### Refine

#### Year of publication

#### Document Type

- Doctoral Thesis (218) (remove)

#### Language

- English (218) (remove)

#### Keywords

#### Faculty / Organisational entity

- Fachbereich Mathematik (218) (remove)

This thesis generalizes the Cohen-Lenstra heuristic for the class groups of real quadratic
number fields to higher class groups. A "good part" of the second class group is defined.
In general this is a non abelian proper factor group of the second class group. Properties
of those groups are described, a probability distribution on the set of those groups is in-
troduced and proposed as generalization of the Cohen-Lenstra heuristic for real quadratic
number fields. The calculation of number field tables which contain information about
higher class groups is explained and the tables are compared to the heuristic. The agree-
ment is close. A program which can create an internet database for number field tables is
presented.

The various uses of fiber-reinforced composites, for example in the enclosures of planes, boats and cars, generates the demand for a detailed analysis of these materials. The final goal is to optimize fibrous materials by the means of “virtual material design”. New fibrous materials are virtually created as realizations of a stochastic model and evaluated with physical simulations. In that way, materials can be optimized for specific use cases, without constructing expensive prototypes or performing mechanical experiments. In order to design a practically fabricable material, the stochastic model is first adapted to an existing material and then slightly modified. The virtual reconstruction of the existing material requires a precise knowledge of the geometry of its microstructure. The first part of this thesis describes a fiber quantification method by the means of local measurements of the fiber radius and orientation. The combination of a sparse chord length transform and inertia moments leads to an efficient and precise new algorithm. It outperforms existing approaches with the possibility to treat different fiber radii within one sample, with high precision in continuous space and comparably fast computing time. This local quantification method can be directly applied on gray value images by adapting the directional distance transforms on gray values. In this work, several approaches of this kind are developed and evaluated. Further characterization of the fiber system requires a segmentation of each single fiber. Using basic morphological operators with specific structuring elements, it is possible to derive a probability for each pixel describing if the pixel belongs to a fiber core in a region without overlapping fibers. Tracking high probabilities leads to a partly reconstruction of the fiber cores in non crossing regions. These core parts are then reconnected over critical regions, if they fulfill certain conditions ensuring the affiliation to the same fiber. In the second part of this work, we develop a new stochastic model for dense systems of non overlapping fibers with a controllable level of bending. Existing approaches in the literature have at least one weakness in either achieving high volume fractions, producing non overlapping fibers, or controlling the bending or the orientation distribution. This gap can be bridged by our stochastic model, which operates in two steps. Firstly, a random walk with the multivariate von Mises-Fisher orientation distribution defines bent fibers. Secondly, a force-biased packing approach arranges them in a non overlapping configuration. Furthermore, we provide the estimation of all parameters needed for the fitting of this model to a real microstructure. Finally, we simulate the macroscopic behavior of different microstructures to derive their mechanical and thermal properties. This part is mostly supported by existing software and serves as a summary of physical simulation applied to random fiber systems. The application on a glass fiber reinforced polymer proves the quality of the reconstruction by our stochastic model, as the effective properties match for both the real microstructure and the realizations of the fitted model. This thesis includes all steps to successfully perform virtual material design on various data sets. With novel and efficient algorithms it contributes to the science of analysis and modeling of fiber reinforced materials.

Numerical Algorithms in Algebraic Geometry with Implementation in Computer Algebra System SINGULAR
(2011)

Polynomial systems arise in many applications: robotics, kinematics, chemical kinetics,
computer vision, truss design, geometric modeling, and many others. Many polynomial
systems have solutions sets, called algebraic varieties, having several irreducible
components. A fundamental problem of the numerical algebraic geometry is to decompose
such an algebraic variety into its irreducible components. The witness point sets are
the natural numerical data structure to encode irreducible algebraic varieties.
Sommese, Verschelde and Wampler represented the irreducible algebraic decomposition of
an affine algebraic variety \(X\) as a union of finite disjoint sets \(\cup_{i=0}^{d}W_i=\cup_{i=0}^{d}\left(\cup_{j=1}^{d_i}W_{ij}\right)\) called numerical irreducible decomposition. The \(W_i\) correspond to the pure i-dimensional components, and the \(W_{ij}\) represent the i-dimensional irreducible components. The numerical irreducible decomposition is implemented in BERTINI.
We modify this concept using partially Gröbner bases, triangular sets, local dimension, and
the so-called zero sum relation. We present in the second chapter the corresponding
algorithms and their implementations in SINGULAR. We give some examples and timings,
which show that the modified algorithms are more efficient if the number of variables is not
too large. For a large number of variables BERTINI is more efficient.
Leykin presented an algorithm to compute the embedded components of an algebraic variety
based on the concept of the deflation of an algebraic variety.
Depending on the modified algorithm mentioned above, we will present in the third chapter an
algorithm and its implementation in SINGULAR to compute the embedded components.
The irreducible decomposition of algebraic varieties allows us to formulate in the fourth
chapter some numerical algebraic algorithms.
In the last chapter we present two SINGULAR libraries. The first library is used to compute
the numerical irreducible decomposition and the embedded components of an algebraic variety.
The second library contains the procedures of the algorithms in the last Chapter to test
inclusion, equality of two algebraic varieties, to compute the degree of a pure i-dimensional
component, and the local dimension.

For computational reasons, the spline interpolation of the Earth's gravitational potential is usually done in a spherical framework. In this work, however, we investigate a spline method with respect to the real Earth. We are concerned with developing the real Earth oriented strategies and methods for the Earth's gravitational potential determination. For this purpose we introduce the reproducing kernel Hilbert space of Newton potentials on and outside given regular surface with reproducing kernel defined as a Newton integral over it's interior. We first give an overview of thus far achieved results considering approximations on regular surfaces using surface potentials (Chapter 3). The main results are contained in the fourth chapter where we give a closer look to the Earth's gravitational potential, the Newton potentials and their characterization in the interior and the exterior space of the Earth. We also present the L2-decomposition for regions in R3 in terms of distributions, as a main strategy to impose the Hilbert space structure on the space of potentials on and outside a given regular surface. The properties of the Newton potential operator are investigated in relation to the closed subspace of harmonic density functions. After these preparations, in the fifth chapter we are able to construct the reproducing kernel Hilbert space of Newton potentials on and outside a regular surface. The spline formulation for the solution to interpolation problems, corresponding to a set of bounded linear functionals is given, and corresponding convergence theorems are proven. The spline formulation reflects the specifics of the Earth's surface, due to the representation of the reproducing kernel (of the solution space) as a Newton integral over the inner space of the Earth. Moreover, the approximating potential functions have the same domain of harmonicity as the actual Earth's gravitational potential, i.e., they are harmonic outside and continuous on the Earth's surface. This is a step forward in comparison to the spherical harmonic spline formulation involving functions harmonic down to the Runge sphere. The sixth chapter deals with the representation of the used kernel in the spherical case. It turns out that in the case of the spherical Earth, this kernel can be considered a kind of generalization to spherically oriented kernels, such as Abel-Poisson or the singularity kernel. We also investigate the existence of the closed expression of the kernel. However, at this point it remains to be unknown to us. So, in Chapter 7, we are led to consider certain discretization methods for integrals over regions in R3, in connection to theory of the multidimensional Euler summation formula for the Laplace operator. We discretize the Newton integral over the real Earth (representing the spline function) and give a priori estimates for approximate integration when using this discretization method. The last chapter summarizes our results and gives some directions for the future research.

In the first part of the thesis we develop the theory of standard bases in free modules over (localized) polynomial rings. Given that linear equations are solvable in the coefficients of the polynomials, we introduce an algorithm to compute standard bases with respect to arbitrary (module) monomial orderings. Moreover, we take special care to principal ideal rings, allowing zero divisors. For these rings we design modified algorithms which are new and much faster than the general ones. These algorithms were motivated by current limitations in formal verification of microelectronic System-on-Chip designs. We show that our novel approach using computational algebra is able to overcome these limitations in important classes of applications coming from industrial challenges.
The second part is based on research in collaboration with Jason Morton, Bernd Sturmfels and Anne Shiu. We devise a general method to describe and compute a certain class of rank tests motivated by statistics. The class of rank tests may loosely be described as being based on computing the number of linear extensions to given partial orders. In order to apply these tests to actual data we developed two algorithms and used our implementations to apply the methodology to gene expression data created at the Stowers Institute for Medical Research. The dataset is concerned with the development of the vertebra. Our rankings proved valuable to the biologists.

Mrázek et al. [14] proposed a unified approach to curve estimation which combines
localization and regularization. In this thesis we will use their approach to study
some asymptotic properties of local smoothers with regularization. In Particular, we
shall discuss the regularized local least squares (RLLS) estimate with correlated errors
(more precisely with stationary time series errors), and then based on this approach
we will discuss the case when the kernel function is dirac function and compare our
smoother with the spline smoother. Finally, we will do some simulation study.

The interest of the exploration of new hydrocarbon fields as well as deep geothermal reservoirs is permanently growing. The analysis of seismic data specific for such exploration projects is very complex and requires the deep knowledge in geology, geophysics, petrology, etc from interpreters, as well as the ability of advanced tools that are able to recover some particular properties. There again the existing wavelet techniques have a huge success in signal processing, data compression, noise reduction, etc. They enable to break complicate functions into many simple pieces at different scales and positions that makes detection and interpretation of local events significantly easier.
In this thesis mathematical methods and tools are presented which are applicable to the seismic data postprocessing in regions with non-smooth boundaries. We provide wavelet techniques that relate to the solutions of the Helmholtz equation. As application we are interested in seismic data analysis. A similar idea to construct wavelet functions from the limit and jump relations of the layer potentials was first suggested by Freeden and his Geomathematics Group.
The particular difficulty in such approaches is the formulation of limit and
jump relations for surfaces used in seismic data processing, i.e., non-smooth
surfaces in various topologies (for example, uniform and
quadratic). The essential idea is to replace the concept of parallel surfaces known for a smooth regular surface by certain appropriate substitutes for non-smooth surfaces.
By using the jump and limit relations formulated for regular surfaces, Helmholtz wavelets can be introduced that recursively approximate functions on surfaces with edges and corners. The exceptional point is that the construction of wavelets allows the efficient implementation in form of
a tree algorithm for the fast numerical computation of functions on the boundary.
In order to demonstrate the
applicability of the Helmholtz FWT, we study a seismic image obtained by the reverse time migration which is based on a finite-difference implementation. In fact, regarding the requirements of such migration algorithms in filtering and denoising the wavelet decomposition is successfully applied to this image for the attenuation of low-frequency
artifacts and noise. Essential feature is the space localization property of
Helmholtz wavelets which numerically enables to discuss the velocity field in
pointwise dependence. Moreover, the multiscale analysis leads us to reveal additional geological information from optical features.

This thesis is devoted to constructive module theory of polynomial
graded commutative algebras over a field.
It treats the theory of Groebner bases (GB), standard bases (SB) and syzygies as well as algorithms
and their implementations.
Graded commutative algebras naturally unify exterior and commutative polynomial algebras.
They are graded non-commutative, associative unital algebras over fields and may contain zero-divisors.
In this thesis
we try to make the most use out of _a priori_ knowledge about
their characteristic (super-commutative) structure
in developing direct symbolic methods, algorithms and implementations,
which are intrinsic to graded commutative algebras and practically efficient.
For our symbolic treatment we represent them as polynomial algebras
and redefine the product rule in order to allow super-commutative structures
and, in particular, to allow zero-divisors.
Using this representation we give a nice characterization
of a GB and an algorithm for its computation.
We can also tackle central localizations of graded commutative algebras by allowing commutative variables to be _local_,
generalizing Mora algorithm (in a similar fashion as G.M.Greuel and G.Pfister by allowing local or mixed monomial orderings)
and working with SBs.
In this general setting we prove a generalized Buchberger's criterion,
which shows that syzygies of leading terms play the utmost important role
in SB and syzygy module computations.
Furthermore, we develop a variation of the La Scala-Stillman free resolution algorithm,
which we can formulate particularly close to our implementation.
On the implementation side
we have further developed the Singular non-commutative subsystem Plural
in order to allow polynomial arithmetic
and more involved non-commutative basic Computer Algebra computations (e.g. S-polynomial, GB)
to be easily implementable for specific algebras.
At the moment graded commutative algebra-related algorithms
are implemented in this framework.
Benchmarks show that our new algorithms and implementation are practically efficient.
The developed framework has a lot of applications in various
branches of mathematics and theoretical physics.
They include computation of sheaf cohomology, coordinate-free verification of affine geometry
theorems and computation of cohomology rings of p-groups, which are partially described in this thesis.

We discuss some first steps towards experimental design for neural network regression which, at present, is too complex to treat fully in general. We encounter two difficulties: the nonlinearity of the models together with the high parameter dimension on one hand, and the common misspecification of the models on the other hand.
Regarding the first problem, we restrict our consideration to neural networks with only one and two neurons in the hidden layer and a univariate input variable. We prove some results regarding locally D-optimal designs, and present a numerical study using the concept of maximin optimal designs.
In respect of the second problem, we have a look at the effects of misspecification on optimal experimental designs.

Intersection Theory on Tropical Toric Varieties and Compactifications of Tropical Parameter Spaces
(2011)

We study toric varieties over the tropical semifield. We define tropical cycles inside these toric varieties and extend the stable intersection of tropical cycles in R^n to these toric varieties. In particular, we show that every tropical cycle can be degenerated into a sum of torus-invariant cycles. This allows us to tropicalize algebraic cycles of toric varieties over an algebraically closed field with non-Archimedean valuation. We see that the tropicalization map is a homomorphism on cycles and an isomorphism on cycle classes. Furthermore, we can use projective toric varieties to compactify known tropical varieties and study their combinatorics. We do this for the tropical Grassmannian in the Plücker embedding and compactify the tropical parameter space of rational degree d curves in tropical projective space using Chow quotients of the tropical Grassmannian.

In this work two main approaches for the evaluation of credit derivatives are analyzed: the copula based approach and the Markov Chain based approach. This work gives the opportunity to use the advantages and avoid disadvantages of both approaches. For example, modeling of contagion effects, i.e. modeling dependencies between counterparty defaults, is complicated under the copula approach. One remedy is to use Markov Chain, where it can be done directly. The work consists of five chapters. The first chapter of this work extends the model for the pricing of CDS contracts presented in the paper by Kraft and Steffensen (2007). In the widely used models for CDS pricing it is assumed that only borrower can default. In our model we assume that each of the counterparties involved in the contract may default. Calculated contract prices are compared with those calculated under usual assumptions. All results are summarized in the form of numerical examples and plots. In the second chapter the copula and its main properties are described. The methods of constructing copulas as well as most common copulas families and its properties are introduced. In the third chapter the method of constructing a copula for the existing Markov Chain is introduced. The cases with two and three counterparties are considered. Necessary relations between the transition intensities are derived to directly find some copula functions. The formulae for default dependencies like Spearman's rho and Kendall's tau for defined copulas are derived. Several numerical examples are presented in which the copulas are built for given Markov Chains. The fourth chapter deals with the approximation of copulas if for a given Markov Chain a copula cannot be provided explicitly. The fifth chapter concludes this thesis.

In the classical Merton investment problem of maximizing the expected utility from terminal wealth and intermediate consumption stock prices are independent of the investor who is optimizing his investment strategy. This is reasonable as long as the considered investor is small and thus does not influence the asset prices. However for an investor whose actions may affect the financial market the framework of the classical investment problem turns out to be inappropriate. In this thesis we provide a new approach to the field of large investor models. We study the optimal investment problem of a large investor in a jump-diffusion market which is in one of two states or regimes. The investor’s portfolio proportions as well as his consumption rate affect the intensity of transitions between the different regimes. Thus the investor is ’large’ in the sense that his investment decisions are interpreted by the market as signals: If, for instance, the large investor holds 25% of his wealth in a certain asset then the market may regard this as evidence for the corresponding asset to be priced incorrectly, and a regime shift becomes likely. More specifically, the large investor as modeled here may be the manager of a big mutual fund, a big insurance company or a sovereign wealth fund, or the executive of a company whose stocks are in his own portfolio. Typically, such investors have to disclose their portfolio allocations which impacts on market prices. But even if a large investor does not disclose his portfolio composition as it is the case of several hedge funds then the other market participants may speculate about the investor’s strategy which finally could influence the asset prices. Since the investor’s strategy only impacts on the regime shift intensities the asset prices do not necessarily react instantaneously. Our model is a generalization of the two-states version of the Bäuerle-Rieder model. Hence as the Bäuerle-Rieder model it is suitable for long investment periods during which market conditions could change. The fact that the investor’s influence enters the intensities of the transitions between the two states enables us to solve the investment problem of maximizing the expected utility from terminal wealth and intermediate consumption explicitly. We present the optimal investment strategy for a large investor with CRRA utility for three different kinds of strategy-dependent regime shift intensities – constant, step and affine intensity functions. In each case we derive the large investor’s optimal strategy in explicit form only dependent on the solution of a system of coupled ODEs of which we show that it admits a unique global solution. The thesis is organized as follows. In Section 2 we repeat the classical Merton investment problem of a small investor who does not influence the market. Further the Bäuerle-Rieder investment problem in which the market states follow a Markov chain with constant transition intensities is discussed. Section 3 introduces the aforementioned investment problem of a large investor. Besides the mathematical framework and the HJB-system we present a verification theorem that is necessary to verify the optimality of the solutions to the investment problem that we derive later on. The explicit derivation of the optimal investment strategy for a large investor with power utility is given in Section 4. For three kinds of intensity functions – constant, step and affine – we give the optimal solution and verify that the corresponding ODE-system admits a unique global solution. In case of the strategy-dependent intensity functions we distinguish three particular kinds of this dependency – portfolio-dependency, consumption-dependency and combined portfolio- and consumption-dependency. The corresponding results for an investor having logarithmic utility are shown in Section 5. In the subsequent Section 6 we consider the special case of a market consisting of only two correlated stocks besides the money market account. We analyze the investor’s optimal strategy when only the position in one of those two assets affects the market state whereas the position in the other asset is irrelevant for the regime switches. Various comparisons of the derived investment problems are presented in Section 7. Besides the comparisons of the particular problems with each other we also dwell on the sensitivity of the solution concerning the parameters of the intensity functions. Finally we consider the loss the large investor had to face if he neglected his influence on the market. In Section 8 we conclude the thesis.

A Multi-Phase Flow Model Incorporated with Population Balance Equation in a Meshfree Framework
(2011)

This study deals with the numerical solution of a meshfree coupled model of Computational Fluid Dynamics (CFD) and Population Balance Equation (PBE) for liquid-liquid extraction columns. In modeling the coupled hydrodynamics and mass transfer in liquid extraction columns one encounters multidimensional population balance equation that could not be fully resolved numerically within a reasonable time necessary for steady state or dynamic simulations. For this reason, there is an obvious need for a new liquid extraction model that captures all the essential physical phenomena and still tractable from computational point of view. This thesis discusses a new model which focuses on discretization of the external (spatial) and internal coordinates such that the computational time is drastically reduced. For the internal coordinates, the concept of the multi-primary particle method; as a special case of the Sectional Quadrature Method of Moments (SQMOM) is used to represent the droplet internal properties. This model is capable of conserving the most important integral properties of the distribution; namely: the total number, solute and volume concentrations and reduces the computational time when compared to the classical finite difference methods, which require many grid points to conserve the desired physical quantities. On the other hand, due to the discrete nature of the dispersed phase, a meshfree Lagrangian particle method is used to discretize the spatial domain (extraction column height) using the Finite Pointset Method (FPM). This method avoids the extremely difficult convective term discretization using the classical finite volume methods, which require a lot of grid points to capture the moving fronts propagating along column height.

A main result of this thesis is a conceptual proof of the fact that the weighted number of tropical curves of given degree and genus, which pass through the right number of general points in the plane (resp., which pass through general points in R^r and represent a given point in the moduli space of genus g curves) is independent of the choices of points. Another main result is a new correspondence theorem between plane tropical cycles and plane elliptic algebraic curves.

In this work, we develop a framework for analyzing an executive’s own- company stockholding and work effort preferences. The executive, character- ized by risk aversion and work effectiveness parameters, invests his personal wealth without constraint in the financial market, including the stock of his own company whose value he can directly influence with work effort. The executive’s utility-maximizing personal investment and work effort strategy is derived in closed form for logarithmic and power utility and for exponential utility for the case of zero interest rates. Additionally, a utility indifference rationale is applied to determine his fair compensation. Being unconstrained by performance contracting, the executive’s work effort strategy establishes a base case for theoretical or empirical assessment of the benefits or otherwise of constraining executives with performance contracting. Further, we consider a highly-qualified individual with respect to her choice between two distinct career paths. She can choose between a mid-level management position in a large company and an executive position within a smaller listed company with the possibility to directly affect the company’s share price. She invests in the financial market including the share of the smaller listed company. The utility maximizing strategy from consumption, investment, and work effort is derived in closed form for logarithmic utility and power utility. Conditions for the individual to pursue her career with the smaller listed company are obtained. The participation constraint is formulated in terms of the salary differential between the two positions. The smaller listed company can offer less salary. The salary shortfall is offset by the possibilityto benefit from her work effort by acquiring own-company shares. This givesinsight into aspects of optimal contract design. Our framework is applicable to the pharmaceutical and financial industry, as well as the IT sector.

This thesis deals with the solution of special problems arising in financial engineering or financial mathematics. The main focus lies on commodity indices. Chapter 1 addresses the important issue for the financial engineering practice of developing well-suited models for certain assets (here: commodity indices). Descriptive analysis of the Dow Jones-UBS commodity index compared to the Standard & Poor 500 stock index provides us with first insights of some features of the corresponding distributions. Statistical tests of normality and mean reversion then helps us in setting up a model for commodity indices. Additionally, chapter 1 encompasses a thorough introduction to commodity investment, history of commodities trading and the most important derivatives, namely futures and European options on futures. Chapter 2 proposes a model for commodity indices and derives fair prices for the most important derivatives in the commodity markets. It is a Heston model supplemented with a stochastic convenience yield. The Heston model belongs to the model class of stochastic volatility models and is currently widely used in stock markets. For the application in the commodity markets the stochastic convenience yield is included in the drift of the instantaneous spot return process. Motivated by the results of chapter 1 it seems reasonable to model the convenience yield by a mean reverting Ornstein-Uhlenbeck process. Since trading desks only apply and consider models with closed form solutions for options I derive such formulas for commodity futures by solving the corresponding partial differential equation. Additionally, semi-closed form formulas for European options on futures are determined. The Cauchy problem with respect to these options is more challenging than the first one. A solution can be provided. Unlike equities, which typically entitle the holder to a continuing stake in a corporation, commodity futures contracts normally specify a certain date for the delivery of the underlying physical commodity. In order to avoid the delivery process and maintain a futures position, nearby contracts must be sold and contracts that have not yet reached the delivery period must be purchased (so called rolling). Optimal trading days for selling and buying futures are determined by applying statistical tests for stochastic dominance. Besides the optimization of the rolling procedure for commodity futures we dedicate ourselves in chapter 3 with the optimization of the weightings of the commodity futures that make up the index. To this end, I apply the Markowitz approach or mean-variance optimization. The mean-variance optimization penalizes up-side and down-side risk equally, whereas most investors do not mind up-side risk. To overcome this, I consider in the next step other risk measures, namely Value-at-Risk and Conditional Value-at-Risk. The Conditional Value-at-Risk is generalized to discontinuous cumulative distribution functions of the loss. For continuous loss distributions, the Conditional Value-at-Risk at a given confidence level is defined as the expected loss exceeding the Value-at-Risk. Loss distributions associated with finite sampling or scenario modeling are, however, discontinuous. Various risk measures involving discontinuous loss distributions shall be introduced and compared. I then apply the theoretical results to the field of portfolio optimization with commodity indices. Furthermore, I uncover graphically the behavior of these risk measures. For this purpose, I consider the risk measures as a function of the confidence level. Based on a special discrete loss distribution, the graphs demonstrate the different properties of these risk measures. The goal of the first section of chapter 4 is to apply the mathematical concept of excursions for the creation of optimal highly automated or algorithmic trading strategies. The idea is to consider the gain of the strategy and the excursion time it takes to realize the gain. In this section I calculate formulas for the Ornstein-Uhlenbeck process. I show that the corresponding formulas can be calculated quite fast since the only function appearing in the formulas is the so called imaginary error function. This function is already implemented in many programs, such as in Maple. My main contribution of this topic is the optimization of the trading strategy for Ornstein-Uhlenbeck processes via the Banach fixed-point theorem. The second section of chapter 4 deals with statistical arbitrage strategies, a long horizon trading opportunity that generates a riskless profit. The results of this section provide an investor with a tool to investigate empirically if some strategies (for example momentum strategies) constitute statistical arbitrage opportunities or not.

The purpose of Exploration in Oil Industry is to "discover" an oil-containing geological formation from exploration data. In the context of this PhD project this oil-containing geological formation plays the role of a geometrical object, which may have any shape. The exploration data may be viewed as a "cloud of points", that is a finite set of points, related to the geological formation surveyed in the exploration experiment. Extensions of topological methodologies, such as homology, to point clouds are helpful in studying them qualitatively and capable of resolving the underlying structure of a data set. Estimation of topological invariants of the data space is a good basis for asserting the global features of the simplicial model of the data. For instance the basic statistical idea, clustering, are correspond to dimension of the zero homology group of the data. A statistics of Betti numbers can provide us with another connectivity information. In this work represented a method for topological feature analysis of exploration data on the base of so called persistent homology. Loosely, this is the homology of a growing space that captures the lifetimes of topological attributes in a multiset of intervals called a barcode. Constructions from algebraic topology empowers to transform the data, to distillate it into some persistent features, and to understand then how it is organized on a large scale or at least to obtain a low-dimensional information which can point to areas of interest. The algorithm for computing of the persistent Betti numbers via barcode is realized in the computer algebra system "Singular" in the scope of the work.

A classical conjecture in the representation theory of finite groups, the McKay conjecture, states that for any finite group and prime number p the number of complex irreducible characters of degree prime to p is equal to the number of complex irreducible characters of degree prime to p of the normalizer of a p-Sylow subgroup. Recently a reduction theorem was proved by Isaacs, Malle and Navarro: If all simple groups are “good”, then the McKay conjecture holds. In this work we are concerned with the problem of goodness for finite groups of Lie type in their defining characteristic. A simple group is called “good” if certain equivariant bijections between the involved character sets exist. We present a structural approach to the construction of such a bijection by utilizing the so-called “Steinberg-Map”. This yields very natural bijections and we prove most of the desired properties.

This thesis deals with the numerical study of multiscale problems arising in the modelling of processes of the flow of fluid in plain and porous media. Many of these processes, governed by partial differential equations, are relevant in engineering, industry, and environmental studies. The overall task of modelling and simulating the filtration-related multiscale processes becomes interdisciplinary as it employs physics, mathematics and computer programming to reach its aim. Keeping the challenges in mind, the main focus is to overcome the limitations of accuracy, speed and memory and to develop novel efficient numerical algorithms which could, in part or whole, be utilized by those working in the field of porous media. This work has essentially four parts. A single grid basic algorithm and a corresponding parallel algorithm to solve the macroscopic Navier-Stokes-Brinkmann model is discussed. An upscaling subgrid algorithm is derived and numerically tested for the same model. Moving a step further in the line of multiscale methods, an iterative Mutliscale Finite Volume (iMSFV) method is developed for the Stokes-Darcy system. Additionally, the last part of the thesis deals with ways to incorporate changes occurring at different (meso) scale level. The flow equations are coupled with the Convection-Diffusion-Reaction (CDR) equation, which models the transport and capturing of particle concentrations. By employing the numerical method for the coupled flow and transport problem, we understand the interplay between the flow velocity and filtration.

This dissertation is intended to give a systematic treatment of hypersurface singularities in arbitrary characteristic which provides the necessary tools, theoretically and computationally, for the purpose of classification. This thesis consists of five chapters: In chapter 1, we introduce the background on isolated hypersurface singularities needed for our work. In chapter 2, we formalize the notions of piecewise-homogeneous grading and we discuss thoroughly non-degeneracy in arbitrary characteristic. Chapter 3 is devoted to determinacy and normal forms of isolated hypersurface singularities. In the first part, we give finite determinacy theorems in arbitrary characteristic with respect to right respectively contact equivalence. Furthermore, we show that "isolated" and finite determinacy properties are equivalent. In the second part, we formalize Arnol'd's key ideas for the computation of normal forms an define the conditions (AA) and (AAC). The last part of Chapter 3 is devoted to the study of normal forms in the general setting of hypersurface singularities imposing neither condition (A) nor Newton-Nondegeneracy. In Chapter 4, we present algorithms which we implement in Singular for the purpose of explicit computation of regular bases and normal forms. In chapter 5, we transfer some classical results on invariants over the field C of complex numbers to algebraically closed fields of characteristic zero known as Lefschetz principle.

Mrázek et al. [25] proposed a unified approach to curve estimation which combines localization and regularization. Franke et al. [10] used that approach to discuss the case of the regularized local least-squares (RLLS) estimate. In this thesis we will use the unified approach of Mrázek et al. to study some asymptotic properties of local smoothers with regularization. In particular, we shall discuss the Huber M-estimate and its limiting cases towards the L2 and the L1 cases. For the regularization part, we will use quadratic regularization. Then, we will define a more general class of regularization functions. Finally, we will do a Monte Carlo simulation study to compare different types of estimates.

Limit theorems constitute a classical and important field in probability theory. In several applications, in particular in demographic or medical contexts, killed Markov processes suggest themselves as models for populations undergoing culling by mortality or other processes. In these situations mathematical research features a general interest in the observable distribution of survivors, which is known as Yaglom limit or quasi-stationary distribution. Previous work often focuses on discrete state spaces, commonly birth-death processes (or with some more flexible localization of the transitions), with killing only on the boundary. The central concerns of this thesis are to describe, for a given class of one dimensional diffusion processes, the quasistationary distributions (if any), and to describe the convergence (or not) of the process conditioned on survival to one of these quasistationary distributions. Rather general diffusion processes on the half-line are considered, where 0 is allowed to be regular or an exit boundary. Very similar techniques are applied in this work in order to derive results on the large time behavior of an exotic measure valued process, which is closely related to so-called point interactions, which have been widely studied in the mathematical physics literature.

Tropical intersection theory
(2010)

This thesis consists of five chapters: Chapter 1 contains the basics of the theory and is essential for the rest of the thesis. Chapters 2-5 are to a large extent independent of each other and can be read separately. - Chapter 1: Foundations of tropical intersection theory In this first chapter we set up the foundations of a tropical intersection theory covering many concepts and tools of its counterpart in algebraic geometry such as affine tropical cycles, Cartier divisors, morphisms of tropical cycles, pull-backs of Cartier divisors, push-forwards of cycles and an intersection product of Cartier divisors and cycles. Afterwards, we generalize these concepts to abstract tropical cycles and introduce a concept of rational equivalence. Finally, we set up an intersection product of cycles and prove that every cycle is rationally equivalent to some affine cycle in the special case that our ambient cycle is R^n. We use this result to show that rational and numerical equivalence agree in this case and prove a tropical Bézout's theorem. - Chapter 2: Tropical cycles with real slopes and numerical equivalence In this chapter we generalize our definitions of tropical cycles to polyhedral complexes with non-rational slopes. We use this new definition to show that if our ambient cycle is a fan then every subcycle is numerically equivalent to some affine cycle. Finally, we restrict ourselves to cycles in R^n that are "generic" in some sense and study the concept of numerical equivalence in more detail. - Chapter 3: Tropical intersection products on smooth varieties We define an intersection product of tropical cycles on tropical linear spaces L^n_k and on other, related fans. Then, we use this result to obtain an intersection product of cycles on any "smooth" tropical variety. Finally, we use the intersection product to introduce a concept of pull-backs of cycles along morphisms of smooth tropical varieties and prove that this pull-back has all expected properties. - Chapter 4: Weil and Cartier divisors under tropical modifications First, we introduce "modifications" and "contractions" and study their basic properties. After that, we prove that under some further assumptions a one-to-one correspondence of Weil and Cartier divisors is preserved by modifications. In particular we can prove that on any smooth tropical variety we have a one-to-one correspondence of Weil and Cartier divisors. - Chapter 5: Chern classes of tropical vector bundles We give definitions of tropical vector bundles and rational sections of tropical vector bundles. We use these rational sections to define the Chern classes of such a tropical vector bundle. Moreover, we prove that these Chern classes have all expected properties. Finally, we classify all tropical vector bundles on an elliptic curve up to isomorphisms.

This thesis deals with the application of binomial option pricing in a single-asset Black-Scholes market and its extension to multi-dimensional situations. Although the binomial approach is, in principle, an efficient method for lower dimensional valuation problems, there are at least two main problems regarding its application: Firstly, traded options often exhibit discontinuities, so that the Berry- Esséen inequality is in general tight; i.e. conventional tree methods converge no faster than with order 1/sqrt(N). Furthermore, they suffer from an irregular convergence behaviour that impedes the possibility to achieve a higher order of convergence via extrapolation methods. Secondly, in multi-asset markets conventional tree construction methods cannot ensure well-defined transition probabilities for arbitrary correlation structures between the assets. As a major aim of this thesis, we present two approaches to get binomial trees into shape in order to overcome the main problems in applications; the optimal drift model for the valuation of single-asset options and the decoupling approach to multi-dimensional option pricing. The new valuation methods are embedded into a self-contained survey of binomial option pricing, which focuses on the convergence behaviour of binomial trees. The optimal drift model is a new one-dimensional binomial scheme that can lead to convergence of order o(1/N) by exploiting the specific structure of the valuation problem under consideration. As a consequence, it has the potential to outperform benchmark algorithms. The decoupling approach is presented as a universal construction method for multi-dimensional trees. The corresponding trees are well-defined for an arbitrary correlation structure of the underlying assets. In addition, they yield a more regular convergence behaviour. In fact, the sawtooth effect can even vanish completely, so that extrapolation can be applied.

This thesis deals with 3 important aspects of optimal investment in real-world financial markets: taxes, crashes, and illiquidity. An introductory chapter reviews the portfolio problem in its historical context and motivates the theme of this work: We extend the standard modelling framework to include specific real-world features and evaluate their significance. In the first chapter, we analyze the optimal portfolio problem with capital gains taxes, assuming that taxes are deferred until the end of the investment horizon. The problem is solved with the help of a modification of the classical martingale method. The second chapter is concerned with optimal asset allocation under the threat of a financial market crash. The investor takes a worst-case attitude towards the crash, so her investment objective is to be best off in the most adverse crash scenario. We first survey the existing literature on the worst-case approach to optimal investment and then present in detail the novel martingale approach to worst-case portfolio optimization. The first part of this chapter is based on joint work with Ralf Korn. In the last chapter, we investigate optimal portfolio decisions in the presence of illiquidity. Illiquidity is understood as a period in which it is impossible to trade on financial markets. We use dynamic programming techniques in combination with abstract convergence results to solve the corresponding optimal investment problem. This chapter is based on joint work with Holger Kraft and Peter Diesinger.

This study deals with the optimal control problems of the glass tube drawing processes where the aim is to control the cross-sectional area (circular) of the tube by using the adjoint variable approach. The process of tube drawing is modeled by four coupled nonlinear partial differential equations. These equations are derived by the axisymmetric Stokes equations and the energy equation by using the approach based on asymptotic expansions with inverse aspect ratio as small parameter. Existence and uniqueness of the solutions of stationary isothermal model is also proved. By defining the cost functional, we formulated the optimal control problem. Then Lagrange functional associated with minimization problem is introduced and the first and the second order optimality conditions are derived. We also proved the existence and uniqueness of the solutions of the stationary isothermal model. We implemented the optimization algorithms based on the steepest descent, nonlinear conjugate gradient, BFGS, and Newton approaches. In the Newton method, CG iterations are introduced to solve the Newton equation. Numerical results are obtained for two different cases. In the first case, the cross-sectional area for the entire time domain is controlled and in the second case, the area at the final time is controlled. We also compared the performance of the optimization algorithms in terms of the solution iterations, functional evaluations and the computation time.

This dissertation deals with two main subjects. Both are strongly related to boundary problems for the Poisson equation and the Laplace equation, respectively. The oblique boundary problem of potential theory as well as the limit formulae and jump relations of potential theory are investigated. We divide this abstract into two parts and start with the oblique boundary problem. Here we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in my diploma thesis. Moreover we prove regularization results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkin approximation. Finally we show that the results are applicable to problems from Geomathematics. Now we come to the limit formulae. There we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. The convergence in Lebesgue spaces for integrable functions is already treated in literature. The achievement of this dissertation is this convergence for the weak derivatives of higher orders. Also the layer functions are elements of Sobolev spaces and the surface is a two dimensional suitable smooth submanifold in the three dimensional space. We are considering the potential of the single layer, the potential of the double layer and their first order normal derivatives. Main tool in the proof in Sobolev norm is the uniform convergence of the tangential derivatives, which is proved with help of some results taken from literature. Additionally, we need a result about the limit formulae in the Lebesgue spaces, which is also taken from literature, and a reduction result for normal derivatives of harmonic functions. Moreover we prove the convergence in the Hölder spaces. Finally we give an application of the limit formulae and jump relations. We generalize a known density of several function systems from Geomathematics in the Lebesgue spaces of square integrable measureable functions, to density in Sobolev spaces, based on the results proved before. Therefore we have prove the limit formula of the single layer potential in dual spaces of Soboelv spaces, where also the layer function is an element of such a distribution space.

This thesis is devoted to the study of tropical curves with emphasis on their enumerative geometry. Major results include a conceptual proof of the fact that the number of rational tropical plane curves interpolating an appropriate number of general points is independent of the choice of points, the computation of intersection products of Psi-classes on the moduli space of rational tropical curves, a computation of the number of tropical elliptic plane curves of given degree and fixed tropical j-invariant as well as a tropical analogue of the Riemann-Roch theorem for algebraic curves. The result are obtained in joint work with Hannah Markwig and/or Andreas Gathmann.

This thesis is devoted to two main topics (accordingly, there are two chapters): In the first chapter, we establish a tropical intersection theory with analogue notions and tools as its algebro-geometric counterpart. This includes tropical cycles, rational functions, intersection products of Cartier divisors and cycles, morphisms, their functors and the projection formula, rational equivalence. The most important features of this theory are the following: - It unifies and simplifies many of the existing results of tropical enumerative geometry, which often contained involved ad-hoc computations. - It is indispensable to formulate and solve further tropical enumerative problems. - It shows deep relations to the intersection theory of toric varieties and connected fields. - The relationship between tropical and classical Gromov-Witten invariants found by Mikhalkin is made plausible from inside tropical geometry. - It is interesting on its own as a subfield of convex geometry. In the second chapter, we study tropical gravitational descendants (i.e. Gromov-Witten invariants with incidence and "Psi-class" factors) and show that many concepts of the classical Gromov-Witten theory such as the famous WDVV equations can be carried over to the tropical world. We use this to extend Mikhalkin's results to a certain class of gravitational descendants, i.e. we show that many of the classical gravitational descendants of P^2 and P^1 x P^1 can be computed by counting tropical curves satisfying certain incidence conditions and with prescribed valences of their vertices. Moreover, the presented theory is not restricted to plane curves and therefore provides an important tool to derive similar results in higher dimensions. A more detailed chapter synopsis can be found at the beginning of each individual chapter.

The thesis at hand deals with the numerical solution of multiscale problems arising in the modeling of processes in fluid and thermo dynamics. Many of these processes, governed by partial differential equations, are relevant in engineering, geoscience, and environmental studies. More precisely, this thesis discusses the efficient numerical computation of effective macroscopic thermal conductivity tensors of high-contrast composite materials. The term "high-contrast" refers to large variations in the conductivities of the constituents of the composite. Additionally, this thesis deals with the numerical solution of Brinkman's equations. This system of equations adequately models viscous flows in (highly) permeable media. It was introduced by Brinkman in 1947 to reduce the deviations between the measurements for flows in such media and the predictions according to Darcy's model.

The goal of this work is the development and investigation of an interdisciplinary and in itself closed hydrodynamic approach to the simulation of dilute and dense granular flow. The definition of “granular flow” is a nontrivial task in itself. We say that it is either the flow of grains in a vacuum or in a fluid. A grain is an observable piece of a certain material, for example stone when we mean the flow of sand. Choosing a hydrodynamic view on granular flow, we treat the granular material as a fluid. A hydrodynamic model is developed, that describes the process of flowing granular material. This is done through a system of partial differential equations and algebraic relations. This system is derived by the kinetic theory of granular gases which is characterized by inelastic collisions extended with approaches from soil mechanics. Solutions to the system have to be obtained to understand the process. The equations are so difficult to solve that an analytical solution is out of reach. So approximate solutions must be obtained. Hence the next step is the choice or development of a numerical algorithm to obtain approximate solutions of the model. Common to every problem in numerical simulation, these two steps do not lead to a result without implementation of the algorithm. Hence the author attempts to present this work in the following frame, to participate in and contribute to the three areas Physics, Mathematics and Software implementation and approach the simulation of granular flow in a combined and interdisciplinary way. This work is structured as follows. A continuum model for granular flow which covers the regime of fast dilute flow as well as slow dense flow up to vanishing velocity is presented in the first chapter. This model is strongly nonlinear in the dependence of viscosity and other coefficients on the hydrodynamic variables and it is singular because some coefficients diverge towards the maximum packing fraction of grains. Hence the second difficulty, the challenging task of numerically obtaining approximate solutions for this model is faced in the second chapter. In the third chapter we aim at the validation of both the model and the numerical algorithm through numerical experiments and investigations and show their application to industrial problems. There we focus intensively on the shear flow experiment from the experimental and analytical work of Bocquet et al. which serves well to demonstrate the algorithm, all boundary conditions involved and provides a setting for analytical studies to compare our results. The fourth chapter rounds up the work with the implementation of both the model and the numerical algorithm in a software framework for the solution of complex rheology problems developed as part of this thesis.

In the context of inverse optimization, inverse versions of maximum flow and minimum cost flow problems have thoroughly been investigated. In these network flow problems there are two important problem parameters: flow capacities of the arcs and costs incurred by sending a unit flow on these arcs. Capacity changes for maximum flow problems and cost changes for minimum cost flow problems have been studied under several distance measures such as rectilinear, Chebyshev, and Hamming distances. This thesis also deals with inverse network flow problems and their counterparts tension problems under the aforementioned distance measures. The major goals are to enrich the inverse optimization theory by introducing new inverse network problems that have not yet been treated in the literature, and to extend the well-known combinatorial results of inverse network flows for more general classes of problems with inherent combinatorial properties such as matroid flows on regular matroids and monotropic programming. To accomplish the first objective, the inverse maximum flow problem under Chebyshev norm is analyzed and the capacity inverse minimum cost flow problem, in which only arc capacities are perturbed, is introduced. In this way, it is attempted to close the gap between the capacity perturbing inverse network problems and the cost perturbing ones. The foremost purpose of studying inverse tension problems on networks is to achieve a well-established generalization of the inverse network problems. Since tensions are duals of network flows, carrying the theoretical results of network flows over to tensions follows quite intuitively. Using this intuitive link between network flows and tensions, a generalization to matroid flows and monotropic programs is built gradually up.

This thesis is devoted to applying symbolic methods to the problems of decoding linear codes and of algebraic cryptanalysis. The paradigm we employ here is as follows. We reformulate the initial problem in terms of systems of polynomial equations over a finite field. The solution(s) of such systems should yield a way to solve the initial problem. Our main tools for handling polynomials and polynomial systems in such a paradigm is the technique of Gröbner bases and normal form reductions. The first part of the thesis is devoted to formulating and solving specific polynomial systems that reduce the problem of decoding linear codes to the problem of polynomial system solving. We analyze the existing methods (mainly for the cyclic codes) and propose an original method for arbitrary linear codes that in some sense generalizes the Newton identities method widely known for cyclic codes. We investigate the structure of the underlying ideals and show how one can solve the decoding problem - both the so-called bounded decoding and more general nearest codeword decoding - by finding reduced Gröbner bases of these ideals. The main feature of the method is that unlike usual methods based on Gröbner bases for "finite field" situations, we do not add the so-called field equations. This tremendously simplifies the underlying ideals, thus making feasible working with quite large parameters of codes. Further we address complexity issues, by giving some insight to the Macaulay matrix of the underlying systems. By making a series of assumptions we are able to provide an upper bound for the complexity coefficient of our method. We address also finding the minimum distance and the weight distribution. We provide solid experimental material and comparisons with some of the existing methods in this area. In the second part we deal with the algebraic cryptanalysis of block iterative ciphers. Namely, we analyze the small-scale variants of the Advanced Encryption Standard (AES), which is a widely used modern block cipher. Here a cryptanalyst composes the polynomial systems which solutions should yield a secret key used by communicating parties in a symmetric cryptosystem. We analyze the systems formulated by researchers for the algebraic cryptanalysis, and identify the problem that conventional systems have many auxiliary variables that are not actually needed for the key recovery. Moreover, having many such auxiliary variables, specific to a given plaintext/ciphertext pair, complicates the use of several pairs which is common in cryptanalysis. We thus provide a new system where the auxiliary variables are eliminated via normal form reductions. The resulting system in key-variables only is then solved. We present experimental evidence that such an approach is quite good for small scaled ciphers. We investigate further our approach and employ the so-called meet-in-the-middle principle to see how far one can go in analyzing just 2-3 rounds of scaled ciphers. Additional "tuning techniques" are discussed together with experimental material. Overall, we believe that the material of this part of the thesis makes a step further in algebraic cryptanalysis of block ciphers.

Continuous stochastic control theory has found many applications in optimal investment. However, it lacks some reality, as it is based on the assumption that interventions are costless, which yields optimal strategies where the controller has to intervene at every time instant. This thesis consists of the examination of two types of more realistic control methods with possible applications. In the first chapter, we study the stochastic impulse control of a diffusion process. We suppose that the controller minimizes expected discounted costs accumulating as running and controlling cost, respectively. Each control action causes costs which are bounded from below by some positive constant. This makes a continuous control impossible as it would lead to an immediate ruin of the controller. We give a rigorous development of the relevant theory, where our guideline is to establish verification and convergence results under minimal assumptions, without focusing on the existence of solutions to the corresponding (quasi-)variational inequalities. If the impulse control problem can be characterized or approximated by (quasi-)variational inequalities, it remains to solve these equations. In Section 1.2, we solve the stochastic impulse control problem for a one-dimensional diffusion process with constant coefficients and convex running costs. Further, in Section 1.3, we solve a particular multi-dimensional example, where the uncontrolled process is given by an at least two-dimensional Brownian motion and the cost functions are rotationally symmetric. By symmetry, this problem can be reduced to a one-dimensional problem. In the last section of the first chapter, we suggest a new impulse control problem, where the controller is in addition allowed to invest his initial capital into a market consisting of a money market account and a risky asset. The costs which arise upon controlling the diffusion process and upon trading in this market have to be paid out of the controller's bond holdings. The aim of the controller is to minimize the running costs, caused by the abstract diffusion process, without getting ruined. The second chapter is based on a paper which is joint work with Holger Kraft and Frank Seifried. We analyze the portfolio decision of an investor trading in a market where the economy switches randomly between two possible states, a normal state where trading takes place continuously, and an illiquidity state where trading is not allowed at all. We allow for jumps in the market prices at the beginning and at the end of a trading interruption. Section 2.1 provides an explicit representation of the investor's portfolio dynamics in the illiquidity state in an abstract market consisting of two assets. In Section 2.2 we specify this market model and assume that the investor maximizes expected utility from terminal wealth. We establish convergence results, if the maximal number of liquidity breakdowns goes to infinity. In the Markovian framework of Section 2.3, we provide the corresponding Hamilton-Jacobi-Bellman equations and prove a verification result. We apply these results to study the portfolio problem for a logarithmic investor and an investor with a power utility function, respectively. Further, we extend this model to an economy with three regimes. For instance, the third state could model an additional financial crisis where trading is still possible, but the excess return is lower and the volatility is higher than in the normal state.

This thesis deals with the following question. Given a moduli space of coherent sheaves on a projective variety with a fixed Hilbert polynomial, to find a natural construction that replaces the subvariety of the sheaves that are not locally free on their support (we call such sheaves singular) by some variety consisting of sheaves that are locally free on their support. We consider this problem on the example of the coherent sheaves on \(\mathbb P_2\) with Hilbert polynomial 3m+1.
Given a singular coherent sheaf \(\mathcal F\) with singular curve C as its support we replace \(\mathcal F\) by locally free sheaves \(\mathcal E\) supported on a reducible curve \(C_0\cup C_1\), where \(C_0\) is a partial normalization of C and \(C_1\) is an extra curve bearing the degree of \(\mathcal E\). These bundles resemble the bundles considered by Nagaraj and Seshadri. Many properties of the singular 3m+1 sheaves are inherited by the new sheaves we introduce in this thesis (we call them R-bundles). We consider R-bundles as natural replacements of the singular sheaves. R-bundles refine the information about 3m+1 sheaves on \(\mathbb P_2\). Namely, for every isomorphism class of singular 3m+1 sheaves there are \(\mathbb P_1\) many equivalence classes of R-bundles. There is a variety \(\tilde M\) of dimension 10 that may be considered as the space of all the isomorphism classes of the non-singular 3m+1 sheaves on \(\mathbb P_2\) together with all the equivalence classes of all R-bundles. This variety is obtained by blowing up the moduli space of 3m+1 sheaves on \(\mathbb P_2\) along the subvariety of singular sheaves. We modify the definition of a 3m+1 family and obtain a notion of a new family over an arbitrary variety S. In particular 3m+1 families of the non-singular sheaves on \(\mathbb P_2\) are families in this sense. New families over one point are either non-singular 3m+1 sheaves or R-bundles. For every variety S we introduce an equivalence relation on the set of all new families over S. The notion of equivalence for families over one point coincides with isomorphism for non-singular 3m+1 sheaves and with equivalence for R-bundles. We obtain a moduli functor \(\tilde{\mathcal M}:(Sch) \rightarrow (Sets)\) that assigns to every variety S the set of the equivalence classes of the new families over S. There is a natural transformation of functors \(\tilde{\mathcal M}\rightarrow \mathcal M\) that establishes a relation between \(\tilde{\mathcal M}\) and the moduli functor \(\mathcal M\) of the 3m+1 moduli problem on \(\mathbb P_2\). There is also a natural transformation \(\tilde{\mathcal M} \rightarrow Hom(\__ ,\tilde M)\), inducing a bijection \(\tilde{\mathcal M}(pt)\cong \tilde M\), which means that \(\tilde M\) is a coarse moduli space of the moduli problem \(\tilde{\mathcal M}\).

This thesis shows an approach to combine the advantages of MBS tyre models and FEM models for the use in full vehicle simulations. The procedure proposed in this thesis aims to describe a nonlinear structure with a Finite Element approach combined with nonlinear model reduction methods. Unlike most model reduction methods - as the frequently used Craig-Bampton approach - the method of Proper Orthogonal Decomposition (POD) offers a projection basis suitable for nonlinear models. For the linear wave equation, the POD method is studied comparing two different choices of snapshot sets. Set 1 consists of deformation snapshots, and set 2 additionally contains velocities and accelerations. An error analysis proves no convergence guarantee for deformations only. For inclusion of derivatives it yields an error bound diminishing for small time steps. The numerical results show a better behaviour for the derivative snapshot method, as long as the sum of the left-over eigenvalues is significant. For the reduction of nonlinear systems - especially when using commercial software - it is necessary to decouple the reduced surrogate system from the full model. To achieve this, a lookup table approach is presented. It makes use of the preceding computation step with the full model necessary to set up the POD basis (training step). The nonlinear term of inner forces and the stiffness matrix are output and stored in a lookup table for the reduced system. Numerical examples include a nonlinear string in Matlab and an airspring computed in Abaqus. Both examples show that effort reductions of two orders of magnitude are possible within a reasonable error tolerance. The lookup approaches perform faster than the Trajectory Piecewise Linear (TPWL) method and produce comparable errors. Furthermore, the Abaqus example shows the influence of training excitation on the quality of the reduced model.

We present a new efficient and robust algorithm for topology optimization of 3D cast parts. Special constraints are fulfilled to make possible the incorporation of a simulation of the casting process into the optimization: In order to keep track of the exact position of the boundary and to provide a full finite element model of the structure in each iteration, we use a twofold approach for the structural update. A level set function technique for boundary representation is combined with a new tetrahedral mesh generator for geometries specified by implicit boundary descriptions. Boundary conditions are mapped automatically onto the updated mesh. For sensitivity analysis, we employ the concept of the topological gradient. Modification of the level set function is reduced to efficient summation of several level set functions, and the finite element mesh is adapted to the modified structure in each iteration of the optimization process. We show that the resulting meshes are of high quality. A domain decomposition technique is used to keep the computational costs of remeshing low. The capabilities of our algorithm are demonstrated by industrial-scale optimization examples.

In this thesis, the coupling of the Stokes equations and the Biot poroelasticity equations for fluid flow normal to porous media is investigated. For that purpose, the transmission conditions across the interfaces between the fluid regions and the porous domain are derived. A proper algorithm is formulated and numerical examples are presented. First, the transmission conditions for the coupling of various physical phenomena are reviewed. For the coupling of free flow with porous media, it has to be distinguished whether the fluid flows tangentially or perpendicularly to the porous medium. This plays an essential role for the formulation of the transmission conditions. In the thesis, the transmission conditions for the coupling of the Stokes equations and the Biot poroelasticity equations for fluid flow normal to the porous medium in one and three dimensions are derived. With these conditions, the continuous fully coupled system of equations in one and three dimensions is formulated. In the one dimensional case the extreme cases, i.e. fluid-fluid interface and fluid impermeable solid interface, are considered. Two chapters of the thesis are devoted to the discretisation of the fully coupled Biot-Stokes system for matching and non-matching grids, respectively. Therefor, operators are introduced that map the internal and boundary variables to the respective domains via Stokes equations, Biot equations and the transmission conditions. The matrix representation of some of these operators is shown. For the non-matching case, a cell-centred grid in the fluid region and a staggered grid in the porous domain are used. Hence, the discretisation is more difficult, since an additional grid on the interface has to be introduced. Corresponding matching functions are needed to transfer the values properly from one domain to the other across the interface. In the end, the iterative solution procedure for the Biot-Stokes system on non-matching grids is presented. For this purpose, a short review of domain decomposition methods is given, which are often the methods of choice for such coupled problems. The iterative solution algorithm is presented, including details like stopping criteria, choice and computation of parameters, formulae for non-dimensionalisation, software and so on. Finally, numerical results for steady state examples, depth filtration and cake filtration examples are presented.

Grey-box modelling deals with models which are able to integrate the following two kinds of information: qualitative (expert) knowledge and quantitative (data) knowledge, with equal importance. The doctoral thesis has two aims: the improvement of an existing neuro-fuzzy approach (LOLIMOT algorithm), and the development of a new model class with corresponding identification algorithm, based on multiresolution analysis (wavelets) and statistical methods. The identification algorithm is able to identify both hidden differential dynamics and hysteretic components. After the presentation of some improvements of the LOLIMOT algorithm based on readily normalized weight functions derived from decision trees, we investigate several mathematical theories, i.e. the theory of nonlinear dynamical systems and hysteresis, statistical decision theory, and approximation theory, in view of their applicability for grey-box modelling. These theories show us directly the way onto a new model class and its identification algorithm. The new model class will be derived from the local model networks through the following modifications: Inclusion of non-Gaussian noise sources; allowance of internal nonlinear differential dynamics represented by multi-dimensional real functions; introduction of internal hysteresis models through two-dimensional "primitive functions"; replacement respectively approximation of the weight functions and of the mentioned multi-dimensional functions by wavelets; usage of the sparseness of the matrix of the wavelet coefficients; and identification of the wavelet coefficients with Sequential Monte Carlo methods. We also apply this modelling scheme to the identification of a shock absorber.

In this thesis, we investigate a statistical model for precipitation time series recorded at a single site. The sequence of observations consists of rainfall amounts aggregated over time periods of fixed duration. As the properties of this sequence depend strongly on the length of the observation intervals, we follow the approach of Rodriguez-Iturbe et. al. [1] and use an underlying model for rainfall intensity in continuous time. In this idealized representation, rainfall occurs in clusters of rectangular cells, and each observations is treated as the sum of cell contributions during a given time period. Unlike the previous work, we use a multivariate lognormal distribution for the temporal structure of the cells and clusters. After formulating the model, we develop a Markov-Chain Monte-Carlo algorithm for fitting it to a given data set. A particular problem we have to deal with is the need to estimate the unobserved intensity process alongside the parameter of interest. The performance of the algorithm is tested on artificial data sets generated from the model. [1] I. Rodriguez-Iturbe, D. R. Cox, and Valerie Isham. Some models for rainfall based on stochastic point processes. Proc. R. Soc. Lond. A, 410:269-288, 1987.

In this thesis we classify simple coherent sheaves on Kodaira fibers of types II, III and IV (cuspidal and tacnode cubic curves and a plane configuration of three concurrent lines). Indecomposable vector bundles on smooth elliptic curves were classified in 1957 by Atiyah. In works of Burban, Drozd and Greuel it was shown that the categories of vector bundles and coherent sheaves on cycles of projective lines are tame. It turns out, that all other degenerations of elliptic curves are vector-bundle-wild. Nevertheless, we prove that the category of coherent sheaves of an arbitrary reduced plane cubic curve, (including the mentioned Kodaira fibers) is brick-tame. The main technical tool of our approach is the representation theory of bocses. Although, this technique was mainly used for purely theoretical purposes, we illustrate its computational potential for investigating tame behavior in wild categories. In particular, it allows to prove that a simple vector bundle on a reduced cubic curve is determined by its rank, multidegree and determinant, generalizing Atiyah's classification. Our approach leads to an interesting class of bocses, which can be wild but are brick-tame.

In this work we study and investigate the minimum width annulus problem (MWAP), the circle center location or circle location problem (CLP) and the point center location or point location problem (PLP) on Rectilinear and Chebyshev planes as well as in networks. The relations between the problems have served as a basis for finding of elegant solution, algorithms for both new and well known problems. So, MWAP was formulated and investigated in Rectilinear space. In contrast to Euclidean metric, MWAP and PLP have at least one common optimal point. Therefore, MWAP on Rectilinear plane was solved in linear time with the help of PLP. Hence, the solution sequence was PLP-->MWAP. It was shown, that MWAP and CLP are equivalent. Thus, CLP can be also solved in linear time. The obtained results were analysed and transfered to Chebyshev metric. After that, the notions of circle, sphere and annulus in networks were introduced. It should be noted that the notion of a circle in a network is different from the notion of a cycle. An O(mn) time algorithm for solution of MWAP was constructed and implemented. The algorithm is based on the fact that the middle point of an edge represents an optimal solution of a local minimum width annulus on this edge. The resulting complexity is better than the complexity O(mn+n^2logn) in unweighted case of the fastest known algorithm for minimizing of the range function, which is mathematically equivalent to MWAP. MWAP in unweighted undirected networks was extended to the MWAP on subsets and to the restricted MWAP. Resulting problems were analysed and solved. Also the p–minimum width annulus problem was formulated and explored. This problem is NP–hard. However, the p–MWAP has been solved in polynomial O(m^2n^3p) time with a natural assumption, that each minimum width annulus covers all vertexes of a network having distances to the central point of annulus less than or equal to the radius of its outer circle. In contrast to the planar case MWAP in undirected unweighted networks have appeared to be a root problem among considered problems. During investigation of properties of circles in networks it was shown that the difference between planar and network circles is significant. This leads to the nonequivalence of CLP and MWAP in the general case. However, MWAP was effectively used in solution procedures for CLP giving the sequence MWAP-->CLP. The complexity of the developed and implemented algorithm is of order O(m^2n^2). It is important to mention that CLP in networks has been formulated for the first time in this work and differs from the well–studied location of cycles in networks. We have constructed an O(mn+n^2logn) algorithm for well–known PLP. The complexity of this algorithm is not worse than the complexity of the currently best algorithms. But the concept of the solution procedure is new – we use MWAP in order to solve PLP building the opposite to the planar case solution sequence MWAP-->PLP and this method has the following advantages: First, the lower bounds LB obtained in the solution procedure are proved to be in any case better than the strongest Halpern’s lower bound. Second, the developed algorithm is so simple that it can be easily applied to complex networks manually. Third, the empirical complexity of the algorithm is equal to O(mn). MWAP was extended to and explored in directed unweighted and weighted networks. The complexity bound O(n^2) of the developed algorithm for finding of the center of a minimum width annulus in the unweighted case does not depend on the number of edges in a network, because the problems can be solved in the order PLP-->MWAP. In the weighted case computational time is of order O(mn^2).

In many medical, financial, industrial, e.t.c. applications of statistics, the model parameters may undergo changes at unknown moment of time. In this thesis, we consider change point analysis in a regression setting for dichotomous responses, i.e. they can be modeled as Bernoulli or 0-1 variables. Applications are widespread including credit scoring in financial statistics and dose-response relations in biometry. The model parameters are estimated using neural network method. We show that the parameter estimates are identifiable up to a given family of transformations and derive the consistency and asymptotic normality of the network parameter estimates using the results in Franke and Neumann Franke Neumann (2000). We use a neural network based likelihood ratio test statistic to detect a change point in a given set of data and derive the limit distribution of the estimator using the results in Gombay and Horvath (1994,1996) under the assumption that the model is properly specified. For the misspecified case, we develop a scaled test statistic for the case of one-dimensional parameter. Through simulation, we show that the sample size, change point location and the size of change influence change point detection. In this work, the maximum likelihood estimation method is used to estimate a change point when it has been detected. Through simulation, we show that change point estimation is influenced by the sample size, change point location and the size of change. We present two methods for determining the change point confidence intervals: Profile log-likelihood ratio and Percentile bootstrap methods. Through simulation, the Percentile bootstrap method is shown to be superior to profile log-likelihood ratio method.

This thesis covers two important fields in financial mathematics, namely the continuous time portfolio optimisation and credit risk modelling. We analyse optimisation problems of portfolios of Call and Put options on the stock and/or the zero coupon bond issued by a firm with default risk. We use the martingale approach for dynamic optimisation problems. Our findings show that the riskier the option gets, the less proportion of his wealth the investor allocates to the risky asset. Further, we analyse the Credit Default Swap (CDS) market quotes on the Eurobonds issued by Turkish sovereign for building the term structure of the sovereign credit risk. Two methods are introduced and compared for bootstrapping the risk-neutral probabilities of default (PD) in an intensity based (or reduced form) credit risk modelling approach. We compare the market-implied PDs with the actual PDs reported by credit rating agencies based on historical experience. Our results highlight the market price of the sovereign credit risk depending on the assigned rating category in the sampling period. Finally, we find an optimal leverage strategy for delivering the payments promised by a Constant Proportion Debt Obligation (CPDO). The problem is solved via the introduction and explicit solution of a stochastic control problem by transforming the related Hamilton-Jacobi-Bellman Equation into its dual. Contrary to the industry practise, the optimal leverage function we derive is a non-linear function of the CPDO asset value. The simulations show promising behaviour of the optimal leverage function compared with the one popular among practitioners.

The nowadays increasing number of fields where large quantities of data are collected generates an emergent demand for methods for extracting relevant information from huge databases. Amongst the various existing data mining models, decision trees are widely used since they represent a good trade-off between accuracy and interpretability. However, one of their main problems is that they are very instable, which complicates the process of the knowledge discovery because the users are disturbed by the different decision trees generated from almost the same input learning samples. In the current work, binary tree classifiers are analyzed and partially improved. The analysis of tree classifiers goes from their topology from the graph theory point of view to the creation of a new tree classification model by means of combining decision trees and soft comparison operators (Mlynski, 2003) with the purpose to not only overcome the well known instability problem of decision trees, but also in order to confer the ability of dealing with uncertainty. In order to study and compare the structural stability of tree classifiers, we propose an instability coefficient which is based on the notion of Lipschitz continuity and offer a metric to measure the proximity between decision trees. This thesis converges towards its main part with the presentation of our model ``Soft Operators Decision Tree\'\' (SODT). Mainly, we describe its construction, application and the consistency of the mathematical formulation behind this. Finally we show the results of the implementation of SODT and compare numerically the stability and accuracy of a SODT and a crisp DT. The numerical simulations support the stability hypothesis and a smaller tendency to overfitting the training data with SODT than with crisp DT is observed. A further aspect of this inclusion of soft operators is that we choose them in a way so that the resulting goodness function (used by this method) is differentiable and thus allows to calculate the best split points by means of gradient descent methods. The main drawback of SODT is the incorporation of the unpreciseness factor, which increases the complexity of the algorithm.

In this dissertation we consider mesoscale based models for flow driven fibre orientation dynamics in suspensions. Models for fibre orientation dynamics are derived for two classes of suspensions. For concentrated suspensions of rigid fibres the Folgar-Tucker model is generalized by incorporating the excluded volume effect. For dilute semi-flexible fibre suspensions a novel moments based description of fibre orientation state is introduced and a model for the flow-driven evolution of the corresponding variables is derived together with several closure approximations. The equation system describing fibre suspension flows, consisting of the incompressible Navier-Stokes equation with an orientation state dependent non-Newtonian constitutive relation and a linear first order hyperbolic system for the fibre orientation variables, has been analyzed, allowing rather general fibre orientation evolution models and constitutive relations. The existence and uniqueness of a solution has been demonstrated locally in time for sufficiently small data. The closure relations for the semiflexible fibre suspension model are studied numerically. A finite volume based discretization of the suspension flow is given and the numerical results for several two and three dimensional domains with different parameter values are presented and discussed.

This dissertation deals with the optimization of the web formation in a spunbond process for the production of artificial fabrics. A mathematical model of the process is presented. Based on the model, two kind of attributes to be optimized are considered, those related with the quality of the fabric and those describing the stability of the production process. The problem falls in the multicriteria and decision making framework. The functions involved on the model of the process are non linear, non convex and non differentiable. A strategy in two steps; exploration and continuation, is proposed to approximate numerically the Pareto frontier and alternative methods are proposed to navigate the set and support the decision making process. The proposed strategy is applied to a particular production process and numerical results are presented.

This thesis is devoted to deal with the stochastic optimization problems in various situations with the aid of the Martingale method. Chapter 2 discusses the Martingale method and its applications to the basic optimization problems, which are well addressed in the literature (for example, [15], [23] and [24]). In Chapter 3, we study the problem of maximizing expected utility of real terminal wealth in the presence of an index bond. Chapter 4, which is a modification of the original research paper joint with Korn and Ewald [39], investigates an optimization problem faced by a DC pension fund manager under inflationary risk. Although the problem is addressed in the context of a pension fund, it presents a way of how to deal with the optimization problem, in the case there is a (positive) endowment. In Chapter 5, we turn to a situation where the additional income, other than the income from returns on investment, is gained by supplying labor. Chapter 6 concerns a situation where the market considered is incomplete. A trick of completing an incomplete market is presented there. The general theory which supports the discussion followed is summarized in the first chapter.

The dissertation deals with the application of Hub Location models in public transport planning. The author proposes new mathematical models along with different solution approaches to solve the instances. Moreover, a novel multi-period formulation is proposed as an extension to the general model. Due to its high complexity heuristic approaches are formulated to find a good solution within a reasonable amount of time.

Nonlinear diffusion filtering of images using the topological gradient approach to edges detection
(2007)

In this thesis, the problem of nonlinear diffusion filtering of gray-scale images is theoretically and numerically investigated. In the first part of the thesis, we derive the topological asymptotic expansion of the Mumford-Shah like functional. We show that the dominant term of this expansion can be regarded as a criterion to edges detection in an image. In the numerical part, we propose the finite volume discretization for the Catté et al. and the Weickert diffusion filter models. The proposed discretization is based on the integro-interpolation method introduced by Samarskii. The numerical schemes are derived for the case of uniform and nonuniform cell-centered grids of the computational domain \(\Omega \subset \mathbb{R}^2\). In order to generate a nonuniform grid, the adaptive coarsening technique is proposed.

A modular level set algorithm is developed to study the interface and its movement for free moving boundary problems. The algorithm is divided into three basic modules : initialization, propagation and contouring. Initialization is the process of finding the signed distance function from closed objects. We discuss here, a methodology to find an accurate signed distance function from a closed, simply connected surface discretized by triangulation. We compute the signed distance function using the direct method and it is stored efficiently in the neighborhood of the interface by a narrow band level set method. A novel approach is employed to determine the correct sign of the distance function at convex-concave junctions of the surface. The accuracy and convergence of the method with respect to the surface resolution is studied. It is shown that the efficient organization of surface and narrow band data structures enables the solution of large industrial problems. We also compare the accuracy of the signed distance function by direct approach with Fast Marching Method (FMM). It is found that the direct approach is more accurate than FMM. Contouring is performed through a variant of the marching cube algorithm used for the isosurface construction from volumetric data sets. The algorithm is designed to keep foreground and background information consistent, contrary to the neutrality principle followed for surface rendering in computer graphics. The algorithm ensures that the isosurface triangulation is closed, non-degenerate and non-ambiguous. The constructed triangulation has desirable properties required for the generation of good volume meshes. These volume meshes are used in the boundary element method for the study of linear electrostatics. For estimating surface properties like interface position, normal and curvature accurately from a discrete level set function, a method based on higher order weighted least squares is developed. It is found that least squares approach is more accurate than finite difference approximation. Furthermore, the method of least squares requires a more compact stencil than those of finite difference schemes. The accuracy and convergence of the method depends on the surface resolution and the discrete mesh width. This approach is used in propagation for the study of mean curvature flow and bubble dynamics. The advantage of this approach is that the curvature is not discretized explicitly on the grid and is estimated on the interface. The method of constant velocity extension is employed for the propagation of the interface. With least squares approach, the mean curvature flow has considerable reduction in mass loss compared to finite difference techniques. In the bubble dynamics, the modules are used for the study of a bubble under the influence of surface tension forces to validate Young-Laplace law. It is found that the order of curvature estimation plays a crucial role for calculating accurate pressure difference between inside and outside of the bubble. Further, we study the coalescence of two bubbles under surface tension force. The application of these modules to various industrial problems is discussed.

The thesis is concerned with multiscale approximation by means of radial basis functions on hierarchically structured spherical grids. A new approach is proposed to construct a biorthogonal system of locally supported zonal functions. By use of this biorthogonal system of locally supported zonal functions, a spherical fast wavelet transform (SFWT) is established. Finally, based on the wavelet analysis, geophysically and geodetically relevant problems involving rotation-invariant pseudodifferential operators are shown to be efficiently and economically solvable.

The desire to model in ever increasing detail geometrical and physical features has lead to a steady increase in the number of points used in field solvers. While many solvers have been ported to parallel machines, grid generators have left behind. Sequential generation of meshes of large size is extremely problematic both in terms of time and memory requirements. Therefore, the need for developing parallel mesh generation technique is well justified. In this work a novel algorithm is presented for automatic parallel generation of tetrahedral computational meshes based on geometrical domain decomposition. It has a potential to remove this bottleneck. Different domain decomposition approaches and criteria have been investigated. Questions regarding time and memory consumption, efficiency of computations and quality of generated surface and volume meshes have been considered. As a result of the work parTgen (partitioner and parallel tetrahedral mesh generator) software package based on the developed algorithm has been created. Several real-life examples of relatively complex structures involving large meshes (of order 10^7-10^8 elements) are given. It has been shown that high mesh quality is achieved. Memory and time consumption are reduced significantly, and parallel algorithm is efficient.

This dissertation is intended to transport the theory of Serre functors into the context of A-infinity-categories. We begin with an introduction to multicategories and closed multicategories, which form a framework in which the theory of A-infinity-categories is developed. We prove that (unital) A-infinity-categories constitute a closed symmetric multicategory. We define the notion of A-infinity-bimodule similarly to Tradler and show that it is equivalent to an A-infinity-functor of two arguments which takes values in the differential graded category of complexes of k-modules, where k is a commutative ground ring. Serre A-infinity-functors are defined via A-infinity-bimodules following ideas of Kontsevich and Soibelman. We prove that a unital closed under shifts A-infinity-category over a field admits a Serre A-infinity-functor if and only if its homotopy category admits an ordinary Serre functor. The proof uses categories and Serre functors enriched in the homotopy category of complexes of k-modules. Another important ingredient is an A-infinity-version of the Yoneda Lemma.

The present work deals with the (global and local) modeling of the windfield on the real topography of Rheinland-Pfalz. Thereby the focus is on the construction of a vectorial windfield from low, irregularly distributed data given on a topographical surface. The developed spline procedure works by means of vectorial (homogeneous, harmonic) polynomials (outer harmonics) which control the oscillation behaviour of the spline interpoland. In the process the characteristic of the spline curvature which defines the energy norm is assumed to be on a sphere inside the Earth interior and not on the Earth’s surface. The numerical advantage of this method arises from the maximum-minimum principle for harmonic functions.

In the theoretical part of this thesis, the difference of the solutions of the elastic and the elastoplastic boundary value problem is analysed, both for linear kinematic and combined linear kinematic and isotropic hardening material. We consider both models in their quasistatic, rate-independent formulation with linearised geometry. The main result of the thesis is, that the differences of the physical obervables (the stresses, strains and displacements) can be expressed as composition of some linear operators and play operators with respect to the exterior forces. Explicit homotopies between both solutions are presented. The main analytical devices are Lipschitz estimates for the stop and the play operator. We present some generalisations of the standard estimates. They allow different input functions, different initial memories and different scalar products. Thereby, the underlying time involving function spaces are the Sobolov spaces of first order with arbitrary integrability exponent between one and infinity. The main results can easily be generalised for the class of continuous functions with bounded total variation. In the practical part of this work, a method to correct the elastic stress tensor over a long time interval at some chosen points of the body is presented and analysed. In contrast to widespread uniaxial corrections (Neuber or ESED), our method takes multiaxiality phenomena like cyclic hardening/softening, ratchetting and non-masing behaviour into account using Jiang's model of elastoplasticity. It can be easily adapted to other constitutive elastoplastic material laws. The theory for our correction model is developped for linear kinematic hardening material, for which error estimated are derived. Our numerical algorithm is very fast and designed for the case that the elastic stress is piecewise linear. The results for the stresses can be significantly improved with Seeger's empirical strain constraint. For the improved model, a simple predictor-correcor algorithm for smooth input loading is established.

The main aim of this work was to obtain an approximate solution of the seismic traveltime tomography problems with the help of splines based on reproducing kernel Sobolev spaces. In order to be able to apply the spline approximation concept to surface wave as well as to body wave tomography problems, the spherical spline approximation concept was extended for the case where the domain of the function to be approximated is an arbitrary compact set in R^n and a finite number of discontinuity points is allowed. We present applications of such spline method to seismic surface wave as well as body wave tomography, and discuss the theoretical and numerical aspects of such applications. Moreover, we run numerous numerical tests that justify the theoretical considerations.

In this thesis, the quasi-static Biot poroelasticity system in bounded multilayered domains in one and three dimensions is studied. In more detail, in the one-dimensional case, a finite volume discretization for the Biot system with discontinuous coefficients is derived. The discretization results in a difference scheme with harmonic averaging of the coefficients. Detailed theoretical analysis of the obtained discrete model is performed. Error estimates, which establish convergence rates for both primary as well as flux unknowns are derived. Besides, modified and more accurate discretizations, which can be applied when the interface position coincides with a grid node, are obtained. These discretizations yield second order convergence of the fluxes of the problem. Finally, the solver for the solution of the produced system of linear equations is developed and extensively tested. A number of numerical experiments, which confirm the theoretical considerations are performed. In the three-dimensional case, the finite volume discretization of the system involves construction of special interpolating polynomials in the dual volumes. These polynomials are derived so that they satisfy the same continuity conditions across the interface, as the original system of PDEs. This technique allows to obtain such a difference scheme, which provides accurate computation of the primary as well as of the flux unknowns, including the points adjacent to the interface. Numerical experiments, based on the obtained discretization, show second order convergence for auxiliary problems with known analytical solutions. A multigrid solver, which incorporates the features of the discrete model, is developed in order to solve efficiently the linear system, produced by the finite volume discretization of the three-dimensional problem. The crucial point is to derive problem-dependent restriction and prolongation operators. Such operators are a well-known remedy for the scalar PDEs with discontinuous coefficients. Here, these operators are derived for the system of PDEs, taking into account interdependence of different unknowns within the system. In the derivation, the interpolating polynomials from the finite volume discretization are employed again, linking thus the discretization and the solution processes. The developed multigrid solver is tested on several model problems. Numerical experiments show that, due to the proper problem-dependent intergrid transfer, the multigrid solver is robust with respect to the discontinuities of the coefficients of the system. In the end, the poroelasticity system with discontinuous coefficients is used to model a real problem. The Biot model, describing this problem, is treated numerically, i.e., discretized by the developed finite volume techniques and then solved by the constructed multigrid solver. Physical characteristics of the process, such as displacement of the skeleton, pressure of the fluid, components of the stress tensor, are calculated and then presented at certain cross-sections.

In the thesis the author presents a mathematical model which describes the behaviour of the acoustical pressure (sound), produced by a bass loudspeaker. The underlying physical propagation of sound is described by the non--linear isentropic Euler system in a Lagrangian description. This system is expanded via asymptotical analysis up to third order in the displacement of the membrane of the loudspeaker. The differential equations which describe the behaviour of the key note and the first order harmonic are compared to classical results. The boundary conditions, which are derived up to third order, are based on the principle that the small control volume sticks to the boundary and is allowed to move only along it. Using classical results of the theory of elliptic partial differential equations, the author shows that under appropriate conditions on the input data the appropriate mathematical problems admit, by the Fredholm alternative, unique solutions. Moreover, certain regularity results are shown. Further, a novel Wave Based Method is applied to solve appropriate mathematical problems. However, the known theory of the Wave Based Method, which can be found in the literature, so far, allowed to apply WBM only in the cases of convex domains. The author finds the criterion which allows to apply the WBM in the cases of non--convex domains. In the case of 2D problems we represent this criterion as a small proposition. With the aid of this proposition one is able to subdivide arbitrary 2D domains such that the number of subdomains is minimal, WBM may be applied in each subdomain and the geometry is not altered, e.g. via polygonal approximation. Further, the same principles are used in the case of 3D problem. However, the formulation of a similar proposition in cases of 3D problems has still to be done. Next, we show a simple procedure to solve an inhomogeneous Helmholtz equation using WBM. This procedure, however, is rather computationally expensive and can probably be improved. Several examples are also presented. We present the possibility to apply the Wave Based Technique to solve steady--state acoustic problems in the case of an unbounded 3D domain. The main principle of the classical WBM is extended to the case of an external domain. Two numerical examples are also presented. In order to apply the WBM to our problems we subdivide the computational domain into three subdomains. Therefore, on the interfaces certain coupling conditions are defined. The description of the optimization procedure, based on the principles of the shape gradient method and level set method, and the results of the optimization finalize the thesis.

In this dissertation we present analysis of macroscopic models for slow dense granular flow. Models are derived from plasticity theory with yield condition and flow rule. Corner stone equations are conservation of mass and conservation of momentum with special constitutive law. Such models are considered in the class of generalised Newtonian fluids, where viscosity depends on the pressure and modulo of the strain-rate tensor. We showed the hyperbolic nature for the evolutionary model in 1D and ill-posed behaviour for 2D and 3D. The steady state equations are always hyperbolic. In the 2D problem we derived a prototype nonlinear backward parabolic equation for the velocity and the similar equation for the shear-rate. Analysis of derived PDE showed the finite blow up time. Blow up time depends on the initial condition. Full 2D and antiplane 3D model were investigated numerically with finite element method. For 2D model we showed the presence of boundary layers. Antiplane 3D model was investigated with the Runge Kutta Discontinuous Galerkin method with mesh addoption. Numerical results confirmed that such a numerical method can be a good choice for the simulations of the slow dense granular flow.

The lattice Boltzmann method (LBM) is a numerical solver for the Navier-Stokes equations, based on an underlying molecular dynamic model. Recently, it has been extended towardsthe simulation of complex fluids. We use the asymptotic expansion technique to investigate the standard scheme, the initialization problem and possible developments towards moving boundary and fluid-structure interaction problems. At the same time, it will be shown how the mathematical analysis can be used to understand and improve the algorithm. First of all, we elaborate the tool "asymptotic analysis", proposing a general formulation of the technique and explaining the methods and the strategy we use for the investigation. A first standard application to the LBM is described, which leads to the approximation of the Navier-Stokes solution starting from the lattice Boltzmann equation. As next, we extend the analysis to investigate origin and dynamics of initial layers. A class of initialization algorithms to generate accurate initial values within the LB framework is described in detail. Starting from existing routines, we will be able to improve the schemes in term of efficiency and accuracy. Then we study the features of a simple moving boundary LBM. In particular, we concentrate on the initialization of new fluid nodes created by the variations of the computational fluid domain. An overview of existing possible choices is presented. Performing a careful analysis of the problem we propose a modified algorithm, which produces satisfactory results. Finally, to set up an LBM for fluid structure interaction, efficient routines to evaluate forces are required. We describe the Momentum Exchange algorithm (MEA). Precise accuracy estimates are derived, and the analysis leads to the construction of an improved method to evaluate the interface stresses. In conclusion, we test the defined code and validate the results of the analysis on several simple benchmarks. From the theoretical point of view, in the thesis we have developed a general formulation of the asymptotic expansion, which is expected to offer a more flexible tool in the investigation of numerical methods. The main practical contribution offered by this work is the detailed analysis of the numerical method. It allows to understand and improve the algorithms, and construct new routines, which can be considered as starting points for future researches.

This thesis introduces so-called cone scalarising functions. They are by construction compatible with a partial order for the outcome space given by a cone. The quality of the parametrisations of the efficient set given by the cone scalarising functions are then investigated. Here, the focus lies on the (weak) efficiency of the generated solutions, the reachability of effiecient points and continuity of the solution set. Based on cone scalarising functions Pareto Navigation a novel, interactive, multiobjective optimisation method is proposed. It changes the ordering cone to realise bounds on partial tradeoffs. Besides, its use of an equality constraint for the changing component of the reference point is a new feature. The efficiency of its solutions, the reachability of efficient solutions and continuity is then analysed. Potential problems are demonstrated using a critical example. Furthermore, the use of Pareto Navigation in a two-phase approach and for nonconvex problems is discussed. Finally, its application for intensity-modulated radiotherapy planning is described. Thereby, its realisation in a graphical user interface is shown.

The topic of this thesis is the coupling of an atomistic and a coarse scale region in molecular dynamics simulations with the focus on the reflection of waves at the interface between the two scales and the velocity of waves in the coarse scale region for a non-equilibrium process. First, two models from the literature for such a coupling, the concurrent coupling of length scales and the bridging scales method are investigated for a one dimensional system with harmonic interaction. It turns out that the concurrent coupling of length scales method leads to the reflection of fine scale waves at the interface, while the bridging scales method gives an approximated system that is not energy conserving. The velocity of waves in the coarse scale region is in both models not correct. To circumvent this problems, we present a coupling based on the displacement splitting of the bridging scales method together with choosing appropriate variables in orthogonal subspaces. This coupling allows the derivation of evolution equations of fine and coarse scale degrees of freedom together with a reflectionless boundary condition at the interface directly from the Lagrangian of the system. This leads to an energy conserving approximated system with a clear separation between modeling errors an errors due to the numerical solution. Possible approximations in the Lagrangian and the numerical computation of the memory integral and other numerical errors are discussed. We further present a method to choose the interpolation from coarse to atomistic scale in such a way, that the fine scale degrees of freedom in the coarse scale region can be neglected. The interpolation weights are computed by comparing the dispersion relations of the coarse scale equations and the fully atomistic system. With this new interpolation weights, the number of degrees of freedom can be drastically reduced without creating an error in the velocity of the waves in the coarse scale region. We give an alternative derivation of the new coupling with the Mori-Zwanzig projection operator formalism, and explain how the method can be extended to non-zero temperature simulations. For the comparison of the results of the approximated with the fully atomistic system, we use a local stress tensor and the energy in the atomistic region. Examples for the numerical solution of the approximated system for harmonic potentials are given in one and two dimensions.

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.

The fast development of the financial markets in the last decade has lead to the creation of a variety of innovative interest rate related products that require advanced numerical pricing methods. Examples in this respect are products with a complicated strong path-dependence such as a Target Redemption Note, a Ratchet Cap, a Ladder Swap and others. On the other side, the usage of the standard in the literature one-factor Hull and White (1990) type of short rate models allows only for a perfect correlation between all continuously compounded spot rates or Libor rates and thus are not suited for pricing innovative products depending on several Libor rates such as for example a "steepener" option. One possible solution to this problem deliver the two-factor short rate models and in this thesis we consider a two-factor Hull and White (1990) type of a short rate process derived from the Heath, Jarrow, Morton (1992) framework by limiting the volatility structure of the forward rate process to a deterministic one. In this thesis, we often choose to use a variety of modified (binomial, trinomial and quadrinomial) tree constructions as a main numerical pricing tool due to their flexibility and fast convergence and (when there is no closed-form solution) compare their results with fine grid Monte Carlo simulations. For the purpose of pricing the already mentioned innovative short-rate related products, in this thesis we offer and examine two different lattice construction methods for the two-factor Hull-White type of a short rate process which are able to deal easily both with modeling of the mean-reversion of the underlying process and with the strong path-dependence of the priced options. Additionally, we prove that the so-called rotated lattice construction method overcomes the typical for the existing two-factor tree constructions problem with obtaining negative "risk-neutral probabilities". With a variety of numerical examples, we show that this leads to a stability in the results especially in cases of high volatility parameters and negative correlation between the base factors (which is typically the case in reality). Further, noticing that Chan et al (1992) and Ritchken and Sankarasubramanian (1995) showed that option prices are sensitive to the level of the short rate volatility, we examine the pricing of European and American options where the short rate process has a volatility structure of a Cheyette (1994) type. In this relation, we examine the application of the two offered lattice construction methods and compare their results with the Monte Carlo simulation ones for a variety of examples. Additionally, for the pricing of American options with the Monte Carlo method we expand and implement the simulation algorithm of Longstaff and Schwartz (2000). With a variety of numerical examples we compare again the stability and the convergence of the different lattice construction methods. Dealing with the problems of pricing strongly path-dependent options, we come across the cumulative Parisian barrier option pricing problem. We notice that in their classical form, the cumulative Parisian barrier options have been priced both analytically (in a quasi closed form) and with a tree approximation (based on the Forward Shooting Grid algorithm, see e.g. Hull and White (1993), Kwok and Lau (2001) and others). However, we offer an additional tree construction method which can be seen as a direct binomial tree integration that uses the analytically calculated conditional survival probabilities. The advantage of the offered method is on one side that the conditional survival probabilities are easier to calculate than the closed-form solution itself and on the other side that this tree construction is very flexible in the sense that it allows easy incorporation of additional features such as e.g a forward starting one. The obtained results are better than the Forward Shooting Grid tree ones and are very close to the analytical quasi closed form solution. Finally, we pay our attention to pricing another type of innovative interest rate alike products - namely the Longevity bond - whose coupon payments depend on the survival function of a given cohort. Due to the lack of a market for mortality, for the pricing of the Longevity bonds we develop (following Korn, Natcheva and Zipperer (2006)) a framework that contains principles from both Insurance and Financial mathematic. Further on, we calibrate the existing models for the stochastic mortality dynamics to historical German data and additionally offer new stochastic extensions of the classical (deterministic) models of mortality such as the Gompertz and the Makeham one. Finally, we compare and analyze the results of the application of all considered models to the pricing of a Longevity bond on the longevity of the German males.

In this thesis, we have dealt with two modeling approaches of the credit risk, namely the structural (firm value) and the reduced form. In the former one, the firm value is modeled by a stochastic process and the first hitting time of this stochastic process to a given boundary defines the default time of the firm. In the existing literature, the stochastic process, triggering the firm value, has been generally chosen as a diffusion process. Therefore, on one hand it is possible to obtain closed form solutions for the pricing problems of credit derivatives and on the other hand the optimal capital structure of a firm can be analysed by obtaining closed form solutions of firm's corporate securities such as; equity value, debt value and total firm value, see Leland(1994). We have extended this approach by modeling the firm value as a jump-diffusion process. The choice of the jump-diffusion process was a crucial step to obtain closed form solutions for corporate securities. As a result, we have chosen a jump-diffusion process with double exponentially distributed jump heights, which enabled us to analyse the effects of jump on the optimal capital structure of a firm. In the second part of the thesis, by following the reduced form models, we have assumed that the default is triggered by the first jump of a Cox process. Further, by following Schönbucher(2005), we have modeled the forward default intensity of a firm as a geometric Brownian motion and derived pricing formulas for credit default swap options in a more general setup than the ones in Schönbucher(2005).

In this thesis diverse problems concerning inflation-linked products are dealt with. To start with, two models for inflation are presented, including a geometric Brownian motion for consumer price index itself and an extended Vasicek model for inflation rate. For both suggested models the pricing formulas of inflation-linked products are derived using the risk-neutral valuation techniques. As a result Black and Scholes type closed form solutions for a call option on inflation index for a Brownian motion model and inflation evolution for an extended Vasicek model as well as for an inflation-linked bond are calculated. These results have been already presented in Korn and Kruse (2004) [17]. In addition to these inflation-linked products, for the both inflation models the pricing formulas of a European put option on inflation, an inflation cap and floor, an inflation swap and an inflation swaption are derived. Consequently, basing on the derived pricing formulas and assuming the geometric Brownian motion process for an inflation index, different continuous-time portfolio problems as well as hedging problems are studied using the martingale techniques as well as stochastic optimal control methods. These utility optimization problems are continuous-time portfolio problems in different financial market setups and in addition with a positive lower bound constraint on the final wealth of the investor. When one summarizes all the optimization problems studied in this work, one will have the complete picture of the inflation-linked market and both counterparts of market-participants, sellers as well as buyers of inflation-linked financial products. One of the interesting results worth mentioning here is naturally the fact that a regular risk-averse investor would like to sell and not buy inflation-linked products due to the high price of inflation-linked bonds for example and an underperformance of inflation-linked bonds compared to the conventional risk-free bonds. The relevance of this observation is proved by investigating a simple optimization problem for the extended Vasicek process, where as a result we still have an underperforming inflation-linked bond compared to the conventional bond. This situation does not change, when one switches to an optimization of expected utility from the purchasing power, because in its nature it is only a change of measure, where we have a different deflator. The negativity of the optimal portfolio process for a normal investor is in itself an interesting aspect, but it does not affect the optimality of handling inflation-linked products compared to the situation not including these products into investment portfolio. In the following, hedging problems are considered as a modeling of the other half of inflation market that is inflation-linked products buyers. Natural buyers of these inflation-linked products are obviously institutions that have payment obligations in the future that are inflation connected. That is why we consider problems of hedging inflation-indexed payment obligations with different financial assets. The role of inflation-linked products in the hedging portfolio is shown to be very important by analyzing two alternative optimal hedging strategies, where in the first one an investor is allowed to trade as inflation-linked bond and in the second one he is not allowed to include an inflation-linked bond into his hedging portfolio. Technically this is done by restricting our original financial market, which is made of a conventional bond, inflation index and a stock correlated with inflation index, to the one, where an inflation index is excluded. As a whole, this thesis presents a wide view on inflation-linked products: inflation modeling, pricing aspects of inflation-linked products, various continuous-time portfolio problems with inflation-linked products as well as hedging of inflation-related payment obligations.

This thesis discusses methods for the classification of finite projective planes via exhaustive search. In the main part the author classifies all projective planes of order 16 admitting a large quasiregular group of collineations. This is done by a complete search using the computer algebra system GAP. Computational methods for the construction of relative difference sets are discussed. These methods are implemented in a GAP-package, which is available separately. As another result --found in cooperation with U. Dempwolff-- the projective planes defined by planar monomials are classified. Furthermore the full automorphism group of the non-translation planes defined by planar monomials are classified.

This thesis deals with modeling aspects of generalized Newtonian and of non-Newtonian fluids, as well as with development and validation of algorithms used in simulation of such fluids. The main contribution in the modeling part are the introduction and analysis of a new model for the generalized Newtonian fluids, where constitutive equation is of an algebraic form. Distinction between shear and extensional viscosities leads to anisotropic viscosity model. It can be considered as a natural extension of the well known (isotropic viscosity) Carreau model, which deals only with shear viscosity properties of the fluid. The proposed model takes additionally into account extensional viscosity properties. Numerical results show that the anisotropic viscosity model gives much better agreement with experimental observations than the isotropic one. Another contribution of the thesis consists of the development and analysis of robust and reliable algorithms for simulation of generalized Newtonian fluids. For such fluids the momentum equations are strongly coupled through mixed derivatives appearing in the viscous term (unlike the case of Newtonian fluids). It is shown in this thesis, that a careful treatment of those derivatives is essential in deriving robust algorithms. A modification of a standard SIMPLE-like algorithm is given, where all the viscous terms from the momentum equations are discretized in an implicit manner. Moreover, it is shown that a block diagonal preconditioner to the viscous operator is good enough to be used in simulations. Furthermore, different solution techniques, namely projection type methods (consists of solving momentum equations and pressure correction equation) and fully coupled methods (momentum and continuity equations are solved together), are compared. It is shown, that explicit discretization of the mixed derivatives lead to stability problems. Further, analytical estimates of eigenvalue distribution for three different preconditioners, applied to the transformed system arising after discretization and linearization of the momentum and continuity equations, are provided. We propose to apply a block Gauss-Seidel preconditioner to the transformed system. The analysis shows, that this preconditioner is able to cluster eigenvalues around unity independent of the transformation step. It is not the case for other preconditioners applied to the transformed system as discussed in the thesis. The block Gauss-Seidel preconditioner has also shown the best behavior (among all preconditioners discussed in the thesis) in numerical experiments. Further contribution consists of comparison and validation of numerical algorithms applied in simulations of non-Newtonian fluids modeled by time integral constitutive equations. Numerical results from simulations of dilute polymer solutions, described by the integral Oldroyd B model, have shown very good quantitative agreement with the results obtained by differential Oldroyd B counterpart in 4:1 planar contraction domain at low Weissenberg numbers. In this case, the Weissenberg number is changed by changing the relaxation time. However, contrary to the differential Oldroyd B model, the integral one allows to perform stable simulations also in the range of high Weissenberg numbers. Moreover, very good agreement with experimental observations has been achieved. Simulations of concentrated polymer solutions (polystyrene and polybutadiene solutions), modeled by the integral Doi Edwards model, supplemented by chain length fluctuations, have shown very good qualitative agreement with the results obtained by its differential approximation in 4:1:4 constriction domain. Again, much higher Weissenberg numbers can be achieved when the integral model is used. Moreover, very good quantitative results with experimental data of polystyrene solution for the first normal stress difference and shear viscosity defined here as the quotient of a shear stress and a shear rate. Finally, comparison of the two methods used for approximating the time integral constitutive equation, namely Deformation Field Method (DFM) and Backward Lagrangian Particle Method (BLPM), is performed. In BLPM the particle paths are recalculated at every time step of the simulations, what has never been tried before. The results have shown, that in the considered geometries both methods give similar results.

For the last decade, optimization of beam orientations in intensity-modulated radiation therapy (IMRT) has been shown to be successful in improving the treatment plan. Unfortunately, the quality of a set of beam orientations depends heavily on its corresponding beam intensity profiles. Usually, a stochastic selector is used for optimizing beam orientation, and then a single objective inverse treatment planning algorithm is used for the optimization of beam intensity profiles. The overall time needed to solve the inverse planning for every random selection of beam orientations becomes excessive. Recently, considerable improvement has been made in optimizing beam intensity profiles by using multiple objective inverse treatment planning. Such an approach results in a variety of beam intensity profiles for every selection of beam orientations, making the dependence between beam orientations and its intensity profiles less important. This thesis takes advantage of this property to accelerate the optimization process through an approximation of the intensity profiles that are used for multiple selections of beam orientations, saving a considerable amount of calculation time. A dynamic algorithm (DA) and evolutionary algorithm (EA), for beam orientations in IMRT planning will be presented. The DA mimics, automatically, the methods of beam's eye view and observer's view which are recognized in conventional conformal radiation therapy. The EA is based on a dose-volume histogram evaluation function introduced as an attempt to minimize the deviation between the mathematical and clinical optima. To illustrate the efficiency of the algorithms they have been applied to different clinical examples. In comparison to the standard equally spaced beams plans, improvements are reported for both algorithms in all the clinical examples even when, for some cases, fewer beams are used. A smaller number of beams is always desirable without compromising the quality of the treatment plan. It results in a shorter treatment delivery time, which reduces potential errors in terms of patient movements and decreases discomfort.

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.

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.)

Over the last decades, mathematical modeling has reached nearly all fields of natural science. The abstraction and reduction to a mathematical model has proven to be a powerful tool to gain a deeper insight into physical and technical processes. The increasing computing power has made numerical simulations available for many industrial applications. In recent years, mathematicians and engineers have turned there attention to model solid materials. New challenges have been found in the simulation of solids and fluid-structure interactions. In this context, it is indispensable to study the dynamics of elastic solids. Elasticity is a main feature of solid bodies while demanding a great deal of the numerical treatment. There exists a multitude of commercial tools to simulate the behavior of elastic solids. Anyhow, the majority of these software packages consider quasi-stationary problems. In the present work, we are interested in highly dynamical problems, e.g. the rotation of a solid. The applicability to free-boundary problems is a further emphasis of our considerations. In the last years, meshless or particle methods have attracted more and more attention. In many fields of numerical simulation these methods are on a par with classical methods or superior to them. In this work, we present the Finite Pointset Method (FPM) which uses a moving least squares particle approximation operator. The application of this method to various industrial problems at the Fraunhofer ITWM has shown that FPM is particularly suitable for highly dynamical problems with free surfaces and strongly changing geometries. Thereby, FPM offers exactly the features that we require for the analysis of the dynamics of solid bodies. In the present work, we provide a numerical scheme capable to simulate the behavior of elastic solids. We present the system of partial differential equations describing the dynamics of elastic solids and show its hyperbolic character. In particular, we focus our attention to the constitutive law for the stress tensor and provide evolution equations for the deviatoric part of the stress tensor in order to circumvent limitations of the classical Hooke's law. Furthermore, we present the basic principle of the Finite Pointset Method. In particular, we provide the concept of upwinding in a given direction as a key ingredient for stabilizing hyperbolic systems. The main part of this work describes the design of a numerical scheme based on FPM and an operator splitting to take the different processes within a solid body into account. Each resulting subsystem is treated separately in an adequate way. Hereby, we introduce the notion of system-inherent directions and dimensional upwinding. Finally, a coupling strategy for the subsystems and results are presented. We close this work with some final conclusions and an outlook on future work.

This work deals with the mathematical modeling and numerical simulation of the dynamics of a curved inertial viscous Newtonian fiber, which is practically applicable to the description of centrifugal spinning processes of glass wool. Neglecting surface tension and temperature dependence, the fiber flow is modeled as a three-dimensional free boundary value problem via instationary incompressible Navier-Stokes equations. From regular asymptotic expansions in powers of the slenderness parameter leading-order balance laws for mass (cross-section) and momentum are derived that combine the unrestricted motion of the fiber center-line with the inner viscous transport. The physically reasonable form of the one-dimensional fiber model results thereby from the introduction of the intrinsic velocity that characterizes the convective terms. For the numerical simulation of the derived model a finite volume code is developed. The results of the numerical scheme for high Reynolds numbers are validated by comparing them with the analytical solution of the inviscid problem. Moreover, the influence of parameters, like viscosity and rotation on the fiber dynamics are investigated. Finally, an application based on industrial data is performed.

In this thesis we have discussed the problem of decomposing an integer matrix \(A\) into a weighted sum \(A=\sum_{k \in {\mathcal K}} \alpha_k Y^k\) of 0-1 matrices with the strict consecutive ones property. We have developed algorithms to find decompositions which minimize the decomposition time \(\sum_{k \in {\mathcal K}} \alpha_k\) and the decomposition cardinality \(|\{ k \in {\mathcal K}: \alpha_k > 0\}|\). In the absence of additional constraints on the 0-1 matrices \(Y^k\) we have given an algorithm that finds the minimal decomposition time in \({\mathcal O}(NM)\) time. For the case that the matrices \(Y^k\) are restricted to shape matrices -- a restriction which is important in the application of our results in radiotherapy -- we have given an \({\mathcal O}(NM^2)\) algorithm. This is achieved by solving an integer programming formulation of the problem by a very efficient combinatorial algorithm. In addition, we have shown that the problem of minimizing decomposition cardinality is strongly NP-hard, even for matrices with one row (and thus for the unconstrained as well as the shape matrix decomposition). Our greedy heuristics are based on the results for the decomposition time problem and produce better results than previously published algorithms.

The aim of the thesis is the numerical investigation of saturated, stationary, incompressible Newtonian flow in porous media when inertia is not negligible. We focus our attention to the Navier-Stokes system with two pressures derived by two-scale homogenization. The thesis is subdivided into five Chapters. After the introductory remarks on porous media, filtration laws and upscaling methods, the first chapter is closed by stating the basic terminology and mathematical fundamentals. In Chapter 2, we start by formulating the Navier-Stokes equations on a periodic porous medium. By two-scale expansions of the velocity and pressure, we formally derive the Navier-Stokes system with two pressures. For the sake of completeness, known existence and uniqueness results are repeated and a convergence proof is given. Finally, we consider Stokes and Navier-Stokes systems with two pressures with respect to their relation to Darcy's law. Chapter 3 and Chapter 4 are devoted to the numerical solution of the nonlinear two pressure system. Therefore, we follow two approaches. The first approach which is developed in Chapter 3 is based on a splitting of the Navier-Stokes system with two pressures into micro and macro problems. The splitting is achieved by Taylor expanding the permeability function or by discretely computing the permeability function. The problems to be solved are a series of Stokes and Navier-Stokes problems on the periodicity cell. The Stokes problems are solved by an Uzawa conjugate gradient method. The Navier-Stokes equations are linearized by a least-squares conjugate gradient method, which leads to the solution of a sequence of Stokes problems. The macro problem consists of solving a nonlinear uniformly elliptic equation of second order. The least-squares linearization is applied to the macro problem leading to a sequence of Poisson problems. All equations will be discretized by finite elements. Numerical results are presented at the end of Chapter 3. The second approach presented in Chapter 4 relies on the variational formulation in a certain Hilbert space setting of the Navier-Stokes system with two pressures. The nonlinear problem is again linearized by the least-squares conjugate gradient method. We obtain a sequence of Stokes systems with two pressures. For the latter systems, we propose a fast solution method which relies on pre-computing Stokes systems on the periodicity cell for finite element basis functions acting as right hand sides. Finally, numerical results are discussed. In Chapter 5 we are concerned with modeling and simulation of the pressing section of a paper machine. We state a two-dimensional model of a press nip which takes into account elasticity and flow phenomena. Nonlinear filtration laws are incorporated into the flow model. We present a numerical solution algorithm and the chapter is closed by a numerical investigation of the model with special focus on inertia effects.

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.

Non-commutative polynomial algebras appear in a wide range of applications, from quantum groups and theoretical physics to linear differential and difference equations. In the thesis, we have developed a framework, unifying many important algebras in the classes of \(G\)- and \(GR\)-algebras and studied their ring-theoretic properties. Let \(A\) be a \(G\)-algebra in \(n\) variables. We establish necessary and sufficient conditions for \(A\) to have a Poincar'e-Birkhoff-Witt (PBW) basis. Further on, we show that besides the existence of a PBW basis, \(A\) shares some other properties with the commutative polynomial ring \(\mathbb{K}[x_1,\ldots,x_n]\). In particular, \(A\) is a Noetherian integral domain of Gel'fand-Kirillov dimension \(n\). Both Krull and global homological dimension of \(A\) are bounded by \(n\); we provide examples of \(G\)-algebras where these inequalities are strict. Finally, we prove that \(A\) is Auslander-regular and a Cohen-Macaulay algebra. In order to perform symbolic computations with modules over \(GR\)-algebras, we generalize Gröbner bases theory, develop and respectively enhance new and existing algorithms. We unite the most fundamental algorithms in a suite of applications, called "Gröbner basics" in the literature. Furthermore, we discuss algorithms appearing in the non-commutative case only, among others two-sided Gröbner bases for bimodules, annihilators of left modules and operations with opposite algebras. An important role in Representation Theory is played by various subalgebras, like the center and the Gel'fand-Zetlin subalgebra. We discuss their properties and their relations to Gröbner bases, and briefly comment some aspects of their computation. We proceed with these subalgebras in the chapter devoted to the algorithmic study of morphisms between \(GR\)-algebras. We provide new results and algorithms for computing the preimage of a left ideal under a morphism of \(GR\)-algebras and show both merits and limitations of several methods that we propose. We use this technique for the computation of the kernel of a morphism, decomposition of a module into central characters and algebraic dependence of pairwise commuting elements. We give an algorithm for computing the set of one-dimensional representations of a \(G\)-algebra \(A\), and prove, moreover, that if the set of finite dimensional representations of \(A\) over a ground field \(K\) is not empty, then the homological dimension of \(A\) equals \(n\). All the algorithms are implemented in a kernel extension Plural of the computer algebra system Singular. We discuss the efficiency of computations and provide a comparison with other computer algebra systems. We propose a collection of benchmarks for testing the performance of algorithms; the comparison of timings shows that our implementation outperforms all of the modern systems with the combination of both broad functionality and fast implementation. In the thesis, there are many new non-trivial examples, and also the solutions to various problems, arising in different fields of mathematics. All of them were obtained with the developed theory and the implementation in Plural, most of them are treated computationally in this thesis for the first time.

Since its invention by Sir Allistair Pilkington in 1952, the float glass process has been used to manufacture long thin flat sheets of glass. Today, float glass is very popular due to its high quality and relatively low production costs. When producing thinner glass the main concern is to retain its optical quality, which can be deteriorated during the manufacturing process. The most important stage of this process is the floating part, hence is considered to be responsible for the loss in the optical quality. A series of investigations performed on the finite products showed the existence of many short wave patterns, which strongly affect the optical quality of the glass. Our work is concerned with finding the mechanism for wave development, taking into account all possible factors. In this thesis, we model the floating part of the process by an theoretical study of the stability of two superposed fluids confined between two infinite plates and subjected to a large horizontal temperature gradient. Our approach is to take into account the mixed convection effects (viscous shear and buoyancy), neglecting on the other hand the thermo-capillarity effects due to the length of our domain and the presence of a small stabilizing vertical temperature gradient. Both fluids are treated as Newtonian with constant viscosity. They are immiscible, incompressible, have very different properties and have a free surface between them. The lower fluid is a liquid metal with a very small kinematic viscosity, whereas the upper fluid is less dense. The two fluids move with different velocities: the speed of the upper fluid is imposed, whereas the lower fluid moves as a result of buoyancy effects. We examine the problem by means of small perturbation analysis, and obtain a system of two Orr-Sommerfeld equations coupled with two energy equations, and general interface and boundary conditions. We solve the system analytically in the long- and short- wave limit, by using asymptotic expansions with respect to the wave number. Moreover, we write the system in the form of a general eigenvalue problem and we solve the system numerically by using Chebyshev spectral methods for fluid dynamics. The results (both analytical and numerical) show the existence of the small-amplitude travelling waves, which move with constant velocity for wave numbers in the intermediate range. We show that the stability of the system is ensured in the long wave limit, a fact which is in agreement with the real float glass process. We analyze the stability for a wide range of wave numbers, Reynolds, Weber and Grashof number, and explain the physical implications on the dynamics of the problem. The consequences of the linear stability results are discussed. In reality in the float glass process, the temperature strongly influences the viscosity of both molten metal and hot glass, which will have direct consequences on the stability of the system. We investigate the linear stability of two superposed fluids with temperature dependent viscosities by considering a different model for the viscosity dependence of each fluid. Although, the temperature-viscosity relationships for glass and metal are more complex than those used in our computations, our intention is to emphasize the effects of this dependence on the stability of the system. It is known from the literature that in the case of one fluid, the heat, which causes viscosity to decrease along the domain, usually destabilizes the flow. For the two superposed fluids problem we investigate this behaviour and discuss the consequences of the linear stability in this new case.

In modern textile manufacturing industries, the function of human eyes to detect disturbances in the production processes which yield defective products is switched to cameras. The camera images are analyzed with various methods to detect these disturbances automatically. There are, however, still problems with in particular semi-regular textures which are typical for weaving patterns. We study three parts of that problem of automatic texture analysis: image smoothing, texture synthesis and defect detection. In image smoothing, we develop a two dimensional kernel smoothing method with locally and directionally adaptive bandwidths allowing correlation in the errors. Two approaches are used in synthesising texture. The first is based on constructing a generalized Ising energy function in the Markov Random Field setup, and for the second, we use two-dimensional periodic bootstrap methods for semi-regular texture synthesis. We treat defect detection as multihypothesis testing problem with the null hypothesis representing the absence of defects and the other hypotheses representing various types of defects. We develop a test based on a nonparametric regression setup, and we use the bootstrap for approximating the distribution of our test statistic.

In the first part of this work, called Simple node singularity, are computed matrix factorizations of all isomorphism classes, up to shiftings, of rank one and two, graded, indecomposable maximal Cohen--Macaulay (shortly MCM) modules over the affine cone of the simple node singularity. The subsection 2.2 contains a description of all rank two graded MCM R-modules with stable sheafification on the projective cone of R, by their matrix factorizations. It is given also a general description of such modules, of any rank, over a projective curve of arithmetic genus 1, using their matrix factorizations. The non-locally free rank two MCM modules are computed using an alghorithm presented in the Introduction of this work, that gives a matrix factorization of any extension of two MCM modules over a hypersurface. In the second part, called Fermat surface, are classified all graded, rank two, MCM modules over the affine cone of the Fermat surface. For the classification of the orientable rank two graded MCM R-modules, is used a description of the orientable modules (over normal rings) with the help of codimension two Gorenstein ideals, realized by Herzog and Kühl. It is proven (in section 4), that they have skew symmetric matrix factorizations (over any normal hypersurface ring). For the classification of the non-orientable rank two MCM R-modules, we use a similar idea as in the case of the orientable ones, only that the ideal is not any more Gorenstein.

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.

It is considered an analytical model of defaultable bond portfolio in terms of its face value process. The face value process dynamically evolves with time and incorporates changes caused by recovery payment on default followed by purchasing of new bonds. The further studies involve properties, distribution and control of the face value process.

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.

An autoregressive-ARCH model with possible exogeneous variables is treated. We estimate the conditional volatility of the model by applying feedforward networks to the residuals and prove consistency and asymptotic normality for the estimates under the rate of feedforward networks complexity. Recurrent neural networks estimates of GARCH and value-at-risk is studied. We prove consistency and asymptotic normality for the recurrent neural networks ARMA estimator under the rate of recurrent networks complexity. We also overcome the estimation problem in stochastic variance models in discrete time by feedforward networks and the introduction of a new distributions on the innovations. We use the method to calculate market risk such as expected shortfall and Value-at risk. We tested this distribution together with other new distributions on the GARCH family models against other common distributions on the financial market such as Normal Inverse Gaussian, normal and the Student's t- distributions. As an application of the models, some German stocks are studied and the different approaches are compared together with the most common method of GARCH(1,1) fit.

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.

This thesis investigates the constrained form of the spherical Minimax location problem and the spherical Weber location problem. Specifically, we consider the problem of locating a new facility on the surface of the unit sphere in the presence of convex spherical polygonal restricted regions and forbidden regions such that the maximum weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized and the sum of the weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized. It is assumed that a forbidden region is an area on the surface of the unit sphere where travel and facility location are not permitted and that distance is measured using the great circle arc distance. We represent a polynomial time algorithm for the spherical Minimax location problem for the special case where all the existing facilities are located on the surface of a hemisphere. Further, we have developed algorithms for spherical Weber location problem using barrier distance on a hemisphere as well as on the unit sphere.

In traditional portfolio optimization under the threat of a crash the investment horizon or time to maturity is neglected. Developing the so-called crash hedging strategies (which are portfolio strategies which make an investor indifferent to the occurrence of an uncertain (down) jumps of the price of the risky asset) the time to maturity turns out to be essential. The crash hedging strategies are derived as solutions of non-linear differential equations which itself are consequences of an equilibrium strategy. Hereby the situation of changing market coefficients after a possible crash is considered for the case of logarithmic utility as well as for the case of general utility functions. A benefit-cost analysis of the crash hedging strategy is done as well as a comparison of the crash hedging strategy with the optimal portfolio strategies given in traditional crash models. Moreover, it will be shown that the crash hedging strategies optimize the worst-case bound for the expected utility from final wealth subject to some restrictions. Another application is to model crash hedging strategies in situations where both the number and the height of the crash are uncertain but bounded. Taking the additional information of the probability of a possible crash happening into account leads to the development of the q-quantile crash hedging strategy.

In the filling process of a car tank, the formation of foam plays an unwanted role, as it may prevent the tank from being completely filled or at least delay the filling. Therefore it is of interest to optimize the geometry of the tank using numerical simulation in such a way that the influence of the foam is minimized. In this dissertation, we analyze the behaviour of the foam mathematically on the mezoscopic scale, that is for single lamellae. The most important goals are on the one hand to gain a deeper understanding of the interaction of the relevant physical effects, on the other hand to obtain a model for the simulation of the decay of a lamella which can be integrated in a global foam model. In the first part of this work, we give a short introduction into the physical properties of foam and find that the Marangoni effect is the main cause for its stability. We then develop a mathematical model for the simulation of the dynamical behaviour of a lamella based on an asymptotic analysis using the special geometry of the lamella. The result is a system of nonlinear partial differential equations (PDE) of third order in two spatial and one time dimension. In the second part, we analyze this system mathematically and prove an existence and uniqueness result for a simplified case. For some special parameter domains the system can be further simplified, and in some cases explicit solutions can be derived. In the last part of the dissertation, we solve the system using a finite element approach and discuss the results in detail.

Nowadays one of the major objectives in geosciences is the determination of the gravitational field of our planet, the Earth. A precise knowledge of this quantity is not just interesting on its own but it is indeed a key point for a vast number of applications. The important question is how to obtain a good model for the gravitational field on a global scale. The only applicable solution - both in costs and data coverage - is the usage of satellite data. We concentrate on highly precise measurements which will be obtained by GOCE (Gravity Field and Steady State Ocean Circulation Explorer, launch expected 2006). This satellite has a gradiometer onboard which returns the second derivatives of the gravitational potential. Mathematically seen we have to deal with several obstacles. The first one is that the noise in the different components of these second derivatives differs over several orders of magnitude, i.e. a straightforward solution of this outer boundary value problem will not work properly. Furthermore we are not interested in the data at satellite height but we want to know the field at the Earth's surface, thus we need a regularization (downward-continuation) of the data. These two problems are tackled in the thesis and are now described briefly. Split Operators: We have to solve an outer boundary value problem at the height of the satellite track. Classically one can handle first order side conditions which are not tangential to the surface and second derivatives pointing in the radial direction employing integral and pseudo differential equation methods. We present a different approach: We classify all first and purely second order operators which fulfill that a harmonic function stays harmonic under their application. This task is done by using modern algebraic methods for solving systems of partial differential equations symbolically. Now we can look at the problem with oblique side conditions as if we had ordinary i.e. non-derived side conditions. The only additional work which has to be done is an inversion of the differential operator, i.e. integration. In particular we are capable to deal with derivatives which are tangential to the boundary. Auto-Regularization: The second obstacle is finding a proper regularization procedure. This is complicated by the fact that we are facing stochastic rather than deterministic noise. The main question is how to find an optimal regularization parameter which is impossible without any additional knowledge. However we could show that with a very limited number of additional information, which are obtainable also in practice, we can regularize in an asymptotically optimal way. In particular we showed that the knowledge of two input data sets allows an order optimal regularization procedure even under the hard conditions of Gaussian white noise and an exponentially ill-posed problem. A last but rather simple task is combining data from different derivatives which can be done by a weighted least squares approach using the information we obtained out of the regularization procedure. A practical application to the downward-continuation problem for simulated gravitational data is shown.

In this dissertation we consider complex, projective hypersurfaces with many isolated singularities. The leading questions concern the maximal number of prescribed singularities of such hypersurfaces in a given linear system, and geometric properties of the equisingular stratum. In the first part a systematic introduction to the theory of equianalytic families of hypersurfaces is given. Furthermore, the patchworking method for constructing hypersurfaces with singularities of prescribed types is described. In the second part we present new existence results for hypersurfaces with many singularities. Using the patchworking method, we show asymptotically proper results for hypersurfaces in P^n with singularities of corank less than two. In the case of simple singularities, the results are even asymptotically optimal. These statements improve all previous general existence results for hypersurfaces with these singularities. Moreover, the results are also transferred to hypersurfaces defined over the real numbers. The last part of the dissertation deals with the Castelnuovo function for studying the cohomology of ideal sheaves of zero-dimensional schemes. Parts of the theory of this function for schemes in P^2 are generalized to the case of schemes on general surfaces in P^3. As an application we show an H^1-vanishing theorem for such schemes.

In this text we survey some large deviation results for diffusion processes. The first chapters present results from the literature such as the Freidlin-Wentzell theorem for diffusions with small noise. We use these results to prove a new large deviation theorem about diffusion processes with strong drift. This is the main result of the thesis. In the later chapters we give another application of large deviation results, namely to determine the exponential decay rate for the Bayes risk when separating two different processes. The final chapter presents techniques which help to experiment with rare events for diffusion processes by means of computer simulations.

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.

In this thesis we propose an efficient method to compute the automorphism group of an arbitrary hyperelliptic function field over a given constant field of odd characteristic as well as over its algebraic extensions. Beside theoretical applications, knowing the automorphism group also is useful in cryptography: The Jacobians of hyperelliptic curves have been suggested by Koblitz as groups for cryptographic purposes, because the discrete logarithm is believed to be hard in this kind of groups. In order to obtain "secure" Jacobians, it is necessary to prevent attacks like Pohlig/Hellman's and Duursma/Gaudry/Morain's. The latter is only feasible, if the corresponding function field has an automorphism of large order. According to a theorem by Madan, automorphisms seem to allow the Pohlig/Hellman attack, too. Hence, the function field of a secure Jacobian will most likely have trivial automorphism group. In other words: Computing the automorphism group of a hyperelliptic function field promises to be a quick test for insecure Jacobians. Let us outline our algorithm for computing the automorphism group Aut(F/k) of a hyperelliptic function field F/k. It is well known that Aut(F/k) is finite. For each possible subgroup U of Aut(F/k), Rolf Brandt has given a normal form for F if k is algebraically closed. Hence our problem reduces to deciding, whether a given hyperelliptic function field F=k(x,y), y^2=D_x has a defining equation of the form given by Brandt. This question can be answered using theorem III.18: We have F=k(t,u), u^2=D_t iff x is a fraction of linear polynomials in t and y=pu, where the factor p is a rational function w.r.t. t which can be determined explicitly from the coefficients of x. This condition can be checked efficiently using Gröbner basis techniques. With additional effort, it is also possible to compute Aut(F/k) if k is not algebraically closed. Investigating a huge number of examples one gets the impression that the above motivation of getting a quick test for insecure Jacobians is partially fulfilled: The computation of automorphism groups is quite fast using the suggested algorithm. Furthermore, fields with nontrivial automorphism groups seem to have insecure Jacobians. Only fields of small characteristic seem to have a reasonable chance of having nontrivial automorphisms. Hence, from a cryptographic point of view, computing Aut(F/k) seems to make sense whenever k has small characteristic.

The question of how to model dependence structures between financial assets was revolutionized since the last decade when the copula concept was introduced in financial research. Even though the concept of splitting marginal behavior and dependence structure (described by a copula) of multidimensional distributions already goes back to Sklar (1955) and Hoeffding (1940), there were very little empirical efforts done to check out the potentials of this approach. The aim of this thesis is to figure out the possibilities of copulas for modelling, estimating and validating purposes. Therefore we extend the class of Archimedean Copulas via a transformation rule to new classes and come up with an explicit suggestion covering the Frank and Gumbel family. We introduce a copula based mapping rule leading to joint independence and as results of this mapping we present an easy method of multidimensional chi²-testing and a new estimate for high dimensional parametric distributions functions. Different ways of estimating the tail dependence coefficient, describing the asymptotic probability of joint extremes, are compared and improved. The limitations of elliptical distributions are carried out and a generalized form of them, preserving their applicability, is developed. We state a method to split a (generalized) elliptical distribution into its radial and angular part. This leads to a positive definite robust estimate of the dispersion matrix (here only given as a theoretical outlook). The impact of our findings is stated by modelling and testing the return distributions of stock- and currency portfolios furthermore of oil related commodities- and LME metal baskets. In addition we show the crash stability of real estate based firms and the existence of nonlinear dependence in between the yield curve.

In this thesis we show that the theory of algebraic correspondences introduced by Deuring in the 1930s can be applied to construct non-trivial homomorphisms between the Jacobi groups of hyperelliptic function fields. Concretely, we deduce algorithms to add and multiply correspondences which perform in a reasonable time if the degrees of the associated divisors of the double field are small. Moreover, we show how to compute the differential matrices associated to prime divisors of the double field for arbitrary genus. These matrices give a representation for the homomorphisms or endomorphisms in the additive group (ring) of matrices which is even faithful if the ground field has characteristic zero. As first examples for non-trivial correspondences we investigate multiplication by m endomorphisms. Afterwards we use factorisations of certain bivariate polynomials to construct prime divisors of the double field that are not equivalent to 0 in a coarser sense. Applying the theory of Deuring, these divisors yield homomorphisms between the Jacobi groups of special classes of hyperelliptic function fields. Finally, we generalise the Richelot isogeny to higher genus and by this way derive a class of hyperelliptic function fields given in terms of their defining polynomials which admit non-trivial homomorphisms. These include homomorphisms between the Jacobi groups of hyperelliptic curves of different as well as of equal genus. In addition we provide an explicit method to construct genus 2 function fields the endomorphism ring of which contains a sqrt(2) multiplication with the help of the Cholesky decomposition of a certain matrix.

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.

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.