Seinen Versuch, den Begriff der negativen Größen in die Weltweisheit einzuführen beginnt der neununddreißigjährige Immanuel Kant mit einer grundsätzlichen Erörterung über einen etwaigen Gebrauch, den man in der Weltweisheit von der Mathematik ma-chen kann. Dabei stellt er die These auf, daß Mathematik grundsätzlich nur auf zweierlei Art in die Philosophie eingreifen könne. Eine erste Möglichkeit sieht Kant in der Nachahmung mathematischer Methoden bei der Darstellung von Philosophie, die andere Möglichkeit besteht für ihn in der konkreten Anwendung mathematischer Theorien in der Naturlehre. Die zuerst genannte Möglichkeit beurteilt Kant ausgesprochen negativ; seine Kritik an dem von Comenius zunächst ganz allgemein formulierten und dann von Christian Wolff insbesondere für die Philosophie favorisierten Programm einer Präsentation der Philosophie nach mathematischem Vorbild einer Darstellung more geometrico demonstrata ist hinlänglich bekannt. Die Verwendung von Mathematik in der Naturlehre sieht Kant zwar durchaus positiv; in den Metaphysischen Anfangsgründen der Naturwissenschaft wird er gut zwei Jahrzehnte später sogar jene berühmte Behauptung hinzufügen, daß in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist. Dennoch weist Kant mit aller Deutlichkeit auf die engen Grenzen des Wirkungsbereichs solcher Anwendungen von Mathematik hin, denn seiner Meinung nach würden aber auch nur die zur Naturlehre gehörigen Einsichten von derartigem mathematischem Zugriff profitieren.
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 [4] and one by Sims [15]. A new approach using prefix string rewriting in free groups is derived from the algebraic method presented by Reinert, Mora and Madlener in [14] 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.
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.
Mathematische Spuren in der Philosophie
(1999)
