\documentclass[%pdflatex,%
               letterpaper,%
               twoside,%
               10pt%
               ]{article}

%\usepackage{MyPaper}

\usepackage[hmargin={1.0in,1.0in},%                   % for use with package titling and twopage option
            vmargin={1.0in,1.0in},%
            %hmargin={0.75in,0.75in},%                   % for use with package titling and twopage option
            %vmargin={0.75in,0.75in},%
            headheight=14pt,%                           % allow header using package fancyhdr
            %twoside,%
            %pdftex%
            ]{geometry}

%\usepackage[dvipsnames,usenames]{color}                 % the color package

% the AMS packages

\usepackage{amsmath}
\numberwithin{equation}{section}                        % number the equation by section in amsmath
\usepackage{amsthm}
\usepackage{amssymb}                                    % mathematical symbols and AMS fonts
\usepackage{bbm}                                        % blackboard bold for numbers
\setlength{\leftmargini}{4em}                           % indentation in enumerate
\renewcommand{\labelenumi}{(\alph{enumi})}              % counters in enumerate
\renewcommand{\labelenumii}{(\arabic{enumii})}              % counters in enumerate
%\setlength{\parskip}{0.5ex plus0.2ex minus0.2ex}

\makeatletter
\let\@upn\@iden
\makeatother
\newtheoremstyle{ToddTheorem}                           % personal theorem style
  {3pt}%      Space above
  {3pt}%      Space below
  {\slshape}%         Body font
  {}%         Indent amount (empty = no indent, \parindent = para indent)
  {\bfseries\upshape}% Thm head font
  {.}%        Punctuation after thm head
  {.5em}%     Space after thm head: " " = normal interword space;
        %       \newline = linebreak
  {}%         Thm head spec (can be left empty, meaning `normal')

\newtheoremstyle{ToddDefinition}                        % personal definition style
  {3pt}%      Space above
  {3pt}%      Space below
  {\upshape}%         Body font
  {}%         Indent amount (empty = no indent, \parindent = para indent)
  {\bfseries\slshape}% Thm head font
  {.}%        Punctuation after thm head
  {.5em}%     Space after thm head: " " = normal interword space;
        %       \newline = linebreak
  {}%         Thm head spec (can be left empty, meaning `normal')

%%%%%% the txfont package  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\usepackage[varg]{txfonts}

% theorem counting with autoref

\makeatletter
\newcommand*{\c@theoremassumption}{\c@theorem}
\newcommand*{\p@theoremassumption}{\p@theorem}
\newcommand*{\thetheoremassumption}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremconjecture}{\c@theorem}
\newcommand*{\p@theoremconjecture}{\p@theorem}
\newcommand*{\thetheoremconjecture}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremcorollary}{\c@theorem}
\newcommand*{\p@theoremcorollary}{\p@theorem}
\newcommand*{\thetheoremcorollary}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremdefinition}{\c@theorem}
\newcommand*{\p@theoremdefinition}{\p@theorem}
\newcommand*{\thetheoremdefinition}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremexample}{\c@theorem}
\newcommand*{\p@theoremexample}{\p@theorem}
\newcommand*{\thetheoremexample}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremfigure}{\c@theorem}
\newcommand*{\p@theoremfigure}{\p@theorem}
\newcommand*{\thetheoremfigure}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremhypothesis}{\c@theorem}
\newcommand*{\p@theoremhypothesis}{\p@theorem}
\newcommand*{\thetheoremhypothesis}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremlemma}{\c@theorem}
\newcommand*{\p@theoremlemma}{\p@theorem}
\newcommand*{\thetheoremlemma}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremproposition}{\c@theorem}
\newcommand*{\p@theoremproposition}{\p@theorem}
\newcommand*{\thetheoremproposition}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremremark}{\c@theorem}
\newcommand*{\p@theoremremark}{\p@theorem}
\newcommand*{\thetheoremremark}{\thetheorem}
\makeatother

\makeatletter
\newcommand*{\c@theoremnotation}{\c@theorem}
\newcommand*{\p@theoremnotation}{\p@theorem}
\newcommand*{\thetheoremnotation}{\thetheorem}
\makeatother

\theoremstyle{ToddTheorem} %
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theoremcorollary]{Corollary}
\newtheorem{lemma}[theoremlemma]{Lemma}
\newtheorem{proposition}[theoremproposition]{Proposition}

\theoremstyle{ToddDefinition} %
\newtheorem{assumption}[theoremassumption]{Assumption}
\newtheorem{conjecture}[theoremconjecture]{Conjecture}
\newtheorem{definition}[theoremdefinition]{Definition}
%\newtheorem{example}[theoremexample]{Example}
\newtheorem*{example}{Example}
\newtheorem{hypothesis}[theoremhypothesis]{Hypothesis}
\newtheorem{remark}[theoremremark]{Remark}
\newtheorem{notation}[theoremnotation]{Notation}

%% theorem counting without autoref
%
%\theoremstyle{ToddTheorem} %
%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{corollary}[theoremcorollary]{Corollary}
%\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{proposition}[theorem]{Proposition}
%
%\theoremstyle{ToddDefinition} %
%\newtheorem{assumption}[theorem]{Assumption}
%\newtheorem{assumption}[theoremassumption]{Assumption}
%\newtheorem{conjecture}[theorem]{Conjecture}
%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{example}[theorem]{Example}
%\newtheorem{hypothesis}[theorem]{Hypothesis}
%\newtheorem{remark}[theorem]{Remark}

\renewcommand{\proofname}{\textsl{\bfseries Proof:}}

% the fancyhdr package

\newcommand{\TheAuthor}{}
\newcommand{\Author}[1]%
        {\renewcommand{\TheAuthor}{#1}}                 % authors
\newcommand{\TheRunningTitle}{}
\newcommand{\RunningTitle}[1]%
        {\renewcommand{\TheRunningTitle}{#1}}           % running title

\usepackage{fancyhdr}                                   % placement of page numbers and headers
\pagestyle{fancy}
%\renewcommand{\sectionmark}[1]%                         % layout of section header
%  {\markright{\thesection.\quad#1}}
%%\renewcommand{\sectionmark}[1]%                         % layout of section header
%%  {\markboth{\thesection.\quad#1}{}}
%\renewcommand{\subsectionmark}[1]%                      % layout of subsection header
%  {\markright{\thesubsection.\quad#1}}
\fancyhead{}                                            % clear all header fields
\fancyhead[CO]%
    {\nouppercase{\upshape\TheAuthor}}                  % display the authors
\fancyhead[CE]%
    {\nouppercase{\sffamily\slshape\TheRunningTitle}}   % display the title
%\fancyhead[CO,CE]{\nouppercase{\textsc\rightmark}}      % first new section on the page displayed
%\fancyhead[RE,LO]{\bfseries\thepage}                    % page number on opposite sides of the page
\fancyhead[RE,RO]{\bfseries\thepage}                    % page number on right side of the page
\fancyfoot{}                                            % clear all footer fields
\renewcommand{\headrulewidth}{0.4pt}                    % head- and foot-rules
\renewcommand{\footrulewidth}{0.4pt}
%\lhead{\nouppercase{\textsc\leftmark}}                  % last section on the page

% the titling package

\usepackage{titling}
\pretitle{\begin{center}\LARGE\sffamily\slshape}        % title format
\posttitle{\par\end{center}\vskip 0.5em}
\predate{\begin{center}\small}                          % date format
\postdate{\par\end{center}\vskip 0.5em}
\thanksmarkseries{fnsymbol}                             % symbols for footnotes

% the abstract package

\usepackage[runin]{abstract}
\setlength{\abstitleskip}{-1.55em}
\renewcommand{\abstractnamefont}{\normalfont\small\bfseries}
\abslabeldelim{.\,}

% table of contents packages

\usepackage[titles]{tocloft}                            % table of contents
\renewcommand{\cftsecfont}{\upshape}                    % section title font
\renewcommand{\cftsubsecfont}{\upshape}
\renewcommand{\cftsubsubsecfont}{\upshape}
\renewcommand{\cftsecaftersnum}{.}                      % period after the section numbers
\renewcommand{\cftsubsecaftersnum}{.}
\renewcommand{\cftsubsubsecaftersnum}{.}
%\renewcommand{\contentsname}{Table of Contents}         % change name of Contents title

% section style package

\usepackage{titlesec}
\titleformat{\section}%
   {\Large\scshape\filcenter}{\thesection.}{1em}{}
\titleformat{\subsection}%
   {\large\sffamily\slshape\filcenter}{\thesubsection.}{1em}{}
\titleformat{\subsubsection}%
   {\normalsize\sffamily\filcenter}{\thesubsubsection.}{1em}{}
\titlespacing*{\section}%
   {0pt}{4.5ex plus 1ex minus .2ex}{3.3ex plus .2ex}
\titlespacing*{\subsection}%
   {0pt}{4.25ex plus 1ex minus .2ex}{3.05ex plus .2ex}
\titlespacing*{\subsubsection}%
   {0pt}{4.25ex plus 1ex minus .2ex}{3.05ex plus .2ex}

% bibliography packages

\usepackage[comma,numbers,square,sort&compress]{natbib} % bibliography style
\usepackage{hypernat}                                   % using natbib with hyperref
%%findme \newcommand{\bibfont}{\small}
%%\newcommand{\citenumfont}{\bfseries}
%%\renewcommand{\bibnumfmt}[1]{[\textbf{#1}]}

% mathematical typesetting packages

%\usepackage{linalgtk}                                   % personal linear algebra style package
%\usepackage[varg]{pxfonts}                              % extension of the mathematical fonts
\usepackage{bm}                                         % bold fonts
\usepackage{mathrsfs}                                   % mathematical script fonts-\mathscr{}
%\usepackage{esdiff}                                     % package to write derivatives
\usepackage{booktabs}                                   % package to customize tables

% graphical packages

%\usepackage{ifpdf}                                      % switch between .pdf mode and .dvi mode
\usepackage[%pdftex%
            ]{graphicx}                                 % the graphics package
%\DeclareGraphicsExtensions{.ps,.pdf,.jpg}%,.eps,.png,.mps}
%\usepackage{rotating}                                   % easily rotate figures
%\usepackage{float}                                      % placement of figures and tables

%% the fancyref package
%
%\usepackage[plain]{fancyref}                            % easy equation and theorem list referencing style
%\usepackage{myfancyref}                                 % personal referencing style

% packages to format the document or ease typesetting

%\usepackage[amsmath,thmmarks,hyperref]{ntheorem}        % personal theorem style - needs ntheorem.cfg
%\usepackage{enumerate,delarray,indentfirst}             % changes to arrays and lists
\usepackage{delarray,indentfirst}                       % changes to arrays and lists
\usepackage[%hang,%
            small,%
            bf]{caption}                     % caption style
\setcaptionmargin{0.5in}
%\renewcommand{\captionfont}{\itshape\small}
%\renewcommand{\captionlabelfont}{\upshape\small}
%\usepackage{framed}                                     % create framed or shaded regions
%\definecolor{shadecolor}{gray}{0.85}

%       environments

\newenvironment{acknowledgment}%
  {\begin{trivlist}\item[]\textbf{Acknowledgments.}}{\end{trivlist}}

% conversion of .dvi to other format (.pdf or .html) packages

%\usepackage{tex4ht}                                     % conversion of .dvi to .html
%\usepackage[letterpaper,%
%            %pdftex,%
%            %pagebackref=true,%                          % cross-referencing of bibliographical entries with page number
%            bookmarks=true,%
%            bookmarksopen=false,%
%            bookmarksnumbered=false,%
%            pdfpagemode=UseOutlines,%
%            pdfstartview={FitH -32768},%
%            hypertexnames=false,%
%            colorlinks=true,%
%            linkcolor=Blue,%
%            citecolor=Blue,%
%            urlcolor=Blue%
%            ]{hyperref}                    % hyperref - cross-referencing in .pdf and .dvi files

% remove hyperref warnings

\makeatletter
\providecommand*{\toclevel@assumption}{0}
\providecommand*{\toclevel@conjecture}{0}
\providecommand*{\toclevel@corollary}{0}
\providecommand*{\toclevel@definition}{0}
\providecommand*{\toclevel@example}{0}
\providecommand*{\toclevel@figure}{0}
\providecommand*{\toclevel@hypothesis}{0}
\providecommand*{\toclevel@lemma}{0}
\providecommand*{\toclevel@proof}{0}
\providecommand*{\toclevel@proposition}{0}
\providecommand*{\toclevel@remark}{0}
\providecommand*{\toclevel@theorem}{0}
\makeatother

% autoref labels

\newcommand{\theoremassumptionautorefname}{Assumption}
\newcommand{\theoremconjectureautorefname}{Conjecture}
\newcommand{\theoremcorollaryautorefname}{Corollary}
\newcommand{\theoremdefinitionautorefname}{Definition}
\newcommand{\theoremexampleautorefname}{Example}
\newcommand{\theoremfigureautorefname}{Figure}
\newcommand{\theoremhypothesisautorefname}{Hypothesis}
\newcommand{\theoremlemmaautorefname}{Lemma}
\newcommand{\theorempropositionautorefname}{Proposition}
\newcommand{\theoremremarkautorefname}{Remark}
\newcommand{\theoremautorefname}{Theorem}
\newcommand{\equationautorefname}{equation}
\newcommand{\sectionautorefname}{Section}
\newcommand{\subsectionautorefname}{\sectionautorefname}
\newcommand{\subsubsectionautorefname}{\sectionautorefname}

% autoref shortcuts

%\newcommand{\aref}[1]{\autoref{#1}}
%\newcommand{\eref}[1]{\hyperref[{#1}]{equation~(\ref*{#1})}}
%\newcommand{\Eref}[1]{\hyperref[{#1}]{Equation~(\ref*{#1})}}
%\newcommand{\eref}[1]{\hyperref[{#1}]{Eq.~(\ref*{#1})}}
\newcommand{\eref}[1]{\hyperref[{#1}]{(\ref*{#1})}}

%\usepackage[final,notref,notcite]{showkeys}             % show labels

%   definitions

\newcommand{\ds}{\displaystyle}

\newcommand{\A}{\mathbb{A}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\N}{\mathbb{N}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}

\def\arg{\mathop\mathrm{arg}\nolimits}
\def\br{\mathop\mathrm{br}\nolimits}
\def\codim{\mathop\mathrm{codim}\nolimits}
\def\cn{\mathop\mathrm{cn}\nolimits}
\def\det{\mathop\mathrm{det}\nolimits}
\def\diag{\mathop\mathrm{diag}\nolimits}
\def\dim{\mathop\mathrm{dim}\nolimits}
\def\dn{\mathop\mathrm{dn}\nolimits}
\def\ess{\mathop\mathrm{ess}\nolimits}
\def\exp{\mathop\mathrm{exp}\nolimits}
\def\fix{\mathop\mathrm{Fix}\nolimits}
\def\gker{\mathop\mathrm{gker}\nolimits}
%\def\id{\mathop\mathrm{id}\nolimits}
%\def\id{\mathop\mathbbm{1}\nolimits}
\def\id{\mathbbm{1}}
\def\ker{\mathop\mathrm{ker}\nolimits}
\def\pt{\mathop\mathrm{pt}\nolimits}
\def\rank{\mathop\mathrm{rank}\nolimits}
\def\rep{\mathop\mathrm{Rep}\nolimits}
\def\res{\mathop\mathrm{Res}\limits}
\def\row{\mathop\mathrm{Rowspace}\nolimits}
\def\sech{\mathop\mathrm{sech}\nolimits}
\def\sgn{\mathop\mathrm{sign}\nolimits}
%\def\skew{\rmi\hspace{0.1em}\ad_J}
\def\skew{\ad_{\rmi J}}
\def\sn{\mathop\mathrm{sn}\nolimits}
\def\SO{\mathop\mathrm{SO}\nolimits}
%\def\span{\mathop\mathrm{span}\nolimits}
\def\Span{\mathop\mathrm{span}\nolimits}
\def\spec{\mathop\mathrm{spec}\nolimits}
\def\trace{\mathop\mathrm{trace}\nolimits}
\def\tr{\mathop\mathrm{T}\nolimits}
\def\Re{\mathop\mathrm{Re}\nolimits}
\def\Im{\mathop\mathrm{Im}\nolimits}

\def\coloneqq{\mathrel{\mathop:}=}

\newcommand{\rma}{\mathrm{a}}
\newcommand{\rmb}{\mathrm{b}}
\newcommand{\rmc}{\mathrm{c}}
\newcommand{\rmd}{\mathrm{d}}
\newcommand{\rme}{\mathrm{e}}
\newcommand{\rmf}{\mathrm{f}}
\newcommand{\rmg}{\mathrm{g}}
\newcommand{\rmh}{\mathrm{h}}
\newcommand{\rmi}{\mathrm{i}}
\newcommand{\rmk}{\mathrm{k}}
\newcommand{\rmm}{\mathrm{m}}
\newcommand{\rmn}{\mathrm{n}}
\newcommand{\rmo}{\mathrm{o}}
\newcommand{\rmp}{\mathrm{p}}
\newcommand{\rmr}{\mathrm{r}}
\newcommand{\rms}{\mathrm{s}}
\newcommand{\rmu}{\mathrm{u}}
\newcommand{\rmv}{\mathrm{v}}
\newcommand{\rmz}{\mathrm{z}}
\newcommand{\rmA}{\mathrm{A}}
\newcommand{\rmD}{\mathrm{D}}
\newcommand{\rmH}{\mathrm{H}}
\newcommand{\rmK}{\mathrm{K}}
\newcommand{\rmN}{\mathrm{N}}
\newcommand{\rmO}{\mathrm{O}}
\newcommand{\rmP}{\mathrm{P}}
\newcommand{\rmR}{\mathrm{R}}
\newcommand{\rmT}{\mathrm{T}}
\newcommand{\rmV}{\mathrm{V}}
\newcommand{\rmZ}{\mathrm{Z}}

\newcommand{\calA}{\mathcal{A}}
\newcommand{\calB}{\mathcal{B}}
\newcommand{\calE}{\mathcal{E}}
\newcommand{\Ham}{\mathcal{H}}
\newcommand{\calH}{\mathcal{H}}
\newcommand{\Kz}{\mathcal{K}}
\newcommand{\calJ}{\mathcal{J}}
\newcommand{\calK}{\mathcal{K}}
\newcommand{\Lz}{\mathcal{L}}
\newcommand{\calL}{\mathcal{L}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\calP}{\mathcal{P}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calS}{\mathcal{S}}
\newcommand{\calU}{\mathcal{U}}
%\newcommand{\Ns}{\mathcal{N}}
\newcommand{\clo}{\mathcal{o}}
\newcommand{\clO}{\mathcal{O}}
\newcommand{\Rg}{\mathcal{R}}
\newcommand{\Ra}{\mathrm{R}}
\newcommand{\Ns}{\mathrm{N}}

\newcommand{\bfE}{\bm{\mathrm{E}}}
\newcommand{\bfY}{\bm{\mathrm{Y}}}
\newcommand{\bfZ}{\bm{\mathrm{Z}}}
\newcommand{\bfa}{\bm{\mathrm{a}}}
\newcommand{\bfc}{\bm{\mathrm{c}}}
\newcommand{\bfd}{\bm{\mathrm{d}}}
\newcommand{\bfe}{\bm{\mathrm{e}}}
\newcommand{\bff}{\bm{\mathrm{f}}}
\newcommand{\bfg}{\bm{\mathrm{g}}}
\newcommand{\bfm}{\bm{\mathrm{m}}}
\newcommand{\bfq}{\bm{\mathrm{q}}}
\newcommand{\bft}{\bm{\mathrm{t}}}
\newcommand{\bfu}{\bm{\mathrm{u}}}
\newcommand{\bfv}{\bm{\mathrm{v}}}
\newcommand{\bfw}{\bm{\mathrm{w}}}
\newcommand{\bfx}{\bm{\mathrm{x}}}
\newcommand{\bfz}{\bm{\mathrm{z}}}

\newcommand{\bfal}{\bm{\alpha}}

%%%%%%% vector notation  %%%%%%%%%%%%%%%%%%%%

\newcommand{\vA}{\bm{\mathit{A}}}
\newcommand{\vB}{\bm{\mathit{B}}}
\newcommand{\vC}{\bm{\mathit{C}}}
\newcommand{\vD}{\bm{\mathit{D}}}
\newcommand{\vE}{\bm{\mathit{E}}}
\newcommand{\vG}{\bm{\mathit{G}}}
\newcommand{\vH}{\bm{\mathit{H}}}
\newcommand{\vI}{\bm{\mathit{I}}}
\newcommand{\vJ}{\bm{\mathit{J}}}
\newcommand{\vL}{\bm{\mathit{L}}}
\newcommand{\vK}{\bm{\mathit{K}}}
\newcommand{\vM}{\bm{\mathit{M}}}
\newcommand{\vN}{\bm{\mathit{N}}}
\newcommand{\vP}{\bm{\mathit{P}}}
\newcommand{\vQ}{\bm{\mathit{Q}}}
\newcommand{\vR}{\bm{\mathit{R}}}
\newcommand{\vS}{\bm{\mathit{S}}}
\newcommand{\vV}{\bm{\mathit{V}}}

\newcommand{\va}{\bm{\mathit{a}}}
\newcommand{\vb}{\bm{\mathit{b}}}
\newcommand{\vc}{\bm{\mathit{c}}}
\newcommand{\vd}{\bm{\mathit{d}}}
\newcommand{\ve}{\bm{\mathit{e}}}
\newcommand{\vf}{\bm{\mathit{f}}}
\newcommand{\vg}{\bm{\mathit{g}}}
\newcommand{\vh}{\bm{\mathit{h}}}
\newcommand{\vo}{\bm{\mathit{o}}}
\newcommand{\vp}{\bm{\mathit{p}}}
\newcommand{\vq}{\bm{\mathit{q}}}
\newcommand{\vr}{\bm{\mathit{r}}}
\newcommand{\vs}{\bm{\mathit{s}}}
\newcommand{\vx}{\bm{\mathit{x}}}
\newcommand{\vu}{\bm{\mathit{u}}}
\newcommand{\vv}{\bm{\mathit{v}}}
\newcommand{\vw}{\bm{\mathit{w}}}
\newcommand{\vy}{\bm{\mathit{y}}}
\newcommand{\vz}{\bm{\mathit{z}}}
\newcommand{\vn}{\bm{\mathit{0}}}

%\newcommand{\vbe}{\vec{\beta}}
%\newcommand{\vde}{\vec{\delta}}
%
%\newcommand{\va}{\vec{a}}
%\newcommand{\vb}{\vec{b}}
%\newcommand{\ve}{\vec{e}}
%\newcommand{\vh}{\vec{h}}
%\newcommand{\vp}{\vec{p}}
%\newcommand{\vs}{\vec{s}}
%\newcommand{\vx}{\vec{x}}
%\newcommand{\vu}{\vec{u}}
%\newcommand{\vv}{\vec{v}}
%\newcommand{\vw}{\vec{w}}
%\newcommand{\vy}{\vec{y}}
%\newcommand{\vn}{\vec{0}}

%%%%%%% Lie group notation  %%%%%%%%%%%%%%%%%%%%

\newcommand{\ad}{\mathrm{ad}}
\newcommand{\Ad}{\mathrm{Ad}}
\newcommand{\cm}{\mathscr{C}}
\newcommand{\gl}{\mathfrak{g}}
\newcommand{\gla}{\mathfrak{g}\mathfrak{l}}
\newcommand{\hl}{\mathfrak{h}}
\renewcommand{\sl}{\mathfrak{s}\mathfrak{l}}
\newcommand{\so}{\mathfrak{s}\mathfrak{o}}
\renewcommand{\sp}{\mathfrak{s}\mathfrak{p}}
\newcommand{\su}{\mathfrak{s}\mathfrak{u}}
\newcommand{\Gl}{\mathfrak{G}}
\newcommand{\Gm}{\mathscr{G}}

\Author{T. Kapitula and Someone Else}
\RunningTitle{Small LaTeX file}

% conversion of .dvi to other format (.pdf or .html) packages

%\usepackage{tex4ht}                                     % conversion of .dvi to .html
\usepackage[letterpaper,%
            %pagebackref=true,%                          % cross-referencing of bibliographical entries with page number
            bookmarks=false,%
            bookmarksopen=false,%
            bookmarksnumbered=false,%
            %pdfpagemode=UseOutlines,%
            %pdfstartview={FitH -32768},%
            hypertexnames=false,%
            colorlinks=true,%
            linkcolor=blue,%
            citecolor=blue,%
            urlcolor=blue]{hyperref}                    % hyperref - cross-referencing in .pdf and .dvi files

%\def\wfig{4.25cm}
%\def\wwfig{5.2cm}
%\def\hfig{2.0cm}

%\usepackage[notref,notcite]{showkeys}
%\usepackage{sidecap}

\begin{document}

\title{Title goes here}
\author{%
        Todd Kapitula %
        \thanks{E-mail: \href{mailto:tmk5@calvin.edu}{tmk5@calvin.edu}} \\
        Department of Mathematics and Statistics \\
        Calvin College \\
        Grand Rapids, MI 49546 %
\and
        Someone else %
        \thanks{E-mail: \href{mailto:me@students.calvin.edu}{me@students.calvin.edu}} \\
        Department of Mathematics \\
        Michigan State University \\
        East Lansing, MI 48824 %
}

% create the title page with an abstract and table of contents

\begin{titlingpage}
\usethanksrule
\setcounter{page}{0}                                    % title page is page 0 in the document
\maketitle
\begin{abstract}
Abstract goes here.
\end{abstract}

\cancelthanksrule
\renewcommand{\footrulewidth}{0.0pt}                    % remove the footrule after the title page

\pdfbookmark[1]{\contentsname}{toc}                     % add the TOC to the bookmarks
\tableofcontents                                        % table of contents for paper
\end{titlingpage}

\section{Introduction}\label{s:intro}

The introduction goes here.

\subsection{First subsection}

Now do some math. First, a numbered equation:
\begin{equation}\label{e:1}
\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1.
\end{equation}
Now, equation \eref{e:1} is the equation for an ellipse. A circle is
described if $a=b$, and a hyperbola is described if \eref{e:1} is
replaced by
\[
\left(\frac{x}{a}\right)^2-\left(\frac{y}{b}\right)^2=\pm1.
\]

Now state a theorem:

\begin{theorem}[Cool Result]\label{thm:1}
Math is fun.
\end{theorem}

\section{A new section}\label{s:2}

Much of this motivational discussion \dots.

%A commented line - will not appear in the output


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% commands to set up the bibliography
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\phantomsection                                         % necessary to add reference section to bookmarks using hyperref
%\addcontentsline{toc}{section}{\refname}                % add reference title to table of contents
%
%\bibliography{../../papers}                             % location of the file papers.bib
%\bibliographystyle{plainnat}                            % bibliography style
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
