Topological groupoids with"approximate" inverses are studied. In the compact case, these"approximate" inverses turn out to be true inverses. Examples of groupoids wL:h"approximate" inverses are given in the section dealing with function spaces. Using the classical construction of Haar as a guide, we succeed in obtaining a (non-trivial) regular, right-invariant measure over a locally compact left group satisfying the conditions: a) open sets are preserved by left translation b) each group component is open. In the section dealing with integrals, we consider a compact metric topological semigroup that is right simple 3nd possesses a right contractive metric (ρ(xz,yz) ≤ ρ(x,y)). It is shown that such a structure always carries a non-trivial right-invariant integral. Throughout the entire development, associativity is invoked only once. The investigation concludes with a section dealing with sufficient conditions under which binary-topological systems become topological groups. A mob-group is defined to be a T_o-space which is also an algebraic group. A theorem in the last section states that a mob-group is a topological group iff given any open set W about the identity, W ∩ W⁻¹ has non-void interior. / Ph. D.

This thesis is an investigation of the concept of paracompactness. It presents the history of paracompactness, analyzes this concept from several diverse points of view and tries to establish the relationship between these different views. The starting point is the work of Tukey [40]. The important problem of the metrization of topological spaces is presented as an application of the concept of paracompactness. / Master of Science

The thesis develops the acoustic theory of the propagation of the shook waves produced by an aircraft in supersonic flight through an atmosphere in which the speed of sound decreases linearly with altitude. The problem is first studied in terms of the geometry of the rays along which the shock wave travels away from its point of origin and into the surrounding atmosphere. The equation of the rays is derived and certain important properties of the rays are discussed. It is shown how these results lead to a systematic graphical procedure for determining the location of the shock wave of a maneuvering aircraft. The theory is then considered in terms of the geometry of the "wave fronts" which represent the instantaneous positions of the individual disturbances created along the flight path. The shape of a wave front and its growth with time are determined. From this the equations for the envelope of a one-parameter family of wave fronts are obtained. The envelope equations are solved in parametric form and several examples are worked out which show some effects of flight maneuvers upon shock wave propagation. / M.S.

Let{(}φ_n(x)} be an orthonormal system in the set of Lebesgue square integrable functions L². Let f𝜖L². The generalized Fourier series of f with respect to {(}φ_n(x)} is the series ∑_n=0^∞ (f, φ_n) φ_n(x), where (f, φ_n) is the inner product of the functions f an φ_n. The e existence of a complete orthonormal system in L² is proven. Conditions for convergence of the generalized Fourier series are presented. A discussion of orthogonal polynomials with special emphasis on the Jacobi polynomial systems is presented. A least squares, differential correction, discrete observation procedure is employed to solve the potential equation with boundary conditions in tenns of three special Jacobi systems. / M.S.

This thesis deals with a particle limit for the n dimensional wave equation and shows that there are asymptotic solutions for certain pulses in the high-frequency limit. These pulses are shown to propagate along rays predicted by geometrical optics. The solutions are computed up to an error which approaches zero as the pulse approaches the particle limit. The method gives a closed solution to the question of where the energy propagates. We assume that the n dimensional space is divided into two halfspaces with two different wave speeds and that these two halfspaces have an interface where the wave speed is not continuous. / M.S.

This thesis constitutes a study in the field of approximation theory and is restricted to sets defined on the complex plane. The main objective is to present means that have been used to determine whether a sequence of polynomials can be considered as being uniformly convergent to a given analytic function and if convergence can be considered stronger than uniform. Background material is given in Chapter I. This includes definitions of point sets and of measures of approximation. Also basic theorems concerning both approximation and conformal mapping are given. In Chapter II properties of conformal mappings are established. The theorems discussed lead to a statement of necessary and sufficient conditions for a mapping of a simply connected region onto the interior of the unit circle to be homeomorphic on the closure of the region. The bulk of the work presented in Chapter II is based on definitions and theorems given by Caratheodory and Markushevich. The last chapter puts to use the theorems given in Chapters I and II to prove Walsh's Theorem and Farrell's Theorem. Other theorems originally presented by Walsh and Mergelyan are also discussed in Chapter III. The thesis concludes with examples of sequences uniformly convergent but not convergent in a given measure of approximations in order to show the reader that the latter property is stronger. / M.S.

This thesis is intended as a self-contained, expository introduction to material found in chapter IV of Bourbaki’s Theorie des Ensembles. With a minimum of external reference, it presents all relevant logical and set-theoretic background material and then develops and extends the notions of "species of structure”, "intrinsic terms”, "canonical mappings", "processes of deduction”, "morphism”, etc. found in this work. / Thesis / Master of Science (MS)

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In meiner Diplomarbeit habe ich mich mit den Organisationsformen im Mathematikunterricht beschäftigt. Ich wollte nicht nur diese Problematik in Betracht nehmen, sondern mich auf den ganzen Bildungsprozess richten. Ich glaube, dass die einzelnen Teile des Bildungsprozesses nicht allein angesehen werden sollen, weil sie sich gegeneinander beeinflussen. Nur gemeinsame harmonische Verflechtung bedeutet einen effektiven und erfolgreichen Bildungsprozess. Den Nachdruck habe ich auf zwei Sachen gelegt. Erstens ist die Didaktik der Mathematik wichtig. Für einen guten Unterricht muss man alle Zusammenhänge gut überlegen, d.h. das Erkenntnissgefüge der Schüler, ihre Denkweise, die Charakteristik der Zielgruppc usw. Zweitens sollte man nicht nur die intellektuelle Seite der Schüler in Einsicht nehmen, sondern sich auf ihre ganzheitliche Persönlichkeit richten. Also auch auf die persönliche und sozialle Entwicklung. Powered by TCPDF (www.tcpdf.org)

Universiẗat, Diss., 2006--Bochum.