### Filtern

#### Erscheinungsjahr

- 2009 (16) (entfernen)

#### Dokumenttyp

- Dissertation (16) (entfernen)

#### Schlagworte

- Algebraische Geometrie (2)
- Derivat <Wertpapier> (2)
- Finanzmathematik (2)
- Numerische Mathematik (2)
- illiquidity (2)
- Advanced Encryption Standard (1)
- Algebraic geometry (1)
- Approximationsalgorithmus (1)
- Arbitrage (1)
- Bewertung (1)

#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (16) (entfernen)

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.

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 \(\tilde{\mathcal M}\).

Das zentrale Thema dieser Arbeit sind vollständig gekoppelte reflektierte Vorwärts-Rückwärts-Stochastische-Differentialgleichungen (FBSDE). Zunächst wird ein Spezialfall, die teilweise gekoppelten FBSDE, betrachtet und deren Verbindung zur Bewertung Amerikanischer Optionen aufgezeigt. Für die Lösung dieser Gleichung wird Monte-Carlo-Simulation benötigt, daher werden verschiedene Varianzreduktionsmaßnahmen erarbeitet und miteinander verglichen. Im Folgenden wird der allgemeinere Fall der vollständig gekoppelten reflektierten FBSDE behandelt. Es wird gezeigt, wie das Problem der Lösung dieser Gleichungen in ein Optimierungsproblem übertragen werden kann und infolgedessen mit numerischen Methoden aus diesem Bereich der Mathematik bearbeitet werden kann. Abschließend folgen Vergleiche der verschiedenen numerischen Ansätze mit bereits existierenden Verfahren.

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.

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.

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.

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.

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 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.

In der Automobilindustrie muss der Nachweis von Bauteilzuverlässigkeiten auf statistischen Verfahren basieren, da die Bauteilfestigkeit und Kundenbeanspruchung streuen. Die bisherigen Vorgehensweisen der Tests führen häufig Fehlentscheidungen bzgl. der Freigabe, was unnötige Design-Änderungen und somit hohe Kosten bedeuten kann. In vorliegender Arbeit wird der Ansatz der partiellen Durchläuferzählung entwickelt, welche die statische Güte der bisherigen Testverfahren (Success Runs) erhöht.