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The thesis discusses discrete-time dynamic flows over a finite time horizon T. These flows take time, called travel time, to pass an arc of the network. Travel times, as well as other network attributes, such as, costs, arc and node capacities, and supply at the source node, can be constant or time-dependent. Here we review results on discrete-time dynamic flow problems (DTDNFP) with constant attributes and develop new algorithms to solve several DTDNFPs with time-dependent attributes. Several dynamic network flow problems are discussed: maximum dynamic flow, earliest arrival flow, and quickest flow problems. We generalize the hybrid capacity scaling and shortest augmenting path algorithmic of the static network flow problem to consider the time dependency of the network attributes. The result is used to solve the maximum dynamic flow problem with time-dependent travel times and capacities. We also develop a new algorithm to solve earliest arrival flow problems with the same assumptions on the network attributes. The possibility to wait (or park) at a node before departing on outgoing arc is also taken into account. We prove that the complexity of new algorithm is reduced when infinite waiting is considered. We also report the computational analysis of this algorithm. The results are then used to solve quickest flow problems. Additionally, we discuss time-dependent bicriteria shortest path problems. Here we generalize the classical shortest path problems in two ways. We consider two - in general contradicting - objective functions and introduce a time dependency of the cost which is caused by a travel time on each arc. These problems have several interesting practical applications, but have not attained much attention in the literature. Here we develop two new algorithms in which one of them requires weaker assumptions as in previous research on the subject. Numerical tests show the superiority of the new algorithms. We then apply dynamic network flow models and their associated solution algorithms to determine lower bounds of the evacuation time, evacuation routes, and maximum capacities of inhabited areas with respect to safety requirements. As a macroscopic approach, our dynamic network flow models are mainly used to produce good lower bounds for the evacuation time and do not consider any individual behavior during the emergency situation. These bounds can be used to analyze existing buildings or help in the design phase of planning a building.

We construct and study two surface measures on the space C([0,1],M) of paths in a compact Riemannian manifold M embedded into the Euclidean space R^n. The first one is induced by conditioning the usual Wiener measure on C([0,T],R^n) to the event that the Brownian particle does not leave the tubular epsilon-neighborhood of M up to time T, and passing to the limit. The second one is defined as the limit of the laws of reflected Brownian motions with reflection on the boundaries of the tubular epsilon-neighborhoods of M. We prove that the both surface measures exist and compare them with the Wiener measure W_M on C([0,T],M). We show that the first one is equivalent to W_M and compute the corresponding density explicitly in terms of the scalar curvature and the mean curvature vector of M. Further, we show that the second surface measure coincides with W_M. Finally, we study the limit behavior of the both surface measures as T tends to infinity.

The central theme in this thesis concerns the development of enhanced methods and algorithms for appraising market and credit risks and their application within the context of standard and more advanced market models. Generally, methods and algorithms for analysing market risk of complex portfolios involve detailed knowledge of option sensitivities, the so-called "Greeks". Based on an analysis of symmetries in financial market models, relations between option sensitivities are obtained, which can be used for the efficient valuation of the Greeks. Mainly, the relations are derived within the Black Scholes model, however, some relations are also valid for more general models, for instance the Heston model. Portfolios are usually influenced by lots of underlyings, so it is necessary to characterise the dependencies of these basic instruments. It is usual to describe such dependencies by correlation matrices. However, estimations of correlation matrices in practice are disturbed by statistical noise and usually have the problem of rank deficiency due to missing data. A fast algorithm is presented which performs a generalized Cholesky decomposition of a perturbed correlation matrix. In contrast to the standard Cholesky algorithm, an advantage of the generalized method is that it works for semi-positive, rank deficient matrices as well. Moreover, it gives an approximative decomposition when the input matrix is indefinite. A comparison with known algorithms with similar features is performed and it turns out, that the new algorithm can be recommended in situations where computation time is the critical issue. The determination of a profit and loss distribution by Fourier inversion of its characteristic function is a powerful tool, but it can break down when the characteristic function is not integrable. In this thesis, methods for Fourier inversion of non-integrable characteristic functions are studied. In this respect, two theorems are obtained which are based on a suitable approximation of the unknown distribution with known density and characteristic function. Further it will be shown, that straightforward Fast Fourier inversion works, when the according density lives on a bounded interval. The above techniques are of crucial importance to determine the profit and loss distribution (P&L) of large portfolios efficiently. The so-called Delta Gamma normal approach has become industrial standard for the estimation of market risk. It is shown, that the performance of the Delta Gamma normal approach can be improved substantially by application of the developed methods. The same optimization procedure also applies to the Delta Gamma Student model. A standard tool for computing the P&L distribution of a loan portfolio is the CreditRisk+ model. Basically, the CreditRisk+ distribution is a discrete distribution which can be computed from its probability generating function. For this a numerically stable method is presented and as an alternative, a new algorithm based on Fourier inversion is proposed. Finally, an extension of the CreditRisk+ model to market risk is developed, which distribution can be obtained efficiently by the presented Fourier inversion methods as well.

Clusters bridge the gap between single atoms or molecules and the condensed phase and it is the challenge of cluster science to obtain a deeper understanding of the molecular foundation of the observed cluster specific properties/reactivities and their dependence on size. The electronic structure of hydrated magnesium monocations [Mg,nH2O]+, n<20, exhibits a strong cluster size dependency. With increasing number of H2O ligands the SOMO evolves from a quasi-valence state (n=3-5), in which the singly occupied molecular orbital (SOMO) is not yet detached from the metal atom and has distinct sp-hybrid character, to a contact ion pair state. For larger clusters (n=17,19) these ion pair states are best described as solvent separated ion pair states, which are formed by a hydrated dication and a hydrated electron. With growing cluster size the SOMO moves away from the magnesium ion to the cluster surface, where it is localized through mutual attractive interactions between the electron density and dangling H-atoms of H2O ligands forming "molecular tweezers" HO-H (e-) H-OH. In case of the hydrated aluminum monocations [Al,nH2O]+,n=20, different isomers of the formal stoichiometry [Al,20H2O]+ were investigated by using gradient-corrected DFT (BLYP) and three different basic structures for [Al,20H2O]+ were identified: (a) [AlI(H2O)20]+ with a threefold coordinated AlI; (b) [HAlIII(OH)(H2O)19]+ with a fourfold coordinated AlIII; (c) [HAlIII(OH)(H2O)19]+ with a fivefold coordinated AlIII. In ground state [AlI(H2O)20]+ (a) which contains aluminum in oxidation state +1 the 3s2 valence electrons remain located at the aluminium monocation. Different than for open shell magnesium monocations no electron transfer into the hydration shell is observed for closed shell AlI. However, clusters of type (a) are high energy isomers (DE»+190 kJ mol-1) and the activation barrier for reaction into cluster type (b) or (c) is only approximately 14 kJ mol-1. The performed ab initio calculations reveal that unlike in [Mg,nH2O]+, n=7-17, for which H atom eliminiation is found to be the result of an intracluster redoxreaction, in [Al,nH2O]+,n=20, H2 is formed in an intracluster acid-base reaction. In [Mg,nH2O]+, n>17, the magnesium dication was found to coexist with a hydrated electron in larger cluster sizes. This proves that intermolecular electron delocalization - previously almost exclusively studied in (H2O)n- and (NH3)n- clusters - can also be an important issue for water clusters doped with an open shell metal cation or a metal anion. Structures and stabilities of hydrated magnesium water cluster anions with the formal stoichiometry [Mg,nH2O]-, n=1-11, were investigated by application of various correlated ab initio methods (MP2, CCSD, CCSD(T)). Metal cations surely have high relevance in numerous biological processes, and as most biological processes take place in aqueous solution hydrated metal ions will be involved. However, in biological systems solvent molecules (i.e. water) compete with different solvated chelate ligands for coordination sites at the metal ion and the solvent and chelate ligands are in mutual interactions with each other and the metal ion. These interactions were investigated for the hydration of ZnII/carnosine complexes by application of FT-ICR-MS, gas-phase H/D exchange experiments and supporting ab initio calculations. In the last chapter of this work the Free Electron Laser IR Multi Photon Dissocition (FEL-IR-MPD) spectra of mass selected cationic niobium acetonitrile complexes with the formal stoichiometry [Nb,nCH3CN]+, n=4-5, in the spectral range 780 – 2500 cm-1 are reported. In case of n=4 the recorded vibrational bands are close to those of the free CH3CN molecule and the experimental spectra do not contain any evident indication of a potential reaction beyond complex formation. By comparison with B3LYP calculated IR absorption spectra the recorded spectra are assigned to high spin (quintet, S=2), planar [NbI(NCCH3)4]+. In [Nb,nCH3CN]+, n=5, new vibrational bands shifted away from those of the acetonitrile monomer are observed between 1300 – 1550 cm-1. These bands are evidence of a chemical modification due to an intramolecular reaction. Screening on the basis of B3LYP calculated IR absorption spectra allow for an assignment of the recorded spectra to the metallacyclic species [NbIII(NCCH3)3(N=C(CH3)C(CH3)=N)]+ (triplet, S=1), which has formed in a internal reductive nitrile coupling reaction from [NbI(NCCH3)5]+. Calculated reaction coordinates explain the experimentally observed differences in reactivity between ground state [NbI(NCCH3)4]+ and [NbI(NCCH3)5]+. The reductive nitrile coupling reaction is exothermic and accessible (Ea=49 kJ mol-1) only in [NbI(NCCH3)5]+, whereas in [NbI(NCCH3)4]+ the reaction is found to be endothermic and retarded by significantly higher activation barriers (Ea>116 kJ mol-1).

In this thesis the combinatorial framework of toric geometry is extended to equivariant sheaves over toric varieties. The central questions are how to extract combinatorial information from the so developed description and whether equivariant sheaves can, like toric varieties, be considered as purely combinatorial objects. The thesis consists of three main parts. In the first part, by systematically extending the framework of toric geometry, a formalism is developed for describing equivariant sheaves by certain configurations of vector spaces. In the second part, homological properties of a certain class of equivariant sheaves are investigated, namely that of reflexive equivariant sheaves. Several kinds of resolutions for these sheaves are constructed which depend only on the configuration of their associated vector spaces. Thus a partially positive answer to the question of combinatorial representability is given. As a particular result, a new way for computing minimal resolutions for Z^n - graded modules over polynomial rings is obtained. In the third part a complete classification of the simplest nontrivial sheaves, equivariant vector bundles of rank two over smooth toric surfaces, is given. A combinatorial characterization is given and parameter spaces (moduli spaces) are constructed which depend only on this characterization. In appendices a outlook on equivariant sheaves and the relation of Chern classes to their combinatorial classification is given, particularly focussing on the case of the projective plane. A classification of equivariant vector bundles of rank three over the projective plane is given.

Semiparametric estimation of conditional quantiles for time series, with applications in finance
(2003)

The estimation of conditional quantiles has become an increasingly important issue in insurance and financial risk management. The stylized facts of financial time series data has rendered direct applications of extreme value theory methodologies, in the estimation of extreme conditional quantiles, inappropriate. On the other hand, quantile regression based procedures work well in nonextreme parts of a given data but breaks down in extreme probability levels. In order to solve this problem, we combine nonparametric regressions for time series and extreme value theory approaches in the estimation of extreme conditional quantiles for financial time series. To do so, a class of time series models that is similar to nonparametric AR-(G)ARCH models but which does not depend on distributional and moments assumptions, is introduced. We discuss estimation procedures for the nonextreme levels using the models and consider the estimates obtained by inverting conditional distribution estimators and by direct estimation using Koenker-Basset (1978) version for kernels. Under some regularity conditions, the asymptotic normality and uniform convergence, with rates, of the conditional quantile estimator for strong mixing time series, are established. We study the estimation of scale function in the introduced models using similar procedures and show that under some regularity conditions, the scale estimate is weakly consistent and asymptotically normal. The application of introduced models in the estimation of extreme conditional quantiles is achieved by augmenting them with methods in extreme value theory. It is shown that the overal extreme conditional quantiles estimator is consistent. A Monte Carlo study is carried out to illustrate the good performance of the estimates and real data are used to demonstrate the estimation of Value-at-Risk and conditional expected shortfall in financial risk management and their multiperiod predictions discussed.

The thesis is concerned with the modelling of ionospheric current systems and induced magnetic fields in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered operator equation. First of all a general multiscale concept for vectorial operator equations between two separable Hilbert spaces is developed in terms of vector kernel functions. The equivalence to the canonical tensorial ansatz is proven and the theory is transferred to the case of multiscale regularization of vectorial inverse problems. As a first application, a special multiresolution analysis of the space of square-integrable vector fields on the sphere, e.g. the Earth’s magnetic field measured on a spherical satellite’s orbit, is presented. By this, a multiscale separation of spherical vector-valued functions with respect to their sources can be established. The vector field is split up into a part induced by sources inside the sphere, a part which is due to sources outside the sphere and a part which is generated by sources on the sphere, i.e. currents crossing the sphere. The multiscale technqiue is tested on a magnetic field data set of the satellite CHAMP and it is shown that crustal field determination can be improved by previously applying our method. In order to reconstruct ionspheric current systems from magnetic field data, an inversion of the Biot-Savart’s law in terms of multiscale regularization is defined. The corresponding operator is formulated and the singular values are calculated. Based on the konwledge of the singular system a regularzation technique in terms of certain product kernels and correponding convolutions can be formed. The method is tested on different simulations and on real magnetic field data of the satellite CHAMP and the proposed satellite mission SWARM.

The goal of this thesis is a physically motivated and thermodynamically consistent formulation of higher gradient inelastic material behavior. Thereby, the influence of the material microstructure is incorporated. Next to theoretical aspects, the thesis is complemented with the algorithmic treatment and numerical implementation of the derived model. Hereby, two major inelastic effects will be addressed: on the one hand elasto-plastic processes and on the other hand damage mechanisms, which will both be modeled within a continuum mechanics framework.

The present thesis deals with coupled steady state laminar flows of isothermal incompressible viscous Newtonian fluids in plain and in porous media. The flow in the pure fluid region is usually described by the (Navier-)Stokes system of equations. The most popular models for the flow in the porous media are those suggested by Darcy and by Brinkman. Interface conditions, proposed in the mathematical literature for coupling Darcy and Navier-Stokes equations, are shortly reviewed in the thesis. The coupling of Navier-Stokes and Brinkman equations in the literature is based on the so called continuous stress tensor interface conditions. One of the main tasks of this thesis is to investigate another type of interface conditions, namely, the recently suggested stress tensor jump interface conditions. The mathematical models based on these interface conditions were not carefully investigated from the mathematical point of view, and also their validity was a subject of discussions. The considerations within this thesis are a step toward better understanding of these interface conditions. Several aspects of the numerical simulations of such coupled flows are considered: -the choice of proper interface conditions between the plain and porous media -analysis of the well-posedness of the arising systems of partial differential equations; -developing numerical algorithm for the stress tensor jump interface conditions, coupling Navier-Stokes equations in the pure liquid media with the Navier-Stokes-Brinkman equations in the porous media; -validation of the macroscale mathematical models on the base of a comparison with the results from a direct numerical simulation of model representative problems, allowing for grid resolution of the pore level geometry; -developing software and performing numerical simulation of 3-D industrial flows, namely of oil flows through car filters.

We present new algorithms and provide an overall framework for the interaction of the classically separate steps of logic synthesis and physical layout in the design of VLSI circuits. Due to the continuous development of smaller sized fabrication processes and the subsequent domination of interconnect delays, the traditional separation of logical and physical design results in increasingly inaccurate cost functions and aggravates the design closure problem. Consequently, the interaction of physical and logical domains has become one of the greatest challenges in the design of VLSI circuits. To address this challenge, we propose different solutions for the control and datapath logic of a design, and show how to combine them to reach design closure.