Thesis (Th. M.)--Westminster Theological Seminary, Philadelphia, 2003. / Includes vita. Includes bibliographical references (leaves 185-188).

Thesis (Th. M.)--Westminster Theological Seminary, Philadelphia, 2003. / Includes vita. Includes bibliographical references (leaves 185-188).

Thesis (Th. M.)--Westminster Theological Seminary, Philadelphia, 2003. / Includes vita. Includes bibliographical references (leaves 185-188).

Uma das questões mais importantes em Teoria de Singularidades é a determinação de condições que garantam a trivialidade topológica em famílias de germes de funções ou aplicações. Neste trabalho é feito um estudo a fim de descrever condições necessárias e suficientes para a trivialidade topológica em famílias de germes de funções com singularidade isolada. Para isto, são apresentados dois métodos. O primeiro é o de campos de vetores controlados, baseado nos trabalhos de Damon-Gaffney e Yoshinaga. O segundo relaciona invariantes associados às famílias de germes de funções com a trivialidade topológica destas. Em ambos os casos, a principal ferramenta é a construção de poliedros de Newton associados às famílias. / One of the most important questions in Theory of Singularities is the determination of conditions that guarantee the topological triviality in families of germs of functions or mappings. In this work a study is made in order to describe necessaries and sufficients conditions for the topological triviality in families of germs of functions with isolated singularity. For this, two methods are presented. The first one is controlled vectors fields method, based on the works of Damon-Gaffney and Yoshinaga. The second relates invariants associated with families of germs of functions with the topological triviality of these. In both cases, the main tool used is the construction of Newton polyhedra associated with families.

No description available.

Our thesis explores aspects of the geometrical work and thought of Isaac Newton in order to better understand and re-evaluate his approach to geometry, and specifically his synthetic methods and the organic description of plane curves. In pursuing this research we study Newton's geometrical work in the context of the changing view of geometry between the late 16th and early 18th centuries, a period defined by the responses of the early modern geometers to a new Latin edition of Pappus' Collectio. By identifying some of the major challenges facing geometers of this period as they attempted to define and practice geometry we are able to contrast Newton's own approach to geometry. The themes emerging from the geometrical thought of early modern geometers provide the mathematical context from which to understand, interpret and re-evaluate the approach taken by Newton. In particular we focus on Newton's profound rejection of the new algebraic Cartesian methods and geometrical philosophies, and the opportunity to focus more clearly on some of his most astonishing geometrical contributions. Our research highlights Newton's geometrical work and examines specific examples of his synthetic methods. In particular we draw attention to the significance of Newton's organic construction and the limitations of Whiteside's observations on this subject. We propose that Newton's organic rulers were genuinely original. We disagree with Whiteside that they were inspired by van Schooten, except in the loosest sense. Further, we argue that Newton's study of singular points by their resolution was new, and that it has been misunderstood by Whiteside in his interpretation of the transformation effected by the rulers. We instead emphasise that it was the standard quadratic transformation. Overall we wish to make better known the importance of geometry in Newton's scientific thought, as well as highlighting the mathematical and historical importance of his organic description of curves as an example of his synthetic approach to geometry. This adds to contemporary discourse surrounding Newton's geometry, and specifically provides a foundation for further research into the implications of Newton's geometrical methods for his successors.

In this dissertation, we consider modified inexact Newton methods applied to second order nonlinear problems. In the implementation of Newton's method applied to problems with a large number of degrees of freedom, it is often necessary to solve the linear Jacobian system iteratively. Although a general theory for the convergence of modified inexact Newton's methods has been developed, its application to nonlinear problems from nonlinear PDE's is far from complete. The case where the nonlinear operator is a zeroth order perturbation of a fixed linear operator was considered in the paper written by Brown et al.. The goal of this dissertation is to show that one can develop modified inexact Newton's methods which converge at a rate independent of the number of unknowns for problems with higher order nonlinearities. To do this, we are required to first, set up the problem on a scale of Hilbert spaces, and second, to devise a special iterative technique which converges in a higher order Sobolev norm, i.e., H1+alpha(omega) \ H1 0(omega) with 0 < alpha < 1/2. We show that the linear system solved in Newton's method can be replaced with one iterative step provided that the initial iterate is close enough. The closeness criteria can be taken independent of the mesh size. In addition, we have the same convergence rates of the method in the norm of H1 0(omega) using the discrete Sobolev inequalities.

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaf 93). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaf 93). Also available on the Internet.

Σκοπός της παρούσης εργασίας είναι η παρουσίαση διαφόρων μεθόδων υπολογισμού των ροπών κατανομής των ριζών των ορθογωνίων πολυωνύμων, ισοδύναμα των αθροισμάτων Newton των ριζών, δηλαδή με, ενός πολυωνύμου βαθμού και των αθροισμάτων Stieltjes που είναι αθροίσματα της μορφής, όπου και είναι οι ρίζες μιας λύσης μιας ομογενούς διαφορικής εξίσωσης δεύτερης τάξης. Κάποια από αυτά τα αθροίσματα βρίσκουν εφαρμογή στο να φράσουμε τις ρίζες κάποιων ειδικών συναρτήσεων, ενώ άλλα χρησιμοποιούνται για την μελέτη της ασυμπτωτικής κατανομής των ριζών των ορθογωνίων πολυωνύμων και για τη μελέτη της μονοτονίας των ριζών. Παρουσιάζουμε δύο μεθόδους υπολογισμού των ροπών της κατανομής των ριζών των ορθογωνίων πολυώνυμων. Στην πρώτη μέθοδο υπολογίζουμε τα αθροίσματα Netwon των ριζών χρησιμοποιώντας τις ιδιοτιμές ενός τριδιαγώνιου πίνακα, ενώ στη δεύτερη μέθοδο ο υπολογισμός των αθροισμάτων Netwon γίνεται μέσω των συντελεστών των διαφορικών εξισώσεων που ικανοποιούν τα πολυώνυμα. Θα πρέπει εδώ να τονίσουμε ότι η δεύτερη μέθοδος μας επιτρέπει να υπολογίσουμε αριθμητικά τα αθροίσματα Netwon. / We present some methods for computing the distirbution of the roots of orthogonal polynomials, equivantly of Newton sums, and we sketch their use in determining some properties of second order ordinary differential equations.