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#### Variations on a Theme: Graph Homomorphisms

(University of Waterloo, 2013-08-30)

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms.
A core of a graph X is a vertex minimal subgraph to which X admits a ...

#### Geometric Ramifications of the Lovász Theta Function and Their Interplay with Duality

(University of Waterloo, 2013-08-30)

The Lovasz theta function and the associated convex sets known as theta bodies are fundamental objects in combinatorial and
semidefinite optimization. They are accompanied by a rich duality theory and
deep connections ...

#### Transmitting Quantum Information Reliably across Various Quantum Channels

(University of Waterloo, 2013-05-01)

Transmitting quantum information across quantum channels is an important task. However quantum information is delicate, and is easily corrupted. We address the task of protecting quantum information from an information ...

#### Algebraic Methods and Monotone Hurwitz Numbers

(University of Waterloo, 2012-09-21)

We develop algebraic methods to solve join-cut equations, which are partial differential equations that arise in the study of permutation factorizations. Using these techniques, we give a detailed study of the recently ...

#### Convex relaxation for the planted clique, biclique, and clustering problems

(University of Waterloo, 2011-08-24)

A clique of a graph G is a set of pairwise adjacent nodes of G. Similarly, a biclique (U, V ) of a bipartite graph G is a pair of disjoint, independent vertex sets such that each node in U is adjacent to every node in V ...

#### Modularity and Structure in Matroids

(University of Waterloo, 2013-04-22)

This thesis concerns sufficient conditions for a matroid to admit one of two types of structural characterization: a representation over a finite field or a description as a frame matroid.
We call a restriction N of a ...

#### Implementing the Schoof-Elkies-Atkin Algorithm with NTL

(University of Waterloo, 2013-04-30)

In elliptic curve cryptography, cryptosystems are based on an additive subgroup of an elliptic curve defined over a finite field, and the hardness of the Elliptic Curve Discrete Logarithm Problem is dependent on the order ...

#### Highly Non-Convex Crossing Sequences

(University of Waterloo, 2012-05-18)

For a given graph, G, the crossing number crₐ(G) denotes the minimum number of edge crossings when a graph is drawn on an orientable surface of genus a. The sequence cr₀(G), cr₁(G), ... is said to be the crossing sequence ...

#### Algebraic Aspects of Multi-Particle Quantum Walks

(University of Waterloo, 2012-12-04)

A continuous time quantum walk consists of a particle moving among the vertices of a graph G. Its movement is governed by the structure of the graph. More formally, the adjacency matrix A is the Hamiltonian that determines ...

#### On the Efficiency and Security of Cryptographic Pairings

(University of Waterloo, 2012-12-19)

Pairing-based cryptography has been employed to obtain several advantageous cryptographic protocols. In particular, there exist several identity-based variants of common cryptographic schemes. The computation of a single ...