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The new international capital standard for credit institutions (“Basel II”) allows banks to use internal rating systems in order to determine the risk weights that are relevant for the calculation of capital charge. Therefore, it is necessary to develop a system that enfolds the main practices and methods existing in the context of credit rating. The aim of this thesis is to give a suggestion of setting up a credit rating system, where the main techniques used in practice are analyzed, presenting some alternatives and considering the problems that can arise from a statistical point of view. Finally, we will set up some guidelines on how to accomplish the challenge of credit scoring. The judgement of the quality of a credit with respect to the probability of default is called credit rating. A method based on a multi-dimensional criterion seems to be natural, due to the numerous effects that can influence this rating. However, owing to governmental rules, the tendency is that typically one-dimensional criteria will be required in the future as a measure for the credit worthiness or for the quality of a credit. The problem as described above can be resolved via transformation of a multi-dimensional data set into a one-dimensional one while keeping some monotonicity properties and also keeping the loss of information (due to the loss of dimensionality) at a minimum level.

Traffic flow on road networks has been a continuous source of challenging mathematical problems. Mathematical modelling can provide an understanding of dynamics of traffic flow and hence helpful in organizing the flow through the network. In this dissertation macroscopic models for the traffic flow in road networks are presented. The primary interest is the extension of the existing macroscopic road network models based on partial differential equations (PDE model). In order to overcome the difficulty of high computational costs of PDE model an ODE model has been introduced. In addition, steady state traffic flow model named as RSA model on road networks has been dicsussed. To obtain the optimal flow through the network cost functionals and corresponding optimal control problems are defined. The solution of these optimization problems provides an information of shortest path through the network subject to road conditions. The resulting constrained optimization problem is solved approximately by solving unconstrained problem invovling exact penalty functions and the penalty parameter. A good estimate of the threshold of the penalty parameter is defined. A well defined algorithm for solving a nonlinear, nonconvex equality and bound constrained optimization problem is introduced. The numerical results on the convergence history of the algorithm support the theoretical results. In addition to this, bottleneck situations in the traffic flow have been treated using a domain decomposition method (DDM). In particular this method could be used to solve the scalar conservation laws with the discontinuous flux functions corresponding to other physical problems too. This method is effective even when the flux function presents more than one discontinuity within the same spatial domain. It is found in the numerical results that the DDM is superior to other schemes and demonstrates good shock resolution.

Wetting of a solid surface with liquids is an important parameter in the chemical engineering process such as distillation, absorption and desorption. The degree of wetting in packed columns mainly contributes in the generating of the effective interfacial area and then enhancing of the heat and mass transfer process. In this work the wetting of solid surfaces was studied in real experimental work and virtually through three dimensional CFD simulations using the multiphase flow VOF model implemented in the commercial software FLUENT. That can be used to simulate the stratified flows [1]. The liquid rivulet flow which is a special case of the film flow and mostly found in packed columns has been discussed. Wetting of a solid flat and wavy metal plate with rivulet liquid flow was simulated and experimentally validated. The local rivulet thickness was measured using an optically assisted mechanical sensor using a needle which is moved perpendicular to the plate surface with a step motor and in the other two directions using two micrometers. The measured and simulated rivulet profiles were compared to some selected theoretical models founded in the literature such as Duffy & Muffatt [2], Towell & Rothfeld [3] and Al-Khalil et al. [4]. The velocity field in a cross section of a rivulet flow and the non-dimensional maximum and mean velocity values for the vertical flat plate was also compared with models from Al-Khalil et al. [4] and Allen & Biggin [5]. Few CFD simulations for the wavy plate case were compared to the experimental findings, and the Towel model for a flat plate [3]. In the second stage of this work 3-D CFD simulations and experimental study has been performed for wetting of a structured packing element and packing sheet consisting of three elements from the type Rombopak 4M, which is a product of the company Kuhni, Switzerland. The hydrodynamics parameters of a packed column, e. i. the degree of wetting, the interfacial area and liquid hold-up have been depicted from the CFD simulations for different liquid systems and liquid loads. Flow patterns on the degree of wetting have been compared to that of the experiments, where the experimental values for the degree of wetting were estimated from the snap shooting of the flow on the packing sheet in a test rig. A new model to describe the hydrodynamics of packed columns equipped with Rombopak 4M was derived with help of the CFD–simulation results. The model predicts the degree of wetting, the specific or interfacial area and liquid hold-up at different flow conditions. This model was compared to Billet & Schultes [6], the SRP model Rocha et al. [7-9], to Shi & Mersmann [10] and others. Since the pressure drop is one of the most important parameter in packed columns especially for vacuum operating columns, few CFD simulations were performed to estimate the dry pressure drop in a structured and flat packing element and were compared to the experimental results. It was found a good agreement from one side, between the experimental and the CFD simulation results, and from the other side between the simulations and theoretical models for the rivulet flow on an inclined plate. The flow patterns and liquid spreading behaviour on the packing element agrees well with the experimental results. The VOF (Volume of Fluid) was found very sensitive to different liquid properties and can be used in optimization of the packing geometries and revealing critical details of wetting and film flow. An extension of this work to perform CFD simulations for the flow inside a block of the packing to get a detailed picture about the interaction between the liquid and packing surfaces is recommended as further perspective.

Uncoupling protein1 (UCP1) in brown adipose tissue was discovered earlier as the main uncoupling source of respiration. We describe the basic facts and a modest contribution of our group to the area of research on mitochondrial uncoupling proteins. After defining the terms uncoupling, leak, proton-mediated uncoupling, we discuss the assumption that due to its low abundance, uncoupling protein 2 (UCP2) can provide only mild uncoupling, i.e. can decrease the proton motive force by several mV only. A fatty acid cycling mechanism is described as a plausible explanation for the protonophoretic function of all uncoupling proteins together with our experiments supporting it. A speculation for the phylogenesis of all uncoupling proteins can be deduced by estimated UCP2 content in several tissues, and details of its activation are explained on the basis of our experiments. In the present study a solubilization and refolding method for UCP2 from inclusion bodies was developed and characterized. As it was known and also demonstrated from previous experiments on UCP1 that fatty acids are substrates, we used the same procedure to study the function of UCP2. Utilizing spin-labelled fatty acids (SLFA) for our experiments we demonstrated the binding of fatty acids to UCP2, and the competition of other natural fatty acids like oleic acid, palmitic acid, arachidonic acid and eicosatrienoic acid to the preformed complex emphasizes the presence of a fatty acid binding site for mitochondrial UCP2. The findings were observed by EPR spectroscopy where the highly immobilized spectra with presence of spin-labelled fatty acid eventually end up as free spin label spectra with a particular concentration of the natural fatty acid added to the UCP2 bound with spin-labelled fatty acid. This fits in significantly with the earlier findings of UCP1 and also leads to assumption of functional explanation about the physiological relevance between the uncoupling proteins functions. The present study, in which representative and sensitive parameters for EPR spectroscopy were established, at the same time describes the concentration effects of fatty acids upon the protein bound with spin-labelled fatty acids which are much of importance in comparison to physiological levels, being in the micromolar range (µM) as compared with milli molar (mM) as for UCP1 previously. In appropriate examples, different fatty acids are used and compared with competitors like alkylsulfonates also emphasizing the function of the protein. And the studies with the effect of nucleotides inhibition demonstrate that there exists a putative binding site for fatty acids. Much significance lies in demonstration with the spin-labelled-ATP studies where competition of ATP to the protein bound to spin-labelled ATP explains about the inhibition effect of nucleotides on the UCP2. So the present study applies different methods for the functional characterization of UCP2. The studies of natural fatty acids and alkylsulfonates with UCP2 bound to spin-labelled fatty acid, and study of nucleotide inhibition on UCP2 are closely related and give the much awaited answer to the question of functional similarities between UCP1 and UCP2. This supports the discussion of many groups which predict the functional similarity between these two proteins based upon sequence homology. Also many attempts have been reported in literature to explain the physiological functional relevance where by this present study can also be added to as we now suppose from the present conclusions of our experiments.

Tropical geometry is a rather new field of algebraic geometry. The main idea is to replace algebraic varieties by certain piece-wise linear objects in R^n, which can be studied with the aid of combinatorics. There is hope that many algebraically difficult operations become easier in the tropical setting, as the structure of the objects seems to be simpler. In particular, tropical geometry shows promise for application in enumerative geometry. Enumerative geometry deals with the counting of geometric objects that are determined by certain incidence conditions. Until around 1990, not many enumerative questions had been answered and there was not much prospect of solving more. But then Kontsevich introduced the moduli space of stable maps which turned out to be a very useful concept for the study of enumerative geometry. A well-known problem of enumerative geometry is to determine the numbers N_cplx(d,g) of complex genus g plane curves of degree d passing through 3d+g-1 points in general position. Mikhalkin has defined the analogous number N_trop(d,g) for tropical curves and shown that these two numbers coincide (Mikhalkin's Correspondence Theorem). Tropical geometry supplies many new ideas and concepts that could be helpful to answer enumerative problems. However, as a rather new field, tropical geometry has to be studied more thoroughly. This thesis is concerned with the ``translation'' of well-known facts of enumerative geometry to tropical geometry. More precisely, the main results of this thesis are: - a tropical proof of the invariance of N_trop(d,g) of the position of the 3d+g-1 points, - a tropical proof for Kontsevich's recursive formula to compute N_trop(d,0) and - a tropical proof of Caporaso's and Harris' algorithm to compute N_trop(d,g). All results were derived in joint work with my advisor Andreas Gathmann. (Note that tropical research is not restricted to the translation of classically well-known facts, there are actually new results shown by means of tropical geometry that have not been known before. For example, Mikhalkin gave a tropical algorithm to compute the Welschinger invariant for real curves. This shows that tropical geometry can indeed be a tool for a better understanding of classical geometry.)

The study provides insights into the dynamic processes of vascular epiphyte vegetation in two host tree species of lowland forest in Panama. Further, a novel approach is presented to examine the possible role of host tree identity in the structuring of vascular epiphyte communities: For three locally common host tree species (Socratea exorrhiza, Marila laxiflora, Perebea xanthochyma) we created null models of the expected epiphyte assemblages assuming that epiphyte colonization reflected random distribution of epiphytes in the forest. In all three tree species, abundances of the majority of epiphyte species (69 – 81 %) were indistinguishable from random, while the remaining species were about equally over- or underrepresented compared to their occurrence in the entire forest plot. Permutations based on the number of colonized trees (reflecting observed spatial patchiness) yielded similar results. Finally, a Canonical Correspondence Analysis also confirmed host-specific differences in epiphyte assemblages. In spite of pronounced preferences of some epiphytes for particular host trees, no epiphyte species was restricted to a single host. We conclude that the epiphytes on a given tree species are not simply a random sample of the local species pool, but there are no indications of host specificity either. To determine the qualitative and quantitative long-term changes in the vascular epiphyte assemblage of the host tree Socratea exorrhiza, in the lowland forest of the San Lorenzo Crane Plot, we followed the fate of the vascular epiphyte assemblage on 99 individuals of this palm species, in three censuses over the course of five years. The composition of the epiphyte assemblage changed little during the course of the study. While the similarity of epiphyte vegetation decreased on single palm individuals through time, the similarity analyzed over all palms increased. Even well-established epiphyte individuals experienced high mortality with only 46 % of the originally mapped individuals surviving the following five years. We found a positive correlation between host tree size and epiphyte richness and detected higher colonization rates of epiphytes per surface area on larger trees. Epiphyte assemblages on single Socratea exorrhiza trees were highly dynamic while the overall composition of the epiphyte vegetation on the host tree species in the study plot was rather stable. We suggest that higher recruitment rates due to localized seed dispersal by already established epiphytes on larger palms promote the colonization of epiphytes on larger palms. Given the known growth rates and mortality rates of the host tree species, the maximum time available for colonization and reproduction of epiphytes on a given Socratea exorrhiza tree is estimated to be about 60 years. Changes in the epiphyte vegetation of c. 1000 individuals of the host tree species Annona glabra at Barro Colorado Island over the course of eight year were documented by means of repeated censuses. Considerable increase in the abundance of the dominating epiphyte species and ongoing colonization of the host tree species suggests that the epiphyte vegetation has not reached a steady state in the maximal 80 years since the establishment of the host tree. Epiphyte species composition as a whole was rather stable. We disentangled the relationship between epiphyte colonization and tree size/available time for colonization with the finding that tree size explained only a low proportion of colonization while other factors like connectivity to dispersal source and time explain may explain a larger part. Epiphyte populations are patchily distributed and examined species exhibit properties of a metapopulation with asynchronous local population growth, high local population turnover, a positive relationship between regional occurrence and patch population size, and negatively correlated relationship between extinction and patch occupancy. The documented metapopulation processes highlight the importance of not colonized suitable habitat for the conservation of epiphytes.

The validity of formulas w.r.t. a specification over first-order logic with a semantics based on all models is semi-decidable. Therefore, we may implement a proof procedure which finds a proof for every valid formula fully automatically. But this semantics often lacks intuition: Some pathological models such as the trivial model may produce unexpected results w.r.t. validity. Instead, we may consider just a class of special models, for instance, the class of all data models. Proofs are then performed using induction. But, inductive validity is not semi-decidable -- even for first-order logic. This theoretical drawback manifests itself in practical limitations: There are theorems that cannot be proved by induction directly but only generalizations can be proved. For their definition, we may have to extend the specification. Therefore, we cannot expect to prove interesting theorems fully automatically. Instead, we have to support user-interaction in a suitable way. In this thesis, we aim at developing techniques that enhance automatic proof control of (inductive) theorem provers and that enable user-interaction in a suitable way. We integrate our new proof techniques into the inductive theorem prover QuodLibet and validate them with various case studies. Essentially, we introduce the following three proof techniques: -We integrate a decision procedure for linear arithmetic into QuodLibet in a close way by defining new inference rules that perform the elementary steps of the decision procedure. This allows us to implement well-known heuristics for automatic proof control. Furthermore, we are able to provide special purpose tactics that support the manual speculation of lemmas if a proof attempt gets stuck. The integration improves the ability of the theorem prover to prove theorems automatically as well as its efficiency. Our approach is competitive with other approaches regarding efficiency; it provides advantages regarding the speculation of lemmas. -The automatic proof control searches for a proof by applying inference rules. The search space is not only infinite, but grows dramatically with the depth of the search. In contrast to this, checking and analyzing performed proofs is very efficient. As the search space also has a high redundancy, it is reasonable to reuse subproofs found during proof search. We define new notions for the contribution of proof steps to a proof. These notions enable the derivation of pruned proofs and the identification of superfluous subformulas in theorems. A proof may be reused in two ways: upward propagation prunes a proof by eliminating superfluous proof steps; sideward reuse closes an open proof obligation by replaying an already found proof. -For interactive theorem provers, it is essential not only to prove automatically as many lemmas as possible but also to restrict proof search in such a way that the proof process stops within a reasonable amount of time. We introduce different markings in the goals to be proved and the lemmas to be applied to restrict proof search in a flexible way: With a forbidden marking, we can simulate well-known approaches for applying conditional lemmas. A mandatory marking provides a new heuristics which is inspired by local contribution of proof steps. With obligatory and generous markings, we can fine-tune the degree of efficiency and extent of proof search manually. With an elaborate case study, we show the benefits of the different techniques, in particular the synergetic effect of their combination.

The primary object of this work is the development of a robust, accurate and efficient time integrator for the dynamics of flexible multibody systems. Particularly a unified framework for the computational dynamics of multibody systems consisting of mass points, rigid bodies and flexible beams forming open kinematic chains or closed loop systems is developed. In addition, it aims at the presentation of (i) a focused survey of the Lagrangian and Hamiltonian formalism for dynamics, (ii) five different methods to enforce constraints with their respective relations, and (iii) three alternative ways for the temporal discretisation of the evolution equations. The relations between the different methods for the constraint enforcement in conjunction with one specific energy-momentum conserving temporal discretisation method are proved and their numerical performances are compared by means of theoretical considerations as well as with the help of numerical examples.

With the burgeoning computing power available, multiscale modelling and simulation has these days become increasingly capable of capturing the details of physical processes on different scales. The mechanical behavior of solids is oftentimes the result of interaction between multiple spatial and temporal scales at different levels and hence it is a typical phenomena of interest exhibiting multiscale characteristic. At the most basic level, properties of solids can be attributed to atomic interactions and crystal structure that can be described on nano scale. Mechanical properties at the macro scale are modeled using continuum mechanics for which we mention stresses and strains. Continuum models, however they offer an efficient way of studying material properties they are not accurate enough and lack microstructural information behind the microscopic mechanics that cause the material to behave in a way it does. Atomistic models are concerned with phenomenon at the level of lattice thereby allowing investigation of detailed crystalline and defect structures, and yet the length scales of interest are inevitably far beyond the reach of full atomistic computation and is rohibitively expensive. This makes it necessary the need for multiscale models. The bottom line and a possible avenue to this end is, coupling different length scales, the continuum and the atomistics in accordance with standard procedures. This is done by recourse to the Cauchy-Born rule and in so doing, we aim at a model that is efficient and reasonably accurate in mimicking physical behaviors observed in nature or laboratory. In this work, we focus on concurrent coupling based on energetic formulations that links the continuum to atomistics. At the atomic scale, we describe deformation of the solid by the displaced positions of atoms that make up the solid and at the continuum level deformation of the solid is described by the displacement field that minimize the total energy. In the coupled model, continuum-atomistic, a continuum formulation is retained as the overall framework of the problem and the atomistic feature is introduced by way of constitutive description, with the Cauchy-Born rule establishing the point of contact. The entire formulation is made in the framework of nonlinear elasticity and all the simulations are carried out within the confines of quasistatic settings. The model gives direct account to measurable features of microstructures developed by crystals through sequential lamination.

Nowadays piezoelectric and ferroelectric materials are becoming more and more an interesting part of smart materials in scientific and engineering applications. Precision machining in manufacturing, micropositioning in metrology, common rail systems with piezo fuel injection control in automobile industry, and ferroelectric random access memories (FRAM) in microelectromechanical systems (MEMS) besides commercial piezo actuators and sensors can be very good examples for the application of piezoceramic and ferroelectric materials. In spite of having good characteristics, piezoelectric and ferroelectric materials have significant nonlinearities, which limit the applications in high performance usage. Domain switching (ferroelastic or ferroelectric) is the main reason for the nonlinearity of ferroelectric materials. External excessive electromechanical loads (mechanical stress and electric field) are driving forces for domain switching. In literature, various important experiments related to the non-linear properties of piezoelectric and ferroelectric materials are reported. Simulations of nonlinear properties of piezoelectric and ferroelectric materials based on physical insights of the material have been performed during the last two decades by using micromechanical and phenomenological models. The most significant experiments and models are deeply discussed in the literature survey. In this thesis the nonlinear behaviour of tetragonal perovskite type piezoceramic materials is simulated theoretically using two and three dimensional micromechanical models which are based on physical insights of the material. In the simulations a bulk piezoceramic material which has numerous grains is considered. Each grain has random orientation in properties of polarization and strain. Randomness of orientations is given by Euler angles equally distributed between \(0\) and \(2\pi\). Each element in the micromechanical model has been assumed to have the same properties of the real piezoelectric grain. In the first part of the simulations, quasi-static characteristics of piezoelectric materials are investigated by applying cyclic, rate independent, bipolar, uni-axial and external electrical loading with an amplitude of 2 kV/mm gradually starting from zero value in virgin state. Moreover, the simulations are undertaken for these materials which are subjected to quasi-static, uni-polar, uni-axial mechanical stress, namely compressive stress. The calculations are performed at each element based on linear constitutive equations, nonlinear domain switching and a probability theory for domain switching. In order to fit the simulations to the experimental data, some parameters such as spontaneous polarization, spontaneous strain, piezoelectric and dielectric constants are chosen from literature. The domain switching of each grain is determined by an electromechanical energy criterion. Depending on the actual energy related to a critical energy a certain probability is introduced for domain switching of the polarization direction. Same energy levels are assumed in the electromechanical energy relation for different types of domain switching like 90º and 180º for perovskite type tetragonal or 70.5º and 109.5º for rhombohedral microstructures. It is assumed that intergranular effects between grains can be modelled by such probability functions phenomenologically. The macroscopic response of the material to the applied electromechanical loading is calculated by using Euler transformations and averaging the individual grains. Properties of piezoelectric materials under fixed mechanical stresses are also investigated by applying constant compressive stress in addition to cyclic electrical loading in the simulations. Compressive stress is applied and kept constant before cyclic bipolar electrical loading is implemented. In the following chapters, a three-dimensional micromechanical model is extended for the simulation of the rate dependent properties of certain perovskite type tetragonal piezoelectric materials. The frequency dependent micromechanical model is now not only based on linear constitutive and nonlinear domain switching but also linear kinetics theories. The material is loaded both electrically and mechanically in separate manner with an alternating electrical voltage and mechanical stress values of various moderate frequencies, which are in the order of 0.01 Hz to 1 Hz. Electromechanical energy equation in combination with a probability function is again used to determine the onset of the domain switching inside the grains. The propagation of the domain wall during the domain switching process in grains is modelled by means of linear kinetics relations after a new domain nucleates. Electric displacement versus electric field hysteresis loops, mechanical strain versus mechanical stress and electric displacement versus mechanical stress for different frequencies and amplitudes of the alternating electric fields and compressive stresses are simulated and presented. A simple micromechanical model without using probabilistic approach is compared with the one that takes it into account. Both models give important insights into the rate dependency of piezoelectric materials, which was observed in some experiments reported in the literature. Intergranular effects are other significant factors for nonlinearities of polycrystalline ferroelectric materials. Even piezoelectric actuators and sensors show nonlinearities when they are operated with electrical loading, which is much lower than the coercive electric field level. Intergranular effects are the main cause of such small hysteresis loops. In the corresponding chapter, two basic field effects which are electrical and mechanical are taken into account for the consideration of intergranular effects micromechanically in the simulations of the two dimensional model. Therefore, a new electromechanical energy equation for the threshold of domain switching is introduced to explain nonlinearities stemming from both domain switching and intergranular effects. The material parameters like coercive electric field and critical spontaneous polarization or strain quantities are not implemented in the electromechanical energy relation. But, this relation contains new parameters which consider both mechanical and electrical field characteristics of neighbouring elements. By using this new model, mechanical strain versus electric field butterfly curves under small electrical loading conditions are also simulated. Hence, a rate dependent concept is applied in butterfly curves by means of linear kinetics model. As a result, the simulations have better matching with corresponding experiments in literature. In the next step, the model can be extended in three dimensional case and the parameters of electromechanical energy relation can be improved in order to get better simulations of nonlinear properties of polycrystalline piezoelectric materials.