Browsing Pure Mathematics by Title
Now showing items 119-138 of 165
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Para-Holomorphic Algebroids and Para-Complex Connections
(University of Waterloo, 2021-12-17)The goal of this paper is to develop the theory of Courant algebroids with integrable para-Hermitian vector bundle structures by invoking the theory of Lie bialgebroids. We consider the case where the underlying manifold ... -
A Parallel Study of the Fock Space Approach to Classical and Free Brownian Motion
(University of Waterloo, 2017-08-28)The purpose of this thesis is to elaborate the similarities between the classical and the free probability by means of developing the chaos decomposition of stochastic integrals driven by Brownian motion and its free ... -
Pick Interpolation and the Distance Formula
(University of Waterloo, 2012-05-09)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 Non-Crossing Partitions of Type B and Applications
(University of Waterloo, 2007-10-19)The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} ... -
Problems in Combinatorial and Analytic Number Theory
(University of Waterloo, 2018-08-01)We focus on three problems in number theory. The first problem studies the random Fibonacci tree, which is an infinite binary tree with non-negative integers at each node. The root consists of the number 1 with a single ... -
The Prouhet-Tarry-Escott problem
(University of Waterloo, 2013-01-15)Given natural numbers n and k, with n>k, the Prouhet-Tarry-Escott (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. ... -
Quantum superchannels on the space of quantum channels
(University of Waterloo, 2023-06-20)Quantum channels, defined as completely-positive and trace-preserving maps on matrix algebras, are an important object in quantum information theory. In this thesis we are concerned with the space of these channels. This ... -
Quaternion Algebras and Quadratic Forms
(University of Waterloo, 2008-05-12)The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, ... -
Ranges of vector states on irreducible operator semigroups
(Springer, 2016)Let 𝜑 be a linear functional of rank one acting on an irreducible semigroup S of operators on a finite- or infinite-dimensional Hilbert space. It is a well-known and simple fact that the range of 𝜑 cannot be a singleton. ... -
Rational approximations on smooth rational surfaces
(University of Waterloo, 2019-08-09)In this thesis, we study a conjecture made by D. McKinnon about rational approximations to rational points in algebraic varieties. The conjecture states that if a rational point P on a variety X lies on a rational curve, ... -
Recurrence in Algebraic Dynamics
(University of Waterloo, 2020-07-28)The Dynamical Mordell--Lang Conjecture states that if a polynomial orbit has infinite intersection with a closed set in an algebraic variety, then the intersection must occur periodically. Although this problem is unsolved ... -
Reducibility of operator semigroups and values of vector states
(Springer, 2017-08-01)Let S be a multiplicative semigroup of bounded linear operators on a complex Hilbert space H, and let Ω be the range of a vector state on S so that Ω = {⟨Sξ, ξ⟩ : S ∈ S} for some fixed unit vector ξ ∈ H. We study the ... -
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 ... -
Regular Dilation on Semigroups
(University of Waterloo, 2018-08-07)Dilation theory originated from Sz.Nagy's celebrated dilation theorem which states that every contractive operator has an isometric dilation. Regular dilation is one of many fruitful directions that aims to generalize ... -
Representations of Operator Algebras
(University of Waterloo, 2012-05-11)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 ... -
Residual finite dimensionality and representations of amenable operator algebras
(Elsevier, 2019-04-15)We consider a version of a famous open problem formulated by Kadison, asking whether bounded representations of operator algebras are automatically completely bounded. We investigate this question in the context of amenable ... -
Semicrossed Products, Dilations, and Jacobson Radicals
(University of Waterloo, 2017-08-18)We compute the C*-envelope of the isometric semicrossed product of a C*-algebra arising from number theory by the multiplicative semigroup of a number ring R, and prove that it is isomorphic to T[R], the left regular ... -
Semistable rank 2 co-Higgs bundles over Hirzebruch surfaces
(University of Waterloo, 2015-08-07)It has been observed by S. Rayan that the complex projective surfaces that potentially admit non-trivial examples of semistable co-Higgs bundles must be found at the lower end of the Enriques-Kodaira classification. ... -
Series-Parallel Posets and Polymorphisms
(University of Waterloo, 2018-09-04)We examine various aspects of the poset retraction problem for series-parallel posets. In particular we show that the poset retraction problem for series-parallel posets that are already solvable in polynomial time are ... -
Settling Time Reducibility Orderings
(University of Waterloo, 2010-04-28)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 ...