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Now showing items 1-10 of 39

#### 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....#### 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>**....#### Intervals with few Prime Numbers

(University of Waterloo, 2004)

In this thesis we discuss some of the tools used in the study of the number of primes in short intervals. In particular, we discuss a large sieve density estimate due to Gallagher and two classical delay equations. ...

#### The Model Theory of Algebraically Closed Fields

(University of Waterloo, 2000)

Model theory can express properties of algebraic subsets of complex n-space. The constructible subsets are precisely the first order definable subsets, and varieties correspond to maximal consistent collections of ...

#### Infinite Sets of D-integral Points on Projective Algebrain Varieties

(University of Waterloo, 2005)

Let

**X**(**K**) ⊂ <strong>P</strong><sup>**n**</sup> (**K**) be a projective algebraic variety over**K**, and let**D**be a subset of <strong>P</strong><sup>**n**</sup><sub>**OK**</sub> such that the codimension of**D**with respect to**X**⊂ <strong>P</strong><sup>**n**</sup><sub>**OK**</sub> is two. We are interested in points**P**on**X**(**K**) with the property that the intersection of the closure of**P**and**D**is empty in <strong>P</strong><sup>**n**</sup><sub>**OK**</sub>, we call such points**D**-integral points on**X**(**K**). First we prove that certain algebraic varieties have infinitely many**D**-integral points. Then we find an explicit description of the complete set of all**D**-integral points in projective n-space over Q for several types of**D**....