Kaiserslautern - Fachbereich Mathematik
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In this thesis we classify simple coherent sheaves on Kodaira fibers of types II, III and IV (cuspidal and tacnode cubic curves and a plane configuration of three concurrent lines). Indecomposable vector bundles on smooth elliptic curves were classified in 1957 by Atiyah. In works of Burban, Drozd and Greuel it was shown that the categories of vector bundles and coherent sheaves on cycles of projective lines are tame. It turns out, that all other degenerations of elliptic curves are vector-bundle-wild. Nevertheless, we prove that the category of coherent sheaves of an arbitrary reduced plane cubic curve, (including the mentioned Kodaira fibers) is brick-tame. The main technical tool of our approach is the representation theory of bocses. Although, this technique was mainly used for purely theoretical purposes, we illustrate its computational potential for investigating tame behavior in wild categories. In particular, it allows to prove that a simple vector bundle on a reduced cubic curve is determined by its rank, multidegree and determinant, generalizing Atiyah's classification. Our approach leads to an interesting class of bocses, which can be wild but are brick-tame.
Nonlinear diffusion filtering of images using the topological gradient approach to edges detection
(2007)
In this thesis, the problem of nonlinear diffusion filtering of gray-scale images is theoretically and numerically investigated. In the first part of the thesis, we derive the topological asymptotic expansion of the Mumford-Shah like functional. We show that the dominant term of this expansion can be regarded as a criterion to edges detection in an image. In the numerical part, we propose the finite volume discretization for the Catté et al. and the Weickert diffusion filter models. The proposed discretization is based on the integro-interpolation method introduced by Samarskii. The numerical schemes are derived for the case of uniform and nonuniform cell-centered grids of the computational domain \(\Omega \subset \mathbb{R}^2\). In order to generate a nonuniform grid, the adaptive coarsening technique is proposed.
The nowadays increasing number of fields where large quantities of data are collected generates an emergent demand for methods for extracting relevant information from huge databases. Amongst the various existing data mining models, decision trees are widely used since they represent a good trade-off between accuracy and interpretability. However, one of their main problems is that they are very instable, which complicates the process of the knowledge discovery because the users are disturbed by the different decision trees generated from almost the same input learning samples. In the current work, binary tree classifiers are analyzed and partially improved. The analysis of tree classifiers goes from their topology from the graph theory point of view to the creation of a new tree classification model by means of combining decision trees and soft comparison operators (Mlynski, 2003) with the purpose to not only overcome the well known instability problem of decision trees, but also in order to confer the ability of dealing with uncertainty. In order to study and compare the structural stability of tree classifiers, we propose an instability coefficient which is based on the notion of Lipschitz continuity and offer a metric to measure the proximity between decision trees. This thesis converges towards its main part with the presentation of our model ``Soft Operators Decision Tree\'\' (SODT). Mainly, we describe its construction, application and the consistency of the mathematical formulation behind this. Finally we show the results of the implementation of SODT and compare numerically the stability and accuracy of a SODT and a crisp DT. The numerical simulations support the stability hypothesis and a smaller tendency to overfitting the training data with SODT than with crisp DT is observed. A further aspect of this inclusion of soft operators is that we choose them in a way so that the resulting goodness function (used by this method) is differentiable and thus allows to calculate the best split points by means of gradient descent methods. The main drawback of SODT is the incorporation of the unpreciseness factor, which increases the complexity of the algorithm.
The thesis is concerned with multiscale approximation by means of radial basis functions on hierarchically structured spherical grids. A new approach is proposed to construct a biorthogonal system of locally supported zonal functions. By use of this biorthogonal system of locally supported zonal functions, a spherical fast wavelet transform (SFWT) is established. Finally, based on the wavelet analysis, geophysically and geodetically relevant problems involving rotation-invariant pseudodifferential operators are shown to be efficiently and economically solvable.