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Metrical Properties of Convex Bodies in Minkowski SpacesAverkov, Gennadiy 27 October 2004 (has links)
The objective of this dissertation is the application of Minkowskian cross-section measures (i.e., section and projection measures in finite-dimensional linear normed spaces over the real
field) to various topics of geometric convexity in Minkowski spaces, such as bodies of constant Minkowskian width, Minkowskian geometry of simplices, geometric inequalities and the corresponding optimization problems for convex bodies. First we examine one-dimensional
Minkowskian cross-section measures deriving (in a unified manner) various properties of these measures. Some of these properties are
extensions of the corresponding Euclidean properties, while others are purely Minkowskian. Further on, we discover some new results
on the geometry of a simplex in Minkowski spaces, involving descriptions of the so-called tangent Minkowskian balls and of simplices with equal Minkowskian heights. We also give some (characteristic) properties of bodies of constant width in Minkowski planes and in higher dimensional Minkowski spaces. This part of investigation has relations to the well known \emph{Borsuk problem} from the combinatorial geometry and to the widely used monotonicity lemma from the theory of Minkowski spaces. Finally, we study bodies of given Minkowskian thickness ($=$ minimal width) having least possible volume. In the planar case a complete
description of this class of bodies is given, while in case of arbitrary dimension sharp estimates for the coefficient in the corresponding geometric inequality are found. / Die Dissertation befasst sich mit Problemen fuer spezielle konvexe Koerper in Minkowski-Raeumen (d.h. in endlich-dimensionalen Banach-Raeumen). Es wurden Klassen der Koerper mit verschiedenen metrischen Eigenschaften betrachtet (z.B., Koerper konstante Breite, reduzierte Koerper, Simplexe mit Inhaltsgleichen Facetten usw.) und einige kennzeichnende und andere Eigenschaften fuer diese Klassen herleitet.
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Higher order asymptotic expansions for weakly correlated random functionsStarkloff, Hans-Jörg 14 June 2004 (has links)
Die vorliegende Arbeit beschäftigt sich mit asymptotischen Entwicklungen höherer Ordnung für zweite Momente von Zufallsvariablen bzw. Zufallsfunktionen, die als lineare Integralfunktionale über schwach abhängige oder
schwach korrelierte Zufallsfunktionen definiert sind. Unter bestimmten Glattheits- und Integrabilitätsbedingungen an die Kernfunktionen
und Regularitätsbedingungen an die Zufallsfunktionen werden
entsprechende asymptotische Entwicklungen
angegeben, außerdem wird auf Abschätzungen der
Genauigkeit eingegangen. Die auftretenden Zufallsfunktionen sind dabei stationäre reell- oder vektorwertige Zufallsprozesse, bestimmte Klassen nichtstationärer Zufallsprozesse und homogene Zufallsfelder. Die Anwendungsmöglichkeit wird an einer Reihe von Beispielen aufgezeigt.
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Aspects of aperiodic order: Spectral theory via dynamical systemsLenz, Daniel 09 June 2005 (has links)
The first part of this work gives an introduction into aperiodic
order in general and the lines of research pursued.
The second part consists of eight
manuscripts.
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Optimal Multilevel Extension OperatorsNepomnyaschikh, Sergey V. 05 September 2005 (has links)
In the present paper we suggest the norm-preserving
explicit operator for the extension of
finite-element functions from boundaries of domains
into the inside. The construction of this operator
is based on the multilevel decomposition of
functions on the boundaries and on the equivalent
norm for this decomposition. The cost of the action
of this operator is proportional to the number of
nodes.
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Hierarchically preconditioned parallel CG-solvers with and without coarse-matrix-solvers inside FEAPMeisel, Mathias, Meyer, Arnd 07 September 2005 (has links)
After some remarks on the parallel implementation of the Finite Element package FEAP, our realisation of the parallel CG-algorithm is sketched. From a technical point of view, a hierarchical preconditioner with and without additional global crosspoint preconditioning is presented. The numerical properties of this preconditioners are discussed and compared to a Schur-complement-preconditioning, using a wide range of data from computations on technical and academic examples from elasticity.
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Solving stable generalized Lyapunov equations with the matrix sign functionBenner, Peter, Quintana-Ortí, Enrique S. 07 September 2005 (has links)
We investigate the numerical solution of the stable generalized Lyapunov equation via the sign function method. This approach has already been proposed to solve standard Lyapunov equations in several publications. The extension to the generalized case is straightforward. We consider some modifications and discuss how to solve generalized Lyapunov equations with semidefinite constant term for the Cholesky factor. The basic computational tools of the method are basic linear algebra operations that can be implemented efficiently on modern computer architectures and in particular on parallel computers. Hence, a considerable speed-up as compared to the Bartels-Stewart and Hammarling's methods is to be expected. We compare the algorithms by performing a variety of numerical tests.
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A Multi-Grid Method for Generalized Lyapunov EquationsPenzl, Thilo 07 September 2005 (has links)
We present a multi-grid method for a class of
structured generalized Lyapunov matrix equations.
Such equations need to be solved in each step of
the Newton method for algebraic Riccati equations,
which arise from linear-quadratic optimal control
problems governed by partial differential equations.
We prove the rate of convergence of the two-grid
method to be bounded independent of the dimension
of the problem under certain assumptions.
The multi-grid method is based on matrix-matrix
multiplications and thus it offers a great
potential for a parallelization. The efficiency
of the method is demonstrated by numerical
experiments.
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Schur-Like Forms for Matrix Lie Groups, Lie Algebras and Jordan AlgebrasAmmar, Gregory, Mehl, Christian, Mehrmann, Volker 09 September 2005 (has links)
We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group. Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these similarity transformations. In general, one cannot achieve diagonal or Schur form, but the form that can be achieved displays the eigenvalues of the matrix. We also discuss matrices in intersections of these classes and their Schur-like forms. Such multistructered matrices arise in applications from quantum physics and quantum chemistry.
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The Anderson Model of Localization: A Challenge for Modern Eigenvalue MethodsElsner, Ulrich, Mehrmann, Volker, Römer, Rudolf A., Schreiber, Michael 09 September 2005 (has links)
We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large, sparse, real, symmetric, and indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the implicitly restarted Arnoldi method coupled with polynomial and several shift-and-invert convergence accelerators as well as with a sparse hybrid tridiagonalization method. We demonstrate that for our problem the Lanczos implementation is faster and more memory efficient than the other approaches. This seemingly innocuous problem presents a major challenge for all modern eigenvalue algorithms.
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DGRSVX and DMSRIC: Fortran 77 subroutines for solving continuous-time matrix algebraic Riccati equations with condition and accuracy estimatesPetkov, P. Hr., Konstantinov, M. M., Mehrmann, V. 12 September 2005 (has links)
We present new Fortran 77 subroutines which implement the Schur method and the
matrix sign function method for the solution of the continuoustime matrix algebraic
Riccati equation on the basis of LAPACK subroutines. In order to avoid some of
the wellknown difficulties with these methods due to a loss of accuracy, we combine
the implementations with block scalings as well as condition estimates and forward
error estimates. Results of numerical experiments comparing the performance of both
methods for more than one hundred well and illconditioned Riccati equations of order
up to 150 are given. It is demonstrated that there exist several classes of examples for
which the matrix sign function approach performs more reliably and more accurately
than the Schur method. In all cases the forward error estimates allow to obtain a reliable
bound on the accuracy of the computed solution.
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