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\title{%
    {Thesis Title}\\
    {\large New York University Tandon School of Engineering}\\
}
\author{Nathaniel T. Stemen}
\date{21 February 2017}

\begin{document}


\begin{center}
    \begin{minipage}{0.75\linewidth}
        \centering
        \textbf{\Large{AN INVESTIGATION OF Q-BALLS}} \\ \vspace{1.cm}

        \textbf{Submitted in Partial Fulfillment} \\ \vspace{0.5cm}

        \textbf{of the Requirements for the} \\ \vspace{0.5cm}

        \textbf{Degree of} \\ \vspace{0.75cm}

        \textbf{BACHELOR OF SCIENCE IN MATHEMATICS AND PHYSICS}\\ \vspace{0.75cm}

        \textbf{at} \\ \vspace{0.5cm}

        \textbf{NEW YORK UNIVERSITY} \\ \vspace{0.5cm}

        \textbf{TANDON SCHOOL OF ENGINEERING} \\ \vspace{0.5cm}

        \textbf{by} \\ \vspace{0.5cm}

        \textbf{Nathaniel T. Stemen} \\ \vspace{0.5cm}

        \textbf{May 2017}
    \end{minipage}
\end{center}
\clearpage

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\noindent

\begin{tabbing}
    Approved by thesis advisor:

    \hspace*{.50in} \=  \\

    \> \\

    \underline{Major}: Mathematics and Physics \\

    \> \\
    \> \\

    \> \rule[0.0in]{2.5in}{0.01in} \\

    \> \textbf{Luciano M. Medina} \\

    \> Professor of Mathematics \\

    \> \\

    \>  Date: \underline{\hspace{2in}} \\
\end{tabbing}


\begin{center}
    \begin{minipage}{\linewidth}
        \centering

        \textbf{AN ABSTRACT} \\ \vspace{1.cm}

        \textbf{AN INVESTIGATION OF Q-BALLS} \\ \vspace{0.5cm}

        \textbf{by} \\ \vspace{0.75cm}

        \textbf{Nathaniel T. Stemen} \\ \vspace{0.75cm}

        \textbf{Advisor: Luciano M. Medina, Ph.D.} \\ \vspace{1.cm}

        {Submitted in Partial Fulfillment of the Requirements} \\ \vspace{0.5cm}

        {for the Degree of Bachelor of Science (Mathematics and Physics)} \\ \vspace{1.cm}

        {May 2017} \\ \vspace{0.5cm}

        {\textbf{Abstract:}}

        {In this thesis we study the dynamics of a complex scalar field with a
            nonrenormalizable potential. In particular we study the existence of the
            non-topological soliton solutions known as Q-Balls. These objects have been
            proposed as candidates to solve the present baryon asymmetry, and arise in
            supersymetric theories, as well as abelian gauge theories, non-abelian gauge
            theores, non-commutative complex scalar field theories, etc. Formulating the
            existence problem using constrained minimization allows for this approach to
            translate easily into the numerical work that follows. With this framework
            the computation of soliton profiles, along with the associated angular
            momentum parameter are achieved. To best understand the work, we present the
            prerequisite classical field theory and an introduction to Q-Balls and
            Q-Vortex solitons.}
    \end{minipage}
\end{center}
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\listoffigures
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\chapter*{Dedication}
I would like to dedicate this thesis to my dog, Jett.

\chapter*{Acknowledgements}
I would like to thank, Dr.\ Luciano M. Medina, the advisor for this work, for
his continued support throughout this research and thesis writing process.
Without his willingness to give so much time and effort to his students, I
would not have learned as much as I did. His continued insight pushed my
understanding and his generous supervision allowed me to take the research in
directions I wished. For his generous nature I am extremely appreciative.

I would also like to thank Professor Joe Esposito who so kindly extended his
time to help build the foundations of the work presented here. Even in the midst
of raising a daughter he found time to extensively answer and ponder any
question I had.

To my fellow peers Shahmir Shahrol, Igor Pshenychny, Shearyar Khan, Kevin Yeh,
and Kye Drageset who provided great conversations about the work, and any field
of mathematics or physics in general, even when they had had enough of Q-Balls.

And finally, to my parents, Charlie, and Diane who provided unwavering support
and patience, even when they maybe shouldn't have.


\tableofcontents

\chapter{Introduction}\label{chap:intro}
\epigraph{I do not keep up with the details of particle physics.}{Murray Gell-Mann\\\textit{Particle Physicist}}
\input{chapters/introduction}

\chapter{Field Theory}\label{chap:fields}
\epigraph{Newton was the greatest genius that ever existed, and the most
    fortunate, for we cannot find more than once a system of the world to
    establish.}{Joseph-Louis Lagrange}
\input{chapters/chapter02}

\chapter{Mathematical Prerequisites}\label{chap:prereqs}
\epigraph{If people do not believe that mathematics is simple, it is only
    because they do not realize how complicated life is.}{John von Neumann}
\input{chapters/chapter03}

\chapter{An introduction to Q-Balls}\label{chap:qballs}
\epigraph{The career of a young theoretical physicist consists of treating the
    harmonic oscillator in ever-increasing levels of abstraction.}{Sidney Coleman\\\textit{Inventor of Q-Balls}}
\input{chapters/chapter04}

\chapter{Numerical Investigations of Q-vortex solutions}\label{chap:numerics}
\epigraph{So, ultimately, in order to understand nature it may be necessary to
    have a deeper understanding of mathematical relationships. But the real reason
    is that the subject is enjoyable\ldots}{Richard Feynman}
\input{chapters/chapter05}

\chapter{Conclusion}\label{chap:conclusion}
\epigraph{Le but de cette thèse est de munir son auteur du titre de Docteur.\\
    The goal of this thesis is to furnish its author with the title of Doctor}{Adrien Douady\\\textit{Beginning of A. Douady's thesis}}
\input{chapters/conclusion}

\appendix
\chapter{Some (possibly) Useful Facts}\label{chap:appendix}
\epigraph{The noblest ambition is that of leaving behind something of permanent
    value.}{G.H. Hardy\\\textit{A Mathematicians Apology}}
\input{chapters/appendix}

\bibliographystyle{unsrt}
\bibliography{refs}

\end{document}