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We study the approximation of functions defined on different domains by trigonometric and transformed trigonometric functions. We investigate which of the many results known from the approximation theory on the d-dimensional torus can be transfered to other domains. We define invertible parameterized transformations and prove conditions under which functions from a weighted Sobolev space can be transformed into functions defined on the torus, that still have a certain degree of Sobolev smoothness and for which we know worst-case upper error bounds. By reverting the initial change of variables we transfer the fast algorithms based on rank-1 lattices used to approximate functions on the torus efficiently over to other domains and obtain adapted FFT algorithms.:1 Introduction 2 Preliminaries and notations 3 Fourier approximation on the torus 4 Torus-to-R d transformation mappings 5 Torus-to-cube transformation mappings 6 Conclusion Alphabetical Index / Wir betrachten die Approximation von Funktionen, die auf verschiedenen Gebieten definiert sind, mittels trigonometrischer und transformierter trigonometrischer Funktionen. Wir untersuchen, welche bisherigen Ergebnisse für die Approximation von Funktionen, die auf einem d-dimensionalen Torus definiert wurden, auf andere Definitionsgebiete übertragen werden können. Dazu definieren wir parametrisierte Transformationsabbildungen und beweisen Bedingungen, bei denen Funktionen aus einem gewichteten Sobolevraum in Funktionen, die auf dem Torus definiert sind, transformiert werden können, die dabei einen gewissen Grad an Sobolevglattheit behalten und für die obere Schranken der Approximationsfehler bewiesen wurden. Durch Umkehrung der ursprünglichen Koordinatentransformation übertragen wir die schnellen Algorithmen, die Rang-1 Gitter Methoden verwenden um Funktionen auf dem Torus effizient zu approximieren, auf andere Definitionsgebiete und erhalten adaptierte FFT Algorithmen.:1 Introduction 2 Preliminaries and notations 3 Fourier approximation on the torus 4 Torus-to-R d transformation mappings 5 Torus-to-cube transformation mappings 6 Conclusion Alphabetical Index

Verallgemeinerungen bekannter Existenz- und Eindeutigkeitsaussagen für gebietsfüllende Kreispackungen. Für jedes beschränkte, einfach zusammenhängende Gebiet und für jeden zulässigen Komplex existiert eine gebietsfüllende, verallgemeinerte Kreispackung, die einer beliebigen der folgenden Normalisierungen genügt. alpha-beta-gamma: drei Randkreise sind je einem Randpunkt (Primende) zugeordnet alpha-gamma: ein Kreis mit fixem Mittelpunkt und ein Randkreis mit zugeordnetem Primende alpha-beta: zwei Kreise mit fixen Mittelpunkten Bedingungen werden angegeben, unter welchen die aufgeführten Normalisierungen eindeutige Lösungen implizieren, welche zudem stetig von den Normalisierungsparametern abhängen. Ist der Alpha-Kreis ein innerer Kreis, dann wird gezeigt, dass die alpha-beta Normalisierung im Allg. keine Eindeutigkeit liefert. Bedingungen werden aufgeführt, die nicht-entartete Lösungen (klassische Kreispackungen) garantieren. Alle Beweise sind möglichst elementar und unabhängig von existierenden Kreispackungs-Ergebnissen. / Existing existence and uniqueness results in the field of domain filling circle packings are generalized. For every bounded, simply connected domain, for every admissible complex, and under any of the following normalizations it is shown that there is a domain filling generalized circle packing. alpha-beta-gamma: three boundary disks are each associated with a boundary point (prime end) alpha-gamma: one disk with fixed center and one boundary disk with associated prime end alpha-beta: two disks with fixed centers Conditions are given under which the stated normalizations yield unique solutions, which then depend continuous on some normalization parameters. For the special case of an interior alpha disk it is shown that the alpha-beta normalization does not yield uniqueness in general. Several conditions are stated that guarantee non-degenerate solutions (classical circle packings). All proofs are kept as elementary as possible and independent of existing circle packing results.

In the first part of this thesis the spectrum of a matrix operator is determined. For this the coefficients of the matrix operator are assumed to satisfy rather general properties which combine smoothness and decay. With this the asymptotics of the eigenfunctions can be determined. This in turn leads to properties of the spectra with the aid of the M-matrix. In the second part it will be shown that if a discrete canonical system has absolutely continuous spectrum of a certain multiplicity, then there is a corresponding number of linearly independent solutions y which are bounded in a weak sense.

The thesis contains 3 repairable systems and 2 replacement systems: First a repairable system is considered with Markovian deterioration and imperfect repair, carried out at fixed times. We look for optimal strategies under certain conditions. Two optimality criteria are considered: expected discounted cost and long-run average cost. Conditions are found under which the optimal policy is a control-limit policy as used by Derman or Ross. We explicitly explain how to derive this optimal policy; numerical examples are given, too. The special case of unbounded cost is also studied. With the first model the state space is numerable but with the second it is not. With the fourth model the system occurs a shock process and is only inspected after such a shock. Models 3 and 5 are replacement systems with Morkovian deterioration and finite state space {0,...,N}. A system in state N is considered to be in a very serious situation. Hence there is the condition, e.g. stipulated by law, that the percentage of all replaced machines in state N in the group of all replaced machines may not be larger than 100 epsilon for a fixed epsilon in [0,1]. We prove that a generalized control limit policy maximizes the expected running time of a machine and we explain explicitly how to derive this optimal policy. Illustrated numerical examples are given.

This thesis examines so called folding neural networks as a mechanism for machine learning. Folding networks form a generalization of partial recurrent neural networks such that they are able to deal with tree structured inputs instead of simple linear lists. In particular, they can handle classical formulas - they were proposed originally for this purpose. After a short explanation of the neural architecture we show that folding networks are well suited as a learning mechanism in principle. This includes three parts: the proof of their universal approximation ability, the aspect of information theoretical learnability, and the examination of the complexity of training. Approximation ability: It is shown that any measurable function can be approximated in probability. Explicit bounds on the number of neurons result if only a finite number of points is dealt with. These bounds are new results in the case of simple recurrent networks, too. Several restrictions occur if a function is to be approximated in the maximum norm. Afterwards, we consider briefly the topic of computability. It is shown that a sigmoidal recurrent neural network can compute any mapping in exponential time. However, if the computation is subject to noise almost the capability of tree automata arises. Information theoretical learnability: This part contains several contributions to distribution dependent learnability: The notation of PAC and PUAC learnability, consistent PAC/ PUAC learnability, and scale sensitive versions are considered. We find equivalent characterizations of these terms and examine their respective relation answering in particular an open question posed by Vidyasagar. It is shown at which level learnability only because of an encoding trick is possible. Two approaches from the literature which can guarantee distribution dependent learnability if the VC dimension of the concept class is infinite are generalized to function classes: The function class is stratified according to the input space or according to a so-called luckiness function which depends on the output of the learning algorithm and the concrete training data. Afterwards, the VC, pseudo-, and fat shattering dimension of folding networks are estimated: We improve some lower bounds for recurrent networks and derive new lower bounds for the pseudodimension and lower and upper bounds for folding networks in general. As a consequence, folding architectures are not distribution independent learnable. Distribution dependent learnability can be guaranteed. Explicit bounds on the number of examples which guarantee valid generalization can be derived using the two approaches mentioned above. We examine in which cases these bounds are polynomial. Furthermore, we construct an explicit example for a learning scenario where an exponential number of examples is necessary. Complexity: It is shown that training a fixed folding architecture with perceptron activation function is polynomial. Afterwards, a decision problem, the so-called loading problem, which is correlated to neural network training is examined. For standard multilayer feed-forward networks the following situations turn out to be NP-hard: Concerning the perceptron activation function, a classical result from the literature, the NP-hardness for varying input dimension, is generalized to arbitrary multilayer architectures. Additionally, NP-hardness can be found if the input dimension is fixed but the number of neurons may vary in at least two hidden layers. Furthermore, the NP-hardness is examined if the number of patterns and number of hidden neurons are correlated. We finish with a generalization of the classical NP result as mentioned above to the sigmoidal activation function which is used in practical applications.

In this thesis scheduling problems with transportation aspects are studied. Classical scheduling models for problems with multiple operations are the so-called shop-scheduling models. In these models jobs consisting of different operations have to be planned on certain machines in such a way that a given objective function is minimized. Each machine may process at most one operation at a time and operations belonging to the same job cannot be processed simultaneously. We generalize these classical shop-scheduling problems by assuming that the jobs additionally have to be transported between the machines. This transportation has to be done by robots which can handle at most one job at a time. Besides transportation times which occur for the jobs during their transport, also empty moving times are considered which arise when a robot moves empty from one machine to another. Two types of problems are distinguished: on the one hand, problems without transportation conflicts (i.e. each transportation can be performed without delay), and on the other hand, problems where transportation conflicts may arise due to a limited capacity of transport robots. In the first part of this thesis several new complexity results are derived for flow-shop problems with a single robot. Since very special cases of these problems are already NP-hard, in the second part of this thesis some techniques are developed for dealing with these hard problems in practice. We concentrate on the job-shop problem with a single robot and the makespan objective. At first we study the subproblem which arises for the robot when some scheduling decisions for the machines have already been made. The resulting single-machine problem can be regarded as a generalization of the traveling salesman problem with time windows where additionally minimal time-lags between certain jobs have to be respected and the makespan has to be minimized. For this single-machine problem we adapt immediate selection techniques used for other scheduling problems and calculate lower bounds based on linear programming and the technique of column generation. On the other hand, to determine upper bounds for the single-machine problem we develop an efficient local search algorithm which finds good solutions in reasonable time. This algorithm is integrated into a local search algorithm for the job-shop problem with a single robot. Finally, the proposed algorithms are tested on different test data and computational results are presented.

The aim of this thesis is to define and study stochastic models of repairable systems and the application of these models to biological systems, especially for cell survival after irradiation with ionizing radiation.

Controlling a high-dimensional dynamical system with continuous state and action spaces in a partially unknown environment like a gas turbine is a challenging problem. So far often hard coded rules based on experts´ knowledge and experience are used. Machine learning techniques, which comprise the field of reinforcement learning, are generally only applied to sub-problems. A reason for this is that most standard RL approaches still fail to produce satisfactory results in those complex environments. Besides, they are rarely data-efficient, a fact which is crucial for most real-world applications, where the available amount of data is limited. In this thesis recurrent neural reinforcement learning approaches to identify and control dynamical systems in discrete time are presented. They form a novel connection between recurrent neural networks (RNN) and reinforcement learning (RL) techniques. RNN are used as they allow for the identification of dynamical systems in form of high-dimensional, non-linear state space models. Also, they have shown to be very data-efficient. In addition, a proof is given for their universal approximation capability of open dynamical systems. Moreover, it is pointed out that they are, in contrast to an often cited statement, well able to capture long-term dependencies. As a first step towards reinforcement learning, it is shown that RNN can well map and reconstruct (partially observable) MDP. In the so-called hybrid RNN approach, the resulting inner state of the network is then used as a basis for standard RL algorithms. The further developed recurrent control neural network combines system identification and determination of an optimal policy in one network. In contrast to most RL methods, it determines the optimal policy directly without making use of a value function. The methods are tested on several standard benchmark problems. In addition, they are applied to different kinds of gas turbine simulations of industrial scale.

This Thesis consists of four independent parts. In the first part I prove that the delooping, i.e.the classifying space, of a grouplike monoid is an $H$-space if and only if its multiplication is a homotopy homomorphism. This is an extension and clarification of a result of Sugawara. Furthermore I prove that the Moore loop space functor and the construction of the classifying space induce an adjunction on the corresponding homotopy categories. In the second part I extend a result of G. Dunn, by proving that the tensorproduct $C_{n_1}\otimes\dots \otimes C_{n_j}$ of little cube operads is a topologically equivalent suboperad of $C_{n_1 \dots n_j}$. In the third part I describe operads as algebras over a certain colored operad. By application of results of Boardman and Vogt I describe a model of the homotopy category of topological operads and algebras over them, as well as a notion of lax operads, i.e. operads whose axioms are weakened up to coherent homotopies. Here the W-construction, a functorial cofibrant replacement for a topological operad, plays a central role. As one application I construct a model for the homotopy category of topological categories. C. Berger claimed to have constructed an operad structure on the permutohedras, whose associated monad is exactly the Milgram-construction of the free two-fold loop space. In the fourth part I prove that this statement is not correct.