@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{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{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} } @article{MatlachDhillonHainetal.2015, author = {Matlach, Juliane and Dhillon, Christine and Hain, Johannes and Schlunck, G{\"u}nther and Grehn, Franz and Klink, Thomas}, title = {Trabeculectomy versus canaloplasty (TVC study) in the treatment of patients with open-angle glaucoma: a prospective randomized clinical trial}, series = {Acta Ophthalmologica}, volume = {93}, journal = {Acta Ophthalmologica}, doi = {10.1111/aos.12722}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-149263}, pages = {753-761}, year = {2015}, abstract = {Purpose: To compare the outcomes of canaloplasty and trabeculectomy in open-angle glaucoma. Methods: This prospective, randomized clinical trial included 62 patients who randomly received trabeculectomy (n = 32) or canaloplasty (n = 30) and were followed up prospectively for 2 years. Primary endpoint was complete (without medication) and qualified success (with or without medication) defined as an intraocular pressure (IOP) of ≤18 mmHg (definition 1) or IOP ≤21 mmHg and ≥20\% IOP reduction (definition 2), IOP ≥5 mmHg, no vision loss and no further glaucoma surgery. Secondary endpoints were the absolute IOP reduction, visual acuity, medication, complications and second surgeries. Results: Surgical treatment significantly reduced IOP in both groups (p < 0.001). Complete success was achieved in 74.2\% and 39.1\% (definition 1, p = 0.01), and 67.7\% and 39.1\% (definition 2, p = 0.04) after 2 years in the trabeculectomy and canaloplasty group, respectively. Mean absolute IOP reduction was 10.8 ± 6.9 mmHg in the trabeculectomy and 9.3 ± 5.7 mmHg in the canaloplasty group after 2 years (p = 0.47). Mean IOP was 11.5 ± 3.4 mmHg in the trabeculectomy and 14.4 ± 4.2 mmHg in the canaloplasty group after 2 years. Following trabeculectomy, complications were more frequent including hypotony (37.5\%), choroidal detachment (12.5\%) and elevated IOP (25.0\%). Conclusions: Trabeculectomy is associated with a stronger IOP reduction and less need for medication at the cost of a higher rate of complications. If target pressure is attainable by moderate IOP reduction, canaloplasty may be considered for its relative ease of postoperative care and lack of complications.}, 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} }