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