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Automated theorem proving is a search problem and, by its undecidability, a very difficult one. The challenge in the development of a practically successful prover is the mapping of the extensively developed theory into a program that runs efficiently on a computer. Starting from a level-based system model for automated theorem provers, in this work we present different techniques that are important for the development of powerful equational theorem provers. The contributions can be divided into three areas: Architecture. We present a novel prover architecture that is based on a set-based compression scheme. With moderate additional computational costs we achieve a substantial reduction of the memory requirements. Further wins are architectural clarity, the easy provision of proof objects, and a new way to parallelize a prover which shows respectable speed-ups in practice. The compact representation paves the way to new applications of automated equational provers in the area of verification systems. Algorithms. To improve the speed of a prover we need efficient solutions for the most time-consuming sub-tasks. We demonstrate improvements of several orders of magnitude for two of the most widely used term orderings, LPO and KBO. Other important contributions are a novel generic unsatisfiability test for ordering constraints and, based on that, a sufficient ground reducibility criterion with an excellent cost-benefit ratio. Redundancy avoidance. The notion of redundancy is of central importance to justify simplifying inferences which are used to prune the search space. In our experience with unfailing completion, the usual notion of redundancy is not strong enough. In the presence of associativity and commutativity, the provers often get stuck enumerating equations that are permutations of each other. By extending and refining the proof ordering, many more equations can be shown redundant. Furthermore, our refinement of the unfailing completion approach allows us to use redundant equations for simplification without the need to consider them for generating inferences. We describe the efficient implementation of several redundancy criteria and experimentally investigate their influence on the proof search. The combination of these techniques results in a considerable improvement of the practical performance of a prover, which we demonstrate with extensive experiments for the automated theorem prover Waldmeister. The progress achieved allows the prover to solve problems that were previously out of reach. This considerably enhances the potential of the prover and opens up the way for new applications.

This thesis contains the mathematical treatment of a special class of analog microelectronic circuits called translinear circuits. The goal is to provide foundations of a new coherent synthesis approach for this class of circuits. The mathematical methods of the suggested synthesis approach come from graph theory, combinatorics, and from algebraic geometry, in particular symbolic methods from computer algebra. Translinear circuits form a very special class of analog circuits, because they rely on nonlinear device models, but still allow a very structured approach to network analysis and synthesis. Thus, translinear circuits play the role of a bridge between the "unknown space" of nonlinear circuit theory and the very well exploited domain of linear circuit theory. The nonlinear equations describing the behavior of translinear circuits possess a strong algebraic structure that is nonetheless flexible enough for a wide range of nonlinear functionality. Furthermore, translinear circuits offer several technical advantages like high functional density, low supply voltage and insensitivity to temperature. This unique profile is the reason that several authors consider translinear networks as the key to systematic synthesis methods for nonlinear circuits. The thesis proposes the usage of a computer-generated catalog of translinear network topologies as a synthesis tool. The idea to compile such a catalog has grown from the observation that on the one hand, the topology of a translinear network must satisfy strong constraints which severely limit the number of "admissible" topologies, in particular for networks with few transistors, and on the other hand, the topology of a translinear network already fixes its essential behavior, at least for static networks, because the so-called translinear principle requires the continuous parameters of all transistors to be the same. Even though the admissible topologies are heavily restricted, it is a highly nontrivial task to compile such a catalog. Combinatorial techniques have been adapted to undertake this task. In a catalog of translinear network topologies, prototype network equations can be stored along with each topology. When a circuit with a specified behavior is to be designed, one can search the catalog for a network whose equations can be matched with the desired behavior. In this context, two algebraic problems arise: To set up a meaningful equation for a network in the catalog, an elimination of variables must be performed, and to test whether a prototype equation from the catalog and a specified equation of desired behavior can be "matched", a complex system of polynomial equations must be solved, where the solutions are restricted to a finite set of integers. Sophisticated algorithms from computer algebra are applied in both cases to perform the symbolic computations. All mentioned algorithms have been implemented using C++, Singular, and Mathematica, and are successfully applied to actual design problems of humidity sensor circuitry at Analog Microelectronics GmbH, Mainz. As result of the research conducted, an exhaustive catalog of all static formal translinear networks with at most eight transistors is available. The application for the humidity sensor system proves the applicability of the developed synthesis approach. The details and implementations of the algorithms are worked out only for static networks, but can easily be adopted for dynamic networks as well. While the implementation of the combinatorial algorithms is stand-alone software written "from scratch" in C++, the implementation of the algebraic algorithms, namely the symbolic treatment of the network equations and the match finding, heavily rely on the sophisticated Gröbner basis engine of Singular and thus on more than a decade of experience contained in a special-purpose computer algebra system. It should be pointed out that the thesis contains the new observation that the translinear loop equations of a translinear network are precisely represented by the toric ideal of the network's translinear digraph. Altogether, this thesis confirms and strengthenes the key role of translinear circuits as systematically designable nonlinear circuits.

Competing Neural Networks as Models for Non Stationary Financial Time Series -Changepoint Analysis-
(2005)

The problem of structural changes (variations) play a central role in many scientific fields. One of the most current debates is about climatic changes. Further, politicians, environmentalists, scientists, etc. are involved in this debate and almost everyone is concerned with the consequences of climatic changes. However, in this thesis we will not move into the latter direction, i.e. the study of climatic changes. Instead, we consider models for analyzing changes in the dynamics of observed time series assuming these changes are driven by a non-observable stochastic process. To this end, we consider a first order stationary Markov Chain as hidden process and define the Generalized Mixture of AR-ARCH model(GMAR-ARCH) which is an extension of the classical ARCH model to suit to model with dynamical changes. For this model we provide sufficient conditions that ensure its geometric ergodic property. Further, we define a conditional likelihood given the hidden process and a pseudo conditional likelihood in turn. For the pseudo conditional likelihood we assume that at each time instant the autoregressive and volatility functions can be suitably approximated by given Feedfoward Networks. Under this setting the consistency of the parameter estimates is derived and versions of the well-known Expectation Maximization algorithm and Viterbi Algorithm are designed to solve the problem numerically. Moreover, considering the volatility functions to be constants, we establish the consistency of the autoregressive functions estimates given some parametric classes of functions in general and some classes of single layer Feedfoward Networks in particular. Beside this hidden Markov Driven model, we define as alternative a Weighted Least Squares for estimating the time of change and the autoregressive functions. For the latter formulation, we consider a mixture of independent nonlinear autoregressive processes and assume once more that the autoregressive functions can be approximated by given single layer Feedfoward Networks. We derive the consistency and asymptotic normality of the parameter estimates. Further, we prove the convergence of Backpropagation for this setting under some regularity assumptions. Last but not least, we consider a Mixture of Nonlinear autoregressive processes with only one abrupt unknown changepoint and design a statistical test that can validate such changes.

Within the last decades, a remarkable development in materials science took place -- nowadays, materials are not only constructed for the use of inert structures but rather designed for certain predefined functions. This innovation was accompanied with the appearance of smart materials with reliable recognition, discrimination and capability of action as well as reaction. Even though ferroelectric materials serve smartly in real applications, they also possess several restrictions at high performance usage. The behavior of these materials is almost linear under the action of low electric fields or low mechanical stresses, but exhibits strong non-linear response under high electric fields or mechanical stresses. High electromechanical loading conditions result in a change of the spontaneous polarization direction with respect to individual domains, which is commonly referred to as domain switching. The aim of the present work is to develop a three-dimensional coupled finite element model, to study the rate-independent and rate-dependent behavior of piezoelectric materials including domain switching based on a micromechanical approach. The proposed model is first elaborated within a two-dimensional finite element setting for piezoelectric materials. Subsequently, the developed two-dimensional model is extended to the three-dimensional case. This work starts with developing a micromechanical model for ferroelectric materials. Ferroelectric materials exhibit ferroelectric domain switching, which refers to the reorientation of domains and occurs under purely electrical loading. For the simulation, a bulk piezoceramic material is considered and each grain is represented by one finite element. In reality, the grains in the bulk ceramics material are randomly oriented. This property is taken into account by applying random orientation as well as uniform distribution for individual elements. Poly-crystalline ferroelectric materials at un-poled virgin state can consequently be characterized by randomly oriented polarization vectors. Energy reduction of individual domains is adopted as a criterion for the initiation of domain switching processes. The macroscopic response of the bulk material is predicted by classical volume-averaging techniques. In general, domain switching does not only depend on external loads but also on neighboring grains, which is commonly denoted as the grain boundary effect. These effects are incorporated into the developed framework via a phenomenologically motivated probabilistic approach by relating the actual energy level to a critical energy level. Subsequently, the order of the chosen polynomial function is optimized so that simulations nicely match measured data. A rate-dependent polarization framework is proposed, which is applied to cyclic electrical loading at various frequencies. The reduction in free energy of a grain is used as a criterion for the onset of the domain switching processes. Nucleation in new grains and propagation of the domain walls during domain switching is modeled by a linear kinetics theory. The simulated results show that for increasing loading frequency the macroscopic coercive field is also increasing and the remanent polarization increases at lower loading amplitudes. The second part of this work is focused on ferroelastic domain switching, which refers to the reorientation of domains under purely mechanical loading. Under sufficiently high mechanical loading, however, the strain directions within single domains reorient with respect to the applied loading direction. The reduction in free energy of a grain is used as a criterion for the domain switching process. The macroscopic response of the bulk material is computed for the hysteresis curve (stress vs strain) whereby uni-axial and quasi-static loading conditions are applied on the bulk material specimen. Grain boundary effects are addressed by incorporating the developed probabilistic approach into this framework and the order of the polynomial function is optimized so that simulations match measured data. Rate dependent domain switching effects are captured for various frequencies and mechanical loading amplitudes by means of the developed volume fraction concept which relates the particular time interval to the switching portion. The final part of this work deals with ferroelectric and ferroelastic domain switching and refers to the reorientation of domains under coupled electromechanical loading. If this free energy for combined electromechanical loading exceeds the critical energy barrier elements are allowed to switch. Firstly, hysteresis and butterfly curves under purely electrical loading are discussed. Secondly, additional mechanical loads in axial and lateral directions are applied to the specimen. The simulated results show that an increasing compressive stress results in enlarged domain switching ranges and that the hysteresis and butterfly curves flatten at higher mechanical loading levels.

This thesis aims at an overall improvement of the diffusion coefficient predictions. For this reason the theoretical determination of diffusion, viscosity, and thermodynamics in liquid systems is discussed. Furthermore, the experimental determination of diffusion coefficients is also part of this work. All investigations presented are carried out for organic binary liquid mixtures. Diffusion coefficient data of 9 highly nonideal binary mixtures are reported over the whole concentration range at various temperatures, (25, 30, and 35) °C. All mixtures investigated in a Taylor dispersion apparatus consist of an alcohol (ethanol, 1-propanol, or 1-butanol) dissolved in hexane, cyclohexane, carbon tetrachloride, or toluene. The uncertainty of the reported data is estimated to be within 310-11 m2s-1. To compute the thermodynamic correction factor an excess Gibbs energy model is required. Therefore, the applicability of COSMOSPACE to binary VLE predictions is thoroughly investigated. For this purpose a new method is developed to determine the required molecular parameters such as segment types, areas, volumes, and interaction parameters. So-called sigma profiles form the basis of this approach which describe the screening charge densities appearing on a molecule’s surface. To improve the prediction results a constrained two-parameter fitting strategy is also developed. These approaches are crucial to guarantee the physical significance of the segment parameters. Finally, the prediction quality of this approach is compared to the findings of the Wilson model, UNIQUAC, and the a priori predictive method COSMO-RS for a broad range of thermodynamic situations. The results show that COSMOSPACE yields results of similar quality compared to the Wilson model, while both perform much better than UNIQUAC and COSMO-RS. Since viscosity influences also the diffusion process, a new mixture viscosity model has been developed on the basis of Eyring’s absolute reaction rate theory. The nonidealities of the mixture are accounted for with the thermodynamically consistent COSMOSPACE approach. The required model and component parameters are derived from sigma-profiles, which form the basis of the a priori predictive method COSMO-RS. To improve the model performance two segment parameters are determined from a least-squares analysis to experimental viscosity data, whereas a constraint optimisation procedure is applied. In this way the parameters retain their physical meaning. Finally, the viscosity calculations of this approach are compared to the findings of the Eyring-UNIQUAC model for a broad range of chemical mixtures. These results show that the new Eyring-COSMOSPACE approach is superior to the frequently employed Eyring-UNIQUAC method. Finally, on the basis of Eyring’s absolute reaction rate theory a new model for the Maxwell-Stefan diffusivity has been developed. This model, an extension of the Vignes equation, describes the concentration dependence of the diffusion coefficient in terms of the diffusivities at infinite dilution and an additional excess Gibbs energy contribution. This energy part allows the explicit consideration of thermodynamic nonidealities within the modelling of this transport property. If the same set of interaction parameters, which has been derived from VLE data, is applied for this part and for the thermodynamic correction, a theoretically sound modelling of VLE and diffusion can be achieved. The influence of viscosity and thermodynamics on the model accuracy is thoroughly investigated. For this purpose diffusivities of 85 binary mixtures consisting of alkanes, cycloalkanes, halogenated alkanes, aromatics, ketones, and alcohols are computed. The average relative deviation between experimental data and computed values is approximately 8 % depending on the choice of the gE-model. These results indicate that this model is superior to some widely used methods. In summary, it can be said that the new approach facilitates the prediction of diffusion coefficients. The final equation is mathematically simple, universally applicable, and the prediction quality is as good as other models recently developed without having to worry about additional parameters, like pure component physical property data, self diffusion coefficients, or mixture viscosities. In contrast to many other models, the influence of the mixture viscosity can be omitted. Though a viscosity model is not required in the prediction of diffusion coefficients with the new equation, the models presented in this work allow a consistent modelling approach of diffusion, viscosity, and thermodynamics in liquid systems.

Fragmentation of tropical rain forests is pervasive and results in various modifications in the ecosystem functioning such as … It has long been noticed that the colony densities of a dominant herbivore in the neotropics - leaf-cutting ant (LCA) - increase in fragmentation-related habitats like forest edges and small fragments, however the reasons for this increase are not clear. The aim of the study was to test the hypothesis that bottom-up control of LCA populations is less effective in fragmented compared to continuous forests and thus explains the increase in LCA colony densities in these habitats. In order to test for less effective bottom-up control, I proposed four working hypotheses. I hypothesized that LCA colonies in fragmented habitats (1) find more palatable vegetation due to low plant defences, (2) forage on few dominant species resulting in a narrow diet breadth, (3) possess small foraging areas and (4) increase herbivory rate at the colony level. The study was conducted in the remnants of the Atlantic rainforest in NE Brazil. Two fragmentation-related forest habitats were included: the edge and a 3500-ha continuous forest and the interior of the 50-ha forest fragment. The interior of the continuous forest served as a control habitat for the study. All working hypotheses can be generally accepted. The results indicate that the abundance of LCA host plant species in the habitats created by forest fragmentation along with weaker chemical defense of those species (especially the lack of terpenoids) allow ants to forage predominantly on palatable species and thus reduce foraging costs on other species. This is supported by narrower ant diet breadth in these habitats. Similarly, small foraging areas in edge habitats and in small forest fragments indicate that there ants do not have to go far to find the suitable host species and thus they save foraging costs. Increased LCA herbivory rates indicate that the damages (i.e., amount of harvested foliage) caused by LCA are more important in fragmentation-related habitats which are more vulnerable to LCA herbivory due to the high availability of palatable plants and a low total amount of foliage (LAI). (1) Few plant defences, (2) narrower ant diet breadth, (3) reduced colony foraging areas, and (4) increased herbivory rates, clearly indicate a weaker bottom-up control for LCA in fragmented habitats. Weak bottom-up control in the fragmentation-related habitats decreases the foraging costs of a LCA colony in these habitats and the colonies might use the surplus of energy resulting from reduced foraging costs to increase the colony growth, the reproduction and turnover. If correct, this explains why fragmented habitats support more LCA colonies at a given time compared to continuous forest habitats. Further studies are urgently needed to estimate LCA colony growth and turnover rates. There are indices that edge effects of forest fragmentation might be more responsible in regulating LCA populations than area or isolation effects. This emphasizes the need to conserve big forest fragments not to fall below a critical size and retain their regular shape. Weak bottom-up control of LCA populations has various consequences on forested ecosystems. I suggest a loop between forest fragmentation and LCA population dynamics: the increased LCA colony densities, along with lower bottom-up control increase LCA herbivory pressure on the forest and thus inevitably amplify the deleterious effects of fragmentation. These effects include direct consequences of leaf removal by ants and various indirect effects on ecosystem functioning. This study contributes to our understanding of how primary fragmentation effects, via the alteration of trophic interactions, may translate into higher order effects on ecosystem functions.

In this dissertation a model of melt spinning (by Doufas, McHugh and Miller) has been investigated. The model (DMM model) which takes into account effects of inertia, air drag, gravity and surface tension in the momentum equation and heat exchange between air and fibre surface, viscous dissipation and crystallization in the energy equation also has a complicated coupling with the microstructure. The model has two parts, before onset of crystallization (BOC) and after onset of crystallization (AOC) with the point of onset of crystallization as the unknown interface. Mathematically the model has been formulated as a Free boundary value problem. Changes have been introduced in the model with respect to the air drag and an interface condition at the free boundary. The mathematical analysis of the nonlinear, coupled free boundary value problem shows that the solution of this problem depends heavily on initial conditions and parameters which renders the global analysis impossible. But by defining a physically acceptable solution, it is shown that for a more restricted set of initial conditions if a unique solution exists for IVP BOC then it is physically acceptable. For this the important property of the positivity of the conformation tensor variables has been proved. Further it is shown that if a physically acceptable solution exists for IVP BOC then under certain conditions it also exists for IVP AOC. This gives an important relation between the initial conditions of IVP BOC and the existence of a physically acceptable solution of IVP AOC. A new investigation has been done for the melt spinning process in the framework of classical mechanics. A Hamiltonian formulation has been done for the melt spinning process for which appropriate Poisson brackets have been derived for the 1-d, elongational flow of a viscoelastic fluid. From the Hamiltonian, cross sectionally averaged balance mass and momentum equations of melt spinning can be derived along with the microstructural equations. These studies show that the complicated problem of melt spinning can also be studied under the framework of classical mechanics. This work provides the basic groundwork on which further investigations on the dynamics of a fibre could be carried out. The Free boundary value problem has been solved numerically using shooting method. Matlab routines have been used to solve the IVPs arising in the problem. Some numerical case studies have been done to study the sensitivity of the ODE systems with respect to the initial guess and parameters. These experiments support the analysis done and throw more light on the stiff nature and ill posedness of the ODE systems. To validate the model, simulations have been performed on sets of data provided by the company. Comparison of numerical results (axial velocity profiles) has been done with the experimental profiles provided by the company. Numerical results have been found to be in excellent agreement with the experimental profiles.

The use of polymers subjected to various tribological situations has become state of
the art. Owing to the advantages of self-lubrication and superior cleanliness, more
and more polymer composites are now being used as sliding elements, which were
formerly composed of metallic materials only. The feature that makes polymer composites
so promising in industrial applications is the opportunity to tailor their properties
with special fillers. The main aim of this study was to strength the importance of
integrating various functional fillers in the design of wear-resistant polymer composites
and to understand the role of fillers in modifying the wear behaviour of the materials.
Special emphasis was focused on enhancement of the wear resistance of
thermosetting and thermoplastic matrix composites by nano-TiO2 particles (with a
diameter of 300nm).
In order to optimize the content of various fillers, the tribological performance of a
series of epoxy-based composites, filled with short carbon fibre (SCF), graphite,
PTFE and nano-TiO2 in different proportions and combinations, was investigated.
The patterns of frictional coefficient, wear resistance and contact temperature were
examined by a pin-on-disc apparatus in a dry sliding condition under different contact
pressures and sliding velocities. The experimental results indicated that the addition
of nano-TiO2 effectively reduced the frictional coefficient, and consequently the contact
temperature, of short-fibre reinforced epoxy composites. Based on scanning
electron microscopy (SEM) and atomic force microscopy (AFM) observations of the
worn surfaces, a positive rolling effect of the nanoparticles between the material pairs
was proposed, which led to remarkable reduction of the frictional coefficient. In particular,
this rolling effect protected the SCF from more severe wear mechanisms, especially
in high sliding pressure and speed situations. As a result, the load carrying capacity of materials was significantly improved. In addition, the different contributions
of two solid lubricants, PTFE powders and graphite flakes, on the tribological
performance of epoxy nanocomposites were compared. It seems that graphite contributes
to the improved wear resistance in general, whereas PTFE can easily form a
transfer film and reduce the wear rate, especially in the running-in period. A combination of SCF and solid lubricants (PTFE and graphite) together with TiO2 nanoparticles
can achieve a synergistic effect on the wear behaviour of materials.
The favourable effect of nanoparticles detected in epoxy composites was also found
in the investigations of thermoplastic, e.g. polyamide (PA) 6,6 matrix. It was found
that nanoparticles could reduce the friction coefficient and wear rate of the PA6,6
composite remarkably, when additionally incorporated with short carbon fibres and
graphite flakes. In particular, the addition of nanoparticles contributed to an obvious
enhancement of the tribological performances of the short-fibre reinforced, hightemperature
resistant polymers, e.g. polyetherimide (PEI), especially under extreme
sliding conditions.
A procedure was proposed in order to correlate the contact temperature and the
wear rate with the frictional dissipated energy. Based on this energy consideration, a
better interpretation of the different performance of distinct tribo-systems is possible.
The validity of the model was illustrated for various sliding tests under different conditions.
Although simple quantitative formulations could not be expected at present, the
study may lead to a fundamental understanding of the mechanisms controlling friction
and wear from a general system point of view. Moreover, using the energybased
models, the artificial neural network (ANN) approach was applied to the experimental
data. The well-trained ANN has the potential to be further used for online
monitoring and prediction of wear progress in practical applications.
Die Verwendung von Polymeren im Hinblick auf verschiedene tribologische Anwendungen
entspricht mittlerweile dem Stand der Technik. Aufgrund der Vorteile von
Selbstschmierung und ausgezeichneter Sauberkeit werden polymere Verbundwerkstoffe
immer mehr als Gleitelemente genutzt, welche früher ausschließlich aus metallischen
Werkstoffen bestanden. Die Besonderheit, die polymere Verbundwerkstoffe
so vielversprechend für industrielle Anwendungen macht, ist die Möglichkeit ihre Eigenschaften
durch Zugabe von speziellen Füllstoffen maßzuschneidern. Das Hauptziel
dieser Arbeit bestand darin, die Wichtigkeit der Integration verschiedener funktionalisierter
Füllstoffe in den Aufbau polymerer Verbundwerkstoffe mit hohem Verschleißwiderstand
aufzuzeigen und die Rolle der Füllstoffe hinsichtlich des Verschleißverhaltens
zu verstehen. Hierbei lag besonderes Augenmerk auf der Verbesserung
des Verschleißwiderstandes bei Verbunden mit duromerer und thermoplastischer
Matrix durch die Präsenz von TiO2-Partikeln (Durchmesser 300nm).
Das tribologische Verhalten epoxidharzbasierter Verbunde, gefüllt mit kurzen Kohlenstofffasern
(SCF), Graphite, PTFE und nano-TiO2 in unterschiedlichen Proportionen
und Kombinationen wurde untersucht, um den jeweiligen Füllstoffgehalt zu optimieren.
Das Verhalten von Reibungskoeffizient, Verschleißwiderstand und Kontakttemperatur
wurde unter Verwendung einer Stift-Scheibe Apparatur bei trockenem
Gleitzustand, verschiedenen Kontaktdrücken und Gleitgeschwindigkeiten erforscht.
Die experimentellen Ergebnisse zeigen, dass die Zugabe von nano-TiO2 in kohlenstofffaserverstärkte
Epoxide den Reibungskoeffizienten und die Kontakttemperatur
herabsetzen können. Basierend auf Aufnahmen der verschlissenen Oberflächen
durch Rasterelektronen- (REM) und Rasterkraftmikroskopie (AFM) trat ein positiver
Rolleffekt der Nanopartikel zwischen den Materialpaaren zum Vorschein, welcher zu
einer beachtlichen Reduktion des Reibungskoeffizienten führte. Dieser Rolleffekt schützte insbesondere die SCF vor schwerwiegenderen Verschleißmechanismen,
speziell bei hohem Gleitdruck und hohen Geschwindigkeiten. Als Ergebnis konnte
die Tragfähigkeit dieser Materialien wesentlich verbessert werden. Zusätzlich wurde
die Wirkung zweier fester Schmierstoffe (PTFE-Pulver und Graphit-Flocken) auf die tribologische Leistungsfähigkeit verglichen. Es scheint, daß Graphit generell zur Verbesserung
des Verschleißwiderstandes beiträgt, wobei PTFE einen Transferfilm bilden
kann und die Verschleißrate insbesondere in der Einlaufphase reduziert. Die
Kombination von SCF und festen Schmierstoffen zusammen mit TiO2-Nanopartikeln
kann einen Synergieeffekt bei dem Verschleißverhalten der Materialien hervorrufen.
Der positive Effekt der Nanopartikel in Duromeren wurde ebenfalls bei den Untersuchungen
von Thermoplasten (PA 66) gefunden. Die Nanopartikel konnten den Reibungskoeffizienten
und die Verschleißrate der PA 66-Verbunde herabsetzen, wobei
zusätzlich Kohlenstofffasern und Graphit-Flocken enthalten waren. Die Zugabe von
Nanopartikeln trug offensichtlich auch zur Verbesserung der tribologischen Leistungsfähigkeit
von SCF-verstärkten, hochtemperaturbeständigen Polymeren (PEI)
insbesondere unter extremen Gleitzuständen, bei. Es wurde eine Methode vorgestellt,
um die Kontakttemperatur und die Verschleißrate mit der durch Reibung dissipierten
Energie zu korrelieren. Diese Energiebetrachtung ermöglicht eine bessere
Interpretation der verschiedenen Eigenschaften von ausgewählten Tribo-Systemen.
Die Gültigkeit dieses Models wurde für mehrere Gleittests unter verschiedenen Bedingungen
erklärt.
Vom generellen Blickpunkt eines tribologischen Systems aus mag diese Arbeit zu
einem fundamentalen Verständnis der Mechanismen führen, welche das Reibungs und Verschleißverhalten kontrollieren, obwohl hier einfache quantitative (mathematische)
Zusammenhänge bisher nicht zu erwarten sind. Der auf energiebasierenden
Modellen fußende Lösungsansatz der neuronalen Netzwerke (ANN) wurde darüber
hinaus auf die experimentellen Datensätze angewendet. Die gut trainierten ANN's
besitzen das Potenzial sie in der praktischen Anwendungen zur Online-
Datenauswertung und zur Vorhersage des Verschleißfortschritts einzusetzen.

We work in the setting of time series of financial returns. Our starting point are the GARCH models, which are very common in practice. We introduce the possibility of having crashes in such GARCH models. A crash will be modeled by drawing innovations from a distribution with much mass on extremely negative events, while in ''normal'' times the innovations will be drawn from a normal distribution. The probability of a crash is modeled to be time dependent, depending on the past of the observed time series and/or exogenous variables. The aim is a splitting of risk into ''normal'' risk coming mainly from the GARCH dynamic and extreme event risk coming from the modeled crashes. We will present several incarnations of this modeling idea and give some basic properties like the conditional first and second moments. For the special case that we just have an ARCH dynamic we can establish geometric ergodicity and, thus, stationarity and mixing conditions. Also in the ARCH case we formulate (quasi) maximum likelihood estimators and can derive conditions for consistency and asymptotic normality of the parameter estimates. In a special case of genuine GARCH dynamic we are able to establish L_1-approximability and hence laws of large numbers for the processes itself. We can formulate a conditional maximum likelihood estimator in this case, but cannot completely establish consistency for them. On the practical side we look for the outcome of estimating models with genuine GARCH dynamic and compare the result to classical GARCH models. We apply the models to Value at Risk estimation and see that in comparison to the classical models many of ours seem to work better although we chose the crash distributions quite heuristically.

In many industrial applications fast and accurate solutions of linear elliptic partial differential equations are needed as one of the building blocks of more complex problems. The domains are often highly complex and meshing turns out to be expensive and difficult to obtain with a sufficient quality. In such cases methods with a regular, not boundary adapted grid offer an attractive alternative. The Explicit Jump Immersed Interface Method is one of these algorithms. The main interest of this work lies in solving the linear elasticity equations. For this purpose the existing EJIIM algorithm has been extended to three dimensions. The Poisson equation is always considered in parallel as the most typical representative of elliptic PDEs. During the work it became clear that EJIIM can have very high computational memory requirements. To overcome this problem an improvement, Reduced EJIIM is proposed. The main theoretical result in this work is the proof of the smoothing property of inverses of elliptic finite difference operators in two and three space dimensions. It is an often observed phenomena that the local truncation error is allowed to be of lower order along some lower dimensional manifold without influencing the global convergence order of the solution.