Latest documents http://opus4.kobv.de/opus4-ubbayreuth/index/index/ Thu, 03 Jan 2013 10:43:10 +0100 Thu, 03 Jan 2013 10:43:10 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/1102 The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the numbers of negative and positive eigenvalues of the associated hermitian form. In this thesis, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert–Morimoto decomposition of quasi-tori as well as the Künneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using L2-methods coupled with the Kazama–Dolbeault isomorphism and Bochner–Kodaira formulas. Tsz On Mario Chan doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/1102 Fri, 01 Mar 2013 10:43:10 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/996 This thesis is devoted to the analysis of a very simple, pointwisely state-constrained optimal control problem of an elliptic partial differential equation. The transfer of an idea from the field of optimal control of ordinary differential equations, which proved fruitful with respect to both theoretical treatment and design of algorithms, is the starting point. On this, the state inequality constraint, which is regarded as an equation inside the active set, is differentiated in order to obtain a control law. A geometrical splitting of the constraints is necessary to carry over this approach to the chosen model problem. The associated assertions are rigorously ensured. The subsequent derivation of a control law in the sense of the abovementioned idea yields an equivalent reformulation of the model problem. The active set appears as an independent and equal optimization variable in this new formulation. Thereby a new class of optimization problem is established, which forms a hybrid of optimal control and shape-/topology optimization: set optimal control. This class is integrated into the very abstract framework of optimization on vector bundles; for that purpose some important notions from the field of calculus on manifolds are introduced and related with shape calculus. First order necessary conditions of the set optimal control problem are derived by means of two different approaches: on the one hand a reduced approach via the elimination of the state variable, which uses a formulation as bilevel optimization problem, is pursued, and on the other hand a formal Lagrange principle is presented. A comparison of the newly obtained optimality conditions with those known form literature yields relations between the Lagrange multipliers; in particular, it becomes apparent that the new approach involves higher regularity. The comparison is embedded to the theory of partial differential-algebraic equations, and it is shown that the new approach yields a reduction of the differential index. Upon investigation of the gradient and the second covariant derivative of the objective functional different Newton- and trial algorithms are presented and discussed in detail. By means of a comparison with the well-established primal-dual active set method different benefits of the new approach become apparent. In particular, the new algorithms can be formulated in function space without any regularization. Some numerical tests illustrate that an efficient and competitive solution of state-constrained optimal control problems is achieved. The whole work gives numerous references to different mathematical disciplines and encourages further investigations. All in all, it should be regarded as a first step towards a more comprehensive perspective on state-constrained optimal control of partial differential equations. Michael Frey doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/996 Thu, 15 Nov 2012 09:09:26 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/953 In Chapter 1 we define period mappings of Hodge-de Rahm type for certain submersive, yet not necessarily locally topologically trivial, morphisms of complex manifolds. Generalizing Griffiths's theory, we interpret the differential of such period mappings as the composition of the Kodaira-Spencer map and a map derived from the sheaf cohomological cup product and the contraction of vector fields with differential forms. In Chapter 2 of the text, we consider a submersive morphism $f\colon X\to S$ of complex spaces which is compactified by a proper, flat, and Kähler morphism $\bar f\colon \bar X\to S$. Taking into account the codimension of $\bar X\setminus X$ in $\bar X$, we draw conclusions about the degeneration behavior of the relative Frölicher spectral sequence of the morphism $f$ and about the local freeness of the modules $\mathrm{R}^qf_*(\Omega^p_f)$; our results can be viewed as relative generalizations of a theorem of Takeo Ohsawa. In our final Chapter 3, we employ the upshots of the preceding two chapters in order to deduce a local Torelli theorem for irreducible symplectic complex spaces. As an application of the local Torelli theorem, we prove that irreducible symplectic complex spaces whose codimension of the singular locus does not deceed $4$ satisfy the so-called Fujiki relation. Tim Kirschner doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/953 Fri, 19 Oct 2012 10:58:25 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/848 We use the middle convolution introduced by Katz to construct a families of lisse sheaves on the affine line without two points. These correspond to continuous representations of the etale fundamental group, which can be specialized to compatible systems of Galois representations. This leads to the second maximally unipotent family. Because of the geometric origin, we can show using a theorem of Barnet-Lamb, Gee, Geraghty and Taylor that they are potentially automorphic. Michael Schulte doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/848 Tue, 14 Aug 2012 11:46:27 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/755 The Tamari lattice, thought as a poset on the set of triangulations of a convex polygon with n vertices, generalizes to the higher Stasheff-Tamari orders on the set of triangulations of a cyclic d-dimensional polytope having n vertices. This survey discusses what is known about these orders, and what one would like to know about them. Jörg Rambau; Victor Reiner preprint http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/755 Tue, 13 Mar 2012 08:36:20 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/754 The Guaranteed Service Model (GSM) computes optimal order-points in multi-echelon inventory control under the assumptions that delivery times can be guaranteed and the demand is bounded. Our new Stochastic Guaranteed Service Model (SGSM) with Recourse covers also scenarios that violate these assumptions. Simulation experiments on real-world data of a large German car manufacturer show that policies based on the SGSM dominate GSM-policies. Jörg Rambau; Konrad Schade preprint http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/754 Tue, 13 Mar 2012 07:58:08 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/752 Diese Arbeit liefert Beiträge zur Optimalen Steuerung partiell-differential algebraischer Gleichungen. Insbesondere werden Zustandsbeschränkungen bei der Optimalen Steuerung gewöhnlicher und partieller Differentialgleichungen sowie gekoppelter Systeme untersucht. Die verschiedenen Konzepte dieser Gebiete werden verglichen, übertragen und eingeordnet. Zentrale Ergebnisse sind die Übertragung der notwendigen Bedingungen nach Bryson, Denham und Dreyfus auf elliptische Optimalsteuerungsprobleme mit punktweisen Zustandsbeschränkungen, die Übertragung von Sprungbedingungen und Maßdarstellungen auf ein ODE-PDE beschränktes Optimalsteuerungsproblem mit Zustandsbeschränkungen bei niederdimensionalen aktiven Mengen, sowie die Entwicklung effizienter numerischer Methoden für komplexe Anwendungsprobleme. Die Beiträge dieser Arbeit gliedern sich in vier Kapitel, deren Aspekte jeweils zusammengefasst werden: Zunächst werden die Grundlagen aus der Optimalen Steuerung gewöhnlicher Differentialgleichungen mit Zustandsbeschränkungen wiederholt. Die beiden geläufigen notwendigen Bedingungen nach Jacobson, Lele und Speyer, sowie nach Bryson, Denham und Dreyfus (BDD-Ansatz) werden erläutert und in den Zusammenhang der Optimalen Steuerung partieller Differentialgleichungen gestellt. Dabei wird der Zusammenhang zwischen den Sprungbedingungen und dem Borel-Maß hergestellt. In Kapitel 2 wird der BDD-Ansatz auf ein Optimalsteuerungsproblem einer elliptischen partiellen Differentialgleichung mit punktweisen Zustandsbeschränkungen und verteilten aktiven Mengen übertragen. Die Idee dieses BDD-Ansatzes ist es, die Zustandsbeschränkung auf der aktiven Menge äquivalent in eine Steuerungs-Zustandsbeschränkung oder ggf. eine reine Steuerungsbeschränkung zu transformieren. Dies erlaubt die Herleitung neuer notwendiger Bedingungen. Durch die Transformation der Zustandsbeschränkungen gewinnen die zugehörigen Lagrange-Multiplikatoren an Regularität. Man erhält aus den neuen notwendigen Bedingungen ein Randwertproblem auf verschiedenen Gebieten mit Übergangsbedingungen. Das Interface zwischen den verschiedenen Gebieten stellt eine Optimierungsvariable dar. Eine notwendige Bedingung am Interface wird mit Techniken der Shapeoptimierung hergeleitet. Das Kapitel 3 behandelt Zustandsbeschränkungen bei gemischten ODE-PDE Problemen: Anhand eines zeitabhängigen Anwendungsproblems - des sogenannten Rocketcars - lässt sich eine vollständige Darstellung des Borel-Maßes auf niederdimensionalen aktiven Mengen angeben. In der Folge lassen sich Sprungbedingungen und weitgehende Regularitätsaussagen herleiten. Die explizite Massdarstellung ermöglicht weiterhin die Formulierung als Mehrpunkt-Anfangsrandwertproblem und den Einsatz angepasster Lösungsmethoden. Kapitel 4 widmet sich schließlich einem komplexen Anwendungsproblem eines OC-PDAE: Ein Brennstoffzellenmodell stellt uns vor ein Optimalsteuerungsproblem eines Systems von partiell-differentiell algebraischen Gleichungen. Es werden notwendige Bedingungen hergeleitet und direkte sowie indirekte (adjungierten-basierte) Methoden der Optimalen Steuerung entwickelt und verglichen. Numerische Experimente bestätigen die Effizienz der vorgestellten Methoden. Insbesondere das indirekte Quasi-Newton-Verfahren erlaubt eine zeitadaptive optimale Steuerung der Brennstoffzellenanlage mit hoher Genauigkeit und unter geringer Rechenzeit. Armin Rund doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/752 Tue, 06 Mar 2012 09:11:16 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/751 This thesis is devoted to study two families of surfaces of general type: extended Burniat surfaces with K^2=3 and Keum-Naie-Mendes Lopes-Pardini surfaces. We focus on the corresponding subsets in the Gieseker moduli space. Extended Burniat surfaces with K^2=3 were constructed by Bauer and Catanese in the course of studying the tertiary Burniat surfaces and they showed that their closure is an irreducible component of the moduli space. We prove here the union of the loci described by them is indeed a full irreducible component. We also study the local deformations of two families of degenerations of the extended Burniat surfaces. Keum-Naie-Mendes Lopes-Pardini surfaces are the surfaces constructed by Mendes Lopes and Paridini, which realize the Keum-Naie surfaces with K^2=3 as degenerations. We reconstruct a subfamily of such surfaces and investigate their deformations. We show that the closure of the corresponding subset of the Keum-Naie-Mendes Lopes-Pardini surfaces is an irreducible component of the moduli space. Yifan Chen doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/751 Tue, 06 Mar 2012 09:01:46 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/735 In free material optimization (FMO), one tries to find the best mechanical structure by minimizing the weight or by maximizing the stiffness with respect to given load cases. Design variables are the material properties represented by elasticity tensors or elementary material matrices, respectively, based on a given finite element discretization. Material properties are as general as possible, i.e., anisotropic, leading to positive definite elasticity tensors, which may be arbitrarily small in case of vanishing material. To guarantee a positive definite global stiffness matrix for computing design constraints, it is required that all iterates of an optimization algorithm retain positive definite tensors. Otherwise, some constraints, e.g., the compliance, cannot be evaluated and the algorithm fails. FMO problems are generalizations of topology optimization problems. The goal of topology optimization is to find the stiffest structure subject to given loads and a limited amount of material. In contrast to FMO the material is explicitly given and cannot vary. Based on a finite element discretization, in each element it is decided whether to use material or not. The regions with vanishing material are interpreted as void. The resulting optimization problem can be solved by numerous efficient nonlinear optimization methods, for example sequential convex programming methods. Sequential convex programming (SCP) formulates separable and strictly convex nonlinear subproblems iteratively by approximating the objective and the constraints. Lower and upper asymptotes are introduced to truncate the feasible region. Due to the special structure, the resulting subproblems can be solved efficiently by appropriate methods, e.g., interior point methods. To ensure global convergence, a line search procedure is introduced. Moreover, an active set strategy is applied to reduce computation time. The iterates of SCP are not guaranteed to be inside the corresponding feasible region described by the constraints. As a consequence it is not able to solve free material optimization problems as the compliance function is only well-defined on the feasible region of some of the constraints. We propose a modification of a SCP method that ensures feasibility with respect to a given set of inequality constraints. The resulting procedure is called feasible sequential convex programming method (SCPF). SCPF expands the resulting subproblems by additional nonlinear constraints, that are passed to the subproblem directly to ensure their feasibility in each iteration step. They are referred as feasibility constraints. In addition, other constraints may be violated within the optimization process. As globalization technique a line search procedure is used to ensure convergence. The resulting subproblems can be solved efficiently taking the sparse structure into account. Moreover, semidefinite constraints have to be replaced by nonlinear ones, such that SCPF is applicable. SCPF successfully solved FMO problems with up to 120.000 variables and 60.000 constraints. Within a theoretical analysis global convergence of SCPF is shown for convex feasibility constraints. Sonja Lehmann doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/735 Tue, 13 Dec 2011 11:19:20 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/702 Diese Dissertation beschäftigt sich mit dem mathematischen Teilgebiet der Gitterbasenreduktion. Aufbauend aus den Erkenntnissen der Diplomarbeit "Gitterbasenreduktion mit Random Sampling" werden verschiedene Modifikationen am ursprünglichen LLL- bzw. BKZ-Verfahren vorgenommen: Es wird der von C. Schnorr entwickelte Ansatz, den LLL-Austauschschritt um Tiefeneinfügungen zu erweitern, aufgegriffen und eine alternative Methode zum Basisaustausch für das BKZ-Verfahren vorgestellt. Ferner werden zwei unterschiedliche Verfahren von A. Wassermann und P. Nguyen zum Abschneiden von Enumerationsbäumen beschrieben. Die Random Sampling Strategie von Schnorr wurde überarbeitet, um ein schlechtes GSA-Verhalten des Gitters zu berücksichtigen und eine neuartige Strategie von Buchmann und Ludwig wurde implementiert, bei der das GSA-Verhalten vollkommen irrelevant ist. Schließlich wird ein grundlegendes, heuristisches Bewertungskonzept für Gittervektoren entwickelt, das im Rahmen eines von T. Vidick und P. Nguyen beschriebenen Siebverfahrens, Anwendung findet. Mit Hinblick auf die Qualität der erreichten Gitter-Reduktion für schwierige Market-Split-Probleme in Dimensionen ≈ 120, liefern diese neuen Methoden hervorragende Ergebnisse in äußerst kurzer Zeit (ca. 5 Stunden auf einem 3 GHz Rechner). Auch für Problemdimensionen > 500 sind die Resultate durchaus noch zufriedenstellend - allerdings ist hierbei der Rechenaufwand (> 7 Tage) nicht mehr zu vernachlässigen. Im Vergleich mit dem kommerziellen Programm CPLEX, das einen völlig anderen Ansatz zur Lösung von ganzzahlig-linearen Gleichungssystemen verfolgt, konnten sogar sehr gute Ergebnisse erzielt werden. Heiko Vogel doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/702 Fri, 05 Aug 2011 11:42:18 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/696 In den Jahren 1968 und 1972 entdeckten Preparata bzw. Kerdock zwei unendliche Serien sehr guter nichtlinearer binärer Codes. Beide umfassen den Nordstrom-Robinson-Code, einen (16, 2^8, 6)-Code, dessen Minimaldistanz die obere Schranke von 5 für lineare binäre Codes gleicher Länge und Kardinalität übertrifft. Lange Zeit war unklar, warum die Codes beider Serien formal dual zueinander sind, d. h. warum ihre Gewichtszähler die MacWilliams-Identität erfüllen. Erst in den neunziger Jahren fand man heraus, dass sie als Bilder linearer Codes über dem Ring Z4 unter der sogenannten Grayabbildung dargestellt werden konnten. Diese Entdeckung löste einerseits das Rätsel und rückte gleichzeitig die Untersuchung linearer Codes über Z4 in den Fokus der Forschung. In den Folgejahren wurden Codes über endlichen Kettenringen als natürliche Verallgemeinerung der klassischen Codes über endlichen Körpern erkannt. Für jeden endlichen Kettenring R ist der Faktorring R/Rad(R) isomorph zu einem endlichen Körper GF(q), und mit Hilfe einer verallgemeinerten Version der Grayabbildung kann jeder R-lineare Code in einen - für gewöhnlich nichtlinearen - Code über GF(q) überführt werden. R-lineare Codes, deren Graybild eine bessere Minimaldistanz aufweist als optimale lineare Codes über GF(q) mit denselben Parametern, nennen wir BTL-Codes (better-than-linear). Ist noch unklar, ob lineare Codes derselben Minimaldistanz über GF(q) existieren, sprechen wir von BTKL-Codes (better-than-known-linear). Im Unterschied zu den umfassenden Tabellen für lineare Codes über Körpern gab es - abgesehen von Z4 - bisher nur wenig vergleichbares Datenmaterial zu linearen Codes über endlichen Kettenringen. Diese Lücke zu schließen und gleichzeitig nach weiteren Beispielen für BTL- und BTKL-Codes zu suchen, waren die Hauptziele der vorliegenden Arbeit. Um dies zu erreichen, wurde ein heuristischer Algorithmus aus meiner Diplomarbeit für die Suche nach guten linearen Codes über endlichen Körpern auf die Situation über endlichen Kettenringen verallgemeinert. Es handelt sich hierbei um einen Greedy-Algorithmus, der versucht, die gewünschten Codes durch schrittweises Erweitern von Generatormatrizen zu konstruieren. Die Entscheidungen in jedem Schritt basieren dabei auf einer von probabilistischen Überlegungen geleiteten Bewertungsfunktion. Eine weitere Verallgemeinerung ermöglichte es außerdem, die Methode auf eine größere Klasse von Problemen anzuwenden. In dieser Arbeit betraf dies im Speziellen die Konstruktion linearer Codes nach der Kramer-Mesner-Methode, also solchen, deren Automorphismengruppe eine bestimmte, vorgeschriebene Untergruppe enthält. Mit Hilfe dieser Verfahren wurde eine Datenbank von mehr als 93.000 linearen Codes mit hoher Minimaldistanz über 24 verschiedenen endlichen Kettenringen aufgebaut. Mehr als 1.200 dieser Codes sind als optimal nachgewiesen. Außerdem wurden mehrere neue BTL- und BTKL-Codes gefunden. Einer von ihnen entpuppte sich als der erste Vertreter einer unendlichen Serie über Z4, für deren beiden Anfangsglieder die BTL-Eigenschaft gezeigt werden konnte. Für einen anderen Code fand sich eine interessante geometrische Interpretation. Die Methoden wurden auch zur Konstruktion klassischer Codes über endlichen Körpern mit vorgeschriebener Automorphismengruppe eingesetzt. Dies führte zur Verbesserung der internationalen Tabellen für die beste bekannte Minimaldistanz an insgesamt 497 Stellen, wobei mindestens 38 der gefundenen Codes optimal sind. Auf Grundlage dieser Ergebnisse ist festzustellen, dass die verallgemeinerte Version des Algorithmus sich als mächtiges Werkzeug für Konstruktionsprobleme der hier vorliegenden Art erwiesen hat. Die erzeugten Tabellen legen außerdem die Vermutung nahe, dass BTL- und BTKL-Codes eher seltene Objekte sind, insbesondere für andere Kettenringe als Z4. Johannes Zwanzger doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/696 Tue, 12 Jul 2011 11:30:41 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/682 In continuously running logistic systems (like in-house pallet transportation systems), finite buffer capacities usually require controls achieving uniformly bounded waiting queues (strong stability). Standard stochastic traffic assumptions (arrival rates below service rates) can, in general, not guarantee these strong stability requirements, no matter which control. Therefore, the worst-case traffic notion of reasonable load was introduced, originally for the analysis of the Online-Dial-a-Ride Problem. A set of requests is reasonable if the requests that are presented in a sufficiently large time period can be served in a time period of at most the same length. The rationale behind this concept is that the occurrence of non-reasonable request sets renders the system overloaded, and capacity should be extended. For reasonable load, there are control policies that can guarantee uniformly bounded flow times, leading to strong stability in many cases. Control policies based on naive eoptimization, however, can in general achieve neither bounded flow times nor strong ability. In this chapter, we review the concept and examples for reasonable load. Moreover, we present new control policies achieving strong stability as well as new elementary examples of request sets where naive reoptimization fails. Sven O. Krumke; Jörg Rambau preprint http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/682 Tue, 31 May 2011 13:18:23 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/678 We develop a method to approximate the value vector of discounted Markov decision problems (MDP) with guaranteed error bounds. It is based on the linear programming characterization of the optimal expected cost. The new idea is to use column generation to dynamically generate only such states that are most relevant for the bounds by incorporating the reduced cost information. The number of states that is sufficient in general and necessary in the worst case to prove such bounds is independent of the cardinality of the state space. Still, in many instances, the column generation algorithm can prove bounds using much fewer states. In this paper, we explain the foundations of the method. Moreover, the method is used to improve the well-known nearest-neighbor policy for the elevator control problem. Stefan Heinz; Jörg Rambau; Andreas Tuchscherer preprint http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/678 Tue, 24 May 2011 11:15:59 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/675 Zusammenfassung Die Arbeit behandelt folgende Fragen aus der Darstellungs- und Invariantentheorie halbeinfacher Liealgebren: Gegeben eine halbeinfache (meist: eine einfache) komplexe endlich dimensionale Liealgebra L . Betrachtet wird das Monoid M = M(L) der Äquivalenzklassen der endlich dimensionalen irreduziblen komplexen Darstellungen von L . M wird identifiziert mit dem Gitter der entsprechenden höchsten Gewichte (bezüglich einer ausgewählten Cartanalgebra von L und einer ausgewählten Basis der zugehörigen Wurzeln). Diese Identifizierung liefert die Monoidstruktur von M . Zu einer Darstellung pi von L kann man die symmetrische Algebra von pi betrachten (als unendlich dimensionale Darstellung, welche die symmetrischen Potenzen von pi als endlich dimensionale direkte Summanden enthält.) Ein höchstes Gewicht von L , das als höchstes Gewicht einer irreduziblen Komponente in einer n-ten symmetrischen Potenz von pi auftritt, sei gutes dominantes Gewicht genannt. Die Menge aller guten dominanten Gewichte bildet ein Untermonoid M(pi) des Monoids M . Solch ein M(pi) besitzt einen natürlich definierten Rang r(pi) , der größer-gleich 1 und kleiner-gleich r ist. (S. Seite 8 der Arbeit in der Einleitung.) Hier ist r der Rang von L , d.h. die Dimension einer Cartanunteralgebra von L . Nun: Eine Darstellung pi von L sei gut genannt, wenn r(pi) kleiner als r ist, und schlecht, wenn r(pi) gleich r ist. In den Paragraphen 4 und 5 der Arbeit werden dann - explizit als L-Darstellungen - die n-ten symmetrischen Potenzen der guten Darstellungen pi beschrieben. Im ersten Paragraphen der Arbeit wird nachgewiesen, dass Darstellungen bis auf wenige Ausnahmen schlecht sind, und es werden Listen von schlechten Darstellungen verifiziert. Letztlich werden die Typen der einfachen Liealgebren - die vier klassischen Reihen und die fünf Ausnahmealgebren - einzeln und individuell abgehandelt. In diesem ersten Paragraphen der Arbeit werden, wie auch später, entscheidend explizite Ausreduzierungen von symmetrischen Potenzen von Darstellungen benutzt, die in ausführlichen Listen zusammengestellt sind, s. Liste 1 und Liste 2 am Ende der Arbeit. (Die Berechnungen wurden mit dem Lie-Berechnungspaket aus [van Leeuwen] gemacht. In Einzelfällen werden auch explizite Ausreduzierungssätze benutzt, weil versucht wird, bei den Beweisen Argumente aus der Darstellungstheorie zu bevorzugen. Gemäß dem ersten Paragraphen sind für die einzelnen Typen einfacher Liealgebren fast alle irreduziblen Darstellungen schlecht. Bei jedem Typ bleibt nur eine kurze Liste von möglicherweise guten Darstellungen übrig. In Paragraph 2 werden nun die Darstellungen in diesen Restlisten als tatsächlich gut nachgewiesen. Paragraph 3 gibt eine Zusammenfassung der guten Darstellung unter einem anderen Gesichtspunkt: Die guten Darstellungen sind geordnet nach ihrem Rang (und nicht nach dem Isomorphietyp der Liealgebra ). In den Paragraphen 4 und 5 wird - für alle guten irreduziblen Darstellungen pi - die genaue Struktur (als vollständig reduzible Darstellung) der symmetrischen Potenzen von pi bestimmt. Benutzt wird dabei auch detailliertere Invariantentheorie und die Kenntnis von Hauptisotropiegruppen bei Darstellungsräumen. Ridvan Güner doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/675 Mon, 16 May 2011 08:47:32 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/661 We study a generalized job-shop problem called the Laser Sharing Problem with fixed tours (LSP-T) where the tasks may need more than one resource simultaneously. This fact will be used to model possible collisions between industrial robots. For three robots we will show that the special case where only one resource is used by more than one robot is already NP-hard. This also implies that one machine scheduling with chained min delay precedence constraints is NP-hard for at least three chains. On the positive side, we present a polynomial algorithm for the two robot case and a pseudo-polynomial algorithm together with an FPTAS for an arbitrary but constant number of robots. This gives a sharp boundary of the complexity status for a constant number of robots. Joachim Schauer; Cornelius Schwarz preprint http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/661 Mon, 04 Apr 2011 09:21:55 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/601 The canonical height is an indispensable tool for the study of the arithmetic of abelian varieties. In this dissertation we investigate methods for the explicit computation of canonical heights on Jacobians of smooth projective curves. Building on an existing algorithm due to Flynn and Smart with modifications by Stoll we generalize efficient methods for the computation of canonical heights on elliptic curves to the case of Jacobian surfaces. The main tools are the explicit theory of the Kummer surface associated to a Jacobian surface, which we develop in full generality, building on earlier work due to Flynn, and a careful study of the local Néron models of the Jacobian. As a first step for a further generalization to Jacobian threefolds of hyperelliptic curves, we completely describe the associated Kummer threefold and conjecture formulas for explicit arithmetic on it, based on experimental data. Assuming the validity of this conjecture, many of the results for Jacobian surfaces can then be generalized. Finally, we use a theorem due to Faltings, Gross and Hriljac which expresses the canonical height on the Jacobian in terms of arithmetic intersection theory on the curve to develop an algorithm for the computation of the canonical height which is applicable in principle to any Jacobian. However, it uses several subroutines and some of these are currently only implemented in the hyperelliptic case, although the theory is available in general. Among the possible applications of the computation of canonical heights are the determination of generators for the Mordell-Weil group of the Jacobian and the computation of its regulator, appearing for instance in the famous Birch and Swinnerton-Dyer conjecture. We illustrate our algorithm with two examples: The regulator of a finite index subgroup of the Mordell-Weil group of the Jacobian of a genus 3 hyperelliptic curve and the non-archimedean part of theregulator computation for the Jacobian of a non-hyperelliptic genus 4 curve, where the remaining computations can be done immediately once the above-mentioned implementations are available. Jan Steffen Müller doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/601 Wed, 19 Jan 2011 07:52:12 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/596 Eine kompakte komplexe Mannigfaltigkeit heißt algebraisch approximierbar, wenn sie beliebig kleine projektive Deformationen besitzt. Eine natürliche Fragestellung ist, ob jede kompakte Kählermannigfaltigkeit algebraisch approximierbar ist. Während dies in Dimension 2 nach den Arbeiten von Kodaira richtig ist, hat Voisin vierdimensionale Gegenbeispiele gefunden. In Dimension 3 ist die Frage noch offen. Ziel der vorliegenden Arbeit ist es, den dreidimensionalen Fall etwas näher zu beleuchten. Dazu wird algebraische Approximierbarkeit zunächst von einem allgemeinen Standpunkt aus betrachtet. Es werden Funktorialitätsfragen untersucht, also der Zusammenhang zwischen algebraischer Approximierbarkeit der Quelle und des Ziels gewisser holomorpher Abbildungen, und Ergebnisse für verschiedene Klassen von Abbildungen erzielt, wie etwa Aufblasungen, endliche Abbildungen, Faserungen und Morikontraktionen. Als Fallstudie einer konkreten Klasse von Kählerdreifaltigkeiten werden anschließend Konikbündel über Kählerflächen untersucht, die in natürlicher Weise in der Moritheorie auftreten. Nach dem Beweis einiger grundlegender Tatsachen über Konikbündel werden ihre Diskriminantenorte genauer untersucht und damit Chernklassenabschätzungen für Konikbündel mit relativer Picardzahl 1 über nichtalgebraischen kompakten Kählerflächen hergeleitet. Unter Verwendung dieser Abschätzungen wird die Existenz infinitesimaler Deformationen solcher Konikbündel gezeigt, was einen wichtigen ersten Schritt zum Beweis der algebraischen Approximierbarkeit darstellt. Ein spezieller Typ von Konikbündeln sind die projektivierten Rang-2-Bündel. Die Periodenabbildung verhilft zu einem guten Verständnis der Deformationstheorie solcher Bündel über K3-Flächen und zweidimensionalen Tori. Konkret werden Fortsetzungssätze für Geradenbündel und Rang-2-Bündel bewiesen, die implizieren, dass jedes projektivierte Rang-2-Bündel über einer K3-Fläche oder einem zweidimensionalen Torus algebraisch approximierbar ist. Durch Untersuchung von Aufblasungen solcher Flächen wird dieses Resultat auf projektivierte Rang-2-Bündel über beliebigen kompakten Kählerflächen mit Kodairadimension 0 ausgedehnt. Schließlich wird die zuvor entwickelte Deformationstheorie für Vektorbündel verwendet, um weitere Approximierbarkeitsergebnisse für Konikbündel über elliptischen Flächen und K3-Flächen zu bekommen. Florian Schrack doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/596 Tue, 21 Dec 2010 10:17:05 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/584 Motivated by the laser sharing problem (LSP) in car body manufacturing, we define the new general routing and scheduling problem (RSP). In the RSP, multiple servers have to visit and process jobs; renewable resources are shared among them. The goal is to find a makespan-minimal scheduled dispatch. We present complexity results as well as a branch-and-bound algorithm for the RSP. This is the first algorithm that is able to solve the LSP for industrially relevant problem scales. Jörg Rambau; Cornelius Schwarz article http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/584 Tue, 05 Oct 2010 09:21:50 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/583 In modern production facilities industrial robots play an important role. When two ore more of them are moving in the same area, care must be taken to avoid collisions between them. Due to expensive equipment costs our approach to handle this is very conservative: Each critical area is modeled as a shared resource where only one robot is allowed to use it at a time. We studied collision avoidance in the context of arc welding robots in car manufacture industry. Here another shared resource comes into place. When using laser welding technology every robot needs to be connected to a laser source supplying it with the necessary energy. Each laser source can be connected to up to six robots but serve only one at a time. An instance of the problem consists of a set of robots, a set of welding task, a number of laser sources, a distance table, collision information and a production cycle time. The goal is to design robot tours covering all task and schedule them resource conflict free such that the makespan does not exceed the cycle time. We propose a general model for integrated routing and scheduling including collision avoidance as well as a branch-and-bound algorithm for it. Computational results on data generated with the robot simulation software KuKa Sim Pro are also provided showing that our algorithm outperforms standard mixed-integer models for our application. Jörg Rambau; Cornelius Schwarz preprint http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/583 Tue, 05 Oct 2010 09:18:51 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/567 In our thesis we treat mainly two topics: the classification of isotrivially fibred surfaces with p_g=q=2, and the construction of new Beauville surfaces. An isotrivially fibred surface is a smooth projective surface endowed with a morphism onto a smooth curve such that all the smooth fibres are isomorphic to each other. The first goal of this thesis is to classify the isotrivially fibred surfaces with p_g=q=2 completing and extending a result by Zucconi. As an important byproduct, we provide new examples of minimal surfaces of general type with p_g=q=2 and K^2=4,5 and the first example with K^2=6. We say that a surface S is isogenous to a product of curves if S = (C times F )/G, for C and F smooth curves and G a finite group acting freely on C times F. Beauville surfaces are a special case of surfaces isogenous to a product. In this thesis we include part of a joint work with Shelly Garion, in which we construct new Beauville surfaces with group G either PSL(2,p^e), or A_n, or S_n, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, and on classical results of Macbeath. The thesis is divided into three chapters, which are subdivided in several sections. In the first chapter we treat the problem of the classification of isotrivially fibred surfaces with p_g=q=2. We start by recalling some basic facts and theorems about fibred surfaces and surfaces isogenous to a higher product of curves. Then we solve the classification problem using techniques coming from both geometry and combinatorial group theory. In the second chapter we deal with Beauville surfaces. First we give a group theoretical characterization of them. Then we enunciate a theorem of Liebeck and Shalev that we use for the construction of Beauville surfaces with group A_n or S_n. Afterwards we also study Beauville surfaces with group PSL(2,p^e). In the third chapter we give a description of the locus, in the moduli space of surfaces of general type, corresponding to the surfaces isogenous to a product with p_g=q=2 described in the first chapter. Indeed, by the results proven by Catanese, this locus is a union of connected components, whose number can be computed using a theorem of Bauer and Catanese. In the same way we are able to provide an asymptotic result about the number of connected components of the moduli space corresponding to certain families of Beauville surfaces with group either PSL(2,p^e), or A_n, or (mathbb{Z}/nmathbb{Z})^2 as p and n go to infinity. Matteo Penegini doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/567 Mon, 26 Jul 2010 12:09:53 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/560 The focus of this thesis lies on examining the solvability of optimal control problems constrained by nonlinear partial differential equations (PDE) and inclusions (PDI). There exist statements on the existence of solutions for optimal control problems with linear and semi-linear PDEs with monotone parts. The theory for non-monotone PDEs resp. the related optimal control problems is, to the author’s knowledge, incomplete regarding important issues. This concerns particularly the case of PDEs containing mappings, which only satisfy boundedness conditions on restricted sets. At first an optimal control problem is considered, which is characterized by a Laplace equation with Dirichlet boundary conditions and a nonlinear non-monotone Nemytskii operator. Under the decisive assumption of the existence of so called sub- and supersolutions for this differential equation and by introducing a truncation operator we can define an auxiliary problem which is characterized by a pseudomonotone operator. Thereby the solution theory for pseudomonotone operators of Brézis (1968) is applicable. Moreover, starting with the definitions of sub- und supersolution it can be shown, that every solution of the auxiliary problem is a solution of the original problem. The choice of a new optimal control problem which substitutes the original optimal control problem is again governed by the properties of the auxiliary operator. The equivalence of the auxiliary problem to the original problem and the existence of at least one solution can be shown. The technique of applying the Theorem of Lax-Milgram on a linearized problem can be adapted to the semi-linear non-monotone case. This procedure is already known from the theory of semi-linear monotone problems. For optimal control problems with quasi-linear differential equations, different methods are required. As in the semi-linear case, the property of pseudomonotonicity plays a key role in proving the existence of a solution of the quasi-linear PDE. In the proof of the existence of a solution for the optimal control problem other properties of the auxiliary operator are exploited. In the elliptic case operators which satisfy the S+ -property are important. In order to utilize this property, a transformation of the operator to some coercive auxiliary operator is necessary. For this reason a term is added, which penalizes the deviation from the admissible set of states. This term is characterized by a factor, which is derived explicitly in this work. The proof of the existence of a solution of the optimal control problem with parabolic equations is based on the definition of an auxiliary operator, coercivity and the S+ -property of operators. The set of solutions of the considered PDE is compact, but the number of solutions and the situation to each other is unknown. This leads to difficulties in deriving necessary optimality conditions. For this reason a direct approach to solve the optimal control problem with semi-linear PDEs is introduced. It is assumed, that the state constraints coincide with the sub- and the supersolution of the PDE with the upper and lower boundary of the control variable. Using an auxiliary operator, this assumption allows the formulation of an equivalent optimal control problem without pointwise state constraints. Through semi-discretization we can define a family of optimal control problems on a finite dimensional state-space. Existence of a subsequence of solutions of these optimal control problems which converges to a solution of the original problem is shown. Another important class of optimal control problems include differential inclusions which are described by multivalued operators. Quasi-linear elliptic inclusions are examined under global as well as local boundedness conditions. Under the assumption of global boundedness the properties of pseudomonotonicity and coercivity for a multivalued auxiliary operator are proven. The existence of at least one solution for the original inclusion follows from the application of a result from Hu and Papageorgiou (1997) on the auxiliary problem. The existence of at least one solution of the optimal control problem is proven by exploiting the coercivity of the multivalued auxiliary operator and the S+ -property of the non-multivalued part of this mapping. In the case of multivalued mappings of Clarke’s gradient type, the existence of at least one solution of the optimal control problem can be shown under local boundedness conditions. Elliptic as well as parabolic quasi-linear inclusions are considered. The proof is again based on coercivity and the S+ -property of the related auxiliary operators and the embedding properties of the spaces. Julia Fischer doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/560 Thu, 24 Jun 2010 09:02:52 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/551 We take a look at certain operators called Cosserat operators and get a compactness result for them leading to several interesting applications. For a more detailed abstract, see the actual abstract at the beginning of the work. Thorsten Riedl doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/551 Wed, 12 May 2010 12:08:54 +0200 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/540 In this thesis, I consider the automorphisms and stable degenerations of surfaces isogenous to a product. First I consider the action of the automorphisms on cohomology in the case where the group G is abelian. It is shown that, if the irregularity of the surface is larger than 1, the action of G on the second cohomology is mostly faithful. For surfaces with irregularity 0 or 1, examples are given. Then I consider the stable degenerations of surfaces isogenous to a product and classify the possible singularities on them. As a result, I show that the Q-Gorenstein deformations of the degenerations with certain singuarities are unobstructed and get some connected components of the moduli space of stable surfaces. Wenfei Liu doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/540 Fri, 19 Mar 2010 11:39:55 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/512 Within the proposed work we consider analytical, conceptional and implementational issues of so called receding horizon controllers in a sampled-data setting. The principle of such a controller is simple: Given the current state of a system we compute an open-loop control which is optimal for a given costfunctional over a fixed prediction horizon. Then, the control is implemented on the first sampling interval and the basic open-loop optimal control problem is shifted forward in time which allows for a repeated evaluation. The contribution of this thesis is threefold: First, we prove estimates for the performance of a receding horizon control, a concept which we call suboptimality degree. These estimate are online computable and can be applied for stabilizing as well as practically stabilizing receding horizon control laws. Moreover, they not only allow for guaranteeing stability of the closed-loop but also for quantifying the loss of performance of the receding horizon control law compared to the infinite horizon control law. Based on these estimates, we introduce adaptation strategies to modify the underlying receding horizon controller in order to guarantee a certain lower bound on the suboptimality degree while reducing the computing cost/time necessary to solve this problem. Within this analysis, the length of the optimization horizon is the parameter we wish to adapt. To this end, we develop and proof several shortening and prolongation strategies which also allow for an effective implementation. Moreover, extensions of our suboptimality estimates to receding horizon controllers with varying optimization horizon are shown. Last, we present details on our implementation of a receding horizon controller PCC2 (http://www.nonlinearmpc.com) which is on the one hand computationally efficient but also allows for easily incorporating our theoretical results. Since a full analysis of such a controller would exceed the scope of this work, we focus on the main aspects of this algorithm using different examples. In particular, we concentrate on the impact of certain choices of parameters on the computing time. We also consider interactions between these parameters to give a guideline to effectively implement and solve further examples. Moreover, we show applicability and effectiveness of our theoretical results using simulations of standard problems for receding horizon controllers. Jürgen Pannek doctoralthesis http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/512 Fri, 20 Nov 2009 09:10:05 +0100 http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/489 The paper contains computational results, the primitive central idempotents of group rings of symmetric and alternating groups of degree smaller or equal 31 in characteristic 3 Harald Meyer other http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/489 Fri, 21 Aug 2009 08:05:03 +0200