-guided support to students in dealing with real-world problems in mathematics lessons is a major challenge for teachers in their professional practice. This paper examines this challenge in the context of simulations and mathematical modelling with digital tools: in addition to mathematical difficulties when autonomously working out individual solutions, students may also experience challenges when using digital tools. These challenges need to be closely examined and diagnosed, and might – if necessary – have to be overcome by intervention in such a way that the students can subsequently continue working independently. Thus, if a difficulty arises in the working process, two knowledge dimensions are necessary in order to provide adapted support to students. For teaching simulations and mathematical modelling with digital tools, more specifically, these knowledge dimensions are: pedagogical content knowledge about simulation and modelling processes supported by digital tools (this includes knowledge about phases and difficulties in the working process) and pedagogical content knowledge about interventions during the mentioned processes (focussing on characteristics of suitable interventions as well as their implementation and effects on the students’ working process). The two knowledge dimensions represent cognitive dispositions as the basis for the conceptualisation and operationalisation of a so-called adaptive intervention competence for teaching simulations and mathematical modelling with digital tools. In our article, we present a domain-specific process model and distinguish different types of teacher interventions. Then we describe the design and content of a university course at two German universities aiming to promote this domain-specific professional adaptive intervention competence, among others. In a study using a quasi-experimental pre-post design (N = 146), we confirm that the structure of cognitive dispositions of adaptive intervention competence for teaching simulations and mathematical modelling with digital tools can be described empirically by a two-dimensional model. In addition, the effectiveness of the course is examined and confirmed quantitatively. Finally, the results are discussed, especially against the background of the sample and the research design, and conclusions are derived for possibilities of promoting professional adaptive intervention competence in university courses. KW - adaptive intervention competence KW - diagnosis KW - simulation KW - mathematical modelling KW - digital tools KW - teacher education KW - pedagogical content knowledge KW - technology Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-323701 SN - 2504-284X VL - 8 ER - TY - JOUR A1 - Roy, S. A1 - Borzì, A. A1 - Habbal, A. T1 - Pedestrian motion modelled by Fokker-Planck Nash games JF - Royal Society Open Science N2 - A new approach to modelling pedestrians' avoidance dynamics based on a Fokker–Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases. KW - Fokker–Planck equation KW - Nash equilibrium KW - pedestrian motion KW - differential games KW - avoidance KW - optimal control Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-170395 VL - 4 IS - 9 ER - TY - JOUR A1 - Almeida, R. A1 - Hristova, S. A1 - Dashkovskiy, S. T1 - Uniform bounded input bounded output stability of fractional‐order delay nonlinear systems with input JF - International Journal of Robust and Nonlinear Control N2 - The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system's dynamic response, and demonstrate the effectiveness of our theoretical results. KW - bounded input bounded output stability KW - Caputo fractional derivative KW - Lyapunov functions KW - Razumikhin method KW - time‐varying delay Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-218554 VL - 31 IS - 1 SP - 225 EP - 249 ER - TY - JOUR A1 - Feireisl, Eduard A1 - Klingenberg, Christian A1 - Markfelder, Simon T1 - On the density of “wild” initial data for the compressible Euler system JF - Calculus of Variations and Partial Differential Equations N2 - We consider a class of “wild” initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural L1-topology and show that its complement is rather large, specifically it is an open dense set. KW - Euler system Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-232607 SN - 0944-2669 VL - 59 ER - TY - JOUR A1 - Lauerbach, Laura A1 - Neukamm, Stefan A1 - Schäffner, Mathias A1 - Schlömerkemper, Anja T1 - Continuum Limit and Homogenization of Stochastic and Periodic Discrete Systems – Fracture in Composite Materials JF - Proceedings in Applied Mathematics & Mechanics N2 - The limiting behaviour of a one‐dimensional discrete system is studied by means of Γ‐convergence. We consider a toy model of a chain of atoms. The interaction potentials are of Lennard‐Jones type and periodically or stochastically distributed. The energy of the system is considered in the discrete to continuum limit, i.e. as the number of atoms tends to infinity. During that limit, a homogenization process takes place. The limiting functional is discussed, especially with regard to fracture. Secondly, we consider a rescaled version of the problem, which yields a limiting energy of Griffith's type consisting of a quadratic integral term and a jump contribution. The periodic case can be found in [8], the stochastic case in [6,7]. KW - discrete systems KW - continuum limit KW - homogenization Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-211835 VL - 19 IS - 1 ER - TY - JOUR A1 - Fall, Mouhammed Moustapha A1 - Minlend, Ignace Aristide A1 - Ratzkin, Jesse T1 - Foliation of an Asymptotically Flat End by Critical Capacitors JF - The Journal of Geometric Analysis N2 - We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace–Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem. KW - asymptotically flat ends KW - foliation KW - over-determined problem Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269997 SN - 1559-002X VL - 32 IS - 2 ER - TY - JOUR A1 - Easton, Andrew A1 - van Dalen, Okki A1 - Goeb, Rainer A1 - Di Bucchianico, Alessandro T1 - Bivariate copula monitoring JF - Quality and Reliability Engineering International N2 - The assumption of multivariate normality underlying the Hotelling T\(^{2}\) chart is often violated for process data. The multivariate dependency structure can be separated from marginals with the help of copula theory, which permits to model association structures beyond the covariance matrix. Copula‐based estimation and testing routines have reached maturity regarding a variety of practical applications. We have constructed a rich design matrix for the comparison of the Hotelling T\(^{2}\) chart with the copula test by Verdier and the copula test by Vuong, which allows for weighting the observations adaptively. Based on the design matrix, we have conducted a large and computationally intensive simulation study. The results show that the copula test by Verdier performs better than Hotelling T\(^{2}\) in a large variety of out‐of‐control cases, whereas the weighted Vuong scheme often fails to provide an improvement. KW - copula KW - multivariate Gaussian distribution KW - multivariate statistical process control (SPC) KW - phase I KW - phase II Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-276501 VL - 38 IS - 3 SP - 1272 EP - 1288 ER - TY - JOUR A1 - Gathungu, Duncan Kioi A1 - Borzì, Alfio T1 - Multigrid Solution of an Elliptic Fredholm Partial Integro-Differential Equation with a Hilbert-Schmidt Integral Operator JF - Applied Mathematics N2 - An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework. KW - elliptic problems KW - finite differences KW - fredholm operator KW - multigrid schemes KW - numerical analysis Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-158525 VL - 8 IS - 7 ER - TY - JOUR A1 - Kienle-Garrido, Melina-Lorén A1 - Breitenbach, Tim A1 - Chudej, Kurt A1 - Borzì, Alfio T1 - Modeling and numerical solution of a cancer therapy optimal control problem JF - Applied Mathematics N2 - A mathematical optimal-control tumor therapy framework consisting of radio- and anti-angiogenesis control strategies that are included in a tumor growth model is investigated. The governing system, resulting from the combination of two well established models, represents the differential constraint of a non-smooth optimal control problem that aims at reducing the volume of the tumor while keeping the radio- and anti-angiogenesis chemical dosage to a minimum. Existence of optimal solutions is proved and necessary conditions are formulated in terms of the Pontryagin maximum principle. Based on this principle, a so-called sequential quadratic Hamiltonian (SQH) method is discussed and benchmarked with an “interior point optimizer―a mathematical programming language” (IPOPT-AMPL) algorithm. Results of numerical experiments are presented that successfully validate the SQH solution scheme. Further, it is shown how to choose the optimisation weights in order to obtain treatment functions that successfully reduce the tumor volume to zero. KW - cancer KW - therapy KW - optimal control problem Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-177152 VL - 9 IS - 8 ER - TY - JOUR A1 - Campana, Francesca Calà A1 - Borzì, Alfio T1 - On the SQH Method for Solving Differential Nash Games JF - Journal of Dynamical and Control Systems N2 - A sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed scheme, and results of numerical experiments are reported that successfully validate its computational performance. KW - successive approximations strategy KW - sequential quadratic hamiltonian method KW - differential nash games KW - pontryagin maximum principle Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269111 SN - 1573-8698 VL - 28 ER - TY - JOUR A1 - Kraus, Daniela A1 - Moucha, Annika A1 - Roth, Oliver T1 - A sharp Bernstein–type inequality and application to the Carleson embedding theorem with matrix weights JF - Analysis and Mathematical Physics N2 - We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil (Int. Math. Res. Not. 2019: 3301–3312, 2019) on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal. KW - Bernstein-type inequality KW - complex polynomials KW - Carleson embedding theorem Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-270485 SN - 1664-235X VL - 12 IS - 1 ER - TY - JOUR A1 - Homburg, Annika A1 - Weiß, Christian H. A1 - Alwan, Layth C. A1 - Frahm, Gabriel A1 - Göb, Rainer T1 - A performance analysis of prediction intervals for count time series JF - Journal of Forecasting N2 - One of the major motivations for the analysis and modeling of time series data is the forecasting of future outcomes. The use of interval forecasts instead of point forecasts allows us to incorporate the apparent forecast uncertainty. When forecasting count time series, one also has to account for the discreteness of the range, which is done by using coherent prediction intervals (PIs) relying on a count model. We provide a comprehensive performance analysis of coherent PIs for diverse types of count processes. We also compare them to approximate PIs that are computed based on a Gaussian approximation. Our analyses rely on an extensive simulation study. It turns out that the Gaussian approximations do considerably worse than the coherent PIs. Furthermore, special characteristics such as overdispersion, zero inflation, or trend clearly affect the PIs' performance. We conclude by presenting two empirical applications of PIs for count time series: the demand for blood bags in a hospital and the number of company liquidations in Germany. KW - coherent forecasting KW - count time series KW - estimation error KW - Gaussian approximation KW - prediction interval Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-217906 VL - 40 IS - 4 SP - 603 EP - 609 ER - TY - JOUR A1 - Mitra, Sourav T1 - Local Existence of Strong Solutions of a Fluid–Structure Interaction Model JF - Journal of Mathematical Fluid Mechanics N2 - We are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system. KW - fluid–structure interaction model KW - compressible fluid KW - Euler–Bernoulli damped beam KW - local existence Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-235097 SN - 1422-6928 VL - 22 ER - TY - JOUR A1 - Natemeyer, Carolin A1 - Wachsmuth, Daniel T1 - A proximal gradient method for control problems with non-smooth and non-convex control cost JF - Computational Optimization and Applications N2 - We investigate the convergence of the proximal gradient method applied to control problems with non-smooth and non-convex control cost. Here, we focus on control cost functionals that promote sparsity, which includes functionals of L\(^{p}\)-type for p\in [0,1). We prove stationarity properties of weak limit points of the method. These properties are weaker than those provided by Pontryagin’s maximum principle and weaker than L-stationarity. KW - sparse control problems KW - proximal gradient method KW - non-smooth and non-convex optimization Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269069 SN - 1573-2894 VL - 80 IS - 2 ER - TY - JOUR A1 - Falk, Michael A1 - Fuller, Timo T1 - New characterizations of multivariate Max-domain of attraction and D-Norms JF - Extremes N2 - In this paper we derive new results on multivariate extremes and D-norms. In particular we establish new characterizations of the multivariate max-domain of attraction property. The limit distribution of certain multivariate exceedances above high thresholds is derived, and the distribution of that generator of a D-norm on R\(^{d}\), whose components sum up to d, is obtained. Finally we introduce exchangeable D-norms and show that the set of exchangeable D-norms is a simplex. KW - extremal exchangeable D-norms KW - multivariate extreme value theory KW - multivariate max-domain of attraction KW - D-norm KW - generator of D-norm KW - multivariate exceedance KW - co-extremality coefficient KW - exchangeable D-norms Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269071 SN - 1572-915X VL - 24 IS - 4 ER - TY - JOUR A1 - Mönius, Katja T1 - Eigenvalues of zero-divisor graphs of finite commutative rings JF - Journal of Algebraic Combinatorics N2 - We investigate eigenvalues of the zero-divisor graph Γ(R) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of Γ(R). The graph Γ(R) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever xy=0. We provide formulas for the nullity of Γ(R), i.e., the multiplicity of the eigenvalue 0 of Γ(R). Moreover, we precisely determine the spectra of \(\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)\) and \(\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)\) for a prime number p. We introduce a graph product ×Γ with the property that Γ(R)≅Γ(R\(_1\))×Γ⋯×ΓΓ(R\(_r\)) whenever R≅R\(_1\)×⋯×R\(_r\). With this product, we find relations between the number of vertices of the zero-divisor graph Γ(R), the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of Γ(R). KW - EJMA-D-19-00287 KW - Zero-divisor graphs KW - Graph eigenvalues KW - Graphnullity KW - Graph products KW - Local rings Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-232792 SN - 0925-9899 VL - 54 ER - TY - JOUR A1 - Kalousek, Martin A1 - Mitra, Sourav A1 - Schlömerkemper, Anja T1 - Existence of weak solutions of diffuse interface models for magnetic fluids JF - Proceedings in Applied Mathematics and Mechanics N2 - In this article we collect some recent results on the global existence of weak solutions for diffuse interface models involving incompressible magnetic fluids. We consider both the cases of matched and unmatched specific densities. For the model involving fluids with identical densities we consider the free energy density to be a double well potential whereas for the unmatched density case it is crucial to work with a singular free energy density. KW - mathematics KW - magnetic fluids KW - diffuse interface models Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-257642 VL - 21 IS - 1 ER - TY - JOUR A1 - Schönlein, Michael T1 - Ensemble reachability of homogenous parameter‐depedent systems JF - Proceedings in Applied Mathematics and Mechanics N2 - In this paper we consider the class (θA, B) of parameter-dependent linear systems given by matrices A ∈ ℂ\(^{nxn}\) and B ∈ ℂ\(^{nxm}\). This class is of interest for several applications and the frequently met task for such systems is to steer the origin toward a given target family f(θ) by using an input that is independent from the parameter. This paper provides a collection of necessary and sufficient conditions for ensemble reachability for these systems. KW - ensemble reachability KW - homogenous parameter-depedent systems KW - mathematics Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-257637 VL - 20 IS - 1 ER - TY - GEN A1 - Breitenbach, Tim T1 - Codes of examples for SQH method N2 - Code examples for the paper "On the SQH Scheme to Solve Nonsmooth PDE Optimal Control Problems" by Tim Breitenbach and Alfio Borzì published in the journal "Numerical Functional Analysis and Optimization", in 2019, DOI: 10.1080/01630563.2019.1599911 KW - Code examples KW - SQH method Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-165669 ER - TY - JOUR A1 - Breitenbach, Tim A1 - Helfrich-Förster, Charlotte A1 - Dandekar, Thomas T1 - An effective model of endogenous clocks and external stimuli determining circadian rhythms JF - Scientific Reports N2 - Circadian endogenous clocks of eukaryotic organisms are an established and rapidly developing research field. To investigate and simulate in an effective model the effect of external stimuli on such clocks and their components we developed a software framework for download and simulation. The application is useful to understand the different involved effects in a mathematical simple and effective model. This concerns the effects of Zeitgebers, feedback loops and further modifying components. We start from a known mathematical oscillator model, which is based on experimental molecular findings. This is extended with an effective framework that includes the impact of external stimuli on the circadian oscillations including high dose pharmacological treatment. In particular, the external stimuli framework defines a systematic procedure by input-output-interfaces to couple different oscillators. The framework is validated by providing phase response curves and ranges of entrainment. Furthermore, Aschoffs rule is computationally investigated. It is shown how the external stimuli framework can be used to study biological effects like points of singularity or oscillators integrating different signals at once. The mathematical framework and formalism is generic and allows to study in general the effect of external stimuli on oscillators and other biological processes. For an easy replication of each numerical experiment presented in this work and an easy implementation of the framework the corresponding Mathematica files are fully made available. They can be downloaded at the following link: https://www.biozentrum.uni-wuerzburg.de/bioinfo/computing/circadian/. KW - computational biology and bioinformatics KW - systems biology Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-261655 VL - 11 IS - 1 ER - TY - THES A1 - Gathungu, Duncan Kioi T1 - On Multigrid and H-Matrix Methods for Partial Integro-Differential Equations T1 - Mehrgitter-Verfahren und hierarchische Matrix-Lösungsverfahren für Klassen von partiellen Integro-Differential-Problemen N2 - The main theme of this thesis is the development of multigrid and hierarchical matrix solution procedures with almost linear computational complexity for classes of partial integro-differential problems. An elliptic partial integro-differential equation, a convection-diffusion partial integro-differential equation and a convection-diffusion partial integro-differential optimality system are investigated. In the first part of this work, an efficient multigrid finite-differences scheme for solving an elliptic Fredholm partial integro-differential equation (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization and a Simpson's quadrature rule to approximate the PIDE problem and a multigrid scheme and a fast multilevel integration method of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework that includes numerical experiments for elliptic PIDE problems with singular kernels. The experience gained in this part of the work is used for the investigation of convection diffusion partial-integro differential equations in the second part of this thesis. Convection-diffusion PIDE problems are discretized using a finite volume scheme referred to as the Chang and Cooper (CC) scheme and a quadrature rule. Also for this class of PIDE problems and this numerical setting, a stability and accuracy analysis of the CC scheme combined with a Simpson's quadrature rule is presented proving second-order accuracy of the numerical solution. To extend and investigate the proposed approximation and solution strategy to the case of systems of convection-diffusion PIDE, an optimal control problem governed by this model is considered. In this case the research focus is the CC-Simpson's discretization of the optimality system and its solution by the proposed multigrid strategy. Second-order accuracy of the optimization solution is proved and results of local Fourier analysis are presented that provide sharp convergence estimates of the optimal computational complexity of the multigrid-fast integration technique. While (geometric) multigrid techniques require ad-hoc implementation depending on the structure of the PIDE problem and on the dimensionality of the domain where the problem is considered, the hierarchical matrix framework allows a more general treatment that exploits the algebraic structure of the problem at hand. In this thesis, this framework is extended to the case of combined differential and integral problems considering the case of a convection-diffusion PIDE. In this case, the starting point is the CC discretization of the convection-diffusion operator combined with the trapezoidal quadrature rule. The hierarchical matrix approach exploits the algebraic nature of the hierarchical matrices for blockwise approximations by low-rank matrices of the sparse convection-diffusion approximation and enables data sparse representation of the fully populated matrix where all essential matrix operations are performed with at most logarithmic optimal complexity. The factorization of part of or the whole coefficient matrix is used as a preconditioner to the solution of the PIDE problem using a generalized minimum residual (GMRes) procedure as a solver. Numerical analysis estimates of the accuracy of the finite-volume and trapezoidal rule approximation are presented and combined with estimates of the hierarchical matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical estimates and the optimal computational complexity of the proposed hierarchical matrix solution procedure. These results include an extension to higher dimensions and an application to the time evolution of the probability density function of a jump diffusion process. N2 - Das Hauptthema dieser Arbeit ist die Entwicklung von Mehrgitter-Verfahren und hierarchischer Matrix-Lösungsverfahren mit nahezu linearer Rechenkomplexität für Klassen von partiellen Integro-Differential-Problemen. Es werden eine elliptische partielle Integro-Differentialgleichung, eine partielle Konvektions-Diffusions-Integro-Differentialgleichung und ein partielles Konvektions-Diffusions-Integro-Differential-Optimalitätssystem untersucht. Im ersten Teil dieser Arbeit wurde ein effizientes Mehrgitter-Finite-Differenzen-Schema zur Lösung einer elliptischen Fredholm partiellen Integro-Differentialgleichungen (PIDE) diskutiert. Dieses Schema kombiniert eine exakte finite Differenzen-Diskretisierung zweiter Ordnung mit einer Quadraturregel von Simpson, um das PIDE-Problem mit einem Mehrgitter-Schema und einer schnellen Multilevel-Integrationsmethode des Fredholm-Operators zu lösen, was eine schnelle Lösung des PIDE-Probleme ermöglicht. Theoretische Abschätzungen der Genauigkeit zweiter Ordnung und Ergebnisse der lokalen Fourier-Analyse der Konvergenz des vorgeschlagenen Mehrgitter-Systems werden präsentiert. Ergebnisse von numerischen Experimenten validieren diese Schätzungen und demonstrieren die optimale rechnerische Komplexität des vorgeschlagenen Frameworks, das numerische Experimente für elliptische PIDE mit singulären Kernen beinhaltet. Die in diesem Teil der Arbeit gewonnenen Erfahrungen werden zur Untersuchung einer partielle Konvektions-Diffusions-Integro-Differentialgleichungen im zweiten Teil verwendet. Konvektions-Diffusions-PIDE-Probleme werden unter Verwendung eines Finite-Volumen-Schemas, das als das Chang- Cooper- (CC-) Schema bezeichnet wird, und einer Quadraturregel diskretisiert. Auch für diese Klasse von PIDE-Problemen und diese numerische Einstellung wird eine Stabilitäts- und Genauigkeitsanalyse des CC-Schemas in Kombination mit einer Quadraturregel von Simpson vorgestellt, die die Genauigkeit der numerischen Lösung zweiter Ordnung beweist. Um die vorgeschlagene Approximations- und Lösungsstrategie auf den Fall von Konvektions-Diffusions-PIDE-Systemen auszudehnen und zu untersuchen, wird ein Optimalsteuerungsproblem mit diesem Modell als Nebenbedingung untersucht. Der Forschungsschwerpunkt liegt dabei auf der Diskretisierung des Optimalitätssystems durch die CC-Simpson-Lösung und dessen Lösung durch die vorgeschlagene Mehrgitter-Strategie. Die Genauigkeit der optimalen Lösung zweiter Ordnung wird bewiesen und es werden Ergebnisse der lokalen Fourier-Analyse präsentiert, die scharfe Konvergenz-Schätzungen der optimalen Berechnungskomplexität der schnellen Mehrgitter Integrationstechnik liefern. Während (geometrische) Mehrgitterverfahren je nach Struktur des PIDE-Problems und der Dimensionalität des Gebietes, in dem das Problem berücksichtigt wird, eine Ad-hoc-Implementierung erfordern, ermöglicht das hierarchische Matrix-Framework eine allgemeinere Behandlung, die die algebraische Struktur des Problems nutzt. In dieser Arbeit wird dieses Verfahren auf den Fall kombinierter Differential- und Integralprobleme im Fall einer Konvektions-Diffusions-PIDE erweitert. In diesem Fall ist der Startpunkt die CC-Diskretisierung des Konvektions-Diffusions-Operators in Kombination mit der Trapez-Quadratur-Regel. Der hierarchische Matrixansatz nutzt die algebraische Natur der hierarchischen Matrizen für blockweise Approximationen durch niedrigrangige Matrizen der dünn besetzten Konvektions-Diffusionsmatrix und ermöglicht eine datenarme Darstellung der vollständig besetzten Matrix, bei der alle wesentlichen Matrixoperationen mit höchstens logarithmisch optimaler Komplexität durchgeführt werden. Die Faktorisierung eines Teils oder der gesamten Koeffizientenmatrix wird als Vorbedingung für die Lösung der PIDE-Probleme unter Verwendung eines verallgemeinerten minimalen Restwert-Verfahrens (GMRes) als Löser verwendet. Eine numerische Analyse der Abschätzungen der Genauigkeit der Finite-Volumen- und Trapezregel-Approximation werden präsentiert und kombiniert mit Abschätzungen der hierarchischen Matrix-Näherung und mit der Genauigkeit der GMRes iterationen kombiniert. Ergebnisse numerischer Experimente werden vorgestellt, die theoretischen Abschätzungen und die optimale rechnerische Komplexität der vorgeschlagenen hierarchischen Matrix Lösungsverfahren erfolgreich validieren. Diese Ergebnisse beinhalten eine Erweiterung auf höhere Dimensionen und eine Anwendung auf die zeitliche Entwicklung der Wahrscheinlichkeitsdichtefunktion des Sprungdiffusionsprozesses. KW - Mehrgitterverfahren KW - Hierarchische Matrix KW - Integrodifferentialgleichung KW - Numerical analysis KW - multigrid KW - hierarchical matrix KW - partial integro-differential equations Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-156430 ER - TY - THES A1 - Technau, Marc T1 - On Beatty sets and some generalisations thereof T1 - Über Beatty-Mengen und einige Verallgemeinerungen dieser N2 - Beatty sets (also called Beatty sequences) have appeared as early as 1772 in the astronomical studies of Johann III Bernoulli as a tool for easing manual calculations and - as Elwin Bruno Christoffel pointed out in 1888 - lend themselves to exposing intricate properties of the real irrationals. Since then, numerous researchers have explored a multitude of arithmetic properties of Beatty sets; the interrelation between Beatty sets and modular inversion, as well as Beatty sets and the set of rational primes, being the central topic of this book. The inquiry into the relation to rational primes is complemented by considering a natural generalisation to imaginary quadratic number fields. N2 - Zu gegebener Beatty-Menge \(\mathscr{B}(\alpha,\beta) = \{ n\alpha+\beta : n\in\mathbb{N} \}\) mit irrationalem \(\alpha>1\) und \(\beta\in\mathbb{R}\), sowie gegebener Primzahl \(p\) und hierzu teilerfremdem \(z\) untersuchen wir das Problem der Auffindung von Punkten \((m,\tilde{m})\) auf der modularen Hyperbel \[ \mathscr{H}_{z,p} = \{(m,\tilde{m}) \in \mathbb{Z}^2\cap[1,p )^2 : m\tilde{m}\equiv z\mod p\} \] mit \(\max\{ m, \tilde{m} \}\) so klein wie möglich, d.h. wir für gewisse \(\alpha\) beweisen nichttriviale Abschätzungen für \[ \min\{ \max\{ m, \tilde{m} \} : (m,\tilde{m})\in\mathscr{H}_{z,p}, \, m\in\mathscr{B}(\alpha,\beta) \}. \] Der Beweis fußt auf neuen Abschätzungen für unvollständige Kloosterman-Summen entlang \(\mathscr{B}(\alpha,\beta)\), welche durch das Speisen einer Methode von Banks und Shparlinski mit neuen Abschätzungen für die periodische Autokorrelation der endlichen Folge \[ 0,\, \operatorname{e}_p(y\overline{1}),\, \operatorname{e}_p(y\overline{2}),\, \ldots,\, \operatorname{e}_p(y\overline{p-1}), \quad \text{with \(y\) indivisible by \(p\)}, \] erhalten werden; (Hierbei bezeichnet \(\overline{m}\) die eindeutige natürliche Zahl \(m'\in[1,p)\) mit \(mm'\equiv 1\bmod p\) und wir schreiben \(\operatorname{e}_p(x) = \exp(2\pi i x/p)\).) Für letzteres adaptieren wir Ideen von Kloosterman. Des weiteren untersuchen wir Mengen der Form \(\{\lfloor m\alpha_1+n\alpha_2+\beta\rfloor : m,n\in\mathbb{N} \}\). Wir zeigen, dass diese stets in einer gewöhnlichen Beatty-Menge \(\mathscr{B}(\tilde{\alpha},\tilde{\beta})\) enthalten sind und geben zulässige Werte für \(\tilde{\alpha}\) und \(\tilde{\beta}\) an. Das Komplement \(\mathscr{C} = \mathscr{B}(\tilde{\alpha},\tilde{\beta}) \setminus \{\lfloor m\alpha_1+n\alpha_2+\beta\rfloor : m,n\in\mathbb{N} \}\) erweist sich als endliche Menge und wir bestimmen obere Schranken für das Supremum von \(\mathscr{C}\). Die Beweise gründen sich auf einfache Verteilungseigenschaften der Folge der Nachkommastellen \(\{n\alpha_1^{-1}\alpha_2\}\), \(n=1,2,\ldots\), sofern \(\alpha_1^{-1}\alpha_2\) irrational ist, und berufen sich anderenfalls auf die Endlichkeit der Frobenius-Zahl einer geeignet gewählten Instanz des Frobeniusschen Münzproblems. Abschließend verallgemeinern wir die Definition von Beatty-Mengen auf imaginär-quadratische Zahlkörper in einer natürlichen Weise. Hat der fragliche Zahlkörper Klassenzahl \(1\), so können wir zeigen, dass diese Beatty-artigen Mengen unendlich viele Primelemente enthalten, sofern der zugehörige Parameter \(\alpha\) nicht im betrachteten Zahlkörper enthalten ist. Für den speziellen Zahlkörper \(\mathbb{Q}(i)\) erhalten wir unter Benutzung des Hurwitzschen Kettenbruch-Algorithmus eine Zahlkörper-Variante eines früheren Resultats von Steuding und dem Autor, welches ein Beatty-Analogon des klassischen Linnikschen Satzes über die kleinste Primzahl in einer arithmetischen Progression darstellt. Die erwähnten Resultate werden durch Zahlkörper-Varianten von klassischen Ergebnissen über die Verteilung von \(\{ p\vartheta \}\), \(p=2,3,5,7,11,\ldots\), \(\vartheta\in\mathbb{R}\setminus\mathbb{Q}\), erhalten; Diese wurden kürzlich von Baier mittels der Harmanschen Siebmethode für \(\mathbb{Q}(i)\) bewiesen. Wir übertragen die zugehörigen Überlegungen auf Zahlkörper mit Klassenzahl \(1\). N2 - For Beatty sets \(\mathscr{B}(\alpha,\beta) = \{ n\alpha+\beta : n\in\mathbb{N} \}\) with irrational \(\alpha>1\) and \(\beta\in\mathbb{R}\), and \(p\) prime and coprime to \(z\), we investigate the problem of detecting points \((m,\tilde{m})\) on the modular hyperbola \[ \mathscr{H}_{z,p} = \{(m,\tilde{m}) \in \mathbb{Z}^2\cap[1,p )^2 : m\tilde{m}\equiv z\mod p\} \] with \(\max\{ m, \tilde{m} \}\) as small as possible, i.e., we obtain non-trivial estimates for \[ \min\{ \max\{ m, \tilde{m} \} : (m,\tilde{m})\in\mathscr{H}_{z,p}, \, m\in\mathscr{B}(\alpha,\beta) \} \] for certain \(\alpha\). The proof rests on new estimates for incomplete Kloosterman sums along \(\mathscr{B}(\alpha,\beta)\) which are in turn obtained on supplying a method due to Banks and Shparlinski with a new estimate for the periodic autocorrelation of the finite sequence \[ 0,\, \operatorname{e}_p(y\overline{1}),\, \operatorname{e}_p(y\overline{2}),\, \ldots,\, \operatorname{e}_p(y\overline{p-1}), \quad \text{with \(y\) indivisible by \(p\)}, \] (\(\overline{m}\) denoting the unique integer \(m'\in[1,p)\) with \(mm'\equiv 1\bmod p\) and \(\operatorname{e}_p(x) = \exp(2\pi i x/p)\), the latter being obtained from adapting an argument due to Kloosterman. Furthermore, we investigate sets of the shape \(\{\lfloor m\alpha_1+n\alpha_2+\beta\rfloor : m,n\in\mathbb{N} \}\). We show that they are always contained in some ordinary Beatty set \(\mathscr{B}(\tilde{\alpha},\tilde{\beta})\) where we give admissible choices for \(\tilde{\alpha}\) and \(\tilde{\beta}\). Their respective complement \(\mathscr{C}\) in this ordinary Beatty set is shown to be finite and bounds for the supremum of \(\mathscr{C}\) are provided. The proofs are based on basic distribution properties of the sequence of fractional parts \(\{n\alpha_1^{-1}\alpha_2\}\), \(n=1,2,\ldots\), when \(\alpha_1^{-1}\alpha_2\) is irrational, and appeal to the finiteness of the Frobenius number associated with a suitably chosen instance of the Frobenius coin problem otherwise. Lastly, we generalise the definition of Beatty sets to imaginary quadratic number fields in a natural fashion. Assuming the number field in question to have class number \(1\), we are able to show that these Beatty-type sets contain infinitely many prime elements provided that the parameter corresponding to \(\alpha\) from above is not contained in the number field. When the number field is \(\mathbb{Q}(i)\), then, using the Hurwitz continued fraction expansion, we obtain a number field analogue of a previous result of Steuding and the author, who gave a Beatty set analogue of Linnik's famous theorem on the least prime number in an arithmetic progression. These results are obtained from number field analogues of classical results about the distribution of \(\{ p\vartheta \}\), \(p=2,3,5,7,11,\ldots\), \(\vartheta\in\mathbb{R}\setminus\mathbb{Q}\), which were worked out recently by Baier for \(\mathbb{Q}(i)\) using Harman's sieve method. We generalise these arguments to imaginary quadratic number fields with class number \(1\). KW - Zahlentheorie KW - Beatty sequence KW - Kloosterman sum KW - prime number KW - distribution modulo one KW - Diophantine approximation KW - imaginary quadratic field Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-163303 SN - 978-3-95826-088-7 (Print) SN - 978-3-95826-089-4 (Online) N1 - Parallel erschienen als Druckausgabe in Würzburg University Press, ISBN 978-3-95826-088-7, 21,80 EUR. PB - Würzburg University Press CY - Würzburg ET - 1. Auflage ER - TY - THES A1 - Barsukow, Wasilij T1 - Low Mach number finite volume methods for the acoustic and Euler equations T1 - Finite Volumen Methoden für den Grenzwert niedriger Machzahlen der akustischen und der Euler-Gleichungen N2 - Finite volume methods for compressible Euler equations suffer from an excessive diffusion in the limit of low Mach numbers. This PhD thesis explores new approaches to overcome this. The analysis of a simpler set of equations that also possess a low Mach number limit is found to give valuable insights. These equations are the acoustic equations obtained as a linearization of the Euler equations. For both systems the limit is characterized by a divergencefree velocity. This constraint is nontrivial only in multiple spatial dimensions. As the Jacobians of the acoustic system do not commute, acoustics cannot be reduced to some kind of multi-dimensional advection. Therefore first an exact solution in multiple spatial dimensions is obtained. It is shown that the low Mach number limit can be interpreted as a limit of long times. It is found that the origin of the inability of a scheme to resolve the low Mach number limit is the lack a discrete counterpart to the limit of long times. Numerical schemes whose discrete stationary states discretize all the analytic stationary states of the PDE are called stationarity preserving. It is shown that for the acoustic equations, stationarity preserving schemes are vorticity preserving and are those that are able to resolve the low Mach limit (low Mach compliant). This establishes a new link between these three concepts. Stationarity preservation is studied in detail for both dimensionally split and multi-dimensional schemes for linear acoustics. In particular it is explained why the same multi-dimensional stencils appear in literature in very different contexts: These stencils are unique discretizations of the divergence that allow for stabilizing stationarity preserving diffusion. Stationarity preservation can also be generalized to nonlinear systems such as the Euler equations. Several ways how such numerical schemes can be constructed for the Euler equations are presented. In particular a low Mach compliant numerical scheme is derived that uses a novel construction idea. Its diffusion is chosen such that it depends on the velocity divergence rather than just derivatives of the different velocity components. This is demonstrated to overcome the low Mach number problem. The scheme shows satisfactory results in numerical simulations and has been found to be stable under explicit time integration. N2 - Finite Volumen Methoden für die kompressiblen Euler-Gleichungen zeigen übermäßige Diffusion im Grenzwert kleiner Machzahlen. Diese Dissertation beschäftigt sich mit neuen Ansätzen, um dieses Problem zu beheben. Die Analyse eines Systems einfacherer Gleichungen, die ebenso einen Grenzwert niedriger Machzahlen haben, liefert wichtige Einsichten. Diese Gleichungen sind die als Linearisierung der Euler-Gleichungen erhaltenen akustischen Gleichungen. Für beide Gleichungssysteme ist der Grenzwert durch ein divergenzfreies Geschwindigkeitsfeld charakterisiert, was nur in mehreren Raumdimensionen nichttrivial ist. Da die Jacobi-Matrizen des akustischen Systems nicht vertauschen, kann Akustik nicht auf irgendeine Art mehrdimensionaler Advektion zurückgeführt werden. Deswegen wird zunächst eine exakte Lösung in mehreren Raumdimensionen gefunden. Es wird gezeigt, dass sich der Grenzwert kleiner Machzahlen als Grenzwert langer Zeiten interpretieren lässt. Als der Ursprung des Versagens eines Schemas im Grenzwert kleiner Machzahlen wird das Fehlen einer diskreten Entsprechung zum Grenzwert langer Zeiten identifiziert. Numerische Schemata, deren diskrete stationäre Zustände alle analytischen stationären Zustände diskretisieren, werden stationaritätserhaltend genannt. Es zeigt sich, dass für die akustischen Gleichungen stationaritätserhaltende Schemata vortizitätserhaltend sind, und gerade diejenigen sind, die auch den Grenzwert kleiner Machzahlen aufzulösen vermögen. Das zeigt eine neue Verbindung zwischen diesen drei Konzepten auf. Erhaltung der Stationarität wird für lineare Akustik im Detail für Schemata studiert, die nach Raumdimensionen aufgeteilt sind, und auch für multi-dimensionale Schemata. Insbesondere wird ein Grund geliefert, warum die gleichen multi-dimensionalen diskreten Operatoren in der Literatur in sehr unterschiedlichen Kontexten auftauchen: Sie sind Diskretisierungen der Divergenz, für die eine stabilisierende, stationaritätserhaltende Diffusion gefunden werden kann. Auch für nichtlineare Gleichungen, wie die Euler-Gleichungen, kann die Erhaltung der Stationarität verallgemeinert werden. Es werden dazu mehrere Wege der Konstruktion numerischer Schemata gezeigt. Insbesondere im Hinblick auf den Grenzwert kleiner Machzahlen wird ein neuartiges Schema hergeleitet, dessen Diffusion so gewählt ist, dass es von der Divergenz der Geschwindigkeit, und nicht bloß von irgendswelchen Ableitungen der Geschwindigkeitskomponenten abhängt. Es wird gezeigt, dass dieses Schema in der Lage ist, den Grenzwert kleiner Machzahlen aufzulösen. Das Schema zeigt zufriedenstellende Resultate in Simulationen und ist stabil unter Verwendung eines expliziten Zeitintegrators. KW - Finite-Volumen-Methode KW - Machzahl KW - finite volume method KW - Euler equations KW - Acoustic equations KW - low Mach number KW - vorticity preserving Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-159965 ER -