Browsing Mathematics (Faculty of) by Subject "Pure Mathematics"
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On Convolution Squares of Singular Measures
(University of Waterloo, 20100825)We prove that if $1 > \alpha > 1/2$, then there exists a probability measure $\mu$ such that the Hausdorff dimension of its support is $\alpha$ and $\mu*\mu$ is a Lipschitz function of class $\alpha1/2$. 
On Diagonal Acts of Monoids
(University of Waterloo, 20070830)In this paper we discuss what is known so far about diagonal acts of monoids. The first results that will be discussed comprise an overview of some work done on determining whether or not the diagonal act can be finitely ... 
On the ErdösTurán conjecture and related results
(University of Waterloo, 20110829)The ErdösTurán Conjecture, posed in 1941 in, states that if a subset B of natural numbers is such that every positive integer n can be written as the sum of a bounded number of terms from B, then the number of such ... 
On the local positivity of line bundles on algebraic surfaces
(University of Waterloo, 20150109)In this thesis, we define a measure for how `positive' an effective line bundle is at a point on a variety, and prove that it is linear on the numerically effective subcone on a smooth, complete, toric surface. 
On the Modular Theory of von Neumann Algebras
(University of Waterloo, 20100917)The purpose of this thesis is to provide an exposition of the \textit{modular theory} of von Neumann algebras. The motivation of the theory is to classify and describe von Neumann algebras which do not admit a trace, and ... 
On the Representation Theory of Semisimple Lie Groups
(University of Waterloo, 20100830)This thesis is an expository account of three central theorems in the representation theory of semisimple Lie groups, namely the theorems of BorelWeilBott, CasselmanOsborne and Kostant. The first of these realizes all ... 
On the Structure of Nonnegative Semigroups of Matrices
(University of Waterloo, 20090818)The results presented here are concerned with questions of decomposability of multiplicative semigroups of matrices with nonnegative entries. Chapter 1 covers some preliminary results which become useful in the remainder ... 
On Transcendence of Irrationals with Noneventually Periodic badic Expansions
(University of Waterloo, 20100518)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 ... 
Operator Spaces and Ideals in Fourier Algebras
(University of Waterloo, 20080811)In this thesis we study ideals in the Fourier algebra, A(G), of a locally compact group G. For a locally compact abelian group G, necessary conditions for a closed ideal in A(G) to be weakly complemented are given, ... 
Operator Theoretic Methods in NevanlinnaPick Interpolation
(University of Waterloo, 20090331)This Master's thesis will develops a modern approach to complex interpolation problems studied by Carath\'{e}odory, Nevanlinna, Pick, and Schur in the early $20^{th}$ century. The fundamental problem to solve is as follows: ... 
Pick Interpolation and the Distance Formula
(University of Waterloo, 20120509)The classical interpolation theorem for the open complex unit disk, due to Nevanlinna and Pick in the early 20th century, gives an elegant criterion for the solvability of the problem as an eigenvalue problem. In the 1960s, ... 
Posets of NonCrossing Partitions of Type B and Applications
(University of Waterloo, 20071019)The thesis is devoted to the study of certain combinatorial objects called \emph{noncrossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{noncrossing partitions} ... 
The ProuhetTarryEscott problem
(University of Waterloo, 20130115)Given natural numbers n and k, with n>k, the ProuhetTarryEscott (PTE) problem asks for distinct subsets of Z, say X={x_1,...,x_n} and Y={y_1,...,y_n}, such that x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for i=1,...,k. ... 
Representations of Operator Algebras
(University of Waterloo, 20120511)The following thesis is divided into two main chapters. In Chapter 2 we study isometric representations of product systems of correspondences over the semigroup 𝐍ᵏ which are minimal dilations of finite dimensional, fully ... 
Semistable rank 2 coHiggs bundles over Hirzebruch surfaces
(University of Waterloo, 20150807)It has been observed by S. Rayan that the complex projective surfaces that potentially admit nontrivial examples of semistable coHiggs bundles must be found at the lower end of the EnriquesKodaira classification. ... 
Settling Time Reducibility Orderings
(University of Waterloo, 20100428)It is known that orderings can be formed with settling time domination and strong settling time domination as relations on c.e. sets. However, it has been shown that no such ordering can be formed when considering computation ... 
Some additive results in F_q[t]
(University of Waterloo, 20150827)We collected several results in integers of additive number theory and translated to results in F_q[t]. The results we collected are related to slim exceptional sets and the asymptotic formula in Waring's problem, a ... 
Some aspects of Cantor sets
(University of Waterloo, 20140328)For every positive, decreasing, summable sequence $a=(a_i)$, we can construct a Cantor set $C_a$ associated with $a$. These Cantor sets are not necessarily selfsimilar. Their dimensional properties and measures have been ... 
Spectral Analysis of Laplacians on Certain Fractals
(University of Waterloo, 20071206)Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a ... 
A survey of Roth's Theorem on progressions of length three
(University of Waterloo, 20111219)For any finite set B and a subset A⊆B, we define the density of A in B to be the value α=A/B. Roth's famous theorem, proven in 1953, states that there is a constant C>0, such that if A⊆{1,...,N} for a positive integer ...