70 search hits
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Stability of flat galaxies
(2008)
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Roman Firt
- In this thesis we investigate the existence and properties of stationary solutions of the flat Vlasov-Poisson system. This system of partial differential equations can be used as a model of extremely flat astronomical objects and is a combination between the two-dimensional motion of particles and the three-dimensional interaction through their gravitational potential.
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Existence and stability of stellardynamic models
(2008)
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Achim Schulze
- We examine existence and stability of stationary solutions to the Vlasov-Poisson system. This system is used in stellardynmaics to describe the evolution of galaxies where collissions are neglected and the evolution is determined by the self-consistent gravitational field which is created by the particles, e.g. the stars . In the first part we examine steady states which decsribe static shells under the influence of a fixed point mass. These solutions can be used as a model for a galaxy with a massive black hole in its center. For the Vlasov--Poisson system under the influence of such a point mass, we prove a global existence result. In the second part, we construct axially symmetric solutions depending on Jacobis integral. The presented results are in accordance with the numerical examinations of the P.O. Vandervoort.
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Higher Order Asymptotics for the MSE of Robust M-Estimators of Location on Shrinking Total Variation Neighborhoods
(2008)
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Matthias Brandl
- In the setup of shrinking neighbourhoods in sample size n about an ideal (“smooth”) central model, Rieder (1994) determines the optimal asymptotic linear estimator w.r.t. the asymptotic MSE evaluated uniformly on these neighbourhoods. We answer the question to which degree the asymptotic optimality carries over to finite sample size in the context of total variation neighbourhoods. In contrast to usual higher order asymptotics, instead of giving approximations to distribution functions (or densities), we expand the risk directly by application of Edgeworth and Taylor expansions. In the context of determining the exact finite sample risk for sample size n>2 M. Kohl showed in Kohl (2005) that the speed of convergence towards the asymptotic risk is faster by an order in case of total variation compared to convex contamination neighbourhoods. M. Kohl conjectured that this is caused by the higher symmetry of total variation neighbourhoods. Looking at the MSE in the asymptotic optimal setup we get the same results as M. Kohl for finite sample risk. Furthermore we can show by direct expansion of the MSE that for a higher speed of convergence symmetry of the ideal distribution F is essential. We confirm our result by a cross-check in the ideal model and illustrate our theoretical investigations for F = N(0;1). We also deal with the question of an actual realization of a least favourable deviation from the ideal model in a finite context. We settle on the symmetric case and monotone odd influence curves of Hampel-type form, and show that for a certain kind of manipulation mechanism we gain our theoretical results up to the desired order. It shows up that we only get access to the results in the finite context if we require the finite sample to attain the minimum and maximum of the given influence curve with a certain probability already. Depending on this probability we derive a lower bound on the sample length n. We substantiate the sufficiently exact algorithm by determining the amount K of observations to be manipulated as well as the bound c on the observations for having maximum influence on the MSE according to the value of the influence curve. We give a restrictive condition on the distribution of K that lets the probability of the (K:n)-quantile exceeding a now concrete bound c be exponentially negligible. The bound c is explicitly calculated for F = N(0;1) and suitable four-point distributions for K are given that satisfy all the previous claimed conditions. Thus we gain an algorithm generating observations from a least favourable distribution out of a finite total variation neighbourhood w.r.t. an asymptotically optimal risk.
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Existence Results for Plasma Physics Models Containing a Fully Coupled Magnetic Field
(2009)
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Martin Seehafer
- The present thesis concern is the initial value problem for three nonlinear systems of partial differential equations: the Vlasov-Darwin system, the Vlasov-Poisswell system and a version of the latter which is called the modified Vlasov-Poisswell system. These equations belong to kinetic theory, which has proved useful when describing large particle systems in different areas of physics such as kinetic theory of gases, the formation of stellar structures or plasma physics. In the present thesis equations originating in plasma physics are considered which describe the evolution of the time dependent density function f(t,x,v) (t - time, x – position, v - particle velocity) of a large ensemble of charged particles in the (x,v)-phase space influenced by the electromagnetic field created by the particles and when neglecting collisions. The focus of the investigation is on existence and uniqueness questions for solutions of the initial value problem, i.e., it is asked whether there exists a solution f of the system under consideration such that f(t=0)=f0 where f0 is a prescribed initial datum. In order to answer this question further properties of solutions such as energy and charge conservation or decay rates must be taken into account. An important issue is, whether - if necessary under additional hypotheses or by weakening the concept of solution - global solutions, i.e., solutions existing for all t>=0, may be obtained. The most important results are a theorem about local existence and uniqueness of classical solutions of the Vlasov-Poisswell system, a global existence result for weak solutions of the modified Vlasov-Poisswell system, and a global existence theorem for classical solutions of the Vlasov-Darwin system under the assumption of smallness of the initial.
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Receding Horizon Control: A Suboptimality-based Approach
(2009)
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Jürgen Pannek
- Within the proposed work we consider analytical, conceptional and implementational issues of so called receding horizon controllers in a sampled-data setting. The principle of such a controller is simple: Given the current state of a system we compute an open-loop control which is optimal for a given costfunctional over a fixed prediction horizon. Then, the control is implemented on the first sampling interval and the basic open-loop optimal control problem is shifted forward in time which allows for a repeated evaluation. The contribution of this thesis is threefold: First, we prove estimates for the performance of a receding horizon control, a concept which we call suboptimality degree. These estimate are online computable and can be applied for stabilizing as well as practically stabilizing receding horizon control laws. Moreover, they not only allow for guaranteeing stability of the closed-loop but also for quantifying the loss of performance of the receding horizon control law compared to the infinite horizon control law. Based on these estimates, we introduce adaptation strategies to modify the underlying receding horizon controller in order to guarantee a certain lower bound on the suboptimality degree while reducing the computing cost/time necessary to solve this problem. Within this analysis, the length of the optimization horizon is the parameter we wish to adapt. To this end, we develop and proof several shortening and prolongation strategies which also allow for an effective implementation. Moreover, extensions of our suboptimality estimates to receding horizon controllers with varying optimization horizon are shown. Last, we present details on our implementation of a receding horizon controller PCC2 (http://www.nonlinearmpc.com) which is on the one hand computationally efficient but also allows for easily incorporating our theoretical results. Since a full analysis of such a controller would exceed the scope of this work, we focus on the main aspects of this algorithm using different examples. In particular, we concentrate on the impact of certain choices of parameters on the computing time. We also consider interactions between these parameters to give a guideline to effectively implement and solve further examples. Moreover, we show applicability and effectiveness of our theoretical results using simulations of standard problems for receding horizon controllers.
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Surfaces Isogenous to a Product: Their Automorphisms and Degenerations
(2010)
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Wenfei Liu
- In this thesis, I consider the automorphisms and stable degenerations of surfaces isogenous to a product. First I consider the action of the automorphisms on cohomology in the case where the group G is abelian. It is shown that, if the irregularity of the surface is larger than 1, the action of G on the second cohomology is mostly faithful. For surfaces with irregularity 0 or 1, examples are given. Then I consider the stable degenerations of surfaces isogenous to a product and classify the possible singularities on them. As a result, I show that the Q-Gorenstein deformations of the degenerations with certain singuarities are unobstructed and get some connected components of the moduli space of stable surfaces.
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Cosserat Operators of Higher Order and Applications
(2010)
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Thorsten Riedl
- We take a look at certain operators called Cosserat operators and get a compactness result for them leading to several interesting applications. For a more detailed abstract, see the actual abstract at the beginning of the work.
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Optimal Control Problems Governed by Nonlinear Partial Differential Equations and Inclusions
(2010)
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Julia Fischer
- The focus of this thesis lies on examining the solvability of optimal control problems constrained by nonlinear partial differential equations (PDE) and inclusions (PDI). There exist statements on the existence of solutions for optimal control problems with linear and semi-linear PDEs with monotone parts. The theory for non-monotone PDEs resp. the related optimal control problems is, to the author’s knowledge, incomplete regarding important issues. This concerns particularly the case of PDEs containing mappings, which only satisfy boundedness conditions on restricted sets. At first an optimal control problem is considered, which is characterized by a Laplace equation with Dirichlet boundary conditions and a nonlinear non-monotone Nemytskii operator. Under the decisive assumption of the existence of so called sub- and supersolutions for this differential equation and by introducing a truncation operator we can define an auxiliary problem which is characterized by a pseudomonotone operator. Thereby the solution theory for pseudomonotone operators of Brézis (1968) is applicable. Moreover, starting with the definitions of sub- und supersolution it can be shown, that every solution of the auxiliary problem is a solution of the original problem. The choice of a new optimal control problem which substitutes the original optimal control problem is again governed by the properties of the auxiliary operator. The equivalence of the auxiliary problem to the original problem and the existence of at least one solution can be shown. The technique of applying the Theorem of Lax-Milgram on a linearized problem can be adapted to the semi-linear non-monotone case. This procedure is already known from the theory of semi-linear monotone problems. For optimal control problems with quasi-linear differential equations, different methods are required. As in the semi-linear case, the property of pseudomonotonicity plays a key role in proving the existence of a solution of the quasi-linear PDE. In the proof of the existence of a solution for the optimal control problem other properties of the auxiliary operator are exploited. In the elliptic case operators which satisfy the S+ -property are important. In order to utilize this property, a transformation of the operator to some coercive auxiliary operator is necessary. For this reason a term is added, which penalizes the deviation from the admissible set of states. This term is characterized by a factor, which is derived explicitly in this work. The proof of the existence of a solution of the optimal control problem with parabolic equations is based on the definition of an auxiliary operator, coercivity and the S+ -property of operators. The set of solutions of the considered PDE is compact, but the number of solutions and the situation to each other is unknown. This leads to difficulties in deriving necessary optimality conditions. For this reason a direct approach to solve the optimal control problem with semi-linear PDEs is introduced. It is assumed, that the state constraints coincide with the sub- and the supersolution of the PDE with the upper and lower boundary of the control variable. Using an auxiliary operator, this assumption allows the formulation of an equivalent optimal control problem without pointwise state constraints. Through semi-discretization we can define a family of optimal control problems on a finite dimensional state-space. Existence of a subsequence of solutions of these optimal control problems which converges to a solution of the original problem is shown. Another important class of optimal control problems include differential inclusions which are described by multivalued operators. Quasi-linear elliptic inclusions are examined under global as well as local boundedness conditions. Under the assumption of global boundedness the properties of pseudomonotonicity and coercivity for a multivalued auxiliary operator are proven. The existence of at least one solution for the original inclusion follows from the application of a result from Hu and Papageorgiou (1997) on the auxiliary problem. The existence of at least one solution of the optimal control problem is proven by exploiting the coercivity of the multivalued auxiliary operator and the S+ -property of the non-multivalued part of this mapping. In the case of multivalued mappings of Clarke’s gradient type, the existence of at least one solution of the optimal control problem can be shown under local boundedness conditions. Elliptic as well as parabolic quasi-linear inclusions are considered. The proof is again based on coercivity and the S+ -property of the related auxiliary operators and the embedding properties of the spaces.
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The classification of isotrivially fibred surfaces with p_g=q=2, and topics on Beauville surfaces
(2010)
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Matteo Penegini
- In our thesis we treat mainly two topics: the classification of isotrivially fibred surfaces with p_g=q=2, and the construction of new Beauville surfaces. An isotrivially fibred surface is a smooth projective surface endowed with a morphism onto a smooth curve such that all the smooth fibres are isomorphic to each other. The first goal of this thesis is to classify the isotrivially fibred surfaces with p_g=q=2 completing and extending a result by Zucconi. As an important byproduct, we provide new examples of minimal surfaces of general type with p_g=q=2 and K^2=4,5 and the first example with K^2=6. We say that a surface S is isogenous to a product of curves if S = (C times F )/G, for C and F smooth curves and G a finite group acting freely on C times F. Beauville surfaces are a special case of surfaces isogenous to a product. In this thesis we include part of a joint work with Shelly Garion, in which we construct new Beauville surfaces with group G either PSL(2,p^e), or A_n, or S_n, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, and on classical results of Macbeath. The thesis is divided into three chapters, which are subdivided in several sections. In the first chapter we treat the problem of the classification of isotrivially fibred surfaces with p_g=q=2. We start by recalling some basic facts and theorems about fibred surfaces and surfaces isogenous to a higher product of curves. Then we solve the classification problem using techniques coming from both geometry and combinatorial group theory. In the second chapter we deal with Beauville surfaces. First we give a group theoretical characterization of them. Then we enunciate a theorem of Liebeck and Shalev that we use for the construction of Beauville surfaces with group A_n or S_n. Afterwards we also study Beauville surfaces with group PSL(2,p^e). In the third chapter we give a description of the locus, in the moduli space of surfaces of general type, corresponding to the surfaces isogenous to a product with p_g=q=2 described in the first chapter. Indeed, by the results proven by Catanese, this locus is a union of connected components, whose number can be computed using a theorem of Bauer and Catanese. In the same way we are able to provide an asymptotic result about the number of connected components of the moduli space corresponding to certain families of Beauville surfaces with group either PSL(2,p^e), or A_n, or (mathbb{Z}/nmathbb{Z})^2 as p and n go to infinity.
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Algebraische Approximation von Kählermannigfaltigkeiten
(2010)
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Florian Schrack
- Eine kompakte komplexe Mannigfaltigkeit heißt algebraisch approximierbar, wenn sie beliebig kleine projektive Deformationen besitzt. Eine natürliche Fragestellung ist, ob jede kompakte Kählermannigfaltigkeit algebraisch approximierbar ist. Während dies in Dimension 2 nach den Arbeiten von Kodaira richtig ist, hat Voisin vierdimensionale Gegenbeispiele gefunden. In Dimension 3 ist die Frage noch offen. Ziel der vorliegenden Arbeit ist es, den dreidimensionalen Fall etwas näher zu beleuchten. Dazu wird algebraische Approximierbarkeit zunächst von einem allgemeinen Standpunkt aus betrachtet. Es werden Funktorialitätsfragen untersucht, also der Zusammenhang zwischen algebraischer Approximierbarkeit der Quelle und des Ziels gewisser holomorpher Abbildungen, und Ergebnisse für verschiedene Klassen von Abbildungen erzielt, wie etwa Aufblasungen, endliche Abbildungen, Faserungen und Morikontraktionen. Als Fallstudie einer konkreten Klasse von Kählerdreifaltigkeiten werden anschließend Konikbündel über Kählerflächen untersucht, die in natürlicher Weise in der Moritheorie auftreten. Nach dem Beweis einiger grundlegender Tatsachen über Konikbündel werden ihre Diskriminantenorte genauer untersucht und damit Chernklassenabschätzungen für Konikbündel mit relativer Picardzahl 1 über nichtalgebraischen kompakten Kählerflächen hergeleitet. Unter Verwendung dieser Abschätzungen wird die Existenz infinitesimaler Deformationen solcher Konikbündel gezeigt, was einen wichtigen ersten Schritt zum Beweis der algebraischen Approximierbarkeit darstellt. Ein spezieller Typ von Konikbündeln sind die projektivierten Rang-2-Bündel. Die Periodenabbildung verhilft zu einem guten Verständnis der Deformationstheorie solcher Bündel über K3-Flächen und zweidimensionalen Tori. Konkret werden Fortsetzungssätze für Geradenbündel und Rang-2-Bündel bewiesen, die implizieren, dass jedes projektivierte Rang-2-Bündel über einer K3-Fläche oder einem zweidimensionalen Torus algebraisch approximierbar ist. Durch Untersuchung von Aufblasungen solcher Flächen wird dieses Resultat auf projektivierte Rang-2-Bündel über beliebigen kompakten Kählerflächen mit Kodairadimension 0 ausgedehnt. Schließlich wird die zuvor entwickelte Deformationstheorie für Vektorbündel verwendet, um weitere Approximierbarkeitsergebnisse für Konikbündel über elliptischen Flächen und K3-Flächen zu bekommen.
