Nonlinear stochastic dynamical systems as ordinary stochastic differential equations and stochastic difference methods are in the center of this presentation in view of the asymptotical behaviour of their moments. We study the exponential p-th mean growth behaviour of their solutions as integration time tends to infinity. For this purpose, the concepts of nonlinear contractivity and stability exponents for moments are introduced as generalizations of well-known moment Lyapunov exponents of linear systems. Under appropriate monotonicity assumptions we gain uniform estimates of these exponents from above and below. Eventually, these concepts are generalized to describe the exponential growth behaviour along certain Lyapunov-type functionals.
The study of queuing theory brings us to the problems of finding to find the limit distribution of the maximal sum of a sequence of random variables and of estimating how close this distribution is to the distribution of the sum.
Rewriting techniques have been applied successfully to various areas of symbolic computation. Here we consider the notion of prefix-rewriting and give a survey on its applications to the subgroup problem in combinatorial group theory. We will see that for certain classes of finitely presented groups finitely generated subgroups can be described through convergent prefix-rewriting systems, which can be obtained from a presentation of the group considered and a set of generators for the subgroup through a specialized Knuth-Bendix style completion procedure. In many instances a finite presentation for the subgroup considered can be constructed from such a convergent prefix-rewriting system, thus solving the subgroup presentation problem. Finally we will see that the classical procedures for computing Nielsen reduced sets of generators for a finitely generated subgroup of a free group and the Todd-Coxeter coset enumeration can be interpreted as particular instances of prefix-completion. Further, both procedures are closely related to the computation of prefix Gr"obner bases for right ideals in free group rings.
Todd and Coxeter's method for enumerating cosets of finitely generated subgroups in finitely presented groups (abbreviated by Tc here) is one famous method from combinatorial group theory for studying the subgroup problem. Since prefix string rewriting is also an appropriate method to study this problem, prefix string rewriting methods have been compared to Tc. We recall and compare two of them briefly, one by Kuhn and Madlener  and one by Sims . A new approach using prefix string rewriting in free groups is derived from the algebraic method presented by Reinert, Mora and Madlener in  which directly emulates Tc. It is extended to free monoids and an algebraic characterization for the "cosets" enumerated in this setting is provided.
Complete presentations provide a natural solution to the word problem in monoids and groups. Here we give a simple way to construct complete presentations for the direct product of groups, when such presentations are available for the factors. Actually, the construction we are referring to is just the classical construction for direct products of groups, which has been known for a long time, but whose completeness-preserving properties had not been detected. Using this result and some known facts about Coxeter groups, we sketch an algorithm to obtain the complete presentation of any finite Coxeter group. A similar application to Abelian and Hamiltonian groups is mentioned.
In der folgenden Arbeit wird das klassische Verschlüsselungsverfahren von Vigenere vorgestellt. Die mathematischen Grundlagen werden präzise formuliert und anschliessend wird die Theorie durch ein praktisches Beispiel erläutert.
Compared to standard numerical methods for hyperbolic systems of conservation laws, Kinetic Schemes model propagation of information by particles instead of waves. In this article, the wave and the particle concept are shown to be closely related. Moreover, a general approach to the construction of Kinetic Schemes for hyperbolic conservation laws is given which summarizes several approaches discussed by other authors. The approach also demonstrates why Kinetic Schemes are particularly well suited for scalar conservation laws and why extensions to general systems are less natural.
Bedingt durch das Wachstum von Informationsnachfrage und -angebot werden effizientere Wege zur Repräsentation von Informationen aller Art benötigt. Dies kann sowohl durch Assimilation und Optimierung der gewählten Datenstruktur und ihrer Repräsentation als auch (additiv) durch Komprimierung derselbigen erreicht werden. Diese Ausarbeitung soll in pragmatischer Art und Weise in das Themengebiet der Komprimierung einführen. Vorgestellt werden insgesamt 3 Stellvertreter aus unterschiedlichen Bereichen : Komprimierung von Texten mittels Huffman-Code, Komprimierung von Bitlisten mittels Laufkomprimierung (RLE-Komprimierung), Komprimierung von - auf dem RGB-Farbmodell basierenden - Grafiken mittels eines eigenen Verfahrens. Während die ersten beiden Verfahren Vertreter verlustfreier Komprimierung sind, ist das Dritte ein Vertreter der verlustbehafteten Komprimierung. Die vorgestellten Verfahren werden zur Arrondierung an konkreten Beispielen eingeübt und schließlich sogar in der Programmiersprache Pascal implementiert. Die konkrete Realisation in einer gegebenen Programmiersprache birgt kanonischerweise die Gefahr, den Blick für das Wesentliche zu verlieren. Deshalb wurde bei der Erstellung dieser Ausarbeitung (im speziellen der Programmieraufgaben) akribisch auf Abstrahierung unnötiger Details geachtet.
A multiscale method is introduced using spherical (vector) wavelets for the computation of the earth's magnetic field within source regions of ionospheric and magnetospheric currents. The considerations are essentially based on two geomathematical keystones, namely (i) the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts and (ii) the Helmholtz decomposition of spherical (tangential) vector fields. Vector wavelets are shown to provide adequate tools for multiscale geomagnetic modelling in form of a multiresolution analysis, thereby completely circumventing the numerical obstacles caused by vector spherical harmonics. The applicability and efficiency of the multiresolution technique is tested with real satellite data.
A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an priori parameter choice strategy, it is shown that the method yields optimal order. Error estimates have also been obtained under stronger assumptions on the the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper includes, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also recent results of Tautenhahn (1996) in the setting Hilbert scales.