@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{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{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{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{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{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{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}
}
@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{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}
}
@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}
}
@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{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}
}
@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{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{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{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{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}
}