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#### Gröbner Bases Theory and The Diamond Lemma

(University of Waterloo, 2006)

Commutative Gröbner bases theory is well known and widely used. In this thesis, we will discuss thoroughly its generalization to noncommutative polynomial ring

**k**<**X**> which is also an associative free algebra. We introduce some results on monomial orders due to John Lawrence and the author. We show that a noncommutative monomial order is a well order while a one-sided noncommutative monomial order may not be. Then we discuss the generalization of polynomial reductions, S-polynomials and the characterizations of noncommutative Gröbner bases. Some results due to Mora are also discussed, such as the generalized Buchberger's algorithm and the solvability of ideal membership problem for homogeneous ideals. At last, we introduce Newman's diamond lemma and Bergman's diamond lemma and show their relations with Gröbner bases theory....#### Counting points of bounded height on del Pezzo surfaces

(University of Waterloo, 2006)

del Pezzo surfaces are isomorphic to either P<sup>1</sup> x P<sup>1</sup> or P<sup>2</sup> blown up <i>a</i> times, where <i>a</i> ranges from 0 to 8. We will look at lines on del Pezzo surfaces isomorphic to P<sup>2</sup> ...

#### Multiplicities of Linear Recurrence Sequences

(University of Waterloo, 2006)

In this report we give an overview of some of the major results concerning the multiplicities of linear recurrence sequences. We first investigate binary recurrence sequences where we exhibit a result due to Beukers and a result due to Brindza, Pintér and Schmidt. We then investigate ternary recurrences and exhibit a result due to Beukers building on work of Beukers and Tijdeman. The last two chapters deal with a very important result due to Schmidt in which we bound the zero-multiplicity of a linear recurrence sequence of order

**t**by a function involving**t**alone. Moreover we improve on Schmidt's bound by making some minor changes to his argument....#### On the Similarity of Operator Algebras to C*-Algebras

(University of Waterloo, 2006)

This is an expository thesis which addresses the requirements for an operator algebra to be similar to a

**C***-algebra. It has been conjectured that this similarity condition is equivalent to either amenability or total reductivity; however, the problem has only been solved for specific types of operators. <br /><br /> We define amenability and total reductivity, as well as present some of the implications of these properties. For the purpose of establishing the desired result in specific cases, we describe the properties of two well-known types of operators, namely the compact operators and quasitriangular operators. Finally, we show that if A is an algebra of compact operators or of triangular operators then A is similar to a**C*** algebra if and only if it has the total reduction property....#### Generalisations of Roth's theorem on finite abelian groups

(University of Waterloo, 2012-12-18)

Roth's theorem, proved by Roth in 1953, states that when A is a subset of the integers [1,N] with A dense enough, A has a three term arithmetic progression (3-AP). Since then the bound originally given by Roth has been ...

#### Reductions and Triangularizations of Sets of Matrices

(University of Waterloo, 2006)

Families of operators that are triangularizable must necessarily satisfy a number of spectral mapping properties. These necessary conditions are often sufficient as well. This thesis investigates such properties in ...

#### Some Functional Equations Connected with the Utility of Gains and Losses

(University of Waterloo, 2002)

The behavioral properties shown by people when they make selections between different choices will be studied. Based on empirical and logical data a mathematical axiomatic model is built. D. Luce is a major ...

#### Non-Isotopic Symplectic Surfaces in Products of Riemann Surfaces

(University of Waterloo, 2006)

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Let Σ

**<sub>g</sub>**be a closed Riemann surface of genus**g**. Generalizing Ivan Smith's construction, for each**g**≥ 1 and**h**≥ 0 we construct an infinite set of infinite families of homotopic but pairwise non-isotopic symplectic surfaces inside the product symplectic manifold Σ**<sub>g</sub>**×Σ**<sub>h</sub>**. In particular, we achieve all positive genera from these families, providing first examples of infinite families of homotopic but pairwise non-isotopic symplectic surfaces of even genera inside Σ**<sub>g</sub>**×Σ**<sub>h</sub>**....#### K-theory for C*-Algebras and for Topological Spaces

(University of Waterloo, 2015-04-27)

K-theory is the study of a collection of abelian groups that are invariant to C*-algebras or to locally compact Hausdorff spaces. These groups are useful for distinguishing C*-algebras and topological spaces, and they are ...

#### On Transcendence of Irrationals with Non-eventually Periodic b-adic Expansions

(University of Waterloo, 2010-05-18)

It is known that almost all numbers are transcendental in the sense of Lebesgue measure. However there is no simple rule to separate transcendental numbers from algebraic numbers. Today research in this direction is about ...