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This thesis considers a model of a scalar partial differential equation in the presence of a singular source term, modeling the interaction between an inviscid fluid represented by the Burgers equation and an arbitrary, finite amount of particles moving inside the fluid, each one acting as a point-wise drag force with a particle related friction constant.
\begin{align*}
\partial_t u + \partial_x (u^2/2) &= \sum_{i \in N(t)} \lambda_i \Big(h_i'(t)-u(t,h_i(t)\Big)\delta(x-h_i(t))
\end{align*}
The model was introduced for the case of a single particle by Lagoutière, Seguin and Takahashi, is a first step towards a better understanding of interaction between fluids and solids on the level of partial differential equations and has the unique property of considering entropy admissible solutions and the interaction with shockwaves.
The model is extended to an arbitrary, finite number of particles and interactions like merging, splitting and crossing of particle paths are considered.
The theory of entropy admissibility is revisited for the cases of interfaces and discontinuous flux conservation laws, existing results are summarized and compared, and adapted for regions of particle interactions. To this goal, the theory of germs introduced by Andreianov, Karlsen and Risebro is extended to this case of non-conservative interface coupling.
Exact solutions for the Riemann Problem of particles drifting apart are computed and analysis on the behavior of entropy solutions across the particle related interfaces is used to determine physically relevant and consistent behavior for merging and splitting of particles. Well-posedness of entropy solutions to the Cauchy problem is proven, using an explicit construction method, L-infinity bounds, an approximation of the particle paths and compactness arguments to obtain existence of entropy solutions. Uniqueness is shown in the class of weak entropy solutions using almost classical Kruzkov-type analysis and the notion of L1-dissipative germs.
Necessary fundamentals of hyperbolic conservation laws, including weak solutions, shocks and rarefaction waves and the Rankine-Hugoniot condition are briefly recapitulated.
Ill-posed optimization problems appear in a wide range of mathematical applications, and their numerical solution requires the use of appropriate regularization techniques. In order to understand these techniques, a thorough analysis is inevitable.
The main subject of this book are quadratic optimal control problems subject to elliptic linear or semi-linear partial differential equations. Depending on the structure of the differential equation, different regularization techniques are employed, and their analysis leads to novel results such as rate of convergence estimates.
Beatty sets (also called Beatty sequences) have appeared as early as 1772 in the astronomical studies of Johann III Bernoulli as a tool for easing manual calculations and - as Elwin Bruno Christoffel pointed out in 1888 - lend themselves to exposing intricate properties of the real irrationals. Since then, numerous researchers have explored a multitude of arithmetic properties of Beatty sets; the interrelation between Beatty sets and modular inversion, as well as Beatty sets and the set of rational primes, being the central topic of this book. The inquiry into the relation to rational primes is complemented by considering a natural generalisation to imaginary quadratic number fields.
The present thesis considers the modelling of gas mixtures via a kinetic description. Fundamentals about the Boltzmann equation for gas mixtures and the BGK approximation are presented. Especially, issues in extending these models to gas mixtures are discussed. A non-reactive two component gas mixture is considered. The two species mixture is modelled by a system of kinetic BGK equations featuring two interaction terms to account for momentum and energy transfer between the two species. The model presented here contains several models from physicists and engineers as special cases. Consistency of this model is proven: conservation properties, positivity of all temperatures and the H-theorem. The form in global equilibrium as Maxwell distributions is specified. Moreover, the usual macroscopic conservation laws can be derived.
In the literature, there is another type of BGK model for gas mixtures developed by Andries, Aoki and Perthame, which contains only one interaction term. In this thesis, the advantages of these two types of models are discussed and the usefulness of the model presented here is shown by using this model to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described in the literature by Dellacherie. In addition, for each of the two models existence and uniqueness of mild solutions is shown. Moreover, positivity of classical solutions is proven.
Then, the model presented here is applied to three physical applications: a plasma consisting of ions and electrons, a gas mixture which deviates from equilibrium and a gas mixture consisting of polyatomic molecules.
First, the model is extended to a model for charged particles. Then, the equations of magnetohydrodynamics are derived from this model. Next, we want to apply this extended model to a mixture of ions and electrons in a special physical constellation which can be found for example in a Tokamak. The mixture is partly in equilibrium in some regions, in some regions it deviates from equilibrium. The model presented in this thesis is taken for this purpose, since it has the advantage to separate the intra and interspecies interactions. Then, a new model based on a micro-macro decomposition is proposed in order to capture the physical regime of being partly in equilibrium, partly not. Theoretical results are presented, convergence rates to equilibrium in the space-homogeneous case and the Landau damping for mixtures, in order to compare it with numerical results.
Second, the model presented here is applied to a gas mixture which deviates from equilibrium such that it is described by Navier-Stokes equations on the macroscopic level. In this macroscopic description it is expected that four physical coefficients will show up, characterizing the physical behaviour of the gases, namely the diffusion coefficient, the viscosity coefficient, the heat conductivity and the thermal diffusion parameter. A Chapman-Enskog expansion of the model presented here is performed in order to capture three of these four physical coefficients. In addition, several possible extensions to an ellipsoidal statistical model for gas mixtures are proposed in order to capture the fourth coefficient. Three extensions are proposed: An extension which is as simple as possible, an intuitive extension copying the one species case and an extension which takes into account the physical motivation of the physicist Holway who invented the ellipsoidal statistical model for one species. Consistency of the extended models like conservation properties, positivity of all temperatures and the H-theorem are proven. The shape of global Maxwell distributions in equilibrium are specified.
Third, the model presented here is applied to polyatomic molecules. A multi component gas mixture with translational and internal energy degrees of freedom is considered. The two species are allowed to have different degrees of freedom in internal energy and are modelled by a system of kinetic ellipsoidal statistical equations. Consistency of this model is shown: conservation properties, positivity of the temperature, H-theorem and the form of Maxwell distributions in equilibrium. For numerical purposes the Chu reduction is applied to the developed model for polyatomic gases to reduce the complexity of the model and an application for a gas consisting of a mono-atomic and a diatomic gas is given.
Last, the limit from the model presented here to the dissipative Euler equations for gas mixtures is proven.
This work is concerned with the numerical approximation of solutions to models that are used to describe atmospheric or oceanographic flows. In particular, this work concen- trates on the approximation of the Shallow Water equations with bottom topography and the compressible Euler equations with a gravitational potential. Numerous methods have been developed to approximate solutions of these models. Of specific interest here are the approximations of near equilibrium solutions and, in the case of the Euler equations, the low Mach number flow regime. It is inherent in most of the numerical methods that the quality of the approximation increases with the number of degrees of freedom that are used. Therefore, these schemes are often run in parallel on big computers to achieve the best pos- sible approximation. However, even on those big machines, the desired accuracy can not be achieved by the given maximal number of degrees of freedom that these machines allow. The main focus in this work therefore lies in the development of numerical schemes that give better resolution of the resulting dynamics on the same number of degrees of freedom, compared to classical schemes.
This work is the result of a cooperation of Prof. Klingenberg of the Institute of Mathe- matics in Wu¨rzburg and Prof. R¨opke of the Astrophysical Institute in Wu¨rzburg. The aim of this collaboration is the development of methods to compute stellar atmospheres. Two main challenges are tackled in this work. First, the accurate treatment of source terms in the numerical scheme. This leads to the so called well-balanced schemes. They allow for an accurate approximation of near equilibrium dynamics. The second challenge is the approx- imation of flows in the low Mach number regime. It is known that the compressible Euler equations tend towards the incompressible Euler equations when the Mach number tends to zero. Classical schemes often show excessive diffusion in that flow regime. The here devel- oped scheme falls into the category of an asymptotic preserving scheme, i.e. the numerical scheme reflects the behavior that is computed on the continuous equations. Moreover, it is shown that the diffusion of the numerical scheme is independent of the Mach number.
In chapter 3, an HLL-type approximate Riemann solver is adapted for simulations of the Shallow Water equations with bottom topography to develop a well-balanced scheme. In the literature, most schemes only tackle the equilibria when the fluid is at rest, the so called Lake at rest solutions. Here a scheme is developed to accurately capture all the equilibria of the Shallow Water equations. Moreover, in contrast to other works, a second order extension is proposed, that does not rely on an iterative scheme inside the reconstruction procedure, leading to a more efficient scheme.
In chapter 4, a Suliciu relaxation scheme is adapted for the resolution of hydrostatic equilibria of the Euler equations with a gravitational potential. The hydrostatic relations are underdetermined and therefore the solutions to that equations are not unique. However, the scheme is shown to be well-balanced for a wide class of hydrostatic equilibria. For specific classes, some quadrature rules are computed to ensure the exact well-balanced property. Moreover, the scheme is shown to be robust, i.e. it preserves the positivity of mass and energy, and stable with respect to the entropy. Numerical results are presented in order to investigate the impact of the different quadrature rules on the well-balanced property.
In chapter 5, a Suliciu relaxation scheme is adapted for the simulations of low Mach number flows. The scheme is shown to be asymptotic preserving and not suffering from excessive diffusion in the low Mach number regime. Moreover, it is shown to be robust under certain parameter combinations and to be stable from an Chapman-Enskog analysis.
Numerical results are presented in order to show the advantages of the new approach.
In chapter 6, the schemes developed in the chapters 4 and 5 are combined in order to investigate the performance of the numerical scheme in the low Mach number regime in a gravitational stratified atmosphere. The scheme is shown the be well-balanced, robust and stable with respect to a Chapman-Enskog analysis. Numerical tests are presented to show the advantage of the newly proposed method over the classical scheme.
In chapter 7, some remarks on an alternative way to tackle multidimensional simulations are presented. However no numerical simulations are performed and it is shown why further research on the suggested approach is necessary.
Finite volume methods for compressible Euler equations suffer from an excessive diffusion in the limit of low Mach numbers. This PhD thesis explores new approaches to overcome this.
The analysis of a simpler set of equations that also possess a low Mach number limit is found to give valuable insights. These equations are the acoustic equations obtained as a linearization of the Euler equations. For both systems the limit is characterized by a divergencefree velocity. This constraint is nontrivial only in multiple spatial dimensions. As the Jacobians of the acoustic system do not commute, acoustics cannot be reduced to some kind of multi-dimensional advection. Therefore first an exact solution in multiple spatial dimensions is obtained. It is shown that the low Mach number limit can be interpreted as a limit of long times.
It is found that the origin of the inability of a scheme to resolve the low Mach number limit is the lack a discrete counterpart to the limit of long times. Numerical schemes whose discrete stationary states discretize all the analytic stationary states of the PDE are called stationarity preserving. It is shown that for the acoustic equations, stationarity preserving schemes are vorticity preserving and are those that are able to resolve the low Mach limit (low Mach compliant). This establishes a new link between these three concepts.
Stationarity preservation is studied in detail for both dimensionally split and multi-dimensional schemes for linear acoustics. In particular it is explained why the same multi-dimensional stencils appear in literature in very different contexts: These stencils are unique discretizations of the divergence that allow for stabilizing stationarity preserving diffusion.
Stationarity preservation can also be generalized to nonlinear systems such as the Euler equations. Several ways how such numerical schemes can be constructed for the Euler equations are presented. In particular a low Mach compliant numerical scheme is derived that uses a novel construction idea. Its diffusion is chosen such that it depends on the velocity divergence rather than just derivatives of the different velocity components. This is demonstrated to overcome the low Mach number problem. The scheme shows satisfactory results in numerical simulations and has been found to be stable under explicit time integration.
An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework.
The main theme of this thesis is the development of multigrid and hierarchical matrix solution procedures with almost linear computational complexity for classes of partial integro-differential problems. An elliptic partial integro-differential equation, a convection-diffusion partial integro-differential equation and a convection-diffusion partial integro-differential optimality system are investigated. In the first part of this work, an efficient multigrid finite-differences scheme for solving an elliptic
Fredholm partial integro-differential equation (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization and a Simpson's quadrature rule to approximate the PIDE problem and a multigrid scheme and a fast
multilevel integration method of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme
are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework that includes numerical experiments for elliptic PIDE problems with singular kernels. The experience gained in this part of the work is used for the investigation of convection diffusion partial-integro differential equations in the second part of this thesis.
Convection-diffusion PIDE problems are discretized using a finite volume scheme referred to as the Chang and Cooper (CC) scheme and a quadrature rule. Also for this class of PIDE problems and this numerical setting, a stability and accuracy analysis of the CC scheme combined with a Simpson's quadrature rule is presented proving second-order accuracy of the numerical solution. To extend and investigate the proposed approximation and solution strategy to the case of systems of convection-diffusion PIDE, an optimal control problem governed by this model is considered. In this case the research focus is the CC-Simpson's discretization of the optimality system and its solution by the proposed multigrid strategy. Second-order accuracy of the optimization solution is proved and results of local Fourier analysis are presented that provide sharp convergence estimates of the optimal computational complexity of the multigrid-fast integration technique.
While (geometric) multigrid techniques require ad-hoc implementation depending on the structure of the PIDE problem and on the dimensionality of the domain where the problem is considered, the hierarchical matrix framework allows a more general treatment that exploits the algebraic structure of the problem at hand. In this thesis, this framework is extended to the case of combined differential and integral problems considering the case of a convection-diffusion PIDE. In this case, the starting point is the CC discretization of the convection-diffusion operator combined with the trapezoidal quadrature rule. The hierarchical matrix approach exploits the algebraic nature of the hierarchical matrices for blockwise approximations by low-rank matrices of the sparse convection-diffusion approximation and enables data sparse representation of the fully populated matrix where all essential matrix operations are performed with at most logarithmic optimal complexity. The factorization of part of or the whole coefficient matrix is used as a preconditioner to the solution of the PIDE problem using a generalized minimum residual (GMRes) procedure as a solver.
Numerical analysis estimates of the accuracy of the
finite-volume and trapezoidal rule approximation are
presented and combined with estimates of the
hierarchical matrix approximation and with the
accuracy of the GMRes iterates. Results of numerical experiments are reported that
successfully validate the theoretical estimates and
the optimal computational complexity of the proposed hierarchical matrix
solution procedure. These results include an extension to higher dimensions and an application to the time evolution of the probability density function of a jump diffusion process.
Über die besondere Bedeutung von Analogiebildungsprozessen beim Lernen im Allgemeinen und beim Lernen von Mathematik im Speziellen besteht ein breiter wissenschaftlicher Konsens. Es liegt deshalb nahe, von einem lernförderlichen Mathematikunterricht zu verlangen, dass er im Bewusstsein dieser Bedeutung entwickelt ist – dass er also einerseits Analogien aufzeigt und sich diese beim Lehren von Mathematik zunutze macht, dass er andererseits aber auch dem Lernenden Gelegenheiten bietet, Analogien zu erkennen und zu entwickeln. Kurz: Die Fähigkeit zum Bilden von Analogien soll durch den Unterricht gezielt gefördert werden.
Um diesem Anspruch gerecht werden zu können, müssen ausreichende Kenntnisse darüber vorliegen, wie Analogiebildungsprozesse beim Lernen von Mathematik und beim Lösen mathematischer Aufgaben ablaufen, wodurch sich erfolgreiche Analogiebildungsprozesse auszeichnen und an welchen Stellen möglicherweise Schwierigkeiten bestehen.
Der Autor zeigt auf, wie Prozesse der Analogiebildung beim Lösen mathematischer Aufgaben initiiert, beobachtet, beschrieben und interpretiert werden können, um auf dieser Grundlage Ansatzpunkte für geeignete Fördermaßnahmen zu identifizieren, bestehende Ideen zur Förderung der Analogiebildungsfähigkeit zu beurteilen und neue Ideen zu entwickeln. Es werden dabei Wege der Analogiebildung nachgezeichnet und untersucht, die auf der Verschränkung zweier Dimensionen der Analogiebildung im Rahmen des zugrundeliegenden theoretischen Modells beruhen. So können verschiedene Vorgehensweisen ebenso kontrastiert werden, wie kritische Punkte im Verlauf eines Analogiebildungsprozesses. Es ergeben sich daraus Unterrichtsvorschläge, die auf den Ideen zum beispielbasierten Lernen aufbauen.
This doctoral thesis provides a classification of equivariant star products (star products together with quantum momentum maps) in terms of equivariant de Rham cohomology. This classification result is then used to construct an analogon of the Kirwan map from which one can directly obtain the characteristic class of certain reduced star products on Marsden-Weinstein reduced symplectic manifolds from the equivariant characteristic class of their corresponding unreduced equivariant star product. From the surjectivity of this map one can conclude that every star product on Marsden-Weinstein reduced symplectic manifolds can (up to equivalence) be obtained as a reduced equivariant star product.
This dissertation is dealing with three mathematical areas, namely polynomial matrices over finite fields, linear systems and coding theory.
Coprimeness properties of polynomial matrices provide criteria for the reachability and observability of interconnected linear systems. Since time-discrete linear systems over finite fields and convolutional codes are basically the same objects, these results could be transfered to criteria for non-catastrophicity of convolutional codes.
We calculate the probability that specially structured polynomial matrices are right prime. In particular, formulas for the number of pairwise coprime polynomials and for the number of mutually left coprime polynomial matrices are calculated. This leads to the probability that a parallel connected linear system is reachable and that a parallel connected convolutional codes is non-catastrophic.
Moreover, the corresponding probabilities are calculated for other networks of linear systems and convolutional codes, such as series connection.
Furthermore, the probabilities that a convolutional codes is MDP and that a clock code is MDS are approximated.
Finally, we consider the probability of finding a solution for a linear network coding problem.
In this work, multi-particle quantum optimal control problems are studied in the framework of time-dependent density functional theory (TDDFT).
Quantum control problems are of great importance in both fundamental research and application of atomic and molecular systems. Typical applications are laser induced chemical reactions, nuclear magnetic resonance experiments, and quantum computing.
Theoretically, the problem of how to describe a non-relativistic system of multiple particles is solved by the Schrödinger equation (SE). However, due to the exponential increase in numerical complexity with the number of particles, it is impossible to directly solve the Schrödinger equation for large systems of interest. An efficient and successful approach to overcome this difficulty is the framework of TDDFT and the use of the time-dependent Kohn-Sham (TDKS) equations therein.
This is done by replacing the multi-particle SE with a set of nonlinear single-particle Schrödinger equations that are coupled through an additional potential.
Despite the fact that TDDFT is widely used for physical and quantum chemical calculation and software packages for its use are readily available, its mathematical foundation is still under active development and even fundamental issues remain unproven today.
The main purpose of this thesis is to provide a consistent and rigorous setting for the TDKS equations and of the related optimal control problems.
In the first part of the thesis, the framework of density functional theory (DFT) and TDDFT are introduced. This includes a detailed presentation of the different functional sets forming DFT. Furthermore, the known equivalence of the TDKS system to the original SE problem is further discussed.
To implement the TDDFT framework for multi-particle computations, the TDKS equations provide one of the most successful approaches nowadays. However, only few mathematical results concerning these equations are available and these results do not cover all issues that arise in the formulation of optimal control problems governed by the TDKS model.
It is the purpose of the second part of this thesis to address these issues such as higher regularity of TDKS solutions and the case of weaker requirements on external (control) potentials that are instrumental for the formulation of well-posed TDKS control problems. For this purpose, in this work, existence and uniqueness of TDKS solutions are investigated in the Galerkin framework and using energy estimates for the nonlinear TDKS equations.
In the third part of this thesis, optimal control problems governed by the TDKS model are formulated and investigated. For this purpose, relevant cost functionals that model the purpose of the control are discussed.
Henceforth, TDKS control problems result from the requirement of optimising the given cost functionals subject to the differential constraint given by the TDKS equations. The analysis of these problems is novel and represents one of the main contributions of the present thesis.
In particular, existence of minimizers is proved and their characterization by TDKS optimality systems is discussed in detail.
To this end, Fréchet differentiability of the TDKS model and of the cost functionals is addressed considering \(H^1\) cost of the control.
This part is concluded by deriving the reduced gradient in the \(L^2\) and \(H^1\) inner product.
While the \(L^2\) optimization is widespread in the literature, the choice of the \(H^1\) gradient is motivated in this work by theoretical consideration and by resulting numerical advantages.
The last part of the thesis is devoted to the numerical approximation of the TDKS optimality systems and to their solution by gradient-based optimization techniques.
For the former purpose, Strang time-splitting pseudo-spectral schemes are discussed including a review of some recent theoretical estimates for these schemes and a numerical validation of these estimates.
For the latter purpose, nonlinear (projected) conjugate gradient methods are implemented and are used to validate the theoretical analysis of this thesis with results of numerical experiments with different cost functional settings.
An explicit Runge-Kutta discontinuous Galerkin (RKDG) method is used to device numerical schemes for both the compressible Euler equations of gas dynamics and the ideal magneto- hydrodynamical (MHD) model. These systems of conservation laws are known to have discontinuous solutions. Discontinuities are the source of spurious oscillations in the solution profile of the numerical approximation, when a high order accurate numerical method is used. Different techniques are reviewed in order to control spurious oscillations. A shock detection technique is shown to be useful in order to determine the regions where the spurious oscillations appear such that a Limiter can be used to eliminate these numeric artifacts. To guarantee the positivity of specific variables like the density and the pressure, a positivity preserving limiter is used. Furthermore, a numerical flux, proven to preserve the entropy stability of the semi-discrete DG scheme for the MHD system is used. Finally, the numerical schemes are implemented using the deal.II C++ libraries in the dflo code. The solution of common test cases show the capability of the method.
A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.
This doctoral thesis is concerned with the mathematical modeling of magnetoelastic materials and the analysis of PDE systems describing these materials and obtained from a variational approach.
The purpose is to capture the behavior of elastic particles that are not only magnetic but exhibit a magnetic domain structure which is well described by the micromagnetic energy and the Landau-Lifshitz-Gilbert equation of the magnetization. The equation of motion for the material’s velocity is derived in a continuum mechanical setting from an energy ansatz. In the modeling process, the focus is on the interplay between Lagrangian and Eulerian coordinate systems to combine elasticity and magnetism in one model without the assumption of small deformations.
The resulting general PDE system is simplified using special assumptions. Existence of weak solutions is proved for two variants of the PDE system, one including gradient flow dynamics on the magnetization, and the other featuring the Landau-Lifshitz-Gilbert equation. The proof is based on a Galerkin method and a fixed point argument. The analysis of the PDE system with the Landau-Lifshitz-Gilbert equation uses a more involved approach to obtain weak solutions based on G. Carbou and P. Fabrie 2001.
Background
HIV-disease progression correlates with immune activation. Here we investigated whether corticosteroid treatment can attenuate HIV disease progression in antiretroviral-untreated patients.
Methods
Double-blind, placebo-controlled randomized clinical trial including 326 HIV-patients in a resource-limited setting in Tanzania (clinicaltrials.gov NCT01299948). Inclusion criteria were a CD4 count above 300 cells/μl, the absence of AIDS-defining symptoms and an ART-naïve therapy status. Study participants received 5 mg prednisolone per day or placebo for 2 years. Primary endpoint was time to progression to an AIDS-defining condition or to a CD4-count below 200 cells/μl.
Results
No significant change in progression towards the primary endpoint was observed in the intent-to-treat (ITT) analysis (19 cases with prednisolone versus 28 cases with placebo, p = 0.1407). In a per-protocol (PP)-analysis, 13 versus 24 study participants progressed to the primary study endpoint (p = 0.0741). Secondary endpoints: Prednisolone-treatment decreased immune activation (sCD14, suPAR, CD38/HLA-DR/CD8+) and increased CD4-counts (+77.42 ± 5.70 cells/μl compared to -37.42 ± 10.77 cells/μl under placebo, p < 0.0001). Treatment with prednisolone was associated with a 3.2-fold increase in HIV viral load (p < 0.0001). In a post-hoc analysis stratifying for sex, females treated with prednisolone progressed significantly slower to the primary study endpoint than females treated with placebo (ITT-analysis: 11 versus 21 cases, p = 0.0567; PP-analysis: 5 versus 18 cases, p = 0.0051): No changes in disease progression were observed in men.
Conclusions
This study could not detect any significant effects of prednisolone on disease progression in antiretroviral-untreated HIV infection within the intent-to-treat population. However, significant effects were observed on CD4 counts, immune activation and HIV viral load. This study contributes to a better understanding of the role of immune activation in the pathogenesis of HIV infection.
First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.
The topic of this thesis is the theoretical and numerical analysis of optimal control problems, whose differential constraints are given by Fokker-Planck models related to jump-diffusion processes. We tackle the issue of controlling a stochastic process by formulating a deterministic optimization problem. The
key idea of our approach is to focus on the probability density function of the process,
whose time evolution is modeled by the Fokker-Planck equation. Our control framework is advantageous since it allows to model the action of the control over the entire range of the process, whose statistics are characterized by the shape of its probability density function.
We first investigate jump-diffusion processes, illustrating their main properties. We define stochastic initial-value problems and present results on the existence and uniqueness of their solutions. We then discuss how numerical solutions of stochastic problems are computed, focusing on the Euler-Maruyama method.
We put our attention to jump-diffusion models with time- and space-dependent coefficients and jumps given by a compound Poisson process. We derive the related Fokker-Planck equations, which take the form of partial integro-differential equations. Their differential term is governed by a parabolic operator, while the nonlocal integral operator is due to the presence of the jumps. The derivation is carried out in two cases. On the one hand, we consider a process with unbounded range. On the other hand, we confine the dynamic of the sample paths to a bounded domain, and thus the behavior of the process in proximity of the boundaries has to be specified. Throughout this thesis, we set the barriers of the domain to be reflecting.
The Fokker-Planck equation, endowed with initial and boundary conditions, gives rise to Fokker-Planck problems. Their solvability is discussed in suitable functional spaces. The properties of their solutions are examined, namely their regularity, positivity and probability mass conservation. Since closed-form solutions to Fokker-Planck problems are usually not available, one has to resort to numerical methods.
The first main achievement of this thesis is the definition and analysis of conservative and positive-preserving numerical methods for Fokker-Planck problems. Our SIMEX1 and SIMEX2 (Splitting-Implicit-Explicit) schemes are defined within the framework given by the method of lines. The differential operator is discretized by a finite volume scheme given by the Chang-Cooper method, while the integral operator is approximated by a mid-point rule. This leads to a large system of ordinary differential equations, that we approximate with the Strang-Marchuk splitting method. This technique decomposes the original problem in a
sequence of different subproblems with simpler structure, which are separately solved and linked to each other through initial conditions and final solutions. After performing the splitting step, we carry out the time integration with first- and second-order time-differencing methods. These steps give rise to the SIMEX1 and SIMEX2 methods, respectively.
A full convergence and stability analysis of our schemes is included. Moreover, we are able to prove that the positivity and the mass conservation of the solution to Fokker-Planck problems are satisfied at the discrete level by the numerical solutions computed with the SIMEX schemes.
The second main achievement of this thesis is the theoretical analysis and the numerical solution of optimal control problems governed by Fokker-Planck models. The field of optimal control deals with finding control functions in such a way that given cost functionals are minimized. Our framework aims at the minimization of the difference between a known sequence of values and the first moment of a jump-diffusion process; therefore, this formulation can also be considered as a parameter estimation problem for stochastic processes. Two cases are discussed, in which the form of the cost functional is continuous-in-time and discrete-in-time, respectively.
The control variable enters the state equation as a coefficient of the Fokker-Planck partial integro-differential operator. We also include in the cost functional a $L^1$-penalization term, which enhances the sparsity of the solution. Therefore, the resulting optimization problem is nonconvex and nonsmooth. We derive the first-order optimality systems satisfied by the optimal solution. The computation of the optimal solution is carried out by means of proximal iterative schemes in an infinite-dimensional framework.
This thesis deals with value sets, i.e. the question of what the set of values that a set of functions can take in a prescribed point looks like.
Interest in such problems has been around for a long time; a first answer was given by the Schwarz lemma in the 19th century, and soon various refinements were proven.
Since the 1930s, a powerful method for solving such problems has been developed, namely Loewner theory. We make extensive use of this tool, as well as variation methods which go back to Schiffer to examine the following questions:
We describe the set of values a schlicht normalised function on the unit disc with prescribed derivative at the origin can take by applying Pontryagin's maximum principle to the radial Loewner equation.
We then determine the value ranges for the set of holomorphic, normalised, and bounded functions that have only real coefficients in their power series expansion around 0, and for the smaller set of functions which are additionally typically real.
Furthermore, we describe the values a univalent self-mapping of the upper half-plane with hydrodynamical normalization which is symmetric with respect to the imaginary axis can take.
Lastly, we give a necessary condition for a schlicht bounded function f on the unit disc to have extremal derivative in a point z where its value f(z) is fixed by using variation methods.
The thesis focuses on the valuation of firms in a system context where cross-holdings of the firms in liabilities and equities are allowed and, therefore, systemic risk can be modeled on a structural level. A main property of such models is that for the determination of the firm values a pricing equilibrium has to be found. While there exists a small but growing amount of research on the existence and the uniqueness of such price equilibria, the literature is still somewhat inconsistent. An example for this fact is that different authors define the underlying financial system on differing ways. Moreover, only few articles pay intense attention on procedures to find the pricing equilibria. In the existing publications, the provided algorithms mainly reflect the individual authors' particular approach to the problem. Additionally, all existing methods do have the drawback of potentially infinite runtime.
For these reasons, the objects of this thesis are as follows. First, a definition of a financial system is introduced in its most general form in Chapter 2. It is shown that under a fairly mild regularity condition the financial system has a unique existing payment equilibrium. In Chapter 3, some extensions and differing definitions of financial systems that exist in literature are presented and it is shown how these models can be embedded into the general model from the proceeding chapter. Second, an overview of existing valuation algorithms to find the equilibrium is given in Chapter 4, where the existing methods are generalized and their corresponding mathematical properties are highlighted. Third, a complete new class of valuation algorithms is developed in Chapter 4 that includes the additional information whether a firm is in default or solvent under a current payment vector. This results in procedures that are able find the solution of the system in a finite number of iteration steps. In Chapter 5, the developed concepts of Chapter 4 are applied to more general financial systems where more than one seniority level of debt is present. Chapter 6 develops optimal starting vectors for non-finite algorithms and Chapter 7 compares the existing and the new developed algorithms concerning their efficiency in an extensive simulation study covering a wide range of possible settings for financial systems.
Extreme value theory aims at modeling extreme but rare events from a probabilistic point of view. It is well-known that so-called generalized Pareto distributions, which are briefly reviewed in Chapter 1, are the only reasonable probability distributions suited for modeling observations above a high threshold, such as waves exceeding the height of a certain dike, earthquakes having at least a certain intensity, and, after applying a simple transformation, share prices falling below some low threshold. However, there are cases for which a generalized Pareto model might fail. Therefore, Chapter 2 derives certain neighborhoods of a generalized Pareto distribution and provides several statistical tests for these neighborhoods, where the cases of observing finite dimensional data and of observing continuous functions on [0,1] are considered. By using a notation based on so-called D-norms it is shown that these tests consistently link both frameworks, the finite dimensional and the functional one. Since the derivation of the asymptotic distributions of the test statistics requires certain technical restrictions, Chapter 3 analyzes these assumptions in more detail. It provides in particular some examples of distributions that satisfy the null hypothesis and of those that do not. Since continuous copula processes are crucial tools for the functional versions of the proposed tests, it is also discussed whether those copula processes actually exist for a given set of data. Moreover, some practical advice is given how to choose the free parameters incorporated in the test statistics. Finally, a simulation study in Chapter 4 compares the in total three different test statistics with another test found in the literature that has a similar null hypothesis. This thesis ends with a short summary of the results and an outlook to further open questions.
The present thesis considers the development and analysis of arbitrary Lagrangian-Eulerian
discontinuous Galerkin (ALE-DG) methods with time-dependent approximation spaces for
conservation laws and the Hamilton-Jacobi equations.
Fundamentals about conservation laws, Hamilton-Jacobi equations and discontinuous Galerkin
methods are presented. In particular, issues in the development of discontinuous Galerkin (DG)
methods for the Hamilton-Jacobi equations are discussed.
The development of the ALE-DG methods based on the assumption that the distribution of
the grid points is explicitly given for an upcoming time level. This assumption allows to construct a time-dependent local affine linear mapping to a reference cell and a time-dependent
finite element test function space. In addition, a version of Reynolds’ transport theorem can be
proven.
For the fully-discrete ALE-DG method for nonlinear scalar conservation laws the geometric
conservation law and a local maximum principle are proven. Furthermore, conditions for slope
limiters are stated. These conditions ensure the total variation stability of the method. In addition, entropy stability is discussed. For the corresponding semi-discrete ALE-DG method,
error estimates are proven. If a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell, the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence for monotone fuxes and the optimal $(k+1)$ convergence for an upwind flux are proven in the $\mathrm{L}^{2}$-norm. The capability of the method is shown by numerical examples for nonlinear conservation laws.
Likewise, for the semi-discrete ALE-DG method for nonlinear Hamilton-Jacobi equations, error
estimates are proven. In the one dimensional case the optimal $\left(k+1\right)$ convergence and in the two dimensional case the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence are proven in the $\mathrm{L}^{2}$-norm, if a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell. For the fullydiscrete method, the geometric conservation is proven and for the piecewise constant forward Euler step the convergence of the method to the unique physical relevant solution is discussed.
Mathematical modelling, simulation, and optimisation are core methodologies for future
developments in engineering, natural, and life sciences. This work aims at applying these
mathematical techniques in the field of biological processes with a focus on the wine
fermentation process that is chosen as a representative model.
In the literature, basic models for the wine fermentation process consist of a system of
ordinary differential equations. They model the evolution of the yeast population number
as well as the concentrations of assimilable nitrogen, sugar, and ethanol. In this thesis,
the concentration of molecular oxygen is also included in order to model the change of
the metabolism of the yeast from an aerobic to an anaerobic one. Further, a more sophisticated
toxicity function is used. It provides simulation results that match experimental
measurements better than a linear toxicity model. Moreover, a further equation for the
temperature plays a crucial role in this work as it opens a way to influence the fermentation
process in a desired way by changing the temperature of the system via a cooling
mechanism. From the view of the wine industry, it is necessary to cope with large scale
fermentation vessels, where spatial inhomogeneities of concentrations and temperature
are likely to arise. Therefore, a system of reaction-diffusion equations is formulated in
this work, which acts as an approximation for a model including computationally very
expensive fluid dynamics.
In addition to the modelling issues, an optimal control problem for the proposed
reaction-diffusion fermentation model with temperature boundary control is presented
and analysed. Variational methods are used to prove the existence of unique weak solutions
to this non-linear problem. In this framework, it is possible to exploit the Hilbert
space structure of state and control spaces to prove the existence of optimal controls.
Additionally, first-order necessary optimality conditions are presented. They characterise
controls that minimise an objective functional with the purpose to minimise the final
sugar concentration. A numerical experiment shows that the final concentration of sugar
can be reduced by a suitably chosen temperature control.
The second part of this thesis deals with the identification of an unknown function
that participates in a dynamical model. For models with ordinary differential equations,
where parts of the dynamic cannot be deduced due to the complexity of the underlying
phenomena, a minimisation problem is formulated. By minimising the deviations of simulation
results and measurements the best possible function from a trial function space
is found. The analysis of this function identification problem covers the proof of the
differentiability of the function–to–state operator, the existence of minimisers, and the
sensitivity analysis by means of the data–to–function mapping. Moreover, the presented
function identification method is extended to stochastic differential equations. Here, the
objective functional consists of the difference of measured values and the statistical expected
value of the stochastic process solving the stochastic differential equation. Using a
Fokker-Planck equation that governs the probability density function of the process, the
probabilistic problem of simulating a stochastic process is cast to a deterministic partial
differential equation. Proofs of unique solvability of the forward equation, the existence of
minimisers, and first-order necessary optimality conditions are presented. The application
of the function identification framework to the wine fermentation model aims at finding
the shape of the toxicity function and is carried out for the deterministic as well as the
stochastic case.
Extreme value theory is concerned with the stochastic modeling of rare and extreme events. While fundamental theories of classical stochastics - such as the laws of small numbers or the central limit theorem - are used to investigate the asymptotic behavior of the sum of random variables, extreme value theory focuses on the maximum or minimum of a set of observations. The limit distribution of the normalized sample maximum among a sequence of independent and identically distributed random variables can be characterized by means of so-called max-stable distributions.
This dissertation concerns with different aspects of the theory of max-stable random vectors and stochastic processes. In particular, the concept of 'differentiability in distribution' of a max-stable process is introduced and investigated. Moreover, 'generalized max-linear models' are introduced in order to interpolate a known max-stable random vector by a max-stable process. Further, the connection between extreme value theory and multivariate records is established. In particular, so-called 'complete' and 'simple' records are introduced as well as it is examined their asymptotic behavior.
Proximal methods are iterative optimization techniques for functionals, J = J1 + J2, consisting of a differentiable part J2 and a possibly nondifferentiable part J1. In this thesis proximal methods for finite- and infinite-dimensional optimization problems are discussed. In finite dimensions, they solve l1- and TV-minimization problems that are effectively applied to image reconstruction in magnetic resonance imaging (MRI). Convergence of these methods in this setting is proved. The proposed proximal scheme is compared to a split proximal scheme and it achieves a better signal-to-noise ratio. In addition, an application that uses parallel imaging is presented.
In infinite dimensions, these methods are discussed to solve nonsmooth linear and bilinear elliptic and parabolic optimal control problems. In particular, fast convergence of these methods is proved. Furthermore, for benchmarking purposes, truncated proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of our proximal schemes that need less computation time than the semismooth Newton method in most cases. Results of numerical experiments are presented that successfully validate the theoretical estimates.
Based on the work of Eisenberg and Noe [2001], Suzuki [2002], Elsinger [2009] and Fischer [2014], we consider a generalization of Merton's asset valuation approach where n firms are linked by cross-ownership of equities and liabilities. Each firm is assumed to have a single outstanding liability, whereas its assets consist of one system-exogenous asset, as well as system-endogenous assets comprising some fraction of other firms' equity and liability, respectively. Following Fischer [2014], one can obtain no-arbitrage prices of equity and the recovery claims of liabilities as solutions of a fixed point problem, and hence obtain no-arbitrage prices of the `firm value' of each firm, which is the value of the firm's liability plus the firm's equity.
In a first step, we consider the two-firm case where explicit formulae for the no-arbitrage prices of the firm values are available (cf. Suzuki [2002]). Since firm values are derivatives of exogenous asset values, the distribution of firm values at maturity can be determined from the distribution of exogenous asset values. The Merton model and most of its known extensions do not account for the cross-ownership structure of the assets owned by the firm. Therefore the assumption of lognormally distributed exogenous assets leads to lognormally distributed firm values in such models, as the values of the liability and the equity add up to the exogenous asset's value (which has lognormal distribution by assumption). Our work therefore starts from lognormally distributed exogenous assets and reveals how cross-ownership, when correctly accounted for in the valuation process, affects the distribution of the firm value, which is not lognormal anymore. In a simulation study we examine the impact of several parameters (amount of cross-ownership of debt and equity, ratio of liabilities to expected exogenous assets value) on the differences between the distribution of firm values obtained from our model and correspondingly matched lognormal distributions. It becomes clear that the assumption of lognormally distributed firm values may lead to both over- and underestimation of the “true" firm values (within the cross-ownership model) and consequently of bankruptcy risk, too.
In a second step, the bankruptcy risk of one firm within the system is analyzed in more detail in a further simulation study, revealing that the correct incorporation of cross-ownership in the valuation procedure is the more important, the tighter the cross-ownership structure between the two firms. Furthermore, depending on the considered type of cross-ownership (debt or equity), the assumption of lognormally distributed firm values is likely to result in an over- resp. underestimation of the actual probability of default. In a similar vein, we consider the Value-at-Risk (VaR) of a firm in the system, which we calculate as the negative α-quantile of the firm value at maturity minus the firm's risk neutral price in t=0, i.e. we consider the (1-α)100%-VaR of the change in firm value. If we let the cross-ownership fractions (i.e. the fraction that one firm holds of another firm's debt or equity) converge to 1 (which is the supremum of the possible values that cross-ownership fractions can take), we can prove that in a system of two firms, the lognormal model will over- resp. underestimate both univariate and bivariate probabilities of default under cross-ownership of debt only resp. cross-ownership of equity only. Furthermore, we provide a formula that allows us to check for an arbitrary scenario of cross-ownership and any non-negative distribution of exogenous assets whether the approximating lognormal model will over- or underestimate the related probability of default of a firm. In particular, any given non-negative distribution of exogenous asset values (non-degenerate in a certain sense) can be transformed into a new, “extreme" distribution of exogenous assets yielding such a low or high actual probability of default that the approximating lognormal model will over- and underestimate this risk, respectively.
After this analysis of the univariate distribution of firm values under cross-ownership in a system of two firms with bivariately lognormally distributed exogenous asset values, we consider the copula of these firm values as a distribution-free measure of the dependency between these firm values. Without cross-ownership, this copula would be the Gaussian copula. Under cross-ownership, we especially consider the behaviour of the copula of firm values in the lower left and upper right corner of the unit square, and depending on the type of cross-ownership and the considered corner, we either obtain error bounds as to how good the copula of firm values under cross-ownership can be approximated with the Gaussian copula, or we see that the copula of firm values can be written as the copula of two linear combinations of exogenous asset values (note that these linear combinations are not lognormally distributed). These insights serve as a basis for our analysis of the tail dependence coefficient of firm values under cross-ownership. Under cross-ownership of debt only, firm values remain upper tail independent, whereas they become perfectly lower tail dependent if the correlation between exogenous asset values exceeds a certain positive threshold, which does not depend on the exact level of cross-ownership. Under cross-ownership of equity only, the situation is reverse in that firm values always remain lower tail independent, but upper tail independence is preserved if and only if the right tail behaviour of both firms’ values is determined by the right tail behaviour of the firms’ own exogenous asset value instead of the respective other firm’s exogenous asset value.
Next, we return to systems of n≥2 firms and analyze sensitivities of no-arbitrage prices of equity and the recovery claims of liabilities with respect to the model parameters. In the literature, such sensitivities are provided with respect to exogenous asset values by Gouriéroux et al. [2012], and we extend the existing results by considering how these no-arbitrage prices depend on the cross-ownership fractions and the level of liabilities. For the former, we can show that all prices are non-decreasing in any cross-ownership fraction in the model, and by use of a version of the Implicit Function Theorem we can also determine exact derivatives. For the latter, we show that the recovery value of debt and the equity value of a firm are non-decreasing and non-increasing in the firm's nominal level of liabilities, respectively, but the firm value is in general not monotone in the firm's level of liabilities. Furthermore, no-arbitrage prices of equity and the recovery claims of liabilities of a firm are in general non-monotone in the nominal level of liabilities of other firms in the system. If we confine ourselves to one type of cross-ownership (i.e. debt or equity), we can derive more precise relationships. All the results can be transferred to risk-neutral prices before maturity.
Finally, following Gouriéroux et al. [2012] and as a kind of extension to the above sensitivity results, we consider how immediate changes in exogenous asset values of one or more firms at maturity affect the financial health of a system of n initially solvent firms. We start with some theoretical considerations on what we call the contagion effect, namely the change in the endogenous asset value of a firm caused by shocks on the exogenous assets of firms within the system. For the two-firm case, an explicit formula is available, making clear that in general (and in particular under cross-ownership of equity only), the effect of contagion can be positive as well as negative, i.e. it can both, mitigate and exacerbate the change in the exogenous asset value of a firm. On the other hand, we cannot generally say that a tighter cross-ownership structure leads to bigger absolute contagion effects. Under cross-ownership of debt only, firms cannot profit from positive shocks beyond the direct effect on exogenous assets, as the contagion effect is always non-positive. Next, we are concerned with spillover effects of negative shocks on a subset of firms to other firms in the system (experiencing non-negative shocks themselves), driving them into default due to large losses in their endogenous asset values. Extending the results of Glasserman and Young [2015], we provide a necessary condition for the shock to cause such an event. This also yields an upper bound for the probability of such an event. We further investigate how the stability of a system of firms exposed to multiple shocks depends on the model parameters in a simulation study. In doing so, we consider three network types (incomplete, core-periphery and ring network) with simultaneous shocks on some of the firms and wiping out a certain percentage of their exogenous assets. Then we analyze for all three types of cross-ownership (debt only, equity only, both debt and equity) how the shock intensity, the shock size, and network parameters as the number of links in the network and the proportion of a firm's debt or equity held within the system of firms influences several output parameters, comprising the total number of defaults and the relative loss in the sum of firm values, among others. Comparing our results to the studies of Nier et al. [2007], Gai and Kapadia [2010] and Elliott et al. [2014], we can only partly confirm their results with respect to the number of defaults. We conclude our work with a theoretical comparison of the complete network (where each firm holds a part of any other firm) and the ring network with respect to the number of defaults caused by a shock on a single firm, as it is done by Allen and Gale [2000]. In line with the literature, we find that under cross-ownership of debt only, complete networks are “robust yet fragile" [Gai and Kapadia, 2010] in that moderate shocks can be completely withstood or drive the firm directly hit by the shock in default, but as soon as the shock exceeds a certain size, all firms are simultaneously in default. In contrast to that, firms default one by one in the ring network, with the first “contagious default" (i.e. a default of a firm not directly hit by the shock) already occurs for smaller shock sizes than under the complete network.
Dysfunction of dopaminergic neurotransmission has been implicated in HIV infection. We showed previously increased dopamine (DA) levels in CSF of therapy-naïve HIV patients and an inverse correlation between CSF DA and CD4 counts in the periphery, suggesting adverse effects of high levels of DA on HIV infection. In the current study including a total of 167 HIV-positive and negative donors from Germany and South Africa (SA), we investigated the mechanistic background for the increase of CSF DA in HIV individuals. Interestingly, we found that the DAT 10/10-repeat allele is present more frequently within HIV individuals than in uninfected subjects. Logistic regression analysis adjusted for gender and ethnicity showed an odds ratio for HIV infection in DAT 10/10 allele carriers of 3.93 (95 % CI 1.72–8.96; p = 0.001, Fishers exact test). 42.6 % HIV-infected patients harbored the DAT 10/10 allele compared to only 10.5 % uninfected DAT 10/10 carriers in SA (odds ratio 6.31), whereas 68.1 versus 40.9 %, respectively, in Germany (odds ratio 3.08). Subjects homozygous for the 10-repeat allele had higher amounts of CSF DA and reduced DAT mRNA expression but similar disease severity compared with those carrying other DAT genotypes. These intriguing and novel findings show the mutual interaction between DA and HIV, suggesting caution in the interpretation of CNS DA alterations in HIV infection solely as a secondary phenomenon to the virus and open the door for larger studies investigating consequences of the DAT functional polymorphism on HIV epidemiology and progression of disease.
Die Arbeit beschäftigt sich mit dem Einsatz von Origami im Schulunterricht. Genauer beschreibt sie eine Unterrichtssequenz zur Flachfaltbarkeit, einem Teilgebiet des mathematischen Papierfaltens, für den Mathematikunterricht in der Oberstufe an Gymnasien und höheren Schulen. Es werden konkrete Handlungsanweisungen sowie Alternativen ausgeführt und begründet und mit vielen Grafiken erläutert. Ferner werden Ziele dieser Unterrichtssequenz gemäß KMK-Bildungsstandards dargelegt. Anschließend wird ein mathematischer Blick auf die Beschäftigung mit der Flachfaltbarkeit sowie eine Einordnung in die aktuelle Forschungslage gegeben.
The first goal of this thesis is to generalize Loewner's famous differential equation to multiply connected domains. The resulting differential equations are known as Komatu--Loewner differential equations. We discuss Komatu--Loewner equations for canonical domains (circular slit disks, circular slit annuli and parallel slit half-planes). Additionally, we give a generalisation to several slits and discuss parametrisations that lead to constant coefficients. Moreover, we compare Komatu--Loewner equations with several slits to single slit Loewner equations.
Finally we generalise Komatu--Loewner equations to hulls satisfying a local growth property.
This thesis deals with the hp-finite element method (FEM) for linear quadratic optimal control problems. Here, a tracking type functional with control costs as regularization shall be minimized subject to an elliptic partial differential equation. In the presence of control constraints, the first order necessary conditions, which are typically used to find optimal solutions numerically, can be formulated as a semi-smooth projection formula. Consequently, optimal solutions may be non-smooth as well. The hp-discretization technique considers this fact and approximates rough functions on fine meshes while using higher order finite elements on domains where the solution is smooth.
The first main achievement of this thesis is the successful application of hp-FEM to two related problem classes: Neumann boundary and interface control problems. They are solved with an a-priori refinement strategy called boundary concentrated (bc) FEM and interface concentrated (ic) FEM, respectively. These strategies generate grids that are heavily refined towards the boundary or interface. We construct an elementwise interpolant that allows to prove algebraic decay of the approximation error for both techniques. Additionally, a detailed analysis of global and local regularity of solutions, which is critical for the speed of convergence, is included. Since the bc- and ic-FEM retain small polynomial degrees for elements touching the boundary and interface, respectively, we are able to deduce novel error estimates in the L2- and L∞-norm. The latter allows an a-priori strategy for updating the regularization parameter in the objective functional to solve bang-bang problems.
Furthermore, we apply the traditional idea of the hp-FEM, i.e., grading the mesh geometrically towards vertices of the domain, for solving optimal control problems (vc-FEM). In doing so, we obtain exponential convergence with respect to the number of unknowns. This is proved with a regularity result in countably normed spaces for the variables of the coupled optimality system.
The second main achievement of this thesis is the development of a fully adaptive hp-interior point method that can solve problems with distributed or Neumann control. The underlying barrier problem yields a non-linear optimality system, which poses a numerical challenge: the numerically stable evaluation of integrals over possibly singular functions in higher order elements. We successfully overcome this difficulty by monitoring the control variable at the integration points and enforcing feasibility in an additional smoothing step. In this work, we prove convergence of an interior point method with smoothing step and derive a-posteriori error estimators. The adaptive mesh refinement is based on the expansion of the solution in a Legendre series. The decay of the coefficients serves as an indicator for smoothness that guides between h- and p-refinement.
The goal of this thesis is to investigate conformal mappings onto circular arc polygon domains, i.e. domains that are bounded by polygons consisting of circular arcs instead of line segments.
Conformal mappings onto circular arc polygon domains contain parameters in addition to the classical parameters of the Schwarz-Christoffel transformation. To contribute to the parameter problem of conformal mappings from the unit disk onto circular arc polygon domains, we investigate two special cases of these mappings. In the first case we can describe the additional parameters if the bounding circular arc polygon is a polygon with straight sides. In the second case we provide an approximation for the additional parameters if the circular arc polygon domain satisfies some symmetry conditions. These results allow us to draw conclusions on the connection between these additional parameters and the classical parameters of the mapping.
For conformal mappings onto multiply connected circular arc polygon domains, we provide an alternative construction of the mapping formula without using the Schottky-Klein prime function. In the process of constructing our main result, mappings for domains of connectivity three or greater, we also provide a formula for conformal mappings onto doubly connected circular arc polygon domains. The comparison of these mapping formulas with already known mappings allows us to provide values for some of the parameters of the mappings onto doubly connected circular arc polygon domains if the image domain is a polygonal domain.
The different components of the mapping formula are constructed by using a slightly modified variant of the Poincaré theta series. This construction includes the design of a function to remove unwanted poles and of different versions of functions that are analytic on the domain of definition of the mapping functions and satisfy some special functional equations.
We also provide the necessary concepts to numerically evaluate the conformal mappings onto multiply connected circular arc polygon domains. As the evaluation of such a map requires the solution of a differential equation, we provide a possible configuration of curves inside the preimage domain to solve the equation along them in addition to a description of the procedure for computing either the formula for the doubly connected case or the case of connectivity three or greater. We also describe the procedures for solving the parameter problem for multiply connected circular arc polygon domains.
The purpose of confidence and prediction intervals is to provide an interval estimation for an unknown distribution parameter or the future value of a phenomenon. In many applications, prior knowledge about the distribution parameter is available, but rarely made use of, unless in a Bayesian framework. This thesis provides exact frequentist confidence intervals of minimal volume exploiting prior information. The scheme is applied to distribution parameters of the binomial and the Poisson distribution. The Bayesian approach to obtain intervals on a distribution parameter in form of credibility intervals is considered, with particular emphasis on the binomial distribution. An application of interval estimation is found in auditing, where two-sided intervals of Stringer type are meant to contain the mean of a zero-inflated population. In the context of time series analysis, covariates are supposed to improve the prediction of future values. Exponential smoothing with covariates as an extension of the popular forecasting method exponential smoothing is considered in this thesis. A double-seasonality version of it is applied to forecast hourly electricity load under the use of meteorological covariates. Different kinds of prediction intervals for exponential smoothing with covariates are formulated.
The subject of this thesis is the rigorous passage from discrete systems to continuum models via variational methods.
The first part of this work studies a discrete model describing a one-dimensional chain of atoms with finite range interactions of Lennard-Jones type. We derive an expansion of the ground state energy using \(\Gamma\)-convergence. In particular, we show that a variant of the Cauchy-Born rule holds true for the model under consideration. We exploit this observation to derive boundary layer energies due to asymmetries of the lattice at the boundary or at cracks of the specimen. Hereby we extend several results obtained previously for models involving only nearest and next-to-nearest neighbour interactions by Braides and Cicalese and Scardia, Schlömerkemper and Zanini.
The second part of this thesis is devoted to the analysis of a quasi-continuum (QC) method. To this end, we consider the discrete model studied in the first part of this thesis as the fully atomistic model problem and construct an approximation based on a QC method. We show that in an elastic setting the expansion by \(\Gamma\)-convergence of the fully atomistic energy and its QC approximation coincide. In the case of fracture, we show that this is not true in general. In the case of only nearest and next-to-nearest neighbour interactions, we give sufficient conditions on the QC approximation such that, also in case of fracture, the minimal energies of the fully atomistic energy and its approximation coincide in the limit.
The thesis ’Hurwitz’s Complex Continued Fractions - A Historical Approach and Modern Perspectives.’ deals with two branches of mathematics: Number Theory and History of Mathematics. On the first glimpse this might be unexpected, however, on the second view this is a very fruitful combination. Doing research in mathematics, it turns out to be very helpful to be aware of the beginnings and development of the corresponding subject.
In the case of Complex Continued Fractions the origins can easily be traced back to the end of the 19th century (see [Perron, 1954, vl. 1, Ch. 46]). One of their godfathers had been the famous mathematician Adolf Hurwitz. During the study of his transformation from real to complex continued fraction theory [Hurwitz, 1888], our attention was arrested by the article ’Ueber eine besondere Art der Kettenbruch-Entwicklung complexer Grössen’ [Hurwitz, 1895] from 1895 of an author called J. Hurwitz. We were not only surprised when we found out that he was the elder unknown brother Julius, furthermore, Julius Hurwitz introduced a complex continued fraction that also appeared (unmentioned) in an ergodic theoretical work from 1985 [Tanaka, 1985]. Those observations formed the Basis of our main research questions:
What is the historical background of Adolf and Julius Hurwitz and their mathematical studies? and What modern perspectives are provided by their complex continued fraction expansions?
In this work we examine complex continued fractions from various viewpoints. After a brief introduction on real continued fractions, we firstly devote ourselves to the lives of the brothers Adolf and Julius Hurwitz. Two excursions on selected historical aspects in respect to their work complete this historical chapter. In the sequel we shed light on Hurwitz’s, Adolf’s as well as Julius’, approaches to complex continued fraction expansions.
Correspondingly, in the following chapter we take a more modern perspective. Highlights are an ergodic theoretical result, namely a variation on the Döblin-Lenstra Conjecture [Bosma et al., 1983], as well as a result on transcendental numbers in tradition of Roth’s theorem [Roth, 1955]. In two subsequent chapters we are concernced with arithmetical properties of complex continued fractions. Firstly, an analogue to Marshall Hall’s Theorem from 1947 [Hall, 1947] on sums of continued fractions is derived. Secondly, a general approach on new types of continued fractions is presented building on the structural properties of lattices. Finally, in the last chapter we take up this approach and obtain an upper bound for the approximation quality of diophantine approximations by quotients of lattice points in the complex plane generalizing a method of Hermann Minkowski, improved by Hilde Gintner [Gintner, 1936], based on ideas from geometry of numbers.
Der Einzug des Rechners in den Mathematikunterricht hat eine Vielzahl neuer Möglichkeiten der Darstellung mit sich gebracht, darunter auch multiple, dynamisch verbundene Repräsentationen mathematischer Probleme. Die Arbeit beantwortet die Frage, ob und wie diese Repräsentationsarten von Schülerinnen und Schüler in Argumentationen genutzt werden. In der empirischen Untersuchung wurde dabei einerseits quantitativ erforscht, wie groß der Einfluss der in der Aufgabenstellung gegebenen Repräsentationsform auf die schriftliche Argumentationen der Schülerinnen und Schüler ist. Andererseits wurden durch eine qualitative Analyse spezifische Nutzungsweisen identifiziert und mittels Toulmins Argumentationsmodell beschrieben. Diese Erkenntnisse wurden genutzt, um Konsequenzen bezüglich der Verwendung von multiplen und/oder dynamischen Repräsentationen im Mathematikunterricht der Sekundarstufe zu formulieren.
The investigation of interacting multi-agent models is a new field of mathematical research with application to the study of behavior in groups of animals or community of people. One interesting feature of multi-agent systems is collective behavior. From the mathematical point of view, one of the challenging issues considering with these dynamical models is development of control mechanisms that are able to influence the time evolution of these systems.
In this thesis, we focus on the study of controllability, stabilization and optimal control problems for multi-agent systems considering three models as follows: The first one is the Hegselmann Krause opinion formation (HK) model. The HK dynamics describes how individuals' opinions are changed by the interaction with others taking place in a bounded domain of confidence. The study of this model focuses on determining feedback controls in order to drive the agents' opinions to reach a desired agreement. The second model is the Heider social balance (HB) model. The HB dynamics explains the evolution of relationships in a social network. One purpose of studying this system is the construction of control function in oder to steer the relationship to reach a friendship state. The third model that we discuss is a flocking model describing collective motion observed in biological systems. The flocking model under consideration includes self-propelling, friction, attraction, repulsion, and alignment features. We investigate a control for steering the flocking system to track a desired trajectory. Common to all these systems is our strategy to add a leader agent that interacts with all other members of the system and includes the control mechanism.
Our control through leadership approach is developed using classical theoretical control methods and a model predictive control (MPC) scheme. To apply the former method, for each model the stability of the corresponding linearized system near consensus is investigated. Further, local controllability is examined. However, only in the
Hegselmann-Krause opinion formation model, the feedback control is determined in order to steer agents' opinions to globally converge to a desired agreement. The MPC approach is an optimal control strategy based on numerical optimization. To apply the MPC scheme, optimal control problems for each model are formulated where the objective functions are different depending on the desired objective of the problem. The first-oder necessary optimality conditions for each problem are presented. Moreover for the numerical treatment, a sequence of open-loop discrete optimality systems is solved by accurate Runge-Kutta schemes, and in the optimization procedure, a nonlinear conjugate gradient solver is implemented. Finally, numerical experiments are performed to investigate the properties of the multi-agent models and demonstrate the ability of the proposed control strategies to drive multi-agent systems to attain a desired consensus and to track a given trajectory.
Background
The prevalence of obesity is rising. Obesity can lead to cardiovascular and ventilatory complications through multiple mechanisms. Cardiac and pulmonary function in asymptomatic subjects and the effect of structured dietary programs on cardiac and pulmonary function is unclear.
Objective
To determine lung and cardiac function in asymptomatic obese adults and to evaluate whether weight loss positively affects functional parameters.
Methods
We prospectively evaluated bodyplethysmographic and echocardiographic data in asymptomatic subjects undergoing a structured one-year weight reduction program.
Results
74 subjects (32 male, 42 female; mean age 42±12 years) with an average BMI 42.5±7.9, body weight 123.7±24.9 kg were enrolled. Body weight correlated negatively with vital capacity (R = −0.42, p<0.001), FEV1 (R = −0.497, p<0.001) and positively with P 0.1 (R = 0.32, p = 0.02) and myocardial mass (R = 0.419, p = 0.002). After 4 months the study subjects had significantly reduced their body weight (−26.0±11.8 kg) and BMI (−8.9±3.8) associated with a significant improvement of lung function (absolute changes: vital capacity +5.5±7.5% pred., p<0.001; FEV1+9.8±8.3% pred., p<0.001, ITGV+16.4±16.0% pred., p<0.001, SR tot −17.4±41.5% pred., p<0.01). Moreover, P0.1/Pimax decreased to 47.7% (p<0.01) indicating a decreased respiratory load. The change of FEV1 correlated significantly with the change of body weight (R = −0.31, p = 0.03). Echocardiography demonstrated reduced myocardial wall thickness (−0.08±0.2 cm, p = 0.02) and improved left ventricular myocardial performance index (−0.16±0.35, p = 0.02). Mitral annular plane systolic excursion (+0.14, p = 0.03) and pulmonary outflow acceleration time (AT +26.65±41.3 ms, p = 0.001) increased.
Conclusion
Even in asymptomatic individuals obesity is associated with abnormalities in pulmonary and cardiac function and increased myocardial mass. All the abnormalities can be reversed by a weight reduction program.
An efficient and accurate computational framework for solving control problems governed by quantum spin systems is presented. Spin systems are extremely important in modern quantum technologies such as nuclear magnetic resonance spectroscopy, quantum imaging and quantum computing. In these applications, two classes of quantum control problems arise: optimal control problems and exact-controllability problems, with a bilinear con- trol structure. These models correspond to the Schrödinger-Pauli equation, describing the time evolution of a spinor, and the Liouville-von Neumann master equation, describing the time evolution of a spinor and a density operator. This thesis focuses on quantum control problems governed by these models. An appropriate definition of the optimiza- tion objectives and of the admissible set of control functions allows to construct controls with specific properties. These properties are in general required by the physics and the technologies involved in quantum control applications. A main purpose of this work is to address non-differentiable quantum control problems. For this reason, a computational framework is developed to address optimal-control prob- lems, with possibly L1 -penalization term in the cost-functional, and exact-controllability problems. In both cases the set of admissible control functions is a subset of a Hilbert space. The bilinear control structure of the quantum model, the L1 -penalization term and the control constraints generate high non-linearities that make difficult to solve and analyse the corresponding control problems. The first part of this thesis focuses on the physical description of the spin of particles and of the magnetic resonance phenomenon. Afterwards, the controlled Schrödinger- Pauli equation and the Liouville-von Neumann master equation are discussed. These equations, like many other controlled quantum models, can be represented by dynamical systems with a bilinear control structure. In the second part of this thesis, theoretical investigations of optimal control problems, with a possible L1 -penalization term in the objective and control constraints, are consid- ered. In particular, existence of solutions, optimality conditions, and regularity properties of the optimal controls are discussed. In order to solve these optimal control problems, semi-smooth Newton methods are developed and proved to be superlinear convergent. The main difficulty in the implementation of a Newton method for optimal control prob- lems comes from the dimension of the Jacobian operator. In a discrete form, the Jacobian is a very large matrix, and this fact makes its construction infeasible from a practical point of view. For this reason, the focus of this work is on inexact Krylov-Newton methods, that combine the Newton method with Krylov iterative solvers for linear systems, and allows to avoid the construction of the discrete Jacobian. In the third part of this thesis, two methodologies for the exact-controllability of quan- tum spin systems are presented. The first method consists of a continuation technique, while the second method is based on a particular reformulation of the exact-control prob- lem. Both these methodologies address minimum L2 -norm exact-controllability problems. In the fourth part, the thesis focuses on the numerical analysis of quantum con- trol problems. In particular, the modified Crank-Nicolson scheme as an adequate time discretization of the Schrödinger equation is discussed, the first-discretize-then-optimize strategy is used to obtain a discrete reduced gradient formula for the differentiable part of the optimization objective, and implementation details and globalization strategies to guarantee an adequate numerical behaviour of semi-smooth Newton methods are treated. In the last part of this work, several numerical experiments are performed to vali- date the theoretical results and demonstrate the ability of the proposed computational framework to solve quantum spin control problems.
In the thesis discrete moments of the Riemann zeta-function and allied Dirichlet series are studied.
In the first part the asymptotic value-distribution of zeta-functions is studied where the samples are taken from a Cauchy random walk on a vertical line inside the critical strip. Building on techniques by Lifshits and Weber analogous results for the Hurwitz zeta-function are derived. Using Atkinson’s dissection this is even generalized to Dirichlet L-functions associated with a primitive character. Both results indicate that the expectation value equals one which shows that the values of these
zeta-function are small on average.
The second part deals with the logarithmic derivative of the Riemann zeta-function on vertical lines and here the samples are with respect to an explicit ergodic transformation. Extending work of Steuding, discrete moments are evaluated and an equivalent formulation for the Riemann Hypothesis in terms of ergodic theory is obtained.
In the third and last part of the thesis, the phenomenon of universality with respect
to stochastic processes is studied. It is shown that certain random shifts of the zeta-function can approximate non-vanishing analytic target functions as good as we please. This result relies on Voronin's universality theorem.
The Cauchy problem for a simplified shallow elastic fluids model, one 3 x 3 system of Temple's type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth (rho - 0). This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for 2 x 2 strictly hyperbolic system and (Heibig, 1994) for n x n strictly hyperbolic system with smooth Riemann invariants.
Several aspects of the stability analysis of large-scale discrete-time systems are considered. An important feature is that the right-hand side does not have have to be continuous.
In particular, constructive approaches to compute Lyapunov functions are derived and applied to several system classes.
For large-scale systems, which are considered as an interconnection of smaller subsystems, we derive a new class of small-gain results, which do not require the subsystems to be robust in some sense. Moreover, we do not only study sufficiency of the conditions, but rather state an assumption under which these conditions are also necessary.
Moreover, gain construction methods are derived for several types of aggregation, quantifying how large a prescribed set of interconnection gains can be in order that a small-gain condition holds.
Analysis of discretization schemes for Fokker-Planck equations and related optimality systems
(2015)
The Fokker-Planck (FP) equation is a fundamental model in thermodynamic kinetic theories and
statistical mechanics.
In general, the FP equation appears in a number of different fields in natural sciences, for instance in solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory. These equations also provide a powerful mean to define
robust control strategies for random models. The FP equations are partial differential equations (PDE) describing the time evolution of the probability density function (PDF) of stochastic processes.
These equations are of different types depending on the underlying stochastic process.
In particular, they are parabolic PDEs for the PDF of Ito processes, and hyperbolic PDEs for piecewise deterministic processes (PDP).
A fundamental axiom of probability calculus requires that the integral of the PDF over all the allowable state space must be equal to one, for all time. Therefore, for the purpose of accurate numerical simulation, a discretized FP equation must guarantee conservativeness of the total probability. Furthermore, since the
solution of the FP equation represents a probability density, any numerical scheme that approximates the FP equation is required to guarantee the positivity of the solution. In addition, an approximation scheme must be accurate and stable.
For these purposes, for parabolic FP equations on bounded domains, we investigate the Chang-Cooper (CC) scheme for space discretization and first- and
second-order backward time differencing. We prove that the resulting
space-time discretization schemes are accurate, conditionally stable, conservative, and preserve positivity.
Further, we discuss a finite difference discretization for the FP system corresponding to a PDP process in a bounded domain.
Next, we discuss FP equations in unbounded domains.
In this case, finite-difference or finite-element methods cannot be applied. By employing a suitable set of basis functions, spectral methods allow to treat unbounded domains. Since FP solutions decay exponentially at infinity, we consider Hermite functions as basis functions, which are Hermite polynomials multiplied by a Gaussian.
To this end, the Hermite spectral discretization is applied
to two different FP equations; the parabolic PDE corresponding to Ito processes, and the system of hyperbolic PDEs corresponding to a PDP process. The resulting discretized schemes are analyzed. Stability and spectral accuracy of the Hermite spectral discretization of the FP problems is proved. Furthermore, we investigate the conservativity of the solutions of FP equations discretized with the Hermite spectral scheme.
In the last part of this thesis, we discuss optimal control problems governed by FP equations on the characterization of their solution by optimality systems. We then investigate the Hermite spectral discretization of FP optimality systems in unbounded domains.
Within the framework of Hermite discretization, we obtain sparse-band systems of ordinary differential equations. We analyze the accuracy of the discretization schemes by showing spectral convergence in approximating the state, the adjoint, and the control variables that appear in the FP optimality systems.
To validate our theoretical estimates, we present results of numerical experiments.
Background
It is hypothesized that because of higher mast cell numbers and mediator release, mastocytosis predisposes patients for systemic immediate-type hypersensitivity reactions to certain drugs including non-steroidal anti-inflammatory drugs (NSAID).
Objective
To clarify whether patients with NSAID hypersensitivity show increased basal serum tryptase levels as sign for underlying mast cell disease.
Methods
As part of our allergy work-up, basal serum tryptase levels were determined in all patients with a diagnosis of NSAID hypersensitivity and the severity of the reaction was graded. Patients with confirmed IgE-mediated hymenoptera venom allergy served as a comparison group.
Results
Out of 284 patients with NSAID hypersensitivity, 26 were identified with basal serum tryptase > 10.0 ng/mL (9.2%). In contrast, significantly (P = .004) more hymenoptera venom allergic patients had elevated tryptase > 10.0 ng/mL (83 out of 484; 17.1%). Basal tryptase > 20.0 ng/mL was indicative for severe anaphylaxis only in venom allergic subjects (29 patients; 4x grade 2 and 25x grade 3 anaphylaxis), but not in NSAID hypersensitive patients (6 patients; 4x grade 1, 2x grade 2).
Conclusions
In contrast to hymenoptera venom allergy, NSAID hypersensitivity do not seem to be associated with elevated basal serum tryptase levels and levels > 20 ng/mL were not related to increased severity of the clinical reaction. This suggests that mastocytosis patients may be treated with NSAID without special precautions.
In this thesis it is shown how the spread of infectious diseases can be described via mathematical models that show the dynamic behavior of epidemics. Ordinary differential equations are used for the modeling process. SIR and SIRS models are distinguished, depending on whether a disease confers immunity to individuals after recovery or not. There are characteristic parameters for each disease like the infection rate or the recovery rate. These parameters indicate how aggressive a disease acts and how long it takes for an individual to recover, respectively. In general the parameters are time-varying and depend on population groups. For this reason, models with multiple subgroups are introduced, and switched systems are used to carry out time-variant parameters.
When investigating such models, the so called disease-free equilibrium is of interest, where no infectives appear within the population. The question is whether there are conditions, under which this equilibrium is stable. Necessary mathematical tools for the stability analysis are presented. The theory of ordinary differential equations, including Lyapunov stability theory, is fundamental. Moreover, convex and nonsmooth analysis, positive systems and differential inclusions are introduced. With these tools, sufficient conditions are given for the disease-free equilibrium of SIS, SIR and SIRS systems to be asymptotically stable.
In this thesis we study smoothness properties of primal and dual gap functions for generalized Nash equilibrium problems (GNEPs) and finite-dimensional quasi-variational inequalities (QVIs). These gap functions are optimal value functions of primal and dual reformulations of a corresponding GNEP or QVI as a constrained or unconstrained optimization problem. Depending on the problem type, the primal reformulation uses regularized Nikaido-Isoda or regularized gap function approaches. For player convex GNEPs and QVIs of the so-called generalized `moving set' type the respective primal gap functions are continuously differentiable. In general, however, these primal gap functions are nonsmooth for both problems. Hence, we investigate their continuity and differentiability properties under suitable assumptions. Here, our main result states that, apart from special cases, all locally minimal points of the primal reformulations are points of differentiability of the corresponding primal gap function.
Furthermore, we develop dual gap functions for a class of GNEPs and QVIs and ensuing unconstrained optimization reformulations of these problems based on an idea by Dietrich (``A smooth dual gap function solution to a class of quasivariational inequalities'', Journal of Mathematical Analysis and Applications 235, 1999, pp. 380--393). For this purpose we rewrite the primal gap functions as a difference of two strongly convex functions and employ the Toland-Singer duality theory. The resulting dual gap functions are continuously differentiable and, under suitable assumptions, have piecewise smooth gradients. Our theoretical analysis is complemented by numerical experiments. The solution methods employed make use of the first-order information established by the aforementioned theoretical investigations.
In attempting to solve the regular inverse Galois problem for arbitrary subfields K of C (particularly for K=Q), a very important result by Fried and Völklein reduces the existence of regular Galois extensions F|K(t) with Galois group G to the existence of K-rational points on components of certain moduli spaces for families of covers of the projective line, known as Hurwitz spaces.
In some cases, the existence of rational points on Hurwitz spaces has been proven by theoretical criteria. In general, however, the question whether a given Hurwitz space has any rational point remains a very difficult problem. In concrete cases, it may be tackled by an explicit computation of a Hurwitz space and the corresponding family of covers.
The aim of this work is to collect and expand on the various techniques that may be used to solve such computational problems and apply them to tackle several families of Galois theoretic interest. In particular, in Chapter 5, we compute explicit curve equations for Hurwitz spaces for certain families of \(M_{24}\) and \(M_{23}\).
These are (to my knowledge) the first examples of explicitly computed Hurwitz spaces of such high genus. They might be used to realize \(M_{23}\) as a regular Galois group over Q if one manages to find suitable points on them.
Apart from the calculation of explicit algebraic equations, we produce complex approximations for polynomials with genus zero ramification of several different ramification types in \(M_{24}\) and \(M_{23}\). These may be used as starting points for similar computations.
The main motivation for these computations is the fact that \(M_{23}\) is currently the only remaining sporadic group that is not known to occur as a Galois group over Q.
We also compute the first explicit polynomials with Galois groups \(G=P\Gamma L_3(4), PGL_3(4), PSL_3(4)\) and \(PSL_5(2)\) over Q(t).
Special attention will be given to reality questions. As an application we compute the first examples of totally real polynomials with Galois groups \(PGL_2(11)\) and \(PSL_3(3)\) over Q.
As a suggestion for further research, we describe an explicit algorithmic version of "Algebraic Patching", following the theory described e.g. by M. Jarden. This could be used to conquer some problems regarding families of covers of genus g>0.
Finally, we present explicit Magma implementations for several of the most important algorithms involved in our computations.
The Riemann zeta-function forms a central object in multiplicative number theory; its value-distribution encodes deep arithmetic properties of the prime numbers. Here, a crucial role is assigned to the analytic behavior of the zeta-function on the so called critical line. In this thesis we study the value-distribution of the Riemann zeta-function near and on the critical line. Amongst others we focus on the following.
PART I: A modified concept of universality, a-points near the critical line and a denseness conjecture attributed to Ramachandra.
The critical line is a natural boundary of the Voronin-type universality property of the Riemann zeta-function. We modify Voronin's concept by adding a scaling factor to the vertical shifts that appear in Voronin's universality theorem and investigate whether this modified concept is appropriate to keep up a certain universality property of the Riemann zeta-function near and on the critical line. It turns out that it is mainly the functional equation of the Riemann zeta-function that restricts the set of functions which can be approximated by this modified concept around the critical line.
Levinson showed that almost all a-points of the Riemann zeta-function lie in a certain funnel-shaped region around the critical line. We complement Levinson's result: Relying on arguments of the theory of normal families and the notion of filling discs, we detect a-points in this region which are very close to the critical line.
According to a folklore conjecture (often attributed to Ramachandra) one expects that the values of the Riemann zeta-function on the critical line lie dense in the complex numbers. We show that there are certain curves which approach the critical line asymptotically and have the property that the values of the zeta-function on these curves are dense in the complex numbers.
Many of our results in part I are independent of the Euler product representation of the Riemann zeta-function and apply for meromorphic functions that satisfy a Riemann-type functional equation in general.
PART II: Discrete and continuous moments.
The Lindelöf hypothesis deals with the growth behavior of the Riemann zeta-function on the critical line. Due to classical works by Hardy and Littlewood, the Lindelöf hypothesis can be reformulated in terms of power moments to the right of the critical line. Tanaka showed recently that the expected asymptotic formulas for these power moments are true in a certain measure-theoretical sense; roughly speaking he omits a set of Banach density zero from the path of integration of these moments. We provide a discrete and integrated version of Tanaka's result and extend it to a large class of Dirichlet series connected to the Riemann zeta-function.
The work at hand studies problems from Loewner theory and is divided into two parts:
In part 1 (chapter 2) we present the basic notions of Loewner theory. Here we use a modern form which was developed by F. Bracci, M. Contreras, S. Díaz-Madrigal et al. and which can be applied to certain higher dimensional complex manifolds.
We look at two domains in more detail: the Euclidean unit ball and the polydisc. Here we consider two classes of biholomorphic mappings which were introduced by T. Poreda and G. Kohr as generalizations of the class S.
We prove a conjecture of G. Kohr about support points of these classes. The proof relies on the observation that the classes describe so called Runge domains, which follows from a result by L. Arosio, F. Bracci and E. F. Wold.
Furthermore, we prove a conjecture of G. Kohr about support points of a class of biholomorphic mappings that comes from applying the Roper-Suffridge extension operator to the class S.
In part 2 (chapter 3) we consider one special Loewner equation: the chordal multiple-slit equation in the upper half-plane.
After describing basic properties of this equation we look at the problem, whether one can choose the coefficient functions in this equation to be constant. D. Prokhorov proved this statement under the assumption that the slits are piecewise analytic. We use a completely different idea to solve the problem in its general form.
As the Loewner equation with constant coefficients holds everywhere (and not just almost everywhere), this result generalizes Loewner’s original idea to the multiple-slit case.
Moreover, we consider the following problems:
• The “simple-curve problem” asks which driving functions describe the growth of simple curves (in contrast to curves that touch itself). We discuss necessary and sufficient conditions, generalize a theorem of J. Lind, D. Marshall and S. Rohde to the multiple-slit equation and we give an example of a set of driving functions which generate simple curves because of a certain self-similarity property.
• We discuss properties of driving functions that generate slits which enclose a given angle with the real axis.
• A theorem by O. Roth gives an explicit description of the reachable set of one point in the radial Loewner equation. We prove the analog for the chordal equation.