52 search hits
-
Derived categories of coherent sheaves on rational homogeneous manifolds
(2005)
-
Christian Böhning
- Abstract. One way to reformulate the celebrated theorem of Beilinson is that $(\mathcal{O}(-n),\dots , \mathcal{O})$ and $(\Omega^n(n), \dots , \Omega^1 (1), \mathcal{O})$ are strong complete exceptional sequences in $D^b(Coh\,\mathbb{P}^n)$, the bounded derived category of coherent sheaves on $\mathbb{P}^n$. In a series of papers M. M. Kapranov generalized this result to flag manifolds of type $A_n$ and quadrics. In another direction, Y. Kawamata has recently proven existence of complete exceptional sequences on toric varieties. Starting point of the present work is a conjecture of F. Catanese which says that on every rational homogeneous manifold $X=G/P$, where $G$ is a connected complex semisimple Lie group and $P\subset G$ a parabolic subgroup, there should exist a complete strong exceptional poset and a bijection of the elements of the poset with the Schubert varieties in $X$ such that the partial order on the poset is the order induced by the Bruhat-Chevalley order. An answer to this question would also be of interest with regard to a conjecture of B. Dubrovin which has its source in considerations concerning a hypothetical mirror partner of a projective variety $Y$: There is a complete exceptional sequence in $D^b(Coh\, Y)$ if and only if the quantum cohomology of $Y$ is generically semisimple (the complete form of the conjecture also makes a prediction about the Gram matrix of such a collection). A proof of this conjecture would also support M. Kontsevich's homological mirror conjecture, one of the most important open problems in applications of complex geometry to physics today. The goal of this work will be to provide further evidence for F. Catanese's conjecture, to clarify some aspects of it and to supply new techniques. In section 2 it is shown among other things that the length of every complete exceptional sequence on $X$ must be the number of Schubert varieties in $X$ and that one can find a complete exceptional sequence on the product of two varieties once one knows such sequences on the single factors, both of which follow from known methods developed by Rudakov, Gorodentsev, Bondal et al. Thus one reduces the problem to the case $X=G/P$ with $G$ simple. Furthermore it is shown that the conjecture holds true for the sequences given by Kapranov for Grassmannians and quadrics. One computes the matrix of the bilinear form on the Grothendieck $K$-group $K_{\circ}(X)$ given by the Euler characteristic with respect to the basis formed by the classes of structure sheaves of Schubert varieties in $X$; this matrix is conjugate to the Gram matrix of a complete exceptional sequence. Section 3 contains a proof of theorem 3.2.7 which gives complete exceptional sequences on quadric bundles over base manifolds on which such sequences are known. This enlarges substantially the class of varieties (in particular rational homogeneous manifolds) on which those sequences are known to exist. In the remainder of section 3 we consider varieties of isotropic flags in a symplectic resp. orthogonal vector space. By a theorem due to Orlov (thm. 3.1.5) one reduces the problem of finding complete exceptional sequences on them to the case of isotropic Grassmannians. For these, theorem 3.3.3 gives generators of the derived category which are homogeneous vector bundles; in special cases those can be used to construct complete exceptional collections. In subsection 3.4 it is shown how one can extend the preceding method to the orthogonal case with the help of theorem 3.2.7. In particular we prove theorem 3.4.1 which gives a generating set for the derived category of coherent sheaves on the Grassmannian of isotropic 3-planes in a 7-dimensional orthogonal vector space. Section 4 is dedicated to providing the geometric motivation of Catanese's conjecture and it contains an alternative approach to the construction of complete exceptional sequences on rational homogeneous manifolds which is based on a theorem of M. Brion (thm. 4.1.1) and cellular resolutions of monomial ideals a la Bayer/Sturmfels. We give a new proof of the theorem of Beilinson on $\mathbb{P}^n$ in order to show that this approach might work in general. We also prove theorem 4.2.5 which gives a concrete description of certain functors that have to be investigated in this approach.
-
Numerical Contributions to the Asymptotic Theory of Robustness
(2005)
-
Matthias Kohl
- In the framework of this dissertation a software package – the R bundle RobASt – by means of the statistics software R has been developed. It includes all robust procedures introduced throughout the thesis. The dissertation itself consists of five parts and starts with a brief motivation, which makes precise why robust statistics is necessary. After that a detailed summary in German and English is given. Part I provides a description of the asymptotic theory of robustness (Chapter 1) which forms the basis of this thesis. It is based on Chapters 4 and 5 of Rieder (1994). Chapter 2 provides supplements to the asymptotic theory of robustness which have proved necessary for this thesis. More precisely, it contains results about: properties of the optimally robust influence curves (ICs), how one should proceed in an optimal way if the neighborhood radius is unknown – as mostly in practice, and the construction of estimates by means of the one-step method. At the end of Chapter 2 convergence of robust models is introduced which is related to the concept of convergence of experiments of Le Cam. Part II deals with optimally robust estimators for some non-standard models in robust statistics. These models are covered by the R package ROptEst which makes use of S4 classes and methods and is part of the R bundle RobASt. More precisely, the binomial (Chapter 3) and Poisson (Chapter 4) model, the exponential scale and Gumbel location model (Chapter 5) as well as the Gamma model (Chapter 6) are investigated. In particular, the binomial and Poisson model are used to study convergence of robust models. Using exponential scale and Gumbel location one can show that there is a connection between certain scale and location models via a log-transformation which also holds for the corresponding optimally robust ICs. Finally, the Gamma model is used to demonstrate how differentiable parameter transformations can be estimated in an optimally robust way. In Part III robust regression with random regressor and unknown error scale (Chapter 7) is treated where it is distinguished between simultaneous and separate estimation. In both cases the optimally robust estimators as well as robust estimators for several narrower classes of M estimators are considered. All these estimators are implemented in the R packages ROptRegTS and RobRex which are part of the R bundle RobASt. Numerical comparisons for several regressor distributions show that the various suboptimal M estimators may have very small but also huge efficiency losses. A further comparison of these and several other well-known robust estimators in case of normal location and scale is made in Chapter 8. These location and scale estimators are implemented in the R package RobLox which is part of the R bundle RobASt. In Part IV (Chapter 9) robust adaptivity in terms of two asymptotic MSE problems is defined. Hence, adaptivity is no longer only a dichotomous criterion but can be evaluated quantitatively in terms of efficiency loss. The various regression and time series models considered include models which are classically as well as robust-adaptive, models which are classically but not robust-adaptive, and finally models which are neither classically nor robust-adaptive. The numerical evaluations show that non-adaptivity depends in a crucial way on the considered model and may be very small in some models (e.g. AR(1) and MA(1)) but may be really huge in other models (e.g. ARCH(1)). Finally, in Part V (Chapter 10 – 12) asymptotic results are compared with their exact finite-sample counterparts. In case of a particular pseudo-loss function in terms of under-/overshoot probabilities an exact finite-sample as well as an asymptotic theory are available. As the analytic evaluation of the finite-sample risk turns out very difficult or even impossible for sample sizes larger than 2, algorithms based on the fast Fourier transform (FFT) have been developed to determine the exact finite-sample distribution of these differently robust estimators. Two interesting findings are: The (first order) asymptotics is too optimistic and the convergence towards the asymptotic values is better in case of total variation than in case of contamination neighborhoods. The appendix of this thesis contains supplementary results on the asymptotic theory of robustness for regression-type models (Appendix A), on the Kronecker product and the vec and vech operators (Appendix B) as well as on the convolution via FFT (Appendix C). Moreover, Appendix D provides a brief description of the R packages distrEx, RandVar, ROptEst and ROptRegTS which are part of the R bundle RobASt.
-
Nilmanifolds: complex structures, geometry and deformations
(2007)
-
Sönke Rollenske
- We consider nilmanifolds with left-invariant complex structure and prove that in the generic case small deformations of such structures are again left-invariant. The relation between nilmanifolds and iterated principal holomorphic torus bundles is clarified and we give criteria under which deformations in the large are again of such type. As an application we obtain a fairly complete picture in dimension three. We show by example that the Frölicher spectral sequence of a nilmanifold may be arbitrarily non degenerate thereby answering a question mentioned in the book of Griffith and Harris. On our way we prove Serre Duality for Lie algebra Dolbeault cohomology and classify complex structures on nilpotent Lie algebras with small commutator subalgebra. MS Subject classification: 32G05; (32G08, 17B30, 53C30, 32C10)
-
Stability of flat galaxies
(2008)
-
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.
-
Existence and stability of stellardynamic models
(2008)
-
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.
-
Higher Order Asymptotics for the MSE of Robust M-Estimators of Location on Shrinking Total Variation Neighborhoods
(2008)
-
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.
-
Existence Results for Plasma Physics Models Containing a Fully Coupled Magnetic Field
(2009)
-
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.
-
Receding Horizon Control: A Suboptimality-based Approach
(2009)
-
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.
-
Surfaces Isogenous to a Product: Their Automorphisms and Degenerations
(2010)
-
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.
-
Cosserat Operators of Higher Order and Applications
(2010)
-
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.
