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\Large{\rm\textbf{Acknowledgements}}
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I would like to first thank my doctoral advisors Professors Donald Kouri and Bernhard Bodmann. I have had the opportunity to work with them for eight years. I started working under their tutelage as an undergraduate in mathematics at the University of Houston. During my undergraduate career, we pursued many different topics, each of which has been very rewarding and heavily impacted this thesis. The work in this thesis began the summer before my senior year and continued while I was pursuing my master's at the University of Waterloo. I returned to the University of Houston for my doctorate to finish the work we began during my undergraduate studies. It has since flourished in ways none of us had ever expected or even dreamed, due in no small part to Professors Kouri and Bodmann. I would not have been able to achieve this doctorate without their mentoring.

I would like to thank my committee members Professors Mehrdad Kalantar and Demetrio Labate from the University of Houston and Professor John Klauder from the University of Florida for their helpful comments and corrections to this thesis.

I would like to give thanks to Nikhil Pandya. Many important insights in this thesis and results not quoted herein came from discussions with him. Taking on a mentor role for Nikhil has fostered much growth and maturity in myself which I hope to continue to hone and refine in the future.

I would like to give special thanks to my fianc\'ee, Angelynn. Through the many difficulties and hurdles throughout my graduate career, she has been my rock and my sounding board. I am very grateful to her for these and so much more.

Finally, I would like to thank my friends Matthew Giambusso, Angela Giambusso, Charles Mills, Liam Bourdo, and Zachary Garvey as well as my family members for pushing me and encouraging me to pursue a doctorate in mathematics and their continued support throughout my graduate career. I would specifically like to thank my uncle Joe Williams Jr. for being my role model and source of inspiration since I was a young child. I would certainly not be the person I am today without his influence.

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