Thesis (Ed.M.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / 2031-01-01

Thesis (Ed.M.)--Boston University

Thesis (Ed.M.)--Boston University

There is no abstract available for this thesis.

The central result presented in this thesis is van der Waerden's theorem on arithmetic progressions. Van der Waerden's theorem guarantees that for any integers k and r, there is an n so that however the set {1, 2, ... , n} is split into r disjoint partition classes, at least one partition class will contain a k-term arithmetic progression. Presented here are a number of variations and generalizations of van der Waerden's theorem that utilize a wide range of techniques from areas of mathematics including combinatorics, number theory, algebra, and topology. / May 2007

We are interested in investigating the number of integral points on quadrics. First, we consider non-degenerate plane conic curves defined over Z. In particular we look at two types of conic sections: hyperbolas with two rational points at infinity, and ellipses. We give upper bounds for the number of integral points on such curves which depends on the number of divisors of the determinant of a given conic. Next we consider quadratic surfaces of the form q(x, y, z) = k, where k is an integer and q is a non-degenerate homogeneous quadratic form defined over Z. We give an upper bound for the number of integral points (x, y, z) with bounded height.

We are interested in investigating the number of integral points on quadrics. First, we consider non-degenerate plane conic curves defined over Z. In particular we look at two types of conic sections: hyperbolas with two rational points at infinity, and ellipses. We give upper bounds for the number of integral points on such curves which depends on the number of divisors of the determinant of a given conic. Next we consider quadratic surfaces of the form q(x, y, z) = k, where k is an integer and q is a non-degenerate homogeneous quadratic form defined over Z. We give an upper bound for the number of integral points (x, y, z) with bounded height.

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves.

Thesis (EdD)-Open University.

Inaug.-diss.--Universität zu Münster. / Lebenslauf. Autographic reproduction of type-written copy; printed t.p. "Die vorliegende arbeit is aus einer bearbeitung der preisaufgabe 'Der gegenwärtige stand des verhältnisses von logik und mathematik, mit besonderer berücksichtigung des Frege-Russellschen aufbaus der arithmetik' hervorgegangen, die die Philosophische und naturwessenschaftliche fakult ät der Imoversotät Münster i. W. ... für das studienjahr 1932-1933 gestellt hatte, und is nach zuerkennung des preises im frühjahr 1933 ... als dissertation. Issued also as Forschungen zur Logistik und Zur grundlegung der exacten wissenschafter, hrsg. von Heinrich Scholz ... hft. 1. "Literatureverzeichnis": p. 78.