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- A bijection between the d-dimensional simplices with distances in {1,2} and the partitions of d+1 (2005)
- We give a construction for the d-dimensional simplices with all distances in {1,2} from the set of partitions of d+1.

- Derived categories of coherent sheaves on rational homogeneous manifolds (2005)
- 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.

- On the characteristic of integral point sets in $\mathbb{E}^m$ (2005)
- We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to provide new exact values for the minimum diameter of integral point sets.

- A note on Erdös-Diophantine graphs and Diophantine carpets (2005)
- A Diophantine figure is a set of points on the integer grid $\mathbb{Z}^{2}$ where all mutual Euclidean distances are integers. We also speak of Diophantine graphs. The vertices are points in $\mathbb{Z}^{2}$ (the coordinates)and the edges are labeled with the distance between the two adjacent vertices, which is integral. In this language a Diophantine figure is a complete Diophantine graph. Two Diophantine graphs are equivalent if they only differ by translation or rotation of vertices. Due to a famous theorem of Erdös and Anning there are complete Diophantine graphs which are not contained in larger ones. We call them Erdös-Diophantine graphs. A special class of Diophantine graphs are Diophantine carpets. These are planar triangulations of a subset of the integer grid. We give an effective construction for Erdös-Diophantine graphs and characterize the chromatic number of Diophantine carpets.

- Numerical Contributions to the Asymptotic Theory of Robustness (2005)
- 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.

- Lq-solutions to the Cosserat spectrum in bounded and exterior domains (2005)
- In the present paper we consider the existence of non-trivial classical and weak Lq-solutions of the Cosserat spectrum $$ Delta U u = a abla Div U u, qquad U u Big|_{partial G}=0 $$ where G is a bounded or an exterior domain with sufficiently smooth boundary. This problem firstly was investigated by Eugene and Francois Cosserat. It is a special case of the Lame equation and describes the displacement of a homogeneous isotropic linear static elastic body without exterior forces. We can prove that a = 1 is an eigenvalue of infinite multiplicity and a = 2 is an accumulation point of eigenvalues of finite multiplicity. E. and F. Cosserat (1900) studied the classical Cosserat spectrum for certain types of domains like a ball, a spherical shell or an ellipsoid. General results are due to Mikhlin (1973), who investigated the Cosserat spectrum for n=3 and q=2, and Kozhevnikov (1993), who treated bounded domains in the case n=3 and q=2. Kozhevnikovs proof is based on the theory of pseudodifferential operators. Faierman, Fries, Mennicken and Möller (2000) gave a direct proof for bounded domains, n>=2 and q=2. Michel Crouzeix 1997 gave a simple proof for bounded domains, n=2,3 and q=2. In this paper we use the idea of Crouzeix to prove the results for bounded and exterior domains, n>=2 and 1<q. For the Lq-solutions of eigenvalues a in R {1,2} we can prove the existence of higher (classical) derivatives. Furthermore they do not depend on q. a =2 is an accumulation point of eigenvalues of the classical Cosserat spectrum, too, and a =1 is also a classical eigenvalue. As an approach we searched for a relationship of Greens function of the Laplacian to the reproducing kernel in Bergman spaces. We couldnt prove that directly. But after solving the Cosserat spectrum in another way we can prove the relationship indirectly.

- A homotopy argument and its applications to the transformation rule for bi-Lipschitz mappings, the Brouwer fixed point theorem and the Brouwer degree (2005)
- The main purpose of the paper is to present an elementary self-contained proof of the change of variables formula for injective, locally bi-Lipschitz mappings. The proof is based on a homotopy argument. Various properties of bi-Lipschitz mappings are studied. As a by-product Lipschitz variants of the classical implicit function theorem and the local diffeomorphism theorem are proved. With the help of the homotopy argument a simple proof is given of Brouwer’s fixed point theorem and the main properties of Brouwer’s degree of mapping.

- Convex hulls of polyominoes (2007)
- In this article we prove a conjecture of Bezdek, Brass, and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of $n$ unit hypercubes in $mathbb{R}^d$. For $d=2$ we enumerate the extremal polyominoes and determine the set of possible areas of the convex hull for each $n$.

- Nilmanifolds: complex structures, geometry and deformations (2007)
- 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)

- Enumeration of integral tetrahedra (2007)
- We determine the numbers of integral tetrahedra with diameter d up to isomorphism for all d<=1000 via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most d in O(d^5) time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral 4x4 matrices with diameter d fulfilling the triangle inequalities we derive an exact formula.