Duisburg, Essen, Universiẗat, Diss., 2005.

No description available.

After an introduction to Brownian motion, Hausdorff dimension, nonstandard analysis and Loeb measure theory, we explore the notion of a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, we find that Hausdorff dimension can be computed through a counting argument rather than the traditional way. This formulation is then applied to obtain simple proofs of some of the dimensional properties of Brownian motion, such as the doubling of the dimension of a set of dimension smaller than 1/2 under Brownian motion, by utilising Anderson's formulation of Brownian motion as a hyperfinite random walk. We also use the technique to refine a theorem of Orey and Taylor's on the Hausdorff dimension of the rapid points of Brownian motion. The result is somewhat stronger than the original. Lastly, we give a corrected proof of Kaufman's result that the rapid points of Brownian motion have similar Hausdorff and Fourier dimensions, implying that they constitute a Salem set. / Mathematical Sciences / D. Phil. (Mathematical Sciences)

University, Diss., 2002--Tübingen.

After an introduction to Brownian motion, Hausdorff dimension, nonstandard analysis and Loeb measure theory, we explore the notion of a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, we find that Hausdorff dimension can be computed through a counting argument rather than the traditional way. This formulation is then applied to obtain simple proofs of some of the dimensional properties of Brownian motion, such as the doubling of the dimension of a set of dimension smaller than 1/2 under Brownian motion, by utilising Anderson's formulation of Brownian motion as a hyperfinite random walk. We also use the technique to refine a theorem of Orey and Taylor's on the Hausdorff dimension of the rapid points of Brownian motion. The result is somewhat stronger than the original. Lastly, we give a corrected proof of Kaufman's result that the rapid points of Brownian motion have similar Hausdorff and Fourier dimensions, implying that they constitute a Salem set. / Mathematical Sciences / D. Phil. (Mathematical Sciences)

No description available.

L’objet de ce travail est l’utilisation de certains formalismes non classiques (analyses non standard, analyses constructives) afin de proposer des bases théoriques nouvelles autour des problèmes de discrétisations d’objets continus. Ceci est fait en utilisant un modèle discret du système des nombres réels appelé droite d’Harthong-Reeb ainsi que la méthode arithmétisation associée qui est un processus de discrétisation des fonctions continues. Cette étude repose sur un cadre arithmétique non standard. Dans un premier temps, nous utilisons une version axiomatique de l’arithmétique non standard. Puis, dans le but d’améliorer le contenu constructif de notre méthode, nous utilisons une autre approche de l’arithmétique non standard découlant de la théorie des Ω-nombres de Laugwitz et Schmieden. Cette seconde approche amène à une représentation discrète et multi-résolution de fonctions continues.Finalement, nous étudions dans quelles mesures, la droite d’Harthong-Reeb satisfait les axiomes de Bridges décrivant le continu constructif. / The aim of this work is to introduce new theoretical basis for the discretization of continuous objects using non classical formalisms. This is done using a discrete model of the continuum called the Harthong-Reeb line together with the related arithmetization method which is a discretisation process of continuous functions. This study stands on a nonstandard arithmetical framework. Firstly, we use an axiomatic version of nonstandard arithmetic. In order to improve the constructive content of our method, the next step is to use another approach of nonstandard arithmetic deriving from the theory of Ω-numbers by Laugwitzand Schmieden. This second approach leads to a discrete multi-resolution representation of continuous functions. Afterwards, we investigate to what extent the Harthong-Reeb line fits Bridges axioms of the constructive continuum.

Univ., Diss.--Cambridge, 1991. / Literaturverz. S. [133] - 138.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22978号 / 理博第4655号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 照井 一成, 教授 長谷川 真人, 准教授 河村 彰星 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This paper presents new concepts in the use of infinite and infinitesimal numbers in real analysis. theory is based upon the hyperreal number system developed by Abraham Robinson in the 1960's in his invention of "nonstandard analysis". paper begins with a short exposition of the construction of the hyperreal nU1l1ber system and the fundamental results of nonstandard analysis which are used throughout the paper. The new theory which is built upon this foundation organizes the set hyperreal numbers through structures which on an infinite base logarithm. Several new relations are introduced whose properties enable the simplification of calculations involving infinite and infinitesimal The paper explores two areas of application of these results to standard problems in elementary calculus. The first is to the evaluation of limits which assume indeterminate forms. The second is to the determination of convergence of infinite series. Both applications provide methods which greatly reduce the amount of con1putation necessary in many situations. / Master of Science