%===================================================
% Packages
%===================================================
\usepackage{amsmath,amssymb,amsthm}
\usepackage{latexsym}
\usepackage[mathscr]{eucal}
\usepackage{graphicx,psfrag}
%\usepackage[all]{xy}
%\usepackage{multicol}
\usepackage{varioref}
%\usepackage{verbatim}
%\usepackage{algorithmic}

%===================================================
% Definitions
%===================================================
\def\+{\phantom{+}}
\def\-{\phantom{-}}
\def\({\left(}
\def\){\right)}
\def\intR{\int_{\mathbb{R}}}
\def\intI{\int_{-\infty}^\infty}
\def\iintI{\iint_{-\infty}^\infty}
\def\R{\mathbb{R}}
\def\Rn{\mathbb{R}^{n}}
\def\Rnn{\mathbb{R}^{n \times n}}
\def\eps{\varepsilon}
\def\QED{\hspace{\stretch{1}}\qedsymbol}
\def\AND{\quad\text{and}\quad}
\def\OR{\quad\text{or}\quad}
\def\AS{\quad\text{as}\quad}
\def\IF{,\quad\text{if }}
\def\WHERE{\quad\text{where}\quad}
\def\xb{\mathbf{x}}
\def\vb{\mathbf{v}}
\def\nospace{\vspace{-12pt}}
\def\grad{\nabla}


%===================================================
% Re-Definitions
%===================================================
\renewcommand\Re{\text{Re}\,}
\renewcommand\Im{\text{Im}\,}

%===================================================
% Make Title Banner
%===================================================
\newcommand{\banner}[1]{
	\begin{center}
		\large #1\\[12pt]
		%\large	Kyle Novak\\[8pt]
		\large \today
	\end{center}
	\vspace{24pt}
}    

\newcommand{\letterbanner}[1]{
	\begin{flushright}%(Kyle Novak)\\ 
	\today\end{flushright}
  #1:\vspace{24pt}
}    
        
%===================================================
% Commands
%===================================================
\newcommand{\D}[2]{\frac{\partial #1}{\partial #2}	}
\newcommand{\dd}[2]{\frac{d #1}{d #2}	}
\newcommand{\splitmat}[2]{	\left(\begin{smallmatrix}#1\end{smallmatrix}\right|\left.\begin{smallmatrix}#2\end{smallmatrix}\right)}
\newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right) }
\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}
\newcommand{\omat}[1]{\begin{matrix}#1\end{matrix}}
\newcommand{\set}[2]{\{\; #1 \mid #2 \;\}}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\mrm}[1]{\mathtt{#1}}
\newcommand{\eqnbox}[1]{\begin{center}\fbox{$\displaystyle{#1}$}\end{center}}
%\newcommand{\ceqnbox}[3]{\begin{center}\colorbox{#1}{\textcolor{#2}{$\displaystyle{#3}$}}\end{center}}
\newcommand{\ceqnbox}[2]{\begin{center}\fcolorbox{#1}{white}{$\displaystyle{#2}$}\end{center}}


%===================================================
% Environment
%===================================================
\newenvironment{rcase}
	{\left.\begin{aligned}}
	{\end{aligned}\right\rbrace}

\newenvironment{doloop}[2]
	{\textbf{for} #1 to #2 \textbf{do} \begin{itemize}}
	{\end{itemize}\textbf{end do}}
\def\state{\item[]}


%===================================================
% Document Parameters
%===================================================
%\setlength{\topmargin}{-0.5in}
%\setlength{\oddsidemargin}{-0.0in}
\setlength{\evensidemargin}{-21.25in}
%\setlength{\hoffset}{0.25pt}
%\setlength{\textwidth}{6in}
%\setlength{\textheight}{8.5in}
%\setlength{\parskip}{12pt}
\setlength{\parindent}{0pt}

%===================================================
% AMS Command: Math Operators
%===================================================
\DeclareMathOperator{\cond}{cond}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\Arg}{Arg}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\sinc}{sinc}
\def\hypergeometric{{}_2\mkern-1mu F_1}
%\DeclareMathOperator{\hypergeometric}{{}_2F_1}


%===================================================
% Latin
%===================================================
\def\etal{\emph{et~al.}}
\def\ie{\emph{i.e.,}}
\def\eg{\emph{e.g.,}}
\def\etc{{etc.}}
\def\vitae{vit\ae{}}
\def\apriori{\emph{a~priori}}
\def\aposteriori{\emph{a~posteriori}}
\def\Schrodinger{Schr\"odinger}