@phdthesis{Ruppert2017, author = {Ruppert, Markus}, title = {Wege der Analogiebildung - Eine qualitative Studie {\"u}ber den Prozess der Analogiebildung beim L{\"o}sen von Aufgaben}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-155910}, school = {Universit{\"a}t W{\"u}rzburg}, pages = {311}, year = {2017}, abstract = {{\"U}ber die besondere Bedeutung von Analogiebildungsprozessen beim Lernen im Allgemeinen und beim Lernen von Mathematik im Speziellen besteht ein breiter wissenschaftlicher Konsens. Es liegt deshalb nahe, von einem lernf{\"o}rderlichen Mathematikunterricht zu verlangen, dass er im Bewusstsein dieser Bedeutung entwickelt ist - dass er also einerseits Analogien aufzeigt und sich diese beim Lehren von Mathematik zunutze macht, dass er andererseits aber auch dem Lernenden Gelegenheiten bietet, Analogien zu erkennen und zu entwickeln. Kurz: Die F{\"a}higkeit zum Bilden von Analogien soll durch den Unterricht gezielt gef{\"o}rdert werden. Um diesem Anspruch gerecht werden zu k{\"o}nnen, m{\"u}ssen ausreichende Kenntnisse dar{\"u}ber vorliegen, wie Analogiebildungsprozesse beim Lernen von Mathematik und beim L{\"o}sen mathematischer Aufgaben ablaufen, wodurch sich erfolgreiche Analogiebildungsprozesse auszeichnen und an welchen Stellen m{\"o}glicherweise Schwierigkeiten bestehen. Der Autor zeigt auf, wie Prozesse der Analogiebildung beim L{\"o}sen mathematischer Aufgaben initiiert, beobachtet, beschrieben und interpretiert werden k{\"o}nnen, um auf dieser Grundlage Ansatzpunkte f{\"u}r geeignete F{\"o}rdermaßnahmen zu identifizieren, bestehende Ideen zur F{\"o}rderung der Analogiebildungsf{\"a}higkeit zu beurteilen und neue Ideen zu entwickeln. Es werden dabei Wege der Analogiebildung nachgezeichnet und untersucht, die auf der Verschr{\"a}nkung zweier Dimensionen der Analogiebildung im Rahmen des zugrundeliegenden theoretischen Modells beruhen. So k{\"o}nnen verschiedene Vorgehensweisen ebenso kontrastiert werden, wie kritische Punkte im Verlauf eines Analogiebildungsprozesses. Es ergeben sich daraus Unterrichtsvorschl{\"a}ge, die auf den Ideen zum beispielbasierten Lernen aufbauen.}, subject = {Analogie}, language = {de} } @phdthesis{Steck2018, author = {Steck, Daniel}, title = {Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-174444}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {This thesis is concerned with a class of general-purpose algorithms for constrained minimization problems, variational inequalities, and quasi-variational inequalities in Banach spaces. A substantial amount of background material from Banach space theory, convex analysis, variational analysis, and optimization theory is presented, including some results which are refinements of those existing in the literature. This basis is used to formulate an augmented Lagrangian algorithm with multiplier safeguarding for the solution of constrained optimization problems in Banach spaces. The method is analyzed in terms of local and global convergence, and many popular problem classes such as nonlinear programming, semidefinite programming, and function space optimization are shown to be included as special cases of the general setting. The algorithmic framework is then extended to variational and quasi-variational inequalities, which include, by extension, Nash and generalized Nash equilibrium problems. For these problem classes, the convergence is analyzed in detail. The thesis then presents a rich collection of application examples for all problem classes, including implementation details and numerical results.}, subject = {Optimierung}, language = {en} } @phdthesis{Forster2016, author = {Forster, Johannes}, title = {Variational Approach to the Modeling and Analysis of Magnetoelastic Materials}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-147226}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2016}, abstract = {This doctoral thesis is concerned with the mathematical modeling of magnetoelastic materials and the analysis of PDE systems describing these materials and obtained from a variational approach. The purpose is to capture the behavior of elastic particles that are not only magnetic but exhibit a magnetic domain structure which is well described by the micromagnetic energy and the Landau-Lifshitz-Gilbert equation of the magnetization. The equation of motion for the material's velocity is derived in a continuum mechanical setting from an energy ansatz. In the modeling process, the focus is on the interplay between Lagrangian and Eulerian coordinate systems to combine elasticity and magnetism in one model without the assumption of small deformations. The resulting general PDE system is simplified using special assumptions. Existence of weak solutions is proved for two variants of the PDE system, one including gradient flow dynamics on the magnetization, and the other featuring the Landau-Lifshitz-Gilbert equation. The proof is based on a Galerkin method and a fixed point argument. The analysis of the PDE system with the Landau-Lifshitz-Gilbert equation uses a more involved approach to obtain weak solutions based on G. Carbou and P. Fabrie 2001.}, subject = {Magnetoelastizit{\"a}t}, language = {en} } @article{SchindeleBorzi2016, author = {Schindele, Andreas and Borz{\`i}, Alfio}, title = {Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional}, series = {Applied Mathematics}, volume = {7}, journal = {Applied Mathematics}, number = {9}, doi = {10.4236/am.2016.79086}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145850}, pages = {967-992}, year = {2016}, abstract = {First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.}, language = {en} } @unpublished{BreitenbachBorzi2019, author = {Breitenbach, Tim and Borz{\`i}, Alfio}, title = {On the SQH scheme to solve non-smooth PDE optimal control problems}, series = {Numerical Functional Analysis and Optimization}, journal = {Numerical Functional Analysis and Optimization}, doi = {10.1080/01630563.2019.1599911}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-180936}, year = {2019}, abstract = {A sequential quadratic Hamiltonian (SQH) scheme for solving different classes of non-smooth and non-convex PDE optimal control problems is investigated considering seven different benchmark problems with increasing difficulty. These problems include linear and nonlinear PDEs with linear and bilinear control mechanisms, non-convex and discontinuous costs of the controls, L\(^1\) tracking terms, and the case of state constraints. The SQH method is based on the characterisation of optimality of PDE optimal control problems by the Pontryagin's maximum principle (PMP). For each problem, a theoretical discussion of the PMP optimality condition is given and results of numerical experiments are presented that demonstrate the large range of applicability of the SQH scheme.}, language = {en} } @phdthesis{Sapozhnikova2018, author = {Sapozhnikova, Kateryna}, title = {Robust Stability of Differential Equations with Maximum}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-173945}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {In this thesis stability and robustness properties of systems of functional differential equations which dynamics depends on the maximum of a solution over a prehistory time interval is studied. Max-operator is analyzed and it is proved that due to its presence such kind of systems are particular case of state dependent delay differential equations with piecewise continuous delay function. They are nonlinear, infinite-dimensional and may reduce to one-dimensional along its solution. Stability analysis with respect to input is accomplished by trajectory estimate and via averaging method. Numerical method is proposed.}, subject = {Differentialgleichung}, language = {en} } @phdthesis{Gaviraghi2017, author = {Gaviraghi, Beatrice}, title = {Theoretical and numerical analysis of Fokker-Planck optimal control problems for jump-diffusion processes}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145645}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2017}, abstract = {The topic of this thesis is the theoretical and numerical analysis of optimal control problems, whose differential constraints are given by Fokker-Planck models related to jump-diffusion processes. We tackle the issue of controlling a stochastic process by formulating a deterministic optimization problem. The key idea of our approach is to focus on the probability density function of the process, whose time evolution is modeled by the Fokker-Planck equation. Our control framework is advantageous since it allows to model the action of the control over the entire range of the process, whose statistics are characterized by the shape of its probability density function. We first investigate jump-diffusion processes, illustrating their main properties. We define stochastic initial-value problems and present results on the existence and uniqueness of their solutions. We then discuss how numerical solutions of stochastic problems are computed, focusing on the Euler-Maruyama method. We put our attention to jump-diffusion models with time- and space-dependent coefficients and jumps given by a compound Poisson process. We derive the related Fokker-Planck equations, which take the form of partial integro-differential equations. Their differential term is governed by a parabolic operator, while the nonlocal integral operator is due to the presence of the jumps. The derivation is carried out in two cases. On the one hand, we consider a process with unbounded range. On the other hand, we confine the dynamic of the sample paths to a bounded domain, and thus the behavior of the process in proximity of the boundaries has to be specified. Throughout this thesis, we set the barriers of the domain to be reflecting. The Fokker-Planck equation, endowed with initial and boundary conditions, gives rise to Fokker-Planck problems. Their solvability is discussed in suitable functional spaces. The properties of their solutions are examined, namely their regularity, positivity and probability mass conservation. Since closed-form solutions to Fokker-Planck problems are usually not available, one has to resort to numerical methods. The first main achievement of this thesis is the definition and analysis of conservative and positive-preserving numerical methods for Fokker-Planck problems. Our SIMEX1 and SIMEX2 (Splitting-Implicit-Explicit) schemes are defined within the framework given by the method of lines. The differential operator is discretized by a finite volume scheme given by the Chang-Cooper method, while the integral operator is approximated by a mid-point rule. This leads to a large system of ordinary differential equations, that we approximate with the Strang-Marchuk splitting method. This technique decomposes the original problem in a sequence of different subproblems with simpler structure, which are separately solved and linked to each other through initial conditions and final solutions. After performing the splitting step, we carry out the time integration with first- and second-order time-differencing methods. These steps give rise to the SIMEX1 and SIMEX2 methods, respectively. A full convergence and stability analysis of our schemes is included. Moreover, we are able to prove that the positivity and the mass conservation of the solution to Fokker-Planck problems are satisfied at the discrete level by the numerical solutions computed with the SIMEX schemes. The second main achievement of this thesis is the theoretical analysis and the numerical solution of optimal control problems governed by Fokker-Planck models. The field of optimal control deals with finding control functions in such a way that given cost functionals are minimized. Our framework aims at the minimization of the difference between a known sequence of values and the first moment of a jump-diffusion process; therefore, this formulation can also be considered as a parameter estimation problem for stochastic processes. Two cases are discussed, in which the form of the cost functional is continuous-in-time and discrete-in-time, respectively. The control variable enters the state equation as a coefficient of the Fokker-Planck partial integro-differential operator. We also include in the cost functional a \$L^1\$-penalization term, which enhances the sparsity of the solution. Therefore, the resulting optimization problem is nonconvex and nonsmooth. We derive the first-order optimality systems satisfied by the optimal solution. The computation of the optimal solution is carried out by means of proximal iterative schemes in an infinite-dimensional framework.}, subject = {Fokker-Planck-Gleichung}, language = {en} } @phdthesis{Tichy2011, author = {Tichy, Diana}, title = {On the Fragility Index}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-73610}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {The Fragility Index captures the amount of risk in a stochastic system of arbitrary dimension. Its main mathematical tool is the asymptotic distribution of exceedance counts within the system which can be derived by use of multivariate extreme value theory. Thereby the basic assumption is that data comes from a distribution which lies in the domain of attraction of a multivariate extreme value distribution. The Fragility Index itself and its extension can serve as a quantitative measure for tail dependence in arbitrary dimensions. It is linked to the well known extremal index for stochastic processes as well the extremal coefficient of an extreme value distribution.}, subject = {Extremwertstatistik}, language = {en} } @phdthesis{Hofmann2012, author = {Hofmann, Martin}, title = {Contributions to Extreme Value Theory in the Space C[0,1]}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-74405}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {We introduce some mathematical framework for extreme value theory in the space of continuous functions on compact intervals and provide basic definitions and tools. Continuous max-stable processes on [0,1] are characterized by their "distribution functions" G which can be represented via a norm on function space, called D-norm. The high conformity of this setup with the multivariate case leads to the introduction of a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. We also introduce the concept of "sojourn time transformation" and compare several types of convergence on function space. Again in complete accordance with the uni- or multivariate case it is now possible to get functional generalized Pareto distributions (GPD) W via W = 1 + log(G) in the upper tail. In particular, this enables us to derive characterizations of the functional domain of attraction condition for copula processes. Moreover, we investigate the sojourn time above a high threshold of a continuous stochastic process. It turns out that the limit, as the threshold increases, of the expected sojourn time given that it is positive, exists if the copula process corresponding to Y is in the functional domain of attraction of a max-stable process. If the process is in a certain neighborhood of a generalized Pareto process, then we can replace the constant threshold by a general threshold function and we can compute the asymptotic sojourn time distribution.}, subject = {Extremwertstatistik}, language = {en} } @techreport{Englert2009, author = {Englert, Stefan}, title = {Mathematica in 15 Minuten (Mathematica Version 6.0)}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-70275}, year = {2009}, abstract = {Mathematica ist ein hervorragendes Programm um mathematische Berechnungen - auch sehr komplexe - auf relativ einfache Art und Weise durchf{\"u}hren zu lassen. Dieses Skript soll eine wirklich kurze Einf{\"u}hrung in Mathematica geben und als Nachschlagewerk einiger g{\"a}ngiger Anwendungen von Mathematica dienen. Dabei wird folgende Grobgliederung verwendet: - Grundlagen: Graphische Oberfl{\"a}che, einfache Berechnungen, Formeleingabe - Bedienung: Vorstellung einiger Kommandos und Einblick in die Funktionsweise - Praxis: Beispielhafte Berechnung einiger Abitur- und {\"U}bungsaufgaben}, subject = {Anwendungssoftware}, language = {de} } @techreport{Englert2012, author = {Englert, Stefan}, title = {Mathematica in 15 Minuten (Mathematica Version 8.0)}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-70287}, year = {2012}, abstract = {Mathematica ist ein hervorragendes Programm um mathematische Berechnungen - auch sehr komplexe - auf relativ einfache Art und Weise durchf{\"u}hren zu lassen. Dieses Skript soll eine wirklich kurze Einf{\"u}hrung in Mathematica geben und als Nachschlagewerk einiger g{\"a}ngiger Anwendungen von Mathematica dienen. Dabei wird folgende Grobgliederung verwendet: - Grundlagen: Graphische Oberfl{\"a}che, einfache Berechnungen, Formeleingabe - Bedienung: Vorstellung einiger Kommandos und Einblick in die Funktionsweise - Praxis: Beispielhafte Berechnung einiger Abitur- und {\"U}bungsaufgaben}, subject = {Anwendungssoftware}, language = {de} } @phdthesis{Dreves2011, author = {Dreves, Axel}, title = {Globally Convergent Algorithms for the Solution of Generalized Nash Equilibrium Problems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-69822}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {Es werden verschiedene Verfahren zur L{\"o}sung verallgemeinerter Nash-Gleichgewichtsprobleme mit dem Schwerpunkt auf deren globaler Konvergenz entwickelt. Ein globalisiertes Newton-Verfahren zur Berechnung normalisierter L{\"o}sungen, ein nichtglattes Optimierungsverfahren basierend auf einer unrestringierten Umformulierung des spieltheoretischen Problems, und ein Minimierungsansatz sowei eine Innere-Punkte-Methode zur L{\"o}sung der gemeinsamen Karush-Kuhn-Tucker-Bedingungen der Spieler werden theoretisch untersucht und numerisch getestet. Insbesondere das Innere-Punkte Verfahren erweist sich als das zur Zeit wohl beste Verfahren zur L{\"o}sung verallgemeinerter Nash-Gleichgewichtsprobleme.}, subject = {Nash-Gleichgewicht}, language = {en} } @phdthesis{Mauder2012, author = {Mauder, Markus}, title = {Time-Optimal Control of the Bi-Steerable Robot: A Case Study in Optimal Control of Nonholonomic Systems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-75036}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {In this thesis, time-optimal control of the bi-steerable robot is addressed. The bi-steerable robot, a vehicle with two independently steerable axles, is a complex nonholonomic system with applications in many areas of land-based robotics. Motion planning and optimal control are challenging tasks for this system, since standard control schemes do not apply. The model of the bi-steerable robot considered here is a reduced kinematic model with the driving velocity and the steering angles of the front and rear axle as inputs. The steering angles of the two axles can be set independently from each other. The reduced kinematic model is a control system with affine and non-affine inputs, as the driving velocity enters the system linearly, whereas the steering angles enter nonlinearly. In this work, a new approach to solve the time-optimal control problem for the bi-steerable robot is presented. In contrast to most standard methods for time-optimal control, our approach does not exclusively rely on discretization and purely numerical methods. Instead, the Pontryagin Maximum Principle is used to characterize candidates for time-optimal solutions. The resultant boundary value problem is solved by optimization to obtain solutions to the path planning problem over a given time horizon. The time horizon is decreased and the path planning is iterated to approximate a time-optimal solution. An optimality condition is introduced which depends on the number of cusps, i.e., reversals of the driving direction of the robot. This optimality condition allows to single out non-optimal solutions with too many cusps. In general, our approach only gives approximations of time-optimal solutions, since only normal regular extremals are considered as solutions to the path planning problem, and the path planning is terminated when an extremal with minimal number of cusps is found. However, for most desired configurations, normal regular extremals with the minimal number of cusps provide time-optimal solutions for the bi-steerable robot. The convergence of the approach is analyzed and its probabilistic completeness is shown. Moreover, simulation results on time-optimal solutions for the bi-steerable robot are presented.}, subject = {Mobiler Roboter}, language = {en} } @phdthesis{Akindeinde2012, author = {Akindeinde, Saheed Ojo}, title = {Numerical Verification of Optimality Conditions in Optimal Control Problems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-76065}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {This thesis is devoted to numerical verification of optimality conditions for non-convex optimal control problems. In the first part, we are concerned with a-posteriori verification of sufficient optimality conditions. It is a common knowledge that verification of such conditions for general non-convex PDE-constrained optimization problems is very challenging. We propose a method to verify second-order sufficient conditions for a general class of optimal control problem. If the proposed verification method confirms the fulfillment of the sufficient condition then a-posteriori error estimates can be computed. A special ingredient of our method is an error analysis for the Hessian of the underlying optimization problem. We derive conditions under which positive definiteness of the Hessian of the discrete problem implies positive definiteness of the Hessian of the continuous problem. The results are complemented with numerical experiments. In the second part, we investigate adaptive methods for optimal control problems with finitely many control parameters. We analyze a-posteriori error estimates based on verification of second-order sufficient optimality conditions using the method developed in the first part. Reliability and efficiency of the error estimator are shown. We illustrate through numerical experiments, the use of the estimator in guiding adaptive mesh refinement.}, subject = {Optimale Kontrolle}, language = {en} } @article{BallesterBolinchesBeidlemanHeinekenetal.2010, author = {Ballester-Bolinches, A. and Beidleman, J. C. and Heineken, H. and Pedraza-Aguilera, M. C.}, title = {Local Classes and Pairwise Mutually Permutable Products of Finite Groups}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-68062}, year = {2010}, abstract = {The main aim of the paper is to present some results about products of pairwise mutually permutable subgroups and local classes.}, subject = {Mathematik}, language = {en} } @article{FerranteWimmer2010, author = {Ferrante, Augusto and Wimmer, Harald K.}, title = {Reachability matrices and cyclic matrices}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-68074}, year = {2010}, abstract = {We study reachability matrices R(A, b) = [b,Ab, . . . ,An-1b], where A is an n × n matrix over a field K and b is in Kn. We characterize those matrices that are reachability matrices for some pair (A, b). In the case of a cyclic matrix A and an n-vector of indeterminates x, we derive a factorization of the polynomial det(R(A, x)).}, subject = {Mathematik}, language = {en} } @book{FalkMarohnMicheletal.2012, author = {Falk, Michael and Marohn, Frank and Michel, Ren{\´e} and Hofmann, Daniel and Macke, Maria and Spachmann, Christoph and Englert, Stefan}, title = {A First Course on Time Series Analysis : Examples with SAS [Version 2012.August.01]}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-72617}, publisher = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {The analysis of real data by means of statistical methods with the aid of a software package common in industry and administration usually is not an integral part of mathematics studies, but it will certainly be part of a future professional work. The present book links up elements from time series analysis with a selection of statistical procedures used in general practice including the statistical software package SAS. Consequently this book addresses students of statistics as well as students of other branches such as economics, demography and engineering, where lectures on statistics belong to their academic training. But it is also intended for the practician who, beyond the use of statistical tools, is interested in their mathematical background. Numerous problems illustrate the applicability of the presented statistical procedures, where SAS gives the solutions. The programs used are explicitly listed and explained. No previous experience is expected neither in SAS nor in a special computer system so that a short training period is guaranteed. This book is meant for a two semester course (lecture, seminar or practical training) where the first three chapters can be dealt within the first semester. They provide the principal components of the analysis of a time series in the time domain. Chapters 4, 5 and 6 deal with its analysis in the frequency domain and can be worked through in the second term. In order to understand the mathematical background some terms are useful such as convergence in distribution, stochastic convergence, maximum likelihood estimator as well as a basic knowledge of the test theory, so that work on the book can start after an introductory lecture on stochastics. Each chapter includes exercises. An exhaustive treatment is recommended. Chapter 7 (case study) deals with a practical case and demonstrates the presented methods. It is possible to use this chapter independent in a seminar or practical training course, if the concepts of time series analysis are already well understood. This book is consecutively subdivided in a statistical part and an SAS-specific part. For better clearness the SAS-specific parts are highlighted. This book is an open source project under the GNU Free Documentation License.}, subject = {Zeitreihenanalyse}, language = {en} } @phdthesis{Schoenlein2012, author = {Sch{\"o}nlein, Michael}, title = {Stability and Robustness of Fluid Networks: A Lyapunov Perspective}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-72235}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {In the verification of positive Harris recurrence of multiclass queueing networks the stability analysis for the class of fluid networks is of vital interest. This thesis addresses stability of fluid networks from a Lyapunov point of view. In particular, the focus is on converse Lyapunov theorems. To gain an unified approach the considerations are based on generic properties that fluid networks under widely used disciplines have in common. It is shown that the class of closed generic fluid network models (closed GFNs) is too wide to provide a reasonable Lyapunov theory. To overcome this fact the class of strict generic fluid network models (strict GFNs) is introduced. In this class it is required that closed GFNs satisfy additionally a concatenation and a lower semicontinuity condition. We show that for strict GFNs a converse Lyapunov theorem is true which provides a continuous Lyapunov function. Moreover, it is shown that for strict GFNs satisfying a trajectory estimate a smooth converse Lyapunov theorem holds. To see that widely used queueing disciplines fulfill the additional conditions, fluid networks are considered from a differential inclusions perspective. Within this approach it turns out that fluid networks under general work-conserving, priority and proportional processor-sharing disciplines define strict GFNs. Furthermore, we provide an alternative proof for the fact that the Markov process underlying a multiclass queueing network is positive Harris recurrent if the associate fluid network defining a strict GFN is stable. The proof explicitely uses the Lyapunov function admitted by the stable strict GFN. Also, the differential inclusions approach shows that first-in-first-out disciplines play a special role.}, subject = {Warteschlangennetz}, language = {en} } @article{ChenchiahSchloemerkemper2012, author = {Chenchiah, Isaac and Schl{\"o}merkemper, Anja}, title = {Non-laminate microstructures in monoclinic-I martensite}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-72134}, year = {2012}, abstract = {We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of 3-tuples of pairwise incompatible strains. Moreover we construct a five-dimensional continuum of T3s and show that its intersection with the boundary of the symmetrised rank-one convex hull is four-dimensional. We also show that there is another kind of monoclinic-I martensite with qualitatively different semi-convex hulls which, so far as we know, has not been experimentally observed. Our strategy is to combine understanding of the algebraic structure of symmetrised rank-one convex cones with knowledge of the faceting structure of the convex polytope formed by the strains.}, subject = {Martensit}, language = {en} } @phdthesis{Tichy2011, author = {Tichy, Michael}, title = {On algebraic aggregation methods in additive preconditioning}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-56541}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {In the following dissertation we consider three preconditioners of algebraic multigrid type, though they are defined for arbitrary prolongation and restriction operators, we consider them in more detail for the aggregation method. The strengthened Cauchy-Schwarz inequality and the resulting angle between the spaces will be our main interests. In this context we will introduce some modifications. For the problem of the one-dimensional convection we obtain perfect theoretical results. Although this is not the case for more complex problems, the numerical results we present will show that the modifications are also useful in these situation. Additionally, we will consider a symmetric problem in the energy norm and present a simple rule for algebraic aggregation.}, subject = {Pr{\"a}konditionierung}, language = {en} } @book{FalkMarohnMicheletal.2011, author = {Falk, Michael and Marohn, Frank and Michel, Ren{\´e} and Hofmann, Daniel and Macke, Maria and Tewes, Bernward and Dinges, Peter and Spachmann, Christoph and Englert, Stefan}, title = {A First Course on Time Series Analysis : Examples with SAS}, organization = {Universit{\"a}t W{\"u}rzburg / Lehrstuhl f{\"u}r Statistik}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-56489}, publisher = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {The analysis of real data by means of statistical methods with the aid of a software package common in industry and administration usually is not an integral part of mathematics studies, but it will certainly be part of a future professional work. The present book links up elements from time series analysis with a selection of statistical procedures used in general practice including the statistical software package SAS. Consequently this book addresses students of statistics as well as students of other branches such as economics, demography and engineering, where lectures on statistics belong to their academic training. But it is also intended for the practician who, beyond the use of statistical tools, is interested in their mathematical background. Numerous problems illustrate the applicability of the presented statistical procedures, where SAS gives the solutions. The programs used are explicitly listed and explained. No previous experience is expected neither in SAS nor in a special computer system so that a short training period is guaranteed. This book is meant for a two semester course (lecture, seminar or practical training) where the first three chapters can be dealt within the first semester. They provide the principal components of the analysis of a time series in the time domain. Chapters 4, 5 and 6 deal with its analysis in the frequency domain and can be worked through in the second term. In order to understand the mathematical background some terms are useful such as convergence in distribution, stochastic convergence, maximum likelihood estimator as well as a basic knowledge of the test theory, so that work on the book can start after an introductory lecture on stochastics. Each chapter includes exercises. An exhaustive treatment is recommended. Chapter 7 (case study) deals with a practical case and demonstrates the presented methods. It is possible to use this chapter independent in a seminar or practical training course, if the concepts of time series analysis are already well understood. This book is consecutively subdivided in a statistical part and an SAS-specific part. For better clearness the SAS-specific parts are highlighted. This book is an open source project under the GNU Free Documentation License.}, subject = {Zeitreihenanalyse}, language = {en} } @phdthesis{Wisheckel2020, author = {Wisheckel, Florian}, title = {Some Applications of D-Norms to Probability and Statistics}, doi = {10.25972/OPUS-21214}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-212140}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {This cumulative dissertation is organized as follows: After the introduction, the second chapter, based on "Asymptotic independence of bivariate order statistics" (2017) by Falk and Wisheckel, is an investigation of the asymptotic dependence behavior of the components of bivariate order statistics. We find that the two components of the order statistics become asymptotically independent for certain combinations of (sequences of) indices that are selected, and it turns out that no further assumptions on the dependence of the two components in the underlying sample are necessary. To establish this, an explicit representation of the conditional distribution of bivariate order statistics is derived. Chapter 3 is from "Conditional tail independence in archimedean copula models" (2019) by Falk, Padoan and Wisheckel and deals with the conditional distribution of an Archimedean copula, conditioned on one of its components. We show that its tails are independent under minor conditions on the generator function, even if the unconditional tails were dependent. The theoretical findings are underlined by a simulation study and can be generalized to Archimax copulas. "Generalized pareto copulas: A key to multivariate extremes" (2019) by Falk, Padoan and Wisheckel lead to Chapter 4 where we introduce a nonparametric approach to estimate the probability that a random vector exceeds a fixed threshold if it follows a Generalized Pareto copula. To this end, some theory underlying the concept of Generalized Pareto distributions is presented first, the estimation procedure is tested using a simulation and finally applied to a dataset of air pollution parameters in Milan, Italy, from 2002 until 2017. The fifth chapter collects some additional results on derivatives of D-norms, in particular a condition for the existence of directional derivatives, and multivariate spacings, specifically an explicit formula for the second-to-last bivariate spacing.}, subject = {Kopula }, language = {en} } @phdthesis{Buchholzer2011, author = {Buchholzer, Hannes}, title = {The Semismooth Newton Method for the Solution of Reactive Transport Problems Including Mineral Precipitation-Dissolution Reactions}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-65342}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {In dieser Arbeit befassen wir uns mit einem reaktiven Transportmodell mit Niederschlags-Aufl{\"o}sung Reaktionen das aus den Geowissenschaften stammt. Es besteht aus PDGs, gew{\"o}hnlichen Differentialgleichungen, algebraischen Gleichungen und Komplementarit{\"a}tsbedingungen. Nach Diskretisation dieses Modells erhalten wir eine großes nichtlineares und nichtglattes Gleichungssystem. Wir l{\"o}sen dieses System mit der semismoothen Newtonverfahren, das von Qi und Sun eingef{\"u}hrt wurde. Der Fokus dieser Arbeit ist in der Anwendung und Konvergenz dieses Algorithmus. Wir zeigen, dass dieser Algorithmus f{\"u}r dieses Problem wohldefiniert ist und sogar lokal quadratisch konvergiert gegen eine BD-regul{\"a}re L{\"o}sung. Wir befassen uns auch mit den dabei entstehenden linearen Gleichungssystemen, die sehr groß und d{\"u}nn besetzt sind, und wie sie effizient gel{\"o}st werden k{\"o}nnen. Ein wichtiger Bestandteil dieser Untersuchung ist die Beschr{\"a}nktheit einer gewissen matrixwertigen Funktion, die in einem eigenen Kapitel gezeigt wird. Als Seitenbetrachtung untersuchen wir wie die extremalen Eigenwerte (und Singul{\"a}rwerte) von gewissen PDE-Operatoren, welche in unserem diskretisierten Modell vorkommen, genau abgesch{\"a}tzt werden k{\"o}nnen.}, subject = {Komplementarit{\"a}tsproblem}, language = {en} } @phdthesis{Schwartz2011, author = {Schwartz, Alexandra}, title = {Mathematical Programs with Complementarity Constraints: Theory, Methods and Applications}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-64891}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {The subject of this thesis are mathematical programs with complementarity conditions (MPCC). At first, an economic example of this problem class is analyzed, the problem of effort maximization in asymmetric n-person contest games. While an analytical solution for this special problem could be derived, this is not possible in general for MPCCs. Therefore, optimality conditions which might be used for numerical approaches where considered next. More precisely, a Fritz-John result for MPCCs with stronger properties than those known so far was derived together with some new constraint qualifications and subsequently used to prove an exact penalty result. Finally, to solve MPCCs numerically, the so called relaxation approach was used. Besides improving the results for existing relaxation methods, a new relaxation with strong convergence properties was suggested and a numerical comparison of all methods based on the MacMPEC collection conducted.}, subject = {Zwei-Ebenen-Optimierung}, language = {en} } @book{Lingel2016, author = {Lingel, Klaus}, title = {Metakognitives Wissen Mathematik - Entwicklung und Zusammenhang mit der Mathematikleistung in der Sekundarstufe I}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-004-7 (print)}, doi = {10.25972/WUP-978-3-95826-005-4}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-103269}, publisher = {W{\"u}rzburg University Press}, pages = {312}, year = {2016}, abstract = {Das Wissen {\"u}ber kognitive Prozesse oder metakognitives Wissen ist seit den 1970er-Jahren Gegenstand der entwicklungspsychologischen Forschung. Im Inhaltsbereich der mathematischen Informationsverarbeitung ist das Konstrukt jedoch - trotz elaborierter theoretischer Modelle {\"u}ber Struktur und Inhalt - empirisch nach wie vor weitgehend unerschlossen. Die vorliegende Studie schließt diese L{\"u}cke, indem sie die Entwicklung des mathematischen metakognitiven Wissens im L{\"a}ngsschnitt untersucht. Dazu wurde nicht nur der Entwicklungsverlauf beschrieben, sondern auch nach den Quellen f{\"u}r die beobachteten individuellen Unterschiede in der Entwicklung gesucht. Auch die aus p{\"a}dagogischen Gesichtspunkten interessanten Zusammenh{\"a}nge zwischen der metakognitiven Wissensentwicklung und der parallel dazu verlaufenden Entwicklung der mathematischen Kompetenzen wurden analysiert.}, subject = {Kognitiver Prozess}, language = {de} } @phdthesis{Ciaramella2015, author = {Ciaramella, Gabriele}, title = {Exact and non-smooth control of quantum spin systems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-118386}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {An efficient and accurate computational framework for solving control problems governed by quantum spin systems is presented. Spin systems are extremely important in modern quantum technologies such as nuclear magnetic resonance spectroscopy, quantum imaging and quantum computing. In these applications, two classes of quantum control problems arise: optimal control problems and exact-controllability problems, with a bilinear con- trol structure. These models correspond to the Schr{\"o}dinger-Pauli equation, describing the time evolution of a spinor, and the Liouville-von Neumann master equation, describing the time evolution of a spinor and a density operator. This thesis focuses on quantum control problems governed by these models. An appropriate definition of the optimiza- tion objectives and of the admissible set of control functions allows to construct controls with specific properties. These properties are in general required by the physics and the technologies involved in quantum control applications. A main purpose of this work is to address non-differentiable quantum control problems. For this reason, a computational framework is developed to address optimal-control prob- lems, with possibly L1 -penalization term in the cost-functional, and exact-controllability problems. In both cases the set of admissible control functions is a subset of a Hilbert space. The bilinear control structure of the quantum model, the L1 -penalization term and the control constraints generate high non-linearities that make difficult to solve and analyse the corresponding control problems. The first part of this thesis focuses on the physical description of the spin of particles and of the magnetic resonance phenomenon. Afterwards, the controlled Schr{\"o}dinger- Pauli equation and the Liouville-von Neumann master equation are discussed. These equations, like many other controlled quantum models, can be represented by dynamical systems with a bilinear control structure. In the second part of this thesis, theoretical investigations of optimal control problems, with a possible L1 -penalization term in the objective and control constraints, are consid- ered. In particular, existence of solutions, optimality conditions, and regularity properties of the optimal controls are discussed. In order to solve these optimal control problems, semi-smooth Newton methods are developed and proved to be superlinear convergent. The main difficulty in the implementation of a Newton method for optimal control prob- lems comes from the dimension of the Jacobian operator. In a discrete form, the Jacobian is a very large matrix, and this fact makes its construction infeasible from a practical point of view. For this reason, the focus of this work is on inexact Krylov-Newton methods, that combine the Newton method with Krylov iterative solvers for linear systems, and allows to avoid the construction of the discrete Jacobian. In the third part of this thesis, two methodologies for the exact-controllability of quan- tum spin systems are presented. The first method consists of a continuation technique, while the second method is based on a particular reformulation of the exact-control prob- lem. Both these methodologies address minimum L2 -norm exact-controllability problems. In the fourth part, the thesis focuses on the numerical analysis of quantum con- trol problems. In particular, the modified Crank-Nicolson scheme as an adequate time discretization of the Schr{\"o}dinger equation is discussed, the first-discretize-then-optimize strategy is used to obtain a discrete reduced gradient formula for the differentiable part of the optimization objective, and implementation details and globalization strategies to guarantee an adequate numerical behaviour of semi-smooth Newton methods are treated. In the last part of this work, several numerical experiments are performed to vali- date the theoretical results and demonstrate the ability of the proposed computational framework to solve quantum spin control problems.}, subject = {Spinsystem}, language = {en} } @phdthesis{Srichan2015, author = {Srichan, Teerapat}, title = {Discrete Moments of Zeta-Functions with respect to random and ergodic transformations}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-118395}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {In the thesis discrete moments of the Riemann zeta-function and allied Dirichlet series are studied. In the first part the asymptotic value-distribution of zeta-functions is studied where the samples are taken from a Cauchy random walk on a vertical line inside the critical strip. Building on techniques by Lifshits and Weber analogous results for the Hurwitz zeta-function are derived. Using Atkinson's dissection this is even generalized to Dirichlet L-functions associated with a primitive character. Both results indicate that the expectation value equals one which shows that the values of these zeta-function are small on average. The second part deals with the logarithmic derivative of the Riemann zeta-function on vertical lines and here the samples are with respect to an explicit ergodic transformation. Extending work of Steuding, discrete moments are evaluated and an equivalent formulation for the Riemann Hypothesis in terms of ergodic theory is obtained. In the third and last part of the thesis, the phenomenon of universality with respect to stochastic processes is studied. It is shown that certain random shifts of the zeta-function can approximate non-vanishing analytic target functions as good as we please. This result relies on Voronin's universality theorem.}, subject = {Riemannsche Zetafunktion}, language = {en} } @phdthesis{Schaeffner2015, author = {Sch{\"a}ffner, Mathias}, title = {Multiscale analysis of non-convex discrete systems via \(\Gamma\)-convergence}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-122349}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {The subject of this thesis is the rigorous passage from discrete systems to continuum models via variational methods. The first part of this work studies a discrete model describing a one-dimensional chain of atoms with finite range interactions of Lennard-Jones type. We derive an expansion of the ground state energy using \(\Gamma\)-convergence. In particular, we show that a variant of the Cauchy-Born rule holds true for the model under consideration. We exploit this observation to derive boundary layer energies due to asymmetries of the lattice at the boundary or at cracks of the specimen. Hereby we extend several results obtained previously for models involving only nearest and next-to-nearest neighbour interactions by Braides and Cicalese and Scardia, Schl{\"o}merkemper and Zanini. The second part of this thesis is devoted to the analysis of a quasi-continuum (QC) method. To this end, we consider the discrete model studied in the first part of this thesis as the fully atomistic model problem and construct an approximation based on a QC method. We show that in an elastic setting the expansion by \(\Gamma\)-convergence of the fully atomistic energy and its QC approximation coincide. In the case of fracture, we show that this is not true in general. In the case of only nearest and next-to-nearest neighbour interactions, we give sufficient conditions on the QC approximation such that, also in case of fracture, the minimal energies of the fully atomistic energy and its approximation coincide in the limit.}, subject = {Gamma-Konvergenz}, language = {en} } @phdthesis{Mohammadi2015, author = {Mohammadi, Masoumeh}, title = {Analysis of discretization schemes for Fokker-Planck equations and related optimality systems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-111494}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {The Fokker-Planck (FP) equation is a fundamental model in thermodynamic kinetic theories and statistical mechanics. In general, the FP equation appears in a number of different fields in natural sciences, for instance in solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory. These equations also provide a powerful mean to define robust control strategies for random models. The FP equations are partial differential equations (PDE) describing the time evolution of the probability density function (PDF) of stochastic processes. These equations are of different types depending on the underlying stochastic process. In particular, they are parabolic PDEs for the PDF of Ito processes, and hyperbolic PDEs for piecewise deterministic processes (PDP). A fundamental axiom of probability calculus requires that the integral of the PDF over all the allowable state space must be equal to one, for all time. Therefore, for the purpose of accurate numerical simulation, a discretized FP equation must guarantee conservativeness of the total probability. Furthermore, since the solution of the FP equation represents a probability density, any numerical scheme that approximates the FP equation is required to guarantee the positivity of the solution. In addition, an approximation scheme must be accurate and stable. For these purposes, for parabolic FP equations on bounded domains, we investigate the Chang-Cooper (CC) scheme for space discretization and first- and second-order backward time differencing. We prove that the resulting space-time discretization schemes are accurate, conditionally stable, conservative, and preserve positivity. Further, we discuss a finite difference discretization for the FP system corresponding to a PDP process in a bounded domain. Next, we discuss FP equations in unbounded domains. In this case, finite-difference or finite-element methods cannot be applied. By employing a suitable set of basis functions, spectral methods allow to treat unbounded domains. Since FP solutions decay exponentially at infinity, we consider Hermite functions as basis functions, which are Hermite polynomials multiplied by a Gaussian. To this end, the Hermite spectral discretization is applied to two different FP equations; the parabolic PDE corresponding to Ito processes, and the system of hyperbolic PDEs corresponding to a PDP process. The resulting discretized schemes are analyzed. Stability and spectral accuracy of the Hermite spectral discretization of the FP problems is proved. Furthermore, we investigate the conservativity of the solutions of FP equations discretized with the Hermite spectral scheme. In the last part of this thesis, we discuss optimal control problems governed by FP equations on the characterization of their solution by optimality systems. We then investigate the Hermite spectral discretization of FP optimality systems in unbounded domains. Within the framework of Hermite discretization, we obtain sparse-band systems of ordinary differential equations. We analyze the accuracy of the discretization schemes by showing spectral convergence in approximating the state, the adjoint, and the control variables that appear in the FP optimality systems. To validate our theoretical estimates, we present results of numerical experiments.}, subject = {Fokker-Planck-Gleichung}, language = {en} } @phdthesis{Wongkaew2015, author = {Wongkaew, Suttida}, title = {On the control through leadership of multi-agent systems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-120914}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {The investigation of interacting multi-agent models is a new field of mathematical research with application to the study of behavior in groups of animals or community of people. One interesting feature of multi-agent systems is collective behavior. From the mathematical point of view, one of the challenging issues considering with these dynamical models is development of control mechanisms that are able to influence the time evolution of these systems. In this thesis, we focus on the study of controllability, stabilization and optimal control problems for multi-agent systems considering three models as follows: The first one is the Hegselmann Krause opinion formation (HK) model. The HK dynamics describes how individuals' opinions are changed by the interaction with others taking place in a bounded domain of confidence. The study of this model focuses on determining feedback controls in order to drive the agents' opinions to reach a desired agreement. The second model is the Heider social balance (HB) model. The HB dynamics explains the evolution of relationships in a social network. One purpose of studying this system is the construction of control function in oder to steer the relationship to reach a friendship state. The third model that we discuss is a flocking model describing collective motion observed in biological systems. The flocking model under consideration includes self-propelling, friction, attraction, repulsion, and alignment features. We investigate a control for steering the flocking system to track a desired trajectory. Common to all these systems is our strategy to add a leader agent that interacts with all other members of the system and includes the control mechanism. Our control through leadership approach is developed using classical theoretical control methods and a model predictive control (MPC) scheme. To apply the former method, for each model the stability of the corresponding linearized system near consensus is investigated. Further, local controllability is examined. However, only in the Hegselmann-Krause opinion formation model, the feedback control is determined in order to steer agents' opinions to globally converge to a desired agreement. The MPC approach is an optimal control strategy based on numerical optimization. To apply the MPC scheme, optimal control problems for each model are formulated where the objective functions are different depending on the desired objective of the problem. The first-oder necessary optimality conditions for each problem are presented. Moreover for the numerical treatment, a sequence of open-loop discrete optimality systems is solved by accurate Runge-Kutta schemes, and in the optimization procedure, a nonlinear conjugate gradient solver is implemented. Finally, numerical experiments are performed to investigate the properties of the multi-agent models and demonstrate the ability of the proposed control strategies to drive multi-agent systems to attain a desired consensus and to track a given trajectory.}, subject = {Mehragentensystem}, language = {en} } @phdthesis{Bauer2015, author = {Bauer, Ulrich Josef}, title = {Conformal Mappings onto Simply and Multiply Connected Circular Arc Polygon Domains}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-123914}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {The goal of this thesis is to investigate conformal mappings onto circular arc polygon domains, i.e. domains that are bounded by polygons consisting of circular arcs instead of line segments. Conformal mappings onto circular arc polygon domains contain parameters in addition to the classical parameters of the Schwarz-Christoffel transformation. To contribute to the parameter problem of conformal mappings from the unit disk onto circular arc polygon domains, we investigate two special cases of these mappings. In the first case we can describe the additional parameters if the bounding circular arc polygon is a polygon with straight sides. In the second case we provide an approximation for the additional parameters if the circular arc polygon domain satisfies some symmetry conditions. These results allow us to draw conclusions on the connection between these additional parameters and the classical parameters of the mapping. For conformal mappings onto multiply connected circular arc polygon domains, we provide an alternative construction of the mapping formula without using the Schottky-Klein prime function. In the process of constructing our main result, mappings for domains of connectivity three or greater, we also provide a formula for conformal mappings onto doubly connected circular arc polygon domains. The comparison of these mapping formulas with already known mappings allows us to provide values for some of the parameters of the mappings onto doubly connected circular arc polygon domains if the image domain is a polygonal domain. The different components of the mapping formula are constructed by using a slightly modified variant of the Poincar{\´e} theta series. This construction includes the design of a function to remove unwanted poles and of different versions of functions that are analytic on the domain of definition of the mapping functions and satisfy some special functional equations. We also provide the necessary concepts to numerically evaluate the conformal mappings onto multiply connected circular arc polygon domains. As the evaluation of such a map requires the solution of a differential equation, we provide a possible configuration of curves inside the preimage domain to solve the equation along them in addition to a description of the procedure for computing either the formula for the doubly connected case or the case of connectivity three or greater. We also describe the procedures for solving the parameter problem for multiply connected circular arc polygon domains.}, subject = {Konforme Abbildungen}, language = {en} } @phdthesis{Hain2015, author = {Hain, Johannes}, title = {Valuation Algorithms for Structural Models of Financial Networks}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-128108}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {The thesis focuses on the valuation of firms in a system context where cross-holdings of the firms in liabilities and equities are allowed and, therefore, systemic risk can be modeled on a structural level. A main property of such models is that for the determination of the firm values a pricing equilibrium has to be found. While there exists a small but growing amount of research on the existence and the uniqueness of such price equilibria, the literature is still somewhat inconsistent. An example for this fact is that different authors define the underlying financial system on differing ways. Moreover, only few articles pay intense attention on procedures to find the pricing equilibria. In the existing publications, the provided algorithms mainly reflect the individual authors' particular approach to the problem. Additionally, all existing methods do have the drawback of potentially infinite runtime. For these reasons, the objects of this thesis are as follows. First, a definition of a financial system is introduced in its most general form in Chapter 2. It is shown that under a fairly mild regularity condition the financial system has a unique existing payment equilibrium. In Chapter 3, some extensions and differing definitions of financial systems that exist in literature are presented and it is shown how these models can be embedded into the general model from the proceeding chapter. Second, an overview of existing valuation algorithms to find the equilibrium is given in Chapter 4, where the existing methods are generalized and their corresponding mathematical properties are highlighted. Third, a complete new class of valuation algorithms is developed in Chapter 4 that includes the additional information whether a firm is in default or solvent under a current payment vector. This results in procedures that are able find the solution of the system in a finite number of iteration steps. In Chapter 5, the developed concepts of Chapter 4 are applied to more general financial systems where more than one seniority level of debt is present. Chapter 6 develops optimal starting vectors for non-finite algorithms and Chapter 7 compares the existing and the new developed algorithms concerning their efficiency in an extensive simulation study covering a wide range of possible settings for financial systems.}, subject = {Risikomanagement}, language = {en} } @phdthesis{Hofmann2009, author = {Hofmann, Daniel}, title = {Characterization of the D-Norm Corresponding to a Multivariate Extreme Value Distribution}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-41347}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {It is well-known that a multivariate extreme value distribution can be represented via the D-Norm. However not every norm yields a D-Norm. In this thesis a necessary and sufficient condition is given for a norm to define an extreme value distribution. Applications of this theorem includes a new proof for the bivariate case, the Pickands dependence function and the nested logistic model. Furthermore the GPD-Flow is introduced and first insights were given such that if it converges it converges against the copula of complete dependence.}, subject = {Kopula }, language = {en} } @phdthesis{Hoheisel2009, author = {Hoheisel, Tim}, title = {Mathematical Programs with Vanishing Constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-40790}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {A new class of optimization problems name 'mathematical programs with vanishing constraints (MPVCs)' is considered. MPVCs are on the one hand very challenging from a theoretical viewpoint, since standard constraint qualifications such as LICQ, MFCQ, or ACQ are most often violated, and hence, the Karush-Kuhn-Tucker conditions do not provide necessary optimality conditions off-hand. Thus, new CQs and the corresponding optimality conditions are investigated. On the other hand, MPVCs have important applications, e.g., in the field of topology optimization. Therefore, numerical algorithms for the solution of MPVCs are designed, investigated and tested for certain problems from truss-topology-optimization.}, subject = {Nichtlineare Optimierung}, language = {en} } @phdthesis{vonHeusinger2009, author = {von Heusinger, Anna}, title = {Numerical Methods for the Solution of the Generalized Nash Equilibrium Problem}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-47662}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {In the generalized Nash equilibrium problem not only the cost function of a player depends on the rival players' decisions, but also his constraints. This thesis presents different iterative methods for the numerical computation of a generalized Nash equilibrium, some of them globally, others locally superlinearly convergent. These methods are based on either reformulations of the generalized Nash equilibrium problem as an optimization problem, or on a fixed point formulation. The key tool for these reformulations is the Nikaido-Isoda function. Numerical results for various problem from the literature are given.}, subject = {Spieltheorie}, language = {en} } @article{FalkMarohn1993, author = {Falk, Michael and Marohn, Frank}, title = {Von Mises condition revisited}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-45790}, year = {1993}, abstract = {It is shown that the rate of convergence in the von Mises conditions of extreme value theory determines the distance of the underlying distribution function F from a generalized Pareto distribution. The distance is measured in terms of the pertaining densities with the limit being ultimately attained if and only if F is ultimately a generalized Pareto distribution. Consequently, the rate of convergence of the extremes in an lid sample, whether in terms of the distribution of the largest order statistics or of corresponding empirical truncated point processes, is determined by the rate of convergence in the von Mises condition. We prove that the converse is also true.}, language = {en} } @article{JanssenMarohn1994, author = {Janssen, A. and Marohn, Frank}, title = {On statistical information of extreme order statistics, local extreme value alternatives, and Poisson point processes}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-45816}, year = {1994}, abstract = {The aim of the present paper is to clarify the role of extreme order statistics in general statistical models. This is done within the general setup of statistical experiments in LeCam's sense. Under the assumption of monotone likelihood ratios, we prove that a sequence of experiments is asymptotically Gaussian if, and only if, a fixed number of extremes asymptotically does not contain any information. In other words: A fixed number of extremes asymptotically contains information iff the Poisson part of the limit experiment is non-trivial. Suggested by this result, we propose a new extreme value model given by local alternatives. The local structure is described by introducing the space of extreme value tangents. It turns out that under local alternatives a new class of extreme value distributions appears as limit distributions. Moreover, explicit representations of the Poisson limit experiments via Poisson point processes are found. As a concrete example nonparametric tests for Frechet type distributions against stochastically larger alternatives are treated. We find asymptotically optimal tests within certain threshold models.}, language = {en} } @inproceedings{Marohn1994, author = {Marohn, Frank}, title = {On testing the exponential and Gumbel distribution}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-45804}, year = {1994}, abstract = {No abstract available}, subject = {Gumbel-Verteilung}, language = {en} } @book{Marohn1990, author = {Marohn, Frank}, title = {On statistical information of extreme order statistics}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-47866}, publisher = {Universit{\"a}t W{\"u}rzburg}, year = {1990}, abstract = {No abstract available}, subject = {Rangstatistik}, language = {en} } @article{FalkMarohn1993, author = {Falk, Michael and Marohn, Frank}, title = {Asymptotically optimal tests for conditional distributions}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-45823}, year = {1993}, abstract = {No abstract available}, language = {en} } @article{Marohn1994, author = {Marohn, Frank}, title = {Asymptotic sufficiency of order statistics for almost regular Weibull type densities}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-45837}, year = {1994}, abstract = {Consider a location family which is defined via a Weibull type density having shape parameter a = 1. We treat the problem, which portion of the order statistics is asymptotically sufficient. It turns out that the intermediate order statistics are relevant.}, language = {en} } @incollection{FalkMarohn1992, author = {Falk, Michael and Marohn, Frank}, title = {Laws of small numbers : Some applications to conditional curve estimation}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-45841}, publisher = {Universit{\"a}t W{\"u}rzburg}, year = {1992}, abstract = {No abstract available}, subject = {Gesetz der kleinen Zahlen}, language = {en} } @article{Marohn1991, author = {Marohn, Frank}, title = {Global sufficiency of extreme order statistics in location models of Weibull type}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-47874}, year = {1991}, abstract = {In Janssen and Reiss (1988) it was shown that in a location model of a Weibull type sample with shape parameter -1 < a < 1 the k(n) lower extremes are asymptotically local sufficient. In the present paper we show that even global sufficiency holds. Moreover, it turns out that convergence of the given statistical experiments in the deficiency metric does not only hold for compact parameter sets but for the whole real line.}, subject = {Extremwertstatistik}, language = {de} } @phdthesis{Schindele2016, author = {Schindele, Andreas}, title = {Proximal methods in medical image reconstruction and in nonsmooth optimal control of partial differential equations}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-136569}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2016}, abstract = {Proximal methods are iterative optimization techniques for functionals, J = J1 + J2, consisting of a differentiable part J2 and a possibly nondifferentiable part J1. In this thesis proximal methods for finite- and infinite-dimensional optimization problems are discussed. In finite dimensions, they solve l1- and TV-minimization problems that are effectively applied to image reconstruction in magnetic resonance imaging (MRI). Convergence of these methods in this setting is proved. The proposed proximal scheme is compared to a split proximal scheme and it achieves a better signal-to-noise ratio. In addition, an application that uses parallel imaging is presented. In infinite dimensions, these methods are discussed to solve nonsmooth linear and bilinear elliptic and parabolic optimal control problems. In particular, fast convergence of these methods is proved. Furthermore, for benchmarking purposes, truncated proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of our proximal schemes that need less computation time than the semismooth Newton method in most cases. Results of numerical experiments are presented that successfully validate the theoretical estimates.}, subject = {Optimale Kontrolle}, language = {en} } @phdthesis{Zott2016, author = {Zott, Maximilian}, title = {Extreme Value Theory in Higher Dimensions - Max-Stable Processes and Multivariate Records}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-136614}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2016}, abstract = {Die Extremwerttheorie behandelt die stochastische Modellierung seltener und extremer Ereignisse. W{\"a}hrend fundamentale Theorien in der klassischen Stochastik, wie etwa die Gesetze der großen Zahlen oder der zentrale Grenzwertsatz das asymptotische Verhalten der Summe von Zufallsvariablen untersucht, liegt in der Extremwerttheorie der Fokus auf dem Maximum oder dem Minimum einer Menge von Beobachtungen. Die Grenzverteilung des normierten Stichprobenmaximums unter einer Folge von unabh{\"a}ngigen und identisch verteilten Zufallsvariablen kann durch sogenannte max-stabile Verteilungen charakterisiert werden. In dieser Dissertation werden verschiedene Aspekte der Theorie der max-stabilen Zufallsvektoren und stochastischen Prozesse behandelt. Insbesondere wird der Begriff der 'Differenzierbarkeit in Verteilung' eines max-stabilen Prozesses eingef{\"u}hrt und untersucht. Ferner werden 'verallgemeinerte max-lineare Modelle' eingef{\"u}hrt, um einen bekannten max-stabilen Zufallsvektor durch einen max-stabilen Prozess zu interpolieren. Dar{\"u}ber hinaus wird der Zusammenhang von extremwerttheoretischen Methoden mit der Theorie der multivariaten Rekorde hergestellt. Insbesondere werden sogenannte 'vollst{\"a}ndige' und 'einfache' Rekorde eingef{\"u}hrt, und deren asymptotisches Verhalten untersucht.}, subject = {Stochastischer Prozess}, language = {en} } @phdthesis{Boehm2015, author = {B{\"o}hm, Christoph}, title = {Loewner equations in multiply connected domains}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-129903}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {The first goal of this thesis is to generalize Loewner's famous differential equation to multiply connected domains. The resulting differential equations are known as Komatu--Loewner differential equations. We discuss Komatu--Loewner equations for canonical domains (circular slit disks, circular slit annuli and parallel slit half-planes). Additionally, we give a generalisation to several slits and discuss parametrisations that lead to constant coefficients. Moreover, we compare Komatu--Loewner equations with several slits to single slit Loewner equations. Finally we generalise Komatu--Loewner equations to hulls satisfying a local growth property.}, subject = {Biholomorphe Abbildung}, language = {en} } @phdthesis{Solak2007, author = {Solak, Ebru}, title = {Almost Completely Decomposable Groups of Type (1,2)}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-24794}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2007}, abstract = {A torsion free abelian group of finite rank is called almost completely decomposable if it has a completely decomposable subgroup of finite index. A p-local, p-reduced almost completely decomposable group of type (1,2) is briefly called a (1,2)-group. Almost completely decomposable groups can be represented by matrices over the ring Z/hZ, where h is the exponent of the regulator quotient. This particular choice of representation allows for a better investigation of the decomposability of the group. Arnold and Dugas showed in several of their works that (1,2)-groups with regulator quotient of exponent at least p^7 allow infinitely many isomorphism types of indecomposable groups. It is not known if the exponent 7 is minimal. In this dissertation, this problem is addressed.}, language = {en} } @phdthesis{Teichert2009, author = {Teichert, Christian}, title = {Globale Minimierung von Linearen Programmen mit Gleichgewichtsrestriktionen und globale Konvergenz eines Filter-SQPEC-Verfahrens f{\"u}r Mathematische Programme mit Gleichgewichtsrestriktionen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-38700}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {Mathematische Programme mit Gleichgewichtsrestriktionen (oder Komplementarit{\"a}tsbedingungen), kurz MPECs, sind als {\"a}ußerst schwere Optimierungsprobleme bekannt. Lokale Minima oder geeignete station{\"a}re Punkte zu finden, ist ein nichttriviales Problem. Diese Arbeit beschreibt, wie man dennoch die spezielle Struktur von MPECs ausnutzen kann und mittels eines Branch-and-Bound-Verfahrens ein globales Minimum von Linearen Programmen mit Gleichgewichtsrestriktionen, kurz LPECs, bekommt. Des Weiteren wird dieser Branch-and-Bound-Algorithmus innerhalb eines Filter-SQPEC-Verfahrens genutzt, um allgemeine MPECs zu l{\"o}sen. F{\"u}r das Filter-SQPEC Verfahren wird ein globaler Konvergenzsatz bewiesen. Außerdem werden f{\"u}r beide Verfahren numerische Resultate angegeben.}, subject = {Nichtlineare Optimierung}, language = {de} } @phdthesis{KuhnAndriotti2009, author = {Kuhn Andriotti, Gustavo}, title = {Prospect Theory Multi-Agent Based Simulations for Non-Rational Route Choice Decision Making Modelling}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-40483}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {Simulations (MASim) and non-rational behaviour. This non-rational behaviour is here based on the Prospect Theory [KT79] (PT), which is compared to the rational behaviour in the Expected Utility Theory [vNM07] (EUT). This model was used to design a modified Q-Learning [Wat89, WD92] algorithm. The PT based Q-Learning was then integrated into a proposed agent architecture. Because much attention is given to a limited interpretation of Simon's definition of bounded-rationality, this interpretation is broadened here. Both theories, rationality and the non-rationality, are compared and the discordance in their results discussed. The main contribution of this work is to show that an alternative is available to the EUT that is more suitable for human decision-makers modelling. The evidences show that rationality is not appropriated for modelling persons. Therefore, instead of fine-tuning the existent model the use of another one is proposed and evaluated. To tackle this, the route choice problem was adopted to perform the experiments. To evaluate the proposed model three traffic scenarios are simulated and their results analysed.}, subject = {Mehragentensystem}, language = {en} } @article{Zillober1993, author = {Zillober, Christian}, title = {A globally convergent version of the method of moving asymptotes}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-31984}, year = {1993}, abstract = {No abstract available}, language = {en} } @article{KredlerZilloberJohannesetal.1993, author = {Kredler, Christian and Zillober, Christian and Johannes, Frank and Sigl, Georg}, title = {An application of preconditioned conjugate gradients to relative placement in chip design}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-31996}, year = {1993}, abstract = {In distance geometry problems and many other applications, we are faced with the optimization of high-dimensional quadratic functions subject to linear equality constraints. A new approach is presented that projects the constraints, preserving sparsity properties of the original quadratic form such that well-known preconditioning techniques for the conjugate gradient method remain applicable. Very-largescale cell placement problems in chip design have been solved successfully with diagonal and incomplete Cholesky preconditioning. Numerical results produced by a FORTRAN 77 program illustrate the good behaviour of the algorithm.}, language = {en} } @inproceedings{Zillober1993, author = {Zillober, Christian}, title = {Sequential convex programming in theory and praxis}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-35513}, year = {1993}, abstract = {In this paper, convex approximation methods, suclt as CONLIN, the method of moving asymptotes (MMA) and a stabilized version of MMA (Sequential Convex Programming), are discussed with respect to their convergence behaviour. In an extensive numerical study they are :finally compared with other well-known optimization methods at 72 examples of sizing problems.}, language = {en} } @phdthesis{Even2009, author = {Even, Nadine}, title = {On Hydrodynamic Limits and Conservation Laws}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-38374}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {No abstract available}, language = {en} } @phdthesis{Wissel2009, author = {Wissel, Julia}, title = {A new biased estimator for multivariate regression models with highly collinear variables}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-36383}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {Es ist wohlbekannt, dass der Kleinste-Quadrate-Sch{\"a}tzer im Falle vorhandener Multikollinearit{\"a}t eine große Varianz besitzt. Eine M{\"o}glichkeit dieses Problem zu umgehen, besteht in der Verwendung von verzerrten Sch{\"a}tzern, z.B den Ridge-Sch{\"a}tzer. In dieser Arbeit wird ein neues Sch{\"a}tzverfahren vorgestellt, dass auf Addition einer kleinen Konstanten omega auf die Regressoren beruht. Der dadurch erzeugte Sch{\"a}tzer wird in Abh{\"a}ngigkeit von omega beschrieben und es wird gezeigt, dass dessen Mean Squared Error kleiner ist als der des Kleinste-Quadrate-Sch{\"a}tzers im Falle von stark korrelierten Regressoren.}, subject = {Starke Kopplung}, language = {en} } @article{LuKlingenbergRendonetal.2014, author = {Lu, Yun-guang and Klingenberg, Christian and Rendon, Leonardo and Zheng, De-Yin}, title = {Global Solutions for a Simplified Shallow Elastic Fluids Model}, series = {Abstract and Applied Analytics}, journal = {Abstract and Applied Analytics}, number = {920248}, issn = {1687-0409}, doi = {10.1155/2014/920248}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-117978}, year = {2014}, abstract = {The Cauchy problem for a simplified shallow elastic fluids model, one 3 x 3 system of Temple's type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth (rho - 0). This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for 2 x 2 strictly hyperbolic system and (Heibig, 1994) for n x n strictly hyperbolic system with smooth Riemann invariants.}, language = {en} } @phdthesis{Geiselhart2015, author = {Geiselhart, Roman}, title = {Advances in the stability analysis of large-scale discrete-time systems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-112963}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2015}, abstract = {Several aspects of the stability analysis of large-scale discrete-time systems are considered. An important feature is that the right-hand side does not have have to be continuous. In particular, constructive approaches to compute Lyapunov functions are derived and applied to several system classes. For large-scale systems, which are considered as an interconnection of smaller subsystems, we derive a new class of small-gain results, which do not require the subsystems to be robust in some sense. Moreover, we do not only study sufficiency of the conditions, but rather state an assumption under which these conditions are also necessary. Moreover, gain construction methods are derived for several types of aggregation, quantifying how large a prescribed set of interconnection gains can be in order that a small-gain condition holds.}, subject = {Ljapunov-Funktion}, language = {en} } @phdthesis{Schleissinger2013, author = {Schleißinger, Sebastian}, title = {Embedding Problems in Loewner Theory}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-96782}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2013}, abstract = {The work at hand studies problems from Loewner theory and is divided into two parts: In part 1 (chapter 2) we present the basic notions of Loewner theory. Here we use a modern form which was developed by F. Bracci, M. Contreras, S. D{\´i}az-Madrigal et al. and which can be applied to certain higher dimensional complex manifolds. We look at two domains in more detail: the Euclidean unit ball and the polydisc. Here we consider two classes of biholomorphic mappings which were introduced by T. Poreda and G. Kohr as generalizations of the class S. We prove a conjecture of G. Kohr about support points of these classes. The proof relies on the observation that the classes describe so called Runge domains, which follows from a result by L. Arosio, F. Bracci and E. F. Wold. Furthermore, we prove a conjecture of G. Kohr about support points of a class of biholomorphic mappings that comes from applying the Roper-Suffridge extension operator to the class S. In part 2 (chapter 3) we consider one special Loewner equation: the chordal multiple-slit equation in the upper half-plane. After describing basic properties of this equation we look at the problem, whether one can choose the coefficient functions in this equation to be constant. D. Prokhorov proved this statement under the assumption that the slits are piecewise analytic. We use a completely different idea to solve the problem in its general form. As the Loewner equation with constant coefficients holds everywhere (and not just almost everywhere), this result generalizes Loewner's original idea to the multiple-slit case. Moreover, we consider the following problems: • The "simple-curve problem" asks which driving functions describe the growth of simple curves (in contrast to curves that touch itself). We discuss necessary and sufficient conditions, generalize a theorem of J. Lind, D. Marshall and S. Rohde to the multiple-slit equation and we give an example of a set of driving functions which generate simple curves because of a certain self-similarity property. • We discuss properties of driving functions that generate slits which enclose a given angle with the real axis. • A theorem by O. Roth gives an explicit description of the reachable set of one point in the radial Loewner equation. We prove the analog for the chordal equation.}, subject = {Biholomorphe Abbildung}, language = {en} } @phdthesis{Reinwand2021, author = {Reinwand, Simon}, title = {Functions of Bounded Variation: Theory, Methods, Applications}, publisher = {Cuvillier-Verlag, G{\"o}ttingen}, isbn = {9783736974036}, doi = {10.25972/OPUS-23515}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-235153}, school = {Universit{\"a}t W{\"u}rzburg}, pages = {326}, year = {2021}, abstract = {Functions of bounded variation are most important in many fields of mathematics. This thesis investigates spaces of functions of bounded variation with one variable of various types, compares them to other classical function spaces and reveals natural "habitats" of BV-functions. New and almost comprehensive results concerning mapping properties like surjectivity and injectivity, several kinds of continuity and compactness of both linear and nonlinear operators between such spaces are given. A new theory about different types of convergence of sequences of such operators is presented in full detail and applied to a new proof for the continuity of the composition operator in the classical BV-space. The abstract results serve as ingredients to solve Hammerstein and Volterra integral equations using fixed point theory. Many criteria guaranteeing the existence and uniqueness of solutions in BV-type spaces are given and later applied to solve boundary and initial value problems in a nonclassical setting. A big emphasis is put on a clear and detailed discussion. Many pictures and synoptic tables help to visualize and summarize the most important ideas. Over 160 examples and counterexamples illustrate the many abstract results and how delicate some of them are.}, subject = {Funktion von beschr{\"a}nkter Variation}, language = {en} } @phdthesis{Nguyen2012, author = {Nguyen, Danh Nam}, title = {Understanding the development of the proving process within a dynamic geometry environment}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-71754}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {Argumentation and proof have played a fundamental role in mathematics education in recent years. The author of this dissertation would like to investigate the development of the proving process within a dynamic geometry system in order to support tertiary students understanding the proving process. The strengths of this dynamic system stimulate students to formulate conjectures and produce arguments during the proving process. Through empirical research, we classified different levels of proving and proposed a methodological model for proving. This methodological model makes a contribution to improve students' levels of proving and develop their dynamic visual thinking. We used Toulmin model of argumentation as a theoretical model to analyze the relationship between argumentation and proof. This research also offers some possible explanation so as to why students have cognitive difficulties in constructing proofs and provides mathematics educators with a deeper understanding on the proving process within a dynamic geometry system.}, subject = {Argumentation}, language = {en} } @article{SchaeffnerSchloemerkemper2018, author = {Sch{\"a}ffner, M. and Schl{\"o}merkemper, A.}, title = {On Lennard-Jones systems with finite range interactions and their asymptotic analysis}, series = {Networks and Heterogeneous Media}, volume = {13}, journal = {Networks and Heterogeneous Media}, number = {1}, doi = {10.3934/nhm.2018005}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-228428}, pages = {95-118}, year = {2018}, abstract = {The aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of Gamma-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formula for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, we provide an explicit expression for the continuum limit in the case of finite range interactions. The obtained homogenization formula is given by the convexification of a Cauchy-Born energy density. Furthermore, we study rescaled energies in which bulk and surface contributions scale in the same way. The related discrete-to-continuum limit yields a rigorous derivation of a one-dimensional version of Griffith' fracture energy and thus generalizes earlier derivations for nearest and next-to-nearest neighbors to the case of finite range interactions. A crucial ingredient to our proofs is a novel decomposition of the energy that allows for re fined estimates.}, language = {en} } @phdthesis{Dirr2001, author = {Dirr, Gunther}, title = {Differentialgleichungen in Fr{\´e}chetr{\"a}umen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-1180417}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2001}, abstract = {Teil 1 der Arbeit beinhaltet eine Zusammenfassung grundlegender funktionalanalytischer Ergebnisse sowie eine Einf{\"u}hrung in die Integral- und Differentialrechnung in Fr{\´e}chetr{\"a}umen. Insbesondere wird in Kapitel 2 eine ausf{\"u}hrliche Darstellung des Lebesgue-Bochner-Integrals auf Fr{\´e}chetr{\"a}umen geliefert. Teil 2 behandelt die Theorie der linearen Differentialgleichungen auf Fr{\´e}chetr{\"a}umen. Dazu werden in Kapitel 3 stark differenzierbare Halbgruppen und deren infinitesimale Generatoren charakterisiert. In Kapitel 4 werden diese Ergebnisse benutzt, um lineare Evolutionsgleichungen (von hyperbolischem oder parabolischem Typ) zu untersuchen. Teil 3 enth{\"a}lt die zentralen Resultate der Arbeit. In Kapitel 5 werden zwei Existenz- und Eindeutigkeitss{\"a}tze f{\"u}r nichtlineare gew{\"o}hnliche Differentialgleichungen in zahmen Fr{\´e}chetr{\"a}umen bewiesen. Kapitel 6 liefert eine Anwendung der Ergebnisse aus Kapitel 5 auf nichtlineare partielle Differentialgleichungen erster Ordnung.}, subject = {Differentialgleichung}, language = {de} } @phdthesis{Dittmann2001, author = {Dittmann, Ulrich}, title = {Coset Types and Tight Subgroups of Almost Completely Decomposable Groups}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-2762}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2001}, abstract = {A completely decomposable group is a direct sum of subgroups of the rationals. An almost completely decomposable group is a torsion free abelian group that contains a completely decomposable group as subgroup of finite index. Tight subgroups are maximal subgroups (with respect to set inclusion) among the completely decomposable subgroups of an almost completely decomposable group. In this dissertation we show an extended version of the theorem of Bezout, give a new criterion for the tightness of a completely decomposable subgroup, derive some conditions under which a tight subgroup is regulating and generalize a theorem of Campagna. We give an example of an almost completely decomposable group, all of whose regulating subgroups do not have a quotient with minimal exponent. We show that among the types of elements of a coset modulo a completely decomposable group there exists a unique maximal type and define this type to be -the- coset type. We give criteria for tightness and regulating in term of coset types as well as a representation of the type subgroups using coset types. We introduce the notion of reducible cosets and show their key role for transitions from one completely decomposable subgroup up to another one containing the first one as a proper subgroup. We give an example of a tight, but not regulating subgroup which contains the regulator. We develop the notion of a fully single covered subset of a lattice, show that V-free implies fully single covered, but not necessarily vice versa, and we define an equivalence relation on the set of all finite subsets of a given lattice. We develop some extension of ordinary Hasse diagrams, and apply the lattice theoretic results on the lattice of types and almost completely decomposable groups.}, subject = {Torsionsfreie Abelsche Gruppe}, language = {en} } @phdthesis{Keilbach2000, author = {Keilbach, Rupert}, title = {Minimalfl{\"a}chen und Bj{\"o}rlingsches Problem in der Relativgeometrie}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-2782}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2000}, abstract = {In dieser Arbeit besch{\"a}ftigen wir uns mit Themen aus der affinen Hyperfl{\"a}chentheorie. Nachdem wir die euklidische Normale, die Blaschkesche Affinnormale, eine gewisse Einparameterfamilie von Relativnormalen und die zentroaffine Normale besprochen und eine neue Einparameterfamilie von Relativnormalen definiert haben, behandeln wir die folgenden drei Schwerpunkte: Zuerst befassen wir uns mit Minimalfl{\"a}chen bez{\"u}glich verschiedener Volumina und der Rolle der jeweiligen Mittleren Kr{\"u}mmung. Wir berechnen die erste und zweite Variation der Volumina, die von den Normalen der erw{\"a}hnten Familien induziert werden. Hierbei stellen wir fest, daß die Mittlere Kr{\"u}mmung nicht immer das Verschwinden der ersten Variation des Volumens anzeigt. Anschließend {\"u}bertragen wir die Begriffe Adjungierte und Assoziierte bei euklidischen Minimalfl{\"a}chen auf Affinminimalfl{\"a}chen: Analog zum euklidischen Fall kann man die Konormale einer Affinminimalfl{\"a}che durch bestimmte ,,harmonische'' Abbildungen darstellen. Wir geben eine Methode an, wie man aus einer gegebenen Affinminimalfl{\"a}che weitere gewinnt, indem man diese Abbildungen entsprechend modifiziert. Schließlich l{\"o}sen wir eine Verallgemeinerung des Bj{\"o}rlingschen Problems f{\"u}r Normalen der oben erw{\"a}hnten Familien: Bei Vorgabe einer Kurve mit zwei Vektorfeldern und der Art der Normalisierung existiert - mit Ausnahmen - je genau eine elliptische und eine hyperbolische Fl{\"a}che in (pseudo-)isothermen Parametern mit folgenden Eigenschaften: Die Kurve ist eine Parameterlinie, die Normale l{\"a}ngs der Kurve stimmt mit dem einen Vektorfeld {\"u}berein, die Konormale mit dem anderen und die Mittlere und Gaußsche Kr{\"u}mmung erf{\"u}llen eine vorgegebene Bedingung.}, subject = {Minimalfl{\"a}che}, language = {de} } @phdthesis{Kessler2000, author = {Keßler, Manuel}, title = {Die Ladyzhenskaya-Konstante in der numerischen Behandlung von Str{\"o}mungsproblemen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-2791}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2000}, abstract = {Charakteristisch f{\"u}r die L{\"o}sbarkeit von elliptischen partiellen Differentialgleichungssystemen mit Nebenbedingungen ist das Auftreten einer inf-sup-Bedingung. Im prototypischen Fall der Stokes-Gleichungen ist diese auch als Ladyzhenskaya-Bedingung bekannt. Die G{\"u}ltigkeit dieser Bedingung, bzw. die Existenz der zugeh{\"o}rigen Konstante ist eine Eigenschaft des Gebietes, innerhalb dessen die Differentialgleichung gel{\"o}st werden soll. W{\"a}hrend die Existenz schon die L{\"o}sbarkeit garantiert, ist beispielsweise f{\"u}r Fehleraussagen bei der numerischen Approximation auch die Gr{\"o}ße der Konstanten sehr wichtig. Insbesondere auch deshalb, weil eine {\"a}hnliche inf-sup-Bedingung auch bei der Diskretisierung mittel Finiter-Elemente-Methoden auftaucht, die hier Babuska-Brezzi-Bedingung heißt. Die Arbeit befaßt sich auf der einen Seite mit einer analytischen Absch{\"a}tzung der Ladyzhenskaya-Konstante f{\"u}r verschiedene Gebiete, wobei {\"A}quivalenzen mit verwandten Problemen aus der komplexen Analysis (Friedrichs-Ungleichung) und der Strukturmechanik (Kornsche Ungleichung) benutzt werden. Ein weiterer Teil befaßt sich mit dem Zusammenhang zwischen kontinuierlicher Ladyzhenskaya- Konstante und diskreter Babuska-Brezzi-Konstante. Die dabei gefundenen Ergebnisse werden mit Hilfe eines dazu entwickelten leistungsf{\"a}higen Finite-Elemente-Programmsystems numerisch verifiziert. Damit k{\"o}nnen erstmals genaue Absch{\"a}tzungen der Konstanten in zwei und drei Dimensionen gefunden werden. Aufbauend auf diesen Resultaten wird ein schneller L{\"o}sungsalgorithmus f{\"u}r die Stokes-Gleichungen vorgeschlagen und anhand von problematischen Gebieten dessen {\"U}berlegenheit gegen{\"u}ber klassischen Verfahren wie beispielsweise der Uzawa-Iteration demonstriert. W{\"a}hrend selbst bei einfachen Geometrien eine Konvergenzbeschleunigung um einen Faktor 5 erwartet werden kann, sind in kritischen F{\"a}llen Faktoren bis zu 1000 m{\"o}glich.}, subject = {Stokes-Gleichung}, language = {de} } @phdthesis{Nahler2001, author = {Nahler, Michael}, title = {Isomorphism classes of almost completely decomposable groups}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-2817}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2001}, abstract = {In this thesis we investigate near-isomorphism classes and isomorphism classes of almost completely decomposable groups. In Chapter 2 we introduce the concept of almost completely decomposable groups and sum up their most important facts. A local group is an almost completely decomposable group with a primary regulator quotient. A uniform group is a rigid local group with a homocyclic regulator quotient. In Chapter 3 a weakening of isomorphism, called type-isomorphism, appears. It is shown that type-isomorphism agrees with Lady's near-isomorphism. By the Main Decomposition Theorem and the Primary Reduction Theorem we are allowed to restrict ourselves on clipped local groups, namely groups without a direct rank-one summand. In Chapter 4 we collect facts of matrices over commutative rings with an identity element. Matrices over the local ring (Z / p^e Z) of residue classes of the rational integers modulo a prime power play an important role. In Chapter 5 we introduce representing matrices of finite essential extensions. Here a normal form for local groups is found by the Gauß algorithm. Uniform groups have representing matrices in Hermite normal form. The classification problems for almost completely decomposable groups up to isomorphism and up to near-isomorphism can be rephrased as equivalence problems for the representing matrices. In Chapter 6 we derive a criterion for the representing matrices of local groups in Gauß normal form. In Chapter 7 we formulate the matrix criterion for uniform groups. Two representing matrices in Hermite normal form describe isomorphic groups if and only if the rest blocks of the representing matrices are T-diagonally equivalent. Starting from a fixed near-isomorphism class in Chapter 8 we investigate isomorphism classes of uniform groups. We count groups and isomorphism classes. In Chapter 9 we specialize on uniform groups of rank 2r with a regulator quotient of rank r such that the rest block of the representing matrix is invertible and normed.}, subject = {Fast vollst{\"a}ndig zerlegbare Gruppe}, language = {en} } @phdthesis{Kramer2004, author = {Kramer, Helmut}, title = {Inzidenzmatrizen endlicher projektiver Ebenen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-11215}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2004}, abstract = {Ziel dieser Arbeit ist eine computerunterst{\"u}tzte Suche nach, bis auf Isomorphie, allen projektiven Ebenen zu einer gegebenen Ordnung durch Berechnung ihrer Inzidenzmatrix. Dies gelingt durch geeignete Vorstrukturierung der Matrix mit Hilfe der Doppelordnung bis Ordnung 9 auf einem aktuellen PC. In diesem Zusammenhang ist insbesondere durch einen gen{\"u}gend schnellen Algorithmus das Problem zu l{\"o}sen, ob zwei Inzidenzmatrizen zu derselben projektiven Ebene geh{\"o}ren. Die besondere Struktur, die die berechneten Beispiele von doppelgeordneten Inzidenzmatrizen der desarguesschen Ebenen aufzeigen, wird zudem durch theoretische {\"U}berlegungen untermauert. In einem letzten Kapitel wird noch eine Verbindung der projektiven Ebenen zu besonderen Blockpl{\"a}nen geschaffen.}, subject = {Projektive Ebene}, language = {de} } @phdthesis{Flegel2005, author = {Flegel, Michael L.}, title = {Constraint qualifications and stationarity concepts for mathematical programs with equilibrium constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-12453}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2005}, abstract = {An exhaustive discussion of constraint qualifications (CQ) and stationarity concepts for mathematical programs with equilibrium constraints (MPEC) is presented. It is demonstrated that all but the weakest CQ, Guignard CQ, are too strong for a discussion of MPECs. Therefore, MPEC variants of all the standard CQs are introduced and investigated. A strongly stationary point (which is simply a KKT-point) is seen to be a necessary first order optimality condition only under the strongest CQs, MPEC-LICQ, MPEC-SMFCQ and Guignard CQ. Therefore a whole set of KKT-type conditions is investigated. A simple approach is given to acquire A-stationarity to be a necessary first order condition under MPEC-Guiganrd CQ. Finally, a whole chapter is devoted to investigating M-stationary, among the strongest stationarity concepts, second only to strong stationarity. It is shown to be a necessary first order condition under MPEC-Guignard CQ, the weakest known CQ for MPECs.}, subject = {Nichtlineare Optimierung}, language = {en} } @book{FalkMarohnMicheletal.2005, author = {Falk, Michael and Marohn, Frank and Michel, Ren{\´e} and Hofmann, Daniel and Macke, Maria and Tewes, Bernward and Dinges, Peter}, title = {A First Course on Time Series Analysis : Examples with SAS}, organization = {Universit{\"a}t W{\"u}rzburg / Lehrstuhl f{\"u}r Statistik , Universit{\"a}t Eichst{\"a}tt/Rechenzentrum}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-12593}, publisher = {Universit{\"a}t W{\"u}rzburg}, year = {2005}, abstract = {The analysis of real data by means of statistical methods with the aid of a software package common in industry and administration usually is not an integral part of mathematics studies, but it will certainly be part of a future professional work. The present book links up elements from time series analysis with a selection of statistical procedures used in general practice including the statistical software package SAS Statistical Analysis System). Consequently this book addresses students of statistics as well as students of other branches such as economics, demography and engineering, where lectures on statistics belong to their academic training. But it is also intended for the practician who, beyond the use of statistical tools, is interested in their mathematical background. Numerous problems illustrate the applicability of the presented statistical procedures, where SAS gives the solutions. The programs used are explicitly listed and explained. No previous experience is expected neither in SAS nor in a special computer system so that a short training period is guaranteed. This book is meant for a two semester course (lecture, seminar or practical training) where the first two chapters can be dealt with in the first semester. They provide the principal components of the analysis of a time series in the time domain. Chapters 3, 4 and 5 deal with its analysis in the frequency domain and can be worked through in the second term. In order to understand the mathematical background some terms are useful such as convergence in distribution, stochastic convergence, maximum likelihood estimator as well as a basic knowledge of the test theory, so that work on the book can start after an introductory lecture on stochastics. Each chapter includes exercises. An exhaustive treatment is recommended. This book is consecutively subdivided in a statistical part and an SAS-specific part. For better clearness the SAS-specific part, including the diagrams generated with SAS, always starts with a computer symbol, representing the beginning of a session at the computer, and ends with a printer symbol for the end of this session. This book is an open source project under the GNU Free Documentation License.}, subject = {Zeitreihenanalyse}, language = {en} } @phdthesis{Trumpf2002, author = {Trumpf, Jochen}, title = {On the geometry and parametrization of almost invariant subspaces and observer theory}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-5034}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2002}, abstract = {In my Ph.D. thesis "On the geometry and parametrization of almost invariant subspaces and observer theory" I consider the set of almost conditioned invariant subspaces of fixed dimension for a given fixed linear finite-dimensional time-invariant observable control system in state space form. Almost conditioned invariant subspaces were introduced by Willems. They generalize the concept of a conditioned invariant subspace requiring the invariance condition to hold only up to an arbitrarily small deviation in the metric of the state space. One of the goals of the theory of almost conditioned invariant subspaces was to identify the subspaces appearing as limits of sequences of conditioned invariant subspaces. An example due to {\"O}zveren, Verghese and Willsky, however, shows that the set of almost conditioned invariant subspaces is not big enough. I address this question in a joint paper with Helmke and Fuhrmann (Towards a compactification of the set of conditioned invariant subspaces, Systems and Control Letters, 48(2):101-111, 2003). Antoulas derived a description of conditioned invariant subspaces as kernels of permuted and truncated reachability matrices of controllable pairs of the appropriate size. This description was used by Helmke and Fuhrmann to construct a diffeomorphism from the set of similarity classes of certain controllable pairs onto the set of tight conditioned invariant subspaces. In my thesis I generalize this result to almost conditioned invariant subspaces describing them in terms of restricted system equivalence classes of controllable triples. Furthermore, I identify the controllable pairs appearing in the kernel representations of conditioned invariant subspaces as being induced by corestrictions of the original system to the subspace. Conditioned invariant subspaces are known to be closely related to partial observers. In fact, a tracking observer for a linear function of the state of the observed system exists if and only if the kernel of that function is conditioned invariant. In my thesis I show that the system matrices of the observers are in fact the corestrictions of the observed system to the kernels of the observed functions. They in turn are closely related to partial realizations. Exploring this connection further, I prove that the set of tracking observer parameters of fixed size, i.e. tracking observers of fixed order together with the functions they are tracking, is a smooth manifold. Furthermore, I construct a vector bundle structure for the set of conditioned invariant subspaces of fixed dimension together with their friends, i.e. the output injections making the subspaces invariant, over that manifold. Willems and Trentelman generalized the concept of a tracking observer by including derivatives of the output of the observed system in the observer equations (PID-observers). They showed that a PID-observer for a linear function of the state of the observed system exists if and only if the kernel of that function is almost conditioned invariant. In my thesis I replace PID-observers by singular systems, which has the advantage that the system matrices of the observers coincide with the matrices appearing in the kernel representations of the subspaces. In a second approach to the parametrization of conditioned invariant subspaces Hinrichsen, M{\"u}nzner and Pr{\"a}tzel-Wolters, Fuhrmann and Helmke and Ferrer, F. Puerta, X. Puerta and Zaballa derived a description of conditioned invariant subspaces in terms of images of block Toeplitz type matrices. They used this description to construct a stratification of the set of conditioned invariant subspaces of fixed dimension into smooth manifolds. These so called Brunovsky strata consist of all the subspaces with fixed restriction indices. They constructed a cell decomposition of the Brunovsky strata into so called Kronecker cells. In my thesis I show that in the tight case this cell decomposition is induced by a Bruhat decomposition of a generalized flag manifold. I identify the adherence order of the cell decomposition as being induced by the reverse Bruhat order.}, subject = {Invarianter Unterraum}, language = {en} } @phdthesis{Vodopivec2005, author = {Vodopivec, Andrija}, title = {Quasibasen abelscher, nichtseparabler p-Gruppen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-15359}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2005}, abstract = {In dieser Arbeit wird der Bau der (abz{\"a}hlbaren) abelschen p-Gruppen untersucht, durch die Betrachtung der dazugeh{\"o}rigen Quasibasen, die als bestimmte erzeugende Systeme der gegebenen p-Gruppe definiert sind. Die Untersuchung wird insbesondere auf die nichtseparablen p-Gruppen und ihre induktiven Quasibasen bezogen.}, subject = {Abelsche p-Gruppe}, language = {de} } @misc{Forster2013, type = {Master Thesis}, author = {Forster, Johannes}, title = {Mathematical Modeling of Complex Fluids}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-83533}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2013}, abstract = {This thesis gives an overview over mathematical modeling of complex fluids with the discussion of underlying mechanical principles, the introduction of the energetic variational framework, and examples and applications. The purpose is to present a formal energetic variational treatment of energies corresponding to the models of physical phenomena and to derive PDEs for the complex fluid systems. The advantages of this approach over force-based modeling are, e.g., that for complex systems energy terms can be established in a relatively easy way, that force components within a system are not counted twice, and that this approach can naturally combine effects on different scales. We follow a lecture of Professor Dr. Chun Liu from Penn State University, USA, on complex fluids which he gave at the University of Wuerzburg during his Giovanni Prodi professorship in summer 2012. We elaborate on this lecture and consider also parts of his work and publications, and substantially extend the lecture by own calculations and arguments (for papers including an overview over the energetic variational treatment see [HKL10], [Liu11] and references therein).}, subject = {Variationsrechnung}, language = {en} } @phdthesis{Meister2006, author = {Meister, Daniel}, title = {The complexity of membership problems for finite recurrent systems and minimal triangulations}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-18837}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2006}, abstract = {The dissertation thesis studies the complexity of membership problems. Generally, membership problems consider the question whether a given object belongs to a set. Object and set are part of the input. The thesis studies the complexity of membership problems for two special kinds of sets. The first problem class asks whether a given natural number belongs to a set of natural numbers. The set of natural numbers is defined via finite recurrent systems: sets are built by iterative application of operations, like union, intersection, complementation and arithmetical operations, to already defined sets. This general problem implies further problems by restricting the set of used operations. The thesis contains completeness results for well-known complexity classes as well as undecidability results for these problems. The second problem class asks whether a given graph is a minimal triangulation of another graph. A graph is a triangulation of another graph, if it is a chordal spanning supergraph of the second graph. If no proper supergraph of the first graph is a triangulation of the second graph, the first graph is a minimal triangulation of the second graph. The complexity of the membership problem for minimal triangulations of several graph classes is investigated. Restricted variants are solved by linear-time algorithms. These algorithms rely on appropriate characterisations of minimal triangulations.}, subject = {Komplexit{\"a}t}, language = {en} } @phdthesis{Klug2006, author = {Klug, Andreas}, title = {Affine-Scaling Methods for Nonlinear Minimization Problems and Nonlinear Systems of Equations with Bound Constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-18851}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2006}, abstract = {In this thesis affine-scaling-methods for two different types of mathematical problems are considered. The first type of problems are nonlinear optimization problems subject to bound constraints. A class of new affine-scaling Newton-type methods is introduced. The methods are shown to be locally quadratically convergent without assuming strict complementarity of the solution. The new methods differ from previous ones mainly in the choice of the scaling matrix. The second type of problems are semismooth system of equations with bound constraints. A new affine-scaling trust-region method for these problems is developed. The method is shown to have strong global and local convergence properties under suitable assumptions. Numerical results are presented for a number of problems arising from different areas.}, subject = {Skalierungsfunktion}, language = {en} } @phdthesis{Michel2006, author = {Michel, Ren{\´e}}, title = {Simulation and Estimation in Multivariate Generalized Pareto Models}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-18489}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2006}, abstract = {The investigation of multivariate generalized Pareto distributions (GPDs) in the framework of extreme value theory has begun only lately. Recent results show that they can, as in the univariate case, be used in Peaks over Threshold approaches. In this manuscript we investigate the definition of GPDs from Section 5.1 of Falk et al. (2004), which does not differ in the area of interest from those of other authors. We first show some theoretical properties and introduce important examples of GPDs. For the further investigation of these distributions simulation methods are an important part. We describe several methods of simulating GPDs, beginning with an efficient method for the logistic GPD. This algorithm is based on the Shi transformation, which was introduced by Shi (1995) and was used in Stephenson (2003) for the simulation of multivariate extreme value distributions of logistic type. We also present nonparametric and parametric estimation methods in GPD models. We estimate the angular density nonparametrically in arbitrary dimension, where the bivariate case turns out to be a special case. The asymptotic normality of the corresponding estimators is shown. Also in the parametric estimations, which are mainly based on maximum likelihood methods, the asymptotic normality of the estimators is shown under certain regularity conditions. Finally the methods are applied to a real hydrological data set containing water discharges of the rivers Altm{\"u}hl and Danube in southern Bavaria.}, subject = {Pareto-Verteilung}, language = {en} } @phdthesis{Petra2006, author = {Petra, Stefania}, title = {Semismooth least squares methods for complementarity problems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-18660}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2006}, abstract = {This thesis is concerned with numerical methods for solving nonlinear and mixed complementarity problems. Such problems arise from a variety of applications such as equilibria models of economics, contact and structural mechanics problems, obstacle problems, discrete-time optimal control problems etc. In this thesis we present a new formulation of nonlinear and mixed complementarity problems based on the Fischer-Burmeister function approach. Unlike traditional reformulations, our approach leads to an over-determined system of nonlinear equations. This has the advantage that certain drawbacks of the Fischer-Burmeister approach are avoided. Among other favorable properties of the new formulation, the natural merit function turns out to be differentiable. To solve the arising over-determined system we use a nonsmooth damped Levenberg-Marquardt-type method and investigate its convergence properties. Under mild assumptions, it can be shown that the global and local fast convergence results are similar to some of the better equation-based method. Moreover, the new method turns out to be significantly more robust than the corresponding equation-based method. For the case of large complementarity problems, however, the performance of this method suffers from the need for solving the arising linear least squares problem exactly at each iteration. Therefore, we suggest a modified version which allows inexact solutions of the least squares problems by using an appropriate iterative solver. Under certain assumptions, the favorable convergence properties of the original method are preserved. As an alternative method for mixed complementarity problems, we consider a box constrained least squares formulation along with a projected Levenberg-Marquardt-type method. To globalize this method, trust region strategies are proposed. Several ingredients are used to improve this approach: affine scaling matrices and multi-dimensional filter techniques. Global convergence results as well as local superlinear/quadratic convergence are shown under appropriate assumptions. Combining the advantages of the new methods, a new software for solving mixed complementarity problems is presented.}, subject = {Komplementarit{\"a}tsproblem}, language = {en} } @phdthesis{Esser2005, author = {Eßer, Wolfram}, title = {Fehlertolerante Volltextsuche in elektronischen Enzyklop{\"a}dien und Heuristiken zur Fehlerratenverbesserung}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-14760}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2005}, abstract = {In der vorliegenden Arbeit wird das Konzept und die praktische Umsetzung einer fehlertoleranten Volltextsuche vorgestellt, welche die unscharfe Recherche nach Suchmustern in umfangreichen, digitalen, enzyklop{\"a}dischen Werken erm{\"o}glichen. Das dabei zur Anwendung kommende neue Verfahren, welches durch Gewichte gesteuert das urspr{\"u}ngliche Benutzer-Suchmuster in seiner Gestalt ver{\"a}ndert (Weighted Pattern Morphing, WPM) und anschließend mit einer nachgeschalteten exakten Volltextsuche sucht, konnte in zahlreichen kommerziellen Anwendungsf{\"a}llen seine Praxistauglichkeit beweisen. Darunter ist die Anwendung zur unscharfen Suche in einer mittelalterlichen, handschriftlichen Chronik besonders interessant, da diese die fr{\"u}hneuhochdeutsche Sprache verwendet und es zur damaligen Zeit noch keine vereinheitlichte Rechtschreibung gab. Aber nicht nur bei der Endbenutzer-Suche kann WPM eingesetzt werden - auch im redaktionellen Umfeld konnten mit dem Verfahren noch mehrere hundert Tippfehler in einem bereits mehrfach lektorierten digitalen Lexikon gefunden werden. Dabei arbeitet das Verfahren deutlich sch{\"a}rfer, als die sonst zur unscharfen Suche (und damit zur Fehler-Suche) verwendete Edit-Distanz. Abschließend wird in der Arbeit noch ein Verfahren vorgestellt, mit dem aus einem 3D-Drahtgitter-Modell und den Faksimile-Scans einer mittelalterlichen Handschrift automatisch ein virtuelles Buch zum Durchbl{\"a}ttern am PC erstellt wurde.}, subject = {Volltextdatenbank}, language = {de} } @phdthesis{Kraus2004, author = {Kraus, Christiane}, title = {On some maximal convergence theorems for real analytic functions in R^N}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-9795}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2004}, abstract = {Ausgangspunkt dieser Arbeit war eine Publikation von D. Braess [Bra01], in der die Approximationsg{\"u}te der Funktionen \$\$ \frac{1}{((x-x_0)^2 + (y-y_0)^2)^s}, \qquad x_0^2 + y_0^2 \ge 1, \quad s \in (0,\infty),\$\$ auf der Einheitskreisscheibe \$x^2+y^2 \le 1\$ durch reelle Polynome untersucht wurde. Braess's Ergebnisse und insbesondere die von ihm angesprochenen offenen Probleme waren von besonderem Interesse, da sie Anlaß zu der Vermutung gaben, dass die klassische Theorie der ``Maximalen Konvergenz'' in Sinne von Walsh auf (zun{\"a}chst) die oben erw{\"a}hnten reell analytischen Funktionen erweitert werden kann. (Die Theorie der Maximalen Konvergenz bringt die Approximationsg{\"u}te einer Funktion auf einer kompakten Menge durch Polynome mit der Analyzit{\"a}t dieser Funktion in Verbindung.) \\ Hauptgegenstand der Arbeit ist die Erweiterung des klassischen ``Maximalen Konvergenz''--Konzeptes auf reell analytische Funktionen in h{\"o}heren Dimensionen. Es werden verschiedene maximale Konvergenzs{\"a}tze sowohl in einer als auch in mehreren Ver{\"a}nderlichen bewiesen. \\ Die Arbeit gliedert sich in drei Hauptteile. \\[2mm] Im ersten Teil wird der theoretische Hintergrund der ``Maximalen Konvergenz'' mit dem Problemkreis von Braess in Zusammenhang gebracht. Es wird gezeigt, dass f{\"u}r betrags-quadratisch holomorphe Funktionen folgender Satz gilt: \\ { \bf {Satz 1}}: Es sei \$g\$ eine holomorphe Funktion auf der abgeschlossenen Einheitskreisscheibe \$\overline{\mathbb{D}}:=\{ z \in \mathbb{C} : |z| \le 1\}\$ und \$F(x,y):= |g(x+iy)|^2\$, \$x,y \in \mathbb{R}\$. Dann gilt: \$\$ \limsup_{n \to \infty} \sqrt[n]{E_n ( \overline{\mathbb{D}},F)} = \frac{1}{\rho}\$\$ genau dann, wenn \$g\$ auf \$ \{ z \in \mathbb{C} : |z| < \rho \}\$ holomorph ist, aber auf keiner echt gr\"o\3eren Kreisscheibe, wobei \$\$ E_n ( \overline{\mathbb{D}},F)= \inf \{ ||F -P_n||_{\overline{\mathbb{D}}}, \, P_n: \mathbb{R}^2 \to \mathbb{R} \mbox{ Polynom vom Grad } \le n \}.\$\$ Dieser Satz beinhaltet nicht nur die Ergebnisse von Braess [Bra01], sondern erweitert ihn, und beantwortet die von Braess aufgeworfenen Fragen vollst{\"a}ndig. Zudem zeigt der Satz die genaue Analogie des klassischen ``Maximalen Konvergenz''--Konzeptes f{\"u}r die Funktionenklasse der betrag--quadratisch holomorphen Funktionen im \$\mathbb{R}^2\$. \\[2mm] In der Literatur gibt es viele Verallgemeinerungen des ``Maximalen Konvergenz''--Begriffes f{\"u}r mehrere komplexe Ver{\"a}nderlichen. Im Hinblick auf die vorliegende Arbeit sind besonders die Artikel [Sic62] und [Sic81] zu erw{\"a}hnen. Diese bereits bekannten Ergebnisse werden im zweiten Teil der Arbeit herangezogen, um den ``Maximalen Konvergenz''--Begriff auf mehrere reelle Ver{\"a}nderlichen zu erweitern. Man beachte, dass der entscheidende Unterschied hier in der polynomialen Approximationsklasse liegt. \\[2mm] Der dritte Teil befaßt sich mit der Verallgemeinerung des Satzes 1 in mehreren Ver{\"a}nderlichen. Eng verbunden mit diesem Problemkreis ist die Charakterisierung einer gewissen Extremalfunktion. Diese Funktion wird zur Bestimmung des Analyzit{\"a}tsbereichs der zu approximierenden Funktion ben{\"o}tigt. Mittels geeigneter Darstellung der Extremalfunktion und Charakterisierung des Analyzit{\"a}tsbereichs gelingt es schließlich, den folgenden Hauptsatz der vorliegenden Arbeit zu beweisen:\\ { \bf { Satz 2}}: Es seien \$g,h\$ holomorphe Funktionen auf der abgeschlossenen Einheitskugel \$\overline{\mathbb{D}}_N:=\{ z \in \mathbb{C}^N : |z| \le 1\}\$ und \$F(x,y):= g(x+iy) \overline{h(x+iy)}\$, \$x,y \in \mathbb{R}^N\$. Dann gilt: \$\$ \limsup_{n \to \infty} \sqrt[n]{E_n ( \overline{\mathbb{D}}_N,F)} = \frac{1}{\rho}\$\$ genau dann, wenn \$g,h\$ auf \${\mathbb{D}}_{N,\rho}:= \{ z \in \mathbb{C}^N : |z| < \rho \}\$ holomorph sind, und mindestens eine der zwei Funktionen \$g,h\$ auf keinem echt gr\"o\3eren Ball als \$\mathbb{D}_{N,\rho}\$ holomorph fortsetzbar ist. Hierbei bezeichnet \$\$ E_n ( \overline{\mathbb{D}}_N,F)= \inf \{ ||F -P_n||_{\overline{\mathbb{D}}_N}, \, P_n: \mathbb{R}^{2N} \to \mathbb{C} \mbox{ Polynom vom Grad } \le n \}.\$\$ \$[\$Bra01\$]\$ Braess, D., {\it Note on the Approximation of Powers of the Distance in Two-Dimensional Domains}, Constructive Approximation (2001), {\bf 17} No. 1, 147-151. \\ \$[\$Sic62\$]\$ Siciak, J., {\it On some extremal functions and their applications in the theory of analytic functions of several complex variables}, Trans. Amer. Math. Soc. (1962), {\bf 105}, 322--357. \\ \$[\$Sic81\$]\$ Siciak, J., {\it Extremal plurisubharmonic functions in \$\mathbb{C}^N\$}, Ann. Pol. Math. (1981), {\bf 39}, 175--211.}, subject = {Reelle Funktion}, language = {de} } @phdthesis{Seider2004, author = {Seider, David}, title = {Solving an eigenvalue problem in laser simulation}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-10057}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2004}, abstract = {In this thesis a new and powerful approach for modeling laser cavity eigenmodes is presented. This approach is based on an eigenvalue problem for singularly perturbed partial differential operators with complex coefficients; such operators have not been investigated in detail until now. The eigenvalue problem is discretized by finite elements, and convergence of the approximate solution is proved by using an abstract convergence theory also developed in this dissertation. This theory for the convergence of an approximate solution of a (quadratic) eigenvalue problem, which particularly can be applied to a finite element discretization, is interesting on its own, since the ideas can conceivably be used to handle equations with a more complex nonlinearity. The discretized eigenvalue problem essentially is solved by preconditioned GMRES, where the preconditioner is constructed according to the underlying physics of the problem. The power and correctness of the new approach for computing laser cavity eigenmodes is clearly demonstrated by successfully simulating a variety of different cavity configurations. The thesis is organized as follows: Chapter 1 contains a short overview on solving the so-called Helmholtz equation with the help of finite elements. The main part of Chapter 2 is dedicated to the analysis of a one-dimensional model problem containing the main idea of a new model for laser cavity eigenmodes which is derived in detail in Chapter 3. Chapter 4 comprises a convergence theory for the approximate solution of quadratic eigenvalue problems. In Chapter 5, a stabilized finite element discretization of the new model is described and its convergence is proved by applying the theory of Chapter 4. Chapter 6 contains computational aspects of solving the resulting system of equations and, finally, Chapter 7 presents numerical results for various configurations, demonstrating the practical relevance of our new approach.}, subject = {Laser}, language = {en} } @phdthesis{Kleinsteuber2005, author = {Kleinsteuber, Martin}, title = {Jacobi-type methods on semisimple Lie algebras : a Lie algebraic approach to numerical linear algebra}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-16454}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2005}, abstract = {Es wird eine Lie-algebraische Verallgemeinerung sowohl des klassischen als auch des Sortier-Jacobi-Verfahrens f{\"u}r das symmetrische Eigenwertproblem behandelt. Der koordinatenfreie Zugang erm{\"o}glicht durch eine neue Betrachtungsweise die Vereinheitlichung strukturierter Eigen- und Singul{\"a}rwertprobleme, darunter bis dato noch nicht betrachtete F{\"a}lle. F{\"u}r beide Verfahren wird lokal quadratische Konvergenz, sowohl f{\"u}r den regul{\"a}ren als auch f{\"u}r den irregul{\"a}ren Fall, gezeigt. Die Analyse und Verallgemeinerung der sog. speziellen Sweeps f{\"u}r das symmetrische Eigenwertproblem f{\"u}hrt zu neuen Sweep-Methoden f{\"u}r strukturierte Eigen- und Singul{\"a}rwertprobleme, die ein besseres Konvergenzverhalten als die bisher bekannten aufweisen.}, subject = {Eigenwert}, language = {en} } @book{FalkMarohnMicheletal.2006, author = {Falk, Michael and Marohn, Frank and Michel, Ren{\´e} and Hofmann, Daniel and Macke, Maria and Tewes, Bernward and Dinges, Peter}, title = {A First Course on Time Series Analysis : Examples with SAS}, organization = {Universit{\"a}t W{\"u}rzburg / Lehrstuhl f{\"u}r Statistik}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-16919}, publisher = {Universit{\"a}t W{\"u}rzburg}, year = {2006}, abstract = {The analysis of real data by means of statistical methods with the aid of a software package common in industry and administration usually is not an integral part of mathematics studies, but it will certainly be part of a future professional work. The present book links up elements from time series analysis with a selection of statistical procedures used in general practice including the statistical software package SAS Statistical Analysis System). Consequently this book addresses students of statistics as well as students of other branches such as economics, demography and engineering, where lectures on statistics belong to their academic training. But it is also intended for the practician who, beyond the use of statistical tools, is interested in their mathematical background. Numerous problems illustrate the applicability of the presented statistical procedures, where SAS gives the solutions. The programs used are explicitly listed and explained. No previous experience is expected neither in SAS nor in a special computer system so that a short training period is guaranteed. This book is meant for a two semester course (lecture, seminar or practical training) where the first two chapters can be dealt with in the first semester. They provide the principal components of the analysis of a time series in the time domain. Chapters 3, 4 and 5 deal with its analysis in the frequency domain and can be worked through in the second term. In order to understand the mathematical background some terms are useful such as convergence in distribution, stochastic convergence, maximum likelihood estimator as well as a basic knowledge of the test theory, so that work on the book can start after an introductory lecture on stochastics. Each chapter includes exercises. An exhaustive treatment is recommended. This book is consecutively subdivided in a statistical part and an SAS-specific part. For better clearness the SAS-specific part, including the diagrams generated with SAS, always starts with a computer symbol, representing the beginning of a session at the computer, and ends with a printer symbol for the end of this session. This book is an open source project under the GNU Free Documentation License.}, subject = {Zeitreihenanalyse}, language = {en} } @phdthesis{Lageman2007, author = {Lageman, Christian}, title = {Convergence of gradient-like dynamical systems and optimization algorithms}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-23948}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2007}, abstract = {This work studies the convergence of trajectories of gradient-like systems. In the first part of this work continuous-time gradient-like systems are examined. Results on the convergence of integral curves of gradient systems to single points of Lojasiewicz and Kurdyka are extended to a class of gradient-like vector fields and gradient-like differential inclusions. In the second part of this work discrete-time gradient-like optimization methods on manifolds are studied. Methods for smooth and for nonsmooth optimization problems are considered. For these methods some convergence results are proven. Additionally the optimization methods for nonsmooth cost functions are applied to sphere packing problems on adjoint orbits.}, subject = {Dynamisches System}, language = {en} } @phdthesis{Moeller2009, author = {M{\"o}ller, Florian}, title = {Exceptional polynomials and monodromy groups in positive characteristic}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-34871}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {We discuss exceptional polynomials, i.e. polynomials over a finite field \$k\$ that induce bijections over infinitely many finite extensions of \$k\$. In the first chapters we give the theoretical background to characterize this class of polynomials with Galois theoretic means. This leads to the notion of arithmetic resp. geometric monodromy groups. In the remaining chapters we restrict our attention to polynomials with primitive affine arithmetic monodromy group. We first classify all exceptional polynomials with the fixed field of the affine kernel of the arithmetic monodromy group being of genus less or equal to 2. Next we show that every full affine group can be realized as the monodromy group of a polynomial. In the remaining chapters we classify affine polynomials of a given degree.}, subject = {Algebraischer Funktionenk{\"o}rper}, language = {en} } @phdthesis{Winkler2008, author = {Winkler, Ralf}, title = {Schwache Randwertprobleme von Systemen elliptischen Charakters auf konischen Gebieten}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-34544}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2008}, abstract = {In der vorliegenden Arbeit werden lineare Systeme elliptischer partieller Differentialgleichungen in schwacher Formulierung auf konischen Gebieten untersucht. Auf einem zun{\"a}chst unbeschr{\"a}nkten Kegelgebiet betrachten wir den Fall beschr{\"a}nkter und nur von den Winkelvariablen abh{\"a}ngiger Koeffizientenfunktionen. Die durch selbige definierte Bilinearform gen{\"u}ge einer G{\aa}rdingschen Ungleichung. In gewichteten Sobolevr{\"a}umen werden Existenz- und Eindeutigkeitsfragen gekl{\"a}rt, wobei das Problem mittels Fouriertransformation auf eine von einem komplexen Parameter abh{\"a}ngige Familie T(·) von Fredholmoperatoren zur{\"u}ckgef{\"u}hrt wird. Unter Anwendung des Residuenkalk{\"u}ls gewinnen wir eine Darstellung der L{\"o}sung in Form einer Zerlegung in einen glatten Anteil einerseits sowie eine endliche Summe von Singul{\"a}rfunktionen andererseits. Durch Abschneidetechniken werden die gewonnenen Erkenntnisse auf den Fall schwach formulierter elliptischer Systeme auf beschr{\"a}nkten Kegelgebieten unter Formulierung in gew{\"o}hnlichen, nicht-gewichteten Sobolevr{\"a}umen angewendet. Die f{\"u}r Regularit{\"a}tsfragen maßgeblichen Eigenwerte der Operatorfunktion T mit minimalem positiven Imagin{\"a}rteil werden im letzten Kapitel der Arbeit am Beispiel der ebenen elastischen Gleichungen numerisch bestimmt.}, subject = {Elliptische Differentialgleichung}, language = {de} } @phdthesis{Baumann2008, author = {Baumann, Markus}, title = {Newton's Method for Path-Following Problems on Manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-28099}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2008}, abstract = {Many optimization problems for a smooth cost function f on a manifold M can be solved by determining the zeros of a vector field F; such as e.g. the gradient F of the cost function f. If F does not depend on additional parameters, numerous zero-finding techniques are available for this purpose. It is a natural generalization however, to consider time-dependent optimization problems that require the computation of time-varying zeros of time-dependent vector fields F(x,t). Such parametric optimization problems arise in many fields of applied mathematics, in particular path-following problems in robotics, recursive eigenvalue and singular value estimation in signal processing, as well as numerical linear algebra and inverse eigenvalue problems in control theory. In the literature, there are already some tracking algorithms for these tasks, but these do not always adequately respect the manifold structure. Hence, available tracking results can often be improved by implementing methods working directly on the manifold. Thus, intrinsic methods are of interests that evolve during the entire computation on the manifold. It is the task of this thesis, to develop such intrinsic zero finding methods. The main results of this thesis are as follows: - A new class of continuous and discrete tracking algorithms is proposed for computing zeros of time-varying vector fields on Riemannian manifolds. This was achieved by studying the newly introduced time-varying Newton Flow and the time-varying Newton Algorithm on Riemannian manifolds. - Convergence analysis is performed on arbitrary Riemannian manifolds. - Concretization of these results on submanifolds, including for a new class of algorithms via local parameterizations. - More specific results in Euclidean space are obtained by considering inexact and underdetermined time-varying Newton Flows. - Illustration of these newly introduced algorithms by examining time-varying tracking tasks in three application areas: Subspace analysis, matrix decompositions (in particular EVD and SVD) and computer vision.}, subject = {Dynamische Optimierung}, language = {en} } @phdthesis{Pechmann2008, author = {Pechmann, Patrick R.}, title = {Penalized Least Squares Methoden mit st{\"u}ckweise polynomialen Funktionen zur L{\"o}sung von partiellen Differentialgleichungen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-28136}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2008}, abstract = {Das Hauptgebiet der Arbeit stellt die Approximation der L{\"o}sungen partieller Differentialgleichungen mit Dirichlet-Randbedingungen durch Splinefunktionen dar. Partielle Differentialgleichungen finden ihre Anwendung beispielsweise in Bereichen der Elektrostatik, der Elastizit{\"a}tstheorie, der Str{\"o}mungslehre sowie bei der Untersuchung der Ausbreitung von W{\"a}rme und Schall. Manche Approximationsaufgaben besitzen keine eindeutige L{\"o}sung. Durch Anwendung der Penalized Least Squares Methode wurde gezeigt, dass die Eindeutigkeit der gesuchten L{\"o}sung von gewissen Minimierungsaufgaben sichergestellt werden kann. Unter Umst{\"a}nden l{\"a}sst sich sogar eine h{\"o}here Stabilit{\"a}t des numerischen Verfahrens gewinnen. F{\"u}r die numerischen Betrachtungen wurde ein umfangreiches, effizientes C-Programm erstellt, welches die Grundlage zur Best{\"a}tigung der theoretischen Voraussagen mit den praktischen Anwendungen bildete.}, subject = {Approximationstheorie}, language = {de} } @phdthesis{Gregor2008, author = {Gregor, Thomas}, title = {{0,1}-Matrices with Rectangular Rule}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-28389}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2008}, abstract = {The incidence matrices of many combinatorial structures satisfy the so called rectangular rule, i.e., the scalar product of any two lines of the matrix is at most 1. We study a class of matrices with rectangular rule, the regular block matrices. Some regular block matrices are submatrices of incidence matrices of finite projective planes. Necessary and sufficient conditions are given for regular block matrices, to be submatrices of projective planes. Moreover, regular block matrices are related to another combinatorial structure, the symmetric configurations. In particular, it turns out, that we may conclude the existence of several symmetric configurations from the existence of a projective plane, using this relationship.}, subject = {Projektive Ebene}, language = {en} } @phdthesis{Jordan2008, author = {Jordan, Jens}, title = {Reachable sets of numerical iteration schemes : a system semigroup approach}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-28416}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2008}, abstract = {We investigate iterative numerical algorithms with shifts as nonlinear discrete-time control systems. Our approach is based on the interpretation of reachable sets as orbits of the system semigroup. In the first part we develop tools for the systematic analysis of the structure of reachable sets of general invertible discrete-time control systems. Therefore we merge classical concepts, such as geometric control theory, semigroup actions and semialgebraic geometry. Moreover, we introduce new concepts such as right divisible systems and the repelling phenomenon. In the second part we apply the semigroup approach to the investigation of concrete numerical iteration schemes. We extend the known results about the reachable sets of classical inverse iteration. Moreover, we investigate the structure of reachable sets and systemgroup orbits of inverse iteration on flag manifolds and Hessenberg varieties, rational iteration schemes, Richardson's method and linear control schemes. In particular we obtain necessary and sufficient conditions for controllability and the appearance of repelling phenomena. Furthermore, a new algorithm for solving linear equations (LQRES) is derived.}, subject = {Nichtlineare Kontrolltheorie}, language = {en} } @phdthesis{Harms2014, author = {Harms, Nadja}, title = {Primal and Dual Gap Functions for Generalized Nash Equilibrium Problems and Quasi-Variational Inequalities}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-106027}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2014}, abstract = {In this thesis we study smoothness properties of primal and dual gap functions for generalized Nash equilibrium problems (GNEPs) and finite-dimensional quasi-variational inequalities (QVIs). These gap functions are optimal value functions of primal and dual reformulations of a corresponding GNEP or QVI as a constrained or unconstrained optimization problem. Depending on the problem type, the primal reformulation uses regularized Nikaido-Isoda or regularized gap function approaches. For player convex GNEPs and QVIs of the so-called generalized `moving set' type the respective primal gap functions are continuously differentiable. In general, however, these primal gap functions are nonsmooth for both problems. Hence, we investigate their continuity and differentiability properties under suitable assumptions. Here, our main result states that, apart from special cases, all locally minimal points of the primal reformulations are points of differentiability of the corresponding primal gap function. Furthermore, we develop dual gap functions for a class of GNEPs and QVIs and ensuing unconstrained optimization reformulations of these problems based on an idea by Dietrich (``A smooth dual gap function solution to a class of quasivariational inequalities'', Journal of Mathematical Analysis and Applications 235, 1999, pp. 380--393). For this purpose we rewrite the primal gap functions as a difference of two strongly convex functions and employ the Toland-Singer duality theory. The resulting dual gap functions are continuously differentiable and, under suitable assumptions, have piecewise smooth gradients. Our theoretical analysis is complemented by numerical experiments. The solution methods employed make use of the first-order information established by the aforementioned theoretical investigations.}, subject = {Nash-Gleichgewicht}, language = {en} } @phdthesis{Oswald2014, author = {Oswald, Nicola}, title = {Hurwitz's Complex Continued Fractions - A Historical Approach and Modern Perspectives}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-106040}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2014}, abstract = {The thesis 'Hurwitz's Complex Continued Fractions - A Historical Approach and Modern Perspectives.' deals with two branches of mathematics: Number Theory and History of Mathematics. On the first glimpse this might be unexpected, however, on the second view this is a very fruitful combination. Doing research in mathematics, it turns out to be very helpful to be aware of the beginnings and development of the corresponding subject. In the case of Complex Continued Fractions the origins can easily be traced back to the end of the 19th century (see [Perron, 1954, vl. 1, Ch. 46]). One of their godfathers had been the famous mathematician Adolf Hurwitz. During the study of his transformation from real to complex continued fraction theory [Hurwitz, 1888], our attention was arrested by the article 'Ueber eine besondere Art der Kettenbruch-Entwicklung complexer Gr{\"o}ssen' [Hurwitz, 1895] from 1895 of an author called J. Hurwitz. We were not only surprised when we found out that he was the elder unknown brother Julius, furthermore, Julius Hurwitz introduced a complex continued fraction that also appeared (unmentioned) in an ergodic theoretical work from 1985 [Tanaka, 1985]. Those observations formed the Basis of our main research questions: What is the historical background of Adolf and Julius Hurwitz and their mathematical studies? and What modern perspectives are provided by their complex continued fraction expansions? In this work we examine complex continued fractions from various viewpoints. After a brief introduction on real continued fractions, we firstly devote ourselves to the lives of the brothers Adolf and Julius Hurwitz. Two excursions on selected historical aspects in respect to their work complete this historical chapter. In the sequel we shed light on Hurwitz's, Adolf's as well as Julius', approaches to complex continued fraction expansions. Correspondingly, in the following chapter we take a more modern perspective. Highlights are an ergodic theoretical result, namely a variation on the D{\"o}blin-Lenstra Conjecture [Bosma et al., 1983], as well as a result on transcendental numbers in tradition of Roth's theorem [Roth, 1955]. In two subsequent chapters we are concernced with arithmetical properties of complex continued fractions. Firstly, an analogue to Marshall Hall's Theorem from 1947 [Hall, 1947] on sums of continued fractions is derived. Secondly, a general approach on new types of continued fractions is presented building on the structural properties of lattices. Finally, in the last chapter we take up this approach and obtain an upper bound for the approximation quality of diophantine approximations by quotients of lattice points in the complex plane generalizing a method of Hermann Minkowski, improved by Hilde Gintner [Gintner, 1936], based on ideas from geometry of numbers.}, subject = {Kettenbruch}, language = {en} } @phdthesis{Staab2002, author = {Staab, Patricia}, title = {Geometrische Eigenschaften von Spiraltypfl{\"a}chen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3727}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2002}, abstract = {Spiraltypfl{\"a}chen sind Minimalfl{\"a}chen des dreidimensionalen euklidischen Raums, die sich durch hohe Symmetrie gegen{\"u}ber komplexen {\"A}hnlichkeitsabbildungen der Minimalkurve auszeichnen. Ihren Namen verdanken Sie folgender Eigenschaft: Sie und ihre komplex Homothetischen sind die einzigen auf Spiralfl{\"a}chen abwickelbaren Minimalfl{\"a}chen. Bekannte Spiraltypfl{\"a}chen sind die Spiralminimalfl{\"a}chen (zugleich Minimal- und Spiralfl{\"a}chen) und die Bourfl{\"a}chen (auf Rotationsfl{\"a}chen abwickelbare Minimalfl{\"a}chen). Das Katenoid und die Enneperfl{\"a}che sind spezielle Bourfl{\"a}chen. In dieser Arbeit werden die Spiraltypfl{\"a}chen auf ihre geometrischen Eigenschaften untersucht. Wir stellen ihre Periodizit{\"a}ten und Symmetrien fest und versuchen, ausgezeichnete Fl{\"a}chenkurven auf ihnen zu finden. Wir verwenden eine globale Weierstraß-Darstellung der Spiraltypfl{\"a}chen. In dieser Darstellung ergeben die Fl{\"a}chen eine Schar mit einem komplexen Scharparameter. Anhand dieser Darstellung leiten wir s{\"a}mtliche Symmetrien der Spiraltypfl{\"a}chen zu linearen {\"A}hnlichkeitsabbildungen der Minimalkurve her. Als Spezialf{\"a}lle erhalten wir die Symmetrien unter Assoziationen und Derivationen (Drehung der Minimalkurve um einen imagin{\"a}ren Drehwinkel), sowie die reellen Symmetrien (Dreh-, Spiegel- und Strecksymmetrien). Unter den Spiraltypfl{\"a}chen gibt es nur zwei translationssymmetrische Fl{\"a}chen. Die Umorientierung einer Spiraltypfl{\"a}che entspricht (bis auf komplexe Homothetie) dem Vorzeichenwechsel des Fl{\"a}chenparameters. Im {\"U}brigen kann durch einfache Spiegelungen an den Koordinatenebenen beziehungsweise Drehungen um die Koordinatenachsen das Vorzeichen von Real- beziehungsweise Imagin{\"a}rteil des Fl{\"a}chenparameters umgekehrt werden. Schließlich stellen wir noch ausgezeichnete Fl{\"a}chenkurven auf den Spiraltypfl{\"a}chen vor: Kr{\"u}mmungslinien, Asymptotenlinien und Geod{\"a}tische, sowie als deren Verallgemeinerungen die Pseudokr{\"u}mmungslinien und Pseudogeod{\"a}tischen.}, subject = {Spiraltypfl{\"a}che}, language = {de} } @phdthesis{Baumeister2004, author = {Baumeister, Joachim}, title = {Agile development of diagnostic knowledge systems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-9698}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2004}, abstract = {The success of diagnostic knowledge systems has been proved over the last decades. Nowadays, intelligent systems are embedded in machines within various domains or are used in interaction with a user for solving problems. However, although such systems have been applied very successfully the development of a knowledge system is still a critical issue. Similarly to projects dealing with customized software at a highly innovative level a precise specification often cannot be given in advance. Moreover, necessary requirements of the knowledge system can be defined not until the project has been started or are changing during the development phase. Many success factors depend on the feedback given by users, which can be provided if preliminary demonstrations of the system can be delivered as soon as possible, e.g., for interactive systems validation the duration of the system dialog. This thesis motivates that classical, document-centered approaches cannot be applied in such a setting. We cope with this problem by introducing an agile process model for developing diagnostic knowledge systems, mainly inspired by the ideas of the eXtreme Programming methodology known in software engineering. The main aim of the presented work is to simplify the engineering process for domain specialists formalizing the knowledge themselves. The engineering process is supported at a primary level by the introduction of knowledge containers, that define an organized view of knowledge contained in the system. Consequently, we provide structured procedures as a recommendation for filling these containers. The actual knowledge is acquired and formalized right from start, and the integration to runnable knowledge systems is done continuously in order to allow for an early and concrete feedback. In contrast to related prototyping approaches the validity and maintainability of the collected knowledge is ensured by appropriate test methods and restructuring techniques, respectively. Additionally, we propose learning methods to support the knowledge acquisition process sufficiently. The practical significance of the process model strongly depends on the available tools supporting the application of the process model. We present the system family d3web and especially the system d3web.KnowME as a highly integrated development environment for diagnostic knowledge systems. The process model and its activities, respectively, are evaluated in two real life applications: in a medical and in an environmental project the benefits of the agile development are clearly demonstrated.}, language = {en} } @phdthesis{Kraus2003, author = {Kraus, Daniela}, title = {Conformal pseudo-metrics and some applications}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-9193}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2003}, abstract = {The point of departure for the present work has been the following free boundary value problem for analytic functions \$f\$ which are defined on a domain \$G \subset \mathbb{C}\$ and map into the unit disk \$\mathbb{D}= \{z \in \mathbb{C} : |z|<1 \}\$. Problem 1: Let \$z_1, \ldots, z_n\$ be finitely many points in a bounded simply connected domain \$G \subset \mathbb{C}\$. Show that there exists a holomorphic function \$f:G \to \mathbb{D}\$ with critical points \$z_j\$ (counted with multiplicities) and no others such that \$\lim_{z \to \xi} \frac{|f'(z)|}{1-|f(z)|^2}=1\$ for all \$\xi \in \partial G\$. If \$G=\mathbb{D}\$, Problem 1 was solved by K?nau [5] in the case of one critical point, and for more than one critical point by Fournier and Ruscheweyh [3]. The method employed by K?nau, Fournier and Ruscheweyh easily extends to more general domains \$G\$, say bounded by a Dini-smooth Jordan curve, but does not work for arbitrary bounded simply connected domains. In this paper we present a new approach to Problem 1, which shows that this boundary value problem is not an isolated question in complex analysis, but is intimately connected to a number of basic open problems in conformal geometry and non-linear PDE. One of our results is a solution to Problem 1 for arbitrary simply connected domains. However, we shall see that our approach has also some other ramifications, for instance to a well-known problem due to Rellich and Wittich in PDE. Roughly speaking, this paper is broken down into two parts. In a first step we construct a conformal metric in a bounded regular domain \$G\subset \mathbb{C}\$ with prescribed non-positive Gaussian curvature \$k(z)\$ and prescribed singularities by solving the first boundary value problem for the Gaussian curvature equation \$\Delta u =-k(z) e^{2u}\$ in \$G\$ with prescribed singularities and continuous boundary data. This is related to the Berger-Nirenberg problem in Riemannian geometry, the question which functions on a surface R can arise as the Gaussian curvature of a Riemannian metric on R. The special case, where \$k(z)=-4\$ and the domain \$G\$ is bounded by finitely many analytic Jordan curves was treated by Heins [4]. In a second step we show every conformal pseudo-metric on a simply connected domain \$G\subseteq \mathbb{C}\$ with constant negative Gaussian curvature and isolated zeros of integer order is the pullback of the hyperbolic metric on \$\mathbb{D}\$ under an analytic map \$f:G \to \mathbb{D}\$. This extends a theorem of Liouville which deals with the case that the pseudo-metric has no zeros at all. These two steps together allow a complete solution of Problem 1. Contents: Chapter I contains the statement of the main results and connects them with some old and new problems in complex analysis, conformal geometry and PDE: the Uniformization Theorem for Riemann surfaces, the problem of Schwarz-Picard, the Berger-Nirenberg problem, Wittich's problem, etc.. Chapter II and III have preparatory character. In Chapter II we recall some basic results about ordinary differential equations in the complex plane. In our presentation we follow Laine [6], but we have reorganized the material and present a self-contained account of the basic features of Riccati, Schwarzian and second order differential equations. In Chapter III we discuss the first boundary value problem for the Poisson equation. We shall need to consider this problem in the most general situation, which does not seem to be covered in a satisfactory way in the existing literature, see [1,2]. In Chapter IV we turn to a discussion of conformal pseudo-metrics in planar domains. We focus on conformal metrics with prescribed singularities and prescribed non-positive Gaussian curvature. We shall establish the existence of such metrics, that is, we solve the corresponding Gaussian curvature equation by making use of the results of Chapter III. In Chapter V we show that every constantly curved pseudo-metric can be represented as the pullback of either the hyperbolic, the euclidean or the spherical metric under an analytic map. This is proved by using the results of Chapter II. Finally we give in Chapter VI some applications of our results. [1,2] Courant, H., Hilbert, D., Methoden der Mathematischen Physik, Erster/ Zweiter Band, Springer-Verlag, Berlin, 1931/1937. [3] Fournier, R., Ruscheweyh, St., Free boundary value problems for analytic functions in the closed unit disk, Proc. Amer. Math. Soc. (1999), 127 no. 11, 3287-3294. [4] Heins, M., On a class of conformal metrics, Nagoya Math. J. (1962), 21, 1-60. [5] K?nau, R., L?gentreue Randverzerrung bei analytischer Abbildung in hyperbolischer und sph?ischer Geometrie, Mitt. Math. Sem. Giessen (1997), 229, 45-53. [6] Laine, I., Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin - New York, 1993.}, subject = {Freies Randwertproblem}, language = {en} } @phdthesis{Nagel2003, author = {Nagel, Christian}, title = {Gl{\"a}ttungsverfahren f{\"u}r semidefinite Programme}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-8099}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2003}, abstract = {In dieser Arbeit werden Algorithmen zur L{\"o}sung von linearen semidefiniten Programmen beschrieben. Unter einer geeigneten Regularit{\"a}tsvoraussetzung ist ein semidefinites Programm {\"a}quivalent zu seinen Optimalit{\"a}tsbedingungen. Die Optimalit{\"a}tsbedingungen bzw. die Zentralen-Pfad-Bedingungen {\"u}berf{\"u}hren wir zun{\"a}chst durch matrixwertige NCP-Funktionen in ein nichtlineares Gleichungssystem. Dieses nichtlineare und teilweise nicht differenzierbare Gleichungssystem l{\"o}sen wir dann mit einem Newton-{\"a}hnlichen Verfahren. Durch die Umformulierung in ein nichtlineares Gleichungssystem muss w{\"a}hrend der Iteration nicht mehr explizit die positive (Semi-)Definitheit der beteiligten Matrizen beachtet werden. Weiter wird gezeigt, dass dieser Ansatz im Gegensatz zu Inneren-Punkte-Methoden sofort symmetrische Suchrichtungen erzeugt. Um globale Konvergenz zu erhalten, werden verschiedene Globalisierungsstrategien (Schrittweitenbestimmung, Trust-Region-Ansatz) untersucht. F{\"u}r das betrachtete Pr{\"a}diktor-Korrektor-Verfahren und das Trust-Region-Verfahren wird lokal superlineare Konvergenz unter strikter Komplementarit{\"a}t und Nichtdegeneriertheit gezeigt. Die theoretische Untersuchung eines nichtglatten Newton-Verfahrens liefert ein lokal quadratisches Konvergenzverhalten ohne strikte Komplementarit{\"a}t, wenn die Nichtdegeneriertheitsvoraussetzung geeignet modifiziert wird.}, subject = {Semidefinite Optimierung}, language = {de} } @phdthesis{Fleischmann2003, author = {Fleischmann, Peter}, title = {Bildung reiner H{\"u}llen in vollst{\"a}ndig zerlegbaren torsionsfreien abelschen Gruppen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-5979}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2003}, abstract = {Reine Untergruppen von vollst{\"a}ndig zerlegbaren torsionsfreien abelschen Gruppen werden Butlergruppen genannt. Eine solche Gruppe l{\"a}ßt sich als endliche Summe von rationalen Rang-1-Gruppen darstellen. Eine solche Darstellung ist nicht eindeutig. Daher werden Methoden entwickelt, die zu einer Darstellung mit reinen Summanden f{\"u}hren. Weiter kann aus dieser Darstellung sowohl die kritische Typenmenge als auch die Typuntergruppen direkt abgelesen werden. Dies vereinfacht die Behandlung von Butlergruppen mit dem Computer und gestattet dar{\"u}berhinaus eine elegantere Darstellung.}, subject = {Torsionsfreie Abelsche Gruppe}, language = {de} } @phdthesis{Preiss2002, author = {Preiß, Martin}, title = {Analytizit{\"a}tseigenschaften gewichteter zentraler Pfade bei monotonen Komplementarit{\"a}tsproblemen und ihre Ausnutzung}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3969}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2002}, abstract = {Die vorliegende Arbeit untersucht die Analytizit{\"a}tseigenschaften unzul{\"a}ssiger Innerer-Punkte Pfade bei monotonen Komplementarit{\"a}tsproblemen und diskutiert m{\"o}gliche algorithmische Anwendungen. In Kapitel 2 werden einige matrixanalytische Konzepte und Resultate zusammengestellt, die f{\"u}r die Beweisf{\"u}hrung in den folgenden Kapiteln ben{\"o}tigt werden. Kapitel 3 gibt eine genaue Definition der Begriffe "monotones lineares Komplementarit{\"a}tsproblem" (LCP) bzw. "semidefinites monotones lineares Komplementarit{\"a}tsproblem" (SDLCP) und zeigt die Grundidee hinter den Innere-Punkte-Verfahren zur L{\"o}sung solcher Probleme. Kapitel 4 beinhaltet die analytischen Hauptresultate f{\"u}r monotone Komplementarit{\"a}tsprobleme. In Abschnitt 4.1 werden einige wohlbekannte Resultate {\"u}ber die Analytizit{\"a}tseigenschaften unzul{\"a}ssiger Innerer-Punkte-Pfade f{\"u}r LCP's wiedergegeben. Diese werden in Abschnitt 4.2 auf den semidefiniten Fall {\"u}bertragen. Unter der Annahme, dass das zugrundeliegende SDLCP eine strikt komplement{\"a}re L{\"o}sung besitzt, wird gezeigt, dass die Inneren-Punkte-Pfade sogar noch im Randpunkt analytisch sind. Kapitel 5 benutzt die Resultate aus Kapitel 4, um die lokal hohe Konvergenzordnung einer Langschrittmethode zur L{\"o}sung von SDLCP's zu zeigen. Kapitel 6 f{\"u}hrt eine neue Methode zur L{\"o}sung von LCP's und SDLCP's mit Hilfe von Inneren-Punkte-Techniken ein. Dabei werden die Pfadfunktionen derart gew{\"a}hlt, dass alle Iterierten auf unzul{\"a}ssigen zentralen Pfaden liegen. Es wird globale und lokale Konvergenz des Verfahrens bewiesen.}, subject = {Innere-Punkte-Methode}, language = {de} } @phdthesis{BletzSiebert2002, author = {Bletz-Siebert, Oliver}, title = {Homogeneous spaces with the cohomology of sphere products and compact quadrangles}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3994}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2002}, abstract = {We consider homogeneous spaces G/H with the same rational homotopy as a product of a 1-sphere and a (m+1)-sphere. We show that these spaces have also the rational cohomology of such a sphere product if H is connected and if the quotient has dimension m+2. Furthermore, we prove that if additionally the fundamental group of G/H is cyclic, then G/H is locally a product of a 1-torus and ofA/H, where A/H is a simply connected rational cohomology (m+1)-sphere (and hence classified). If H fails to be connected, then with U as the connected component of H the G-action on the covering space G/U of G/H has connected stabilizers, and the results apply to G/U. To show that under the assumptions above every natural number may be realized as the order of the group of connected components of H we calculate the cohomology of certain homogeneous spaces. We also determine the rational cohomology of the fibre bundle U-->G-->G/U if G/H meets the assumptions above. This is done by considering the respective Leray-Serre spectral sequence. The structure of the cohomology of U-->G-->G/U then gives a second proof for the structure of compact connected Lie groups acting transitively on spaces with the rational homotopy of a product of a 1-sphere and a (m+1)-sphere. Since a quotient of a homogeneous space with the same rational homotopy or cohomology as a product of a 1-sphere and a (m+1)-sphere is not simply connected, there often arises the question whether or not a considered fibre bundle or fibration is orientable. A large amount of space will therefore be given to the problem of showing that certain fibrations are orientable. For compact connected (m+2)-manifolds with cyclic fundamental groups and with the rational homotopy of a product of a 1-sphere and a (m+1)-sphere we show the following: if a connected Lie group acts transitively on the manifold, then the maximal compact subgroups are either transitive, or their orbits are simply connected rational cohomology spheres of codimension 1. Homogeneous spaces with the same rational cohomology or homotopy as a a product of a 1-sphere and a (m+1)-sphere play a role in the study of different types of geometrical objects. They appear for example as focal manifolds of isoparametric hypersurfaces with four distinct principal curvatures. Further examples of such spaces are the point spaces and the line spaces of compact connected generalized quadrangles. We determine the isometry groups of isoparametric hypersurfaces with 4 principal curvatures of multiplicities 1 and m which are transitive on the focal manifold with non-trivial fundamental group. Buildings were introduced by Jacques Tits to give interpretations of simple groups of Lie type. They are a far-reaching generalization of projective spaces, in particular a generalization of projective planes. There is another generalization of projective planes called generalized polygons. A projective plane is the same as a generalized triangle. The generalized polygons are also contained in the class of buildings: they are the buildings of rank 2. To compact quadrangles one can assign a pair of natural numbers called the topological parameters of the quadrangles. We treat the case k=1. It turns out that there are no other point-transitive compact connected Lie groups for (1,m)-quadrangles than the ones for the real orthogonal quadrangles. Furthermore, we solve the problem of three infinite series of group actions which Kramer left as open problems; there are no quadrangles with the homogeneous spaces in question as point spaces (up to maybe a finite number of small parameters in one of the three series).}, subject = {Homogener Raum}, language = {en} } @phdthesis{Grahl2002, author = {Grahl, J{\"u}rgen}, title = {Blochsches Prinzip, L{\"u}ckenreihen und Semidualit{\"a}t}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3477}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2002}, abstract = {Ein bekanntes heuristisches Prinzip von A. Bloch beschreibt die Korrespondenz zwischen Kriterien f{\"u}r die Konstanz ganzer Funktionen und Normalit{\"a}tskriterien. In der vorliegenden Dissertation untersuchen wir die G{\"u}ltigkeit des Blochschen Prinzip bei L{\"u}ckenreihenproblemen sowie Zusammenh{\"a}nge zwischen Normalit{\"a}tsfragen und der Semidualit{\"a}t von einer bzw. von zwei Funktionen. Die ersten beiden Kapitel stellen die im folgenden ben{\"o}tigten Hilfsmittel aus der Nevanlinnaschen Wertverteilungstheorie und der Normalit{\"a}tstheorie bereit. Im dritten Kapitel beweisen wir ein neues Normalit{\"a}tskriterium f{\"u}r Familien holomorpher Funktionen, f{\"u}r die ein Differentialpolynom einer bestimmten Gestalt nullstellenfrei ist. Dies verallgemeinert fr{\"u}here Resultate von Hayman, Drasin, Langley und Chen \& Hua. Kapitel 4 ist dem Beweis eines unserer im folgenden wichtigsten Hilfsmittel gewidmet: eines tiefliegenden Konvergenzsatzes von H. Cartan {\"u}ber Familien von p-Tupeln holomorpher nullstellenfreier Funktionen, welche einer linearen Relation unterliegen. In Kapitel 5 werden die Konzepte der Dualit{\"a}t und Semidualit{\"a}t eingef{\"u}hrt und die Verbindung zu Normalit{\"a}tsfragen diskutiert. Die neuen Ergebnisse {\"u}ber L{\"u}ckenreihen finden sich im sechsten Kapitel. Der Schwerpunkt liegt hierbei zum einen auf sog. AP-L{\"u}ckenreihen, zum anderen auf allgemeinen Konstruktionsverfahren, mit denen sich neue semiduale L{\"u}ckenstrukturen aus bereits bekannten gewinnen lassen. Zahlreiche unserer Beweise beruhen wesentlich auf dem Satz von Cartan aus Kapitel 4. Im siebten Kapitel erweitern wir unsere Semidualit{\"a}tsuntersuchungen auf Mengen aus zwei Funktionen. Wir ziehen Normalit{\"a}tskriterien (vor allem das in Kapitel 3 bewiesene sowie den Satz von Cartan) heran, um spezielle Mengen als nichtsemidual zu identifizieren. Zuletzt konstruieren wir ein Beispiel einer semidualen Menge aus zwei Funktionen.}, subject = {Blochsches Prinzip}, language = {de} } @phdthesis{Wolfrom2002, author = {Wolfrom, Martin}, title = {Isoparametric hypersurfaces with a homogeneous focal manifold}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3505}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2002}, abstract = {The classification of isoparametric hypersurfaces in spheres with a homogeneous focal manifold is a project that has been started by Linus Kramer. It extends results by E. Cartan and Hsiang and Lawson. Kramer does most part of this classification in his Habilitationsschrift. In particular he obtains a classification for the cases where the homogeneous focal manifold is at least 2-connected. Results of E. Cartan, Dorfmeister and Neher, and Takagi also solve parts of the classification problem. This thesis completes the classification. We classify all closed isoparametric hypersurfaces in spheres with g>2 distinct principal curvatures one of whose multiplicities is 2 such that the lower dimensional focal manifold is homogeneous. The methods are essentially the same as in Kramer's 'Habilitationsschrift'. The cohomology of the focal manifolds in question is known. This leads to two topological classification problems, which are also solved in this thesis. We classify simply connected homogeneous spaces of compact Lie groups with the same integral cohomology ring as a product of spheres S^2 x S^m and m odd on the one hand and a truncated polynomial ring Q[a]/(a^m) with one generator of even degree and m > 1 as its rational cohomology ring on the other hand.}, subject = {Isoparametrische Hyperfl{\"a}che}, language = {en} } @phdthesis{Joachim2004, author = {Joachim, Silvia}, title = {Regulatorketten in Butlergruppen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-10438}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2004}, abstract = {Die fast vollst{\"a}ndig zerlegbaren Gruppen bilden eine Teilklasse der Butlergruppen. Das Konzept des Regulators, d.h. der Durchschnitt aller regulierenden Untergruppen, ist unverzichtbar f{\"u}r fast vollst{\"a}ndig zerlegbare Gruppen. Dieses Konzept l{\"a}sst sich in nat{\"u}rlicher Weise auf die ganze Klasse der Butlergruppen fortsetzen. Allerdings l{\"a}sst sich die Regulatorbildung im allgemeineren Fall der Butlergruppen a priori iterieren. Damit stellt sich erst einmal die Frage, ob es {\"u}berhaupt Butlergruppen gibt mit Regulatorketten, der L{\"a}nge gr{\"o}ßer als 1. Ein erstes Beispiel der L{\"a}nge 2 wurde 1997 von Lehrmann und Mutzbauer konstruiert. In dieser Dissertation wurden mit konzeptionell neuen Techniken Butlergruppen mit beliebiger vorgegebener endlicher Kettenl{\"a}nge angegeben. Grunds{\"a}tzliche Schwierigkeiten bei diesem Unterfangen resultieren aus dem Fehlen, bzw. der Unm{\"o}glichkeit, einer kanonischen Darstellung von Butlergruppen. Man verwendet die allseits gebrauchte Summendarstellung f{\"u}r Butlergruppen. Genau an dieser Stelle bedarf es v{\"o}llig neuer Methoden, verglichen mit den fast vollst{\"a}ndig zerlegbaren Gruppen mit ihrer kanonischen Regulatordarstellung. Alle Teilaufgaben bei der anstehenden Konstruktion von Butlergruppen, die f{\"u}r fast vollst{\"a}ndig zerlegbare Gruppen Standard sind, werden hierbei problematisch, u.a. die Bildung reiner H{\"u}llen, die Bestimmung regulierender Untergruppen und die Regulatorbildung.}, subject = {Butlergruppe}, language = {de} } @phdthesis{Kurniawan2009, author = {Kurniawan, Indra}, title = {Controllability Aspects of the Lindblad-Kossakowski Master Equation : A Lie-Theoretical Approach}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-48815}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {One main task, which is considerably important in many applications in quantum control, is to explore the possibilities of steering a quantum system from an initial state to a target state. This thesis focuses on fundamental control-theoretical issues of quantum dynamics described by the Lindblad-Kossakowski master equation which arises as a bilinear control system on some underlying real vector spaces, e.g controllability aspects and the structure of reachable sets. Based on Lie-algebraic methods from nonlinear control theory, the thesis presents a unified approach to control problems of finite dimensional closed and open quantum systems. In particular, a simplified treatment for controllability of closed quantum systems as well as new accessibility results for open quantum systems are obtained. The main tools to derive the results are the well-known classifications of all matrix Lie groups which act transitively on Grassmann manifolds, and respectively, on real vector spaces without the origin. It is also shown in this thesis that accessibiity of the Lindblad-Kossakowski master equation is a generic property. Moreover, based on the theoretical accessibility results, an algorithm is developed to decide when the Lindblad-Kossakowski master equation is accessible.}, subject = {Kontrolltheorie}, language = {en} }