
%\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath,bm}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsfonts}         
\usepackage{fancybox}   

%\usepackage[numbers]{natbib}



\usepackage[usenames,dvipsnames]{xcolor}%http://en.wikibooks.org/wiki/LaTeX/Colors

%%from  Rudro Rana Biswas
\usepackage[pagebackref,  % this puts links to the page numbers where refs appear
%pdftex, 
bookmarks={false}, pdfauthor={John McGreevy}, pdftitle={Yay, physics!}]{hyperref}
\hypersetup{colorlinks=true, 
linkcolor=BrickRed, 
citecolor=Violet, 
filecolor=OliveGreen, 
urlcolor=RoyalBlue, 
filebordercolor={.8 .8 1}, 
urlbordercolor={.8 .8 0}}%http://en.wikibooks.org/wiki/LaTeX/Hyperlinks
%\usepackage[all]{hypcap}
\usepackage{soul}%strike through text \st{} .  not in math mode.
\setstcolor{Red}
%\usepackage{wrapfig}
%\usepackage{hyperref}


%\usepackage{wrapfig}
%\usepackage{hyperref}

%%%%% some colors


\definecolor{darkgreen}{rgb}{0,0.4,0}
\definecolor{darkred}{rgb}{0.4,0,0}
\definecolor{darkblue}{rgb}{0,0,0.4}
\definecolor{lightblue}{rgb}{.6,.6,0.9}
\newcommand{\ttref}[1]{\textcolor{darkgreen}{\tiny{#1}}}
%\newcommand{\blackcom}[1]{{\footnotesize{#1}}}
\newcommand{\greencom}[1]{\textcolor{darkgreen}{\footnotesize{#1}}}
\newcommand{\redcom}[1]{\textcolor{darkred}{\footnotesize{#1}}}
\newcommand{\bluecom}[1]{\textcolor{darkblue}{\footnotesize{#1}}}
\newcommand{\tttref}[2]{\textcolor{darkgreen}{
{#1, }\texttt{hep-th/{#2}}}}
\newcommand{\qcenter}[1]{\begin{center}{#1}\end{center}}
\def\bl{\textcolor{darkblue}}
\def\bluet{\textcolor{darkblue}}
\def\greent{\textcolor{darkgreen}}
\def\redt{\textcolor{darkred}}


\newcommand{\blackcom}[1]{{\footnotesize{#1}}}


\newcommand{\cobl}{\color{darkblue}}

\newcommand{\cor}{\color{red}}
\newcommand{\cog}{\color{darkgreen}}
\newcommand{\cob}{\color{black}}

\definecolor{uglybrown}{rgb}{0.8,  0.7,  0.5}

\definecolor{palatinatepurple}{rgb}{0.41, 0.16, 0.38}
\definecolor{celebrationcolor}{rgb}{0.75,  0.0,  0.9}


\definecolor{Mathematica1}{rgb}{0.368417, 0.506779, 0.709798}
\definecolor{Mathematica2}{rgb}{0.880722, 0.611041, 0.142051}
\definecolor{Mathematica3}{rgb}{0.560181, 0.691569, 0.194885}



\definecolor{shadecolor}{rgb}{0.90,0.90,0.90}
\definecolor{DVcolor}{rgb}{0.95,  0.5,  0.2}
\definecolor{lightbluemuons}{rgb}{0.0,.65,1.0}

\definecolor{chartreuse}{rgb}{0.70, 1.00, 0.00}
\newcommand{\cogr}{\color{chartreuse}}
\newcommand{\colbl}{\color{lightbluemuons}}


%%%%%%%%%%%%%%  TIKZ


\usepackage{tikz}
\usetikzlibrary{shapes}
\usetikzlibrary{trees}
\usetikzlibrary{snakes}
\usetikzlibrary{matrix,arrows} 				% For commutative diagram
											% http://www.felixl.de/commu.pdf
\usetikzlibrary{positioning}				% For "above of=" commands
\usetikzlibrary{calc,through}				% For coordinates
\usetikzlibrary{decorations.pathreplacing}  % For curly braces
\usepackage[tikz]{bclogo} 					% For cute logo boxes
\usepackage{pgffor}							% For repeating patterns

%\usetikzlibrary{decorations.pathmorphing}	% For Feynman Diagrams
\usetikzlibrary{decorations.markings}
\usetikzlibrary{intersections}				% For interesections

\usetikzlibrary{arrows,decorations.pathmorphing,backgrounds,positioning,fit,petri,automata,shadows,calendar,mindmap, graphs}
\usetikzlibrary{arrows.meta,bending}


% from flip tanedo.
% Based on:
% http://www.texample.net/tikz/examples/feynman-diagram/

\tikzset{
	% >=stealth', %% more traditional arrows, I don't like them
    vector/.style={decorate, decoration={snake,amplitude=1.5pt}, draw},
    fermion/.style={postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{>}}}},
    fermionbar/.style={draw, postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{<}}}},
    fermionnoarrow/.style={},
    gluon/.style={decorate,
        decoration={coil,amplitude=4pt, segment length=5pt}},
    scalar/.style={dashed, postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{>}}}},
    scalarbar/.style={dashed, postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{<}}}},
    scalarnoarrow/.style={dashed,draw},
%
%%% 	Special vectors (when you need to fine-tune wiggles)
%	provector/.style={decorate, decoration={snake,amplitude=2.5pt}, draw},
%	antivector/.style={decorate, decoration={snake,amplitude=-2.5pt}, draw},
%	    electron/.style={draw=black, postaction={decorate},
%        decoration={markings,mark=at position .55 with {\arrow[draw=black]{>}}}},
%	bigvector/.style={decorate, decoration={snake,amplitude=4pt}, draw},
	vectorscalar/.style={loosely dotted,draw=black, postaction={decorate}},
}

\def\centerarc[#1](#2)(#3:#4:#5)% Syntax: [draw options] (center) (initial angle:final angle:radius)
    { \draw[#1] ($(#2)+({#5*cos(#3)},{#5*sin(#3)})$) arc (#3:#4:#5); }





%%%%%%%%%%%%%%%


\usepackage{enumitem}

\usepackage{slashed}

\usepackage[font=small,labelfont=bf]{caption}
\DeclareCaptionFont{tiny}{\tiny}
%\captionsetup{font=tiny}
\captionsetup{font=footnotesize}

%\usepackage{yfonts} 
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basenam	e #1 .tif`.png}

%\usepackage[bookmarks=true,colorlinks,linkcolor=red,urlcolor=blue,citecolor=blue]{hyperref}
%\usepackage[usenames]{color}

%\usepackage[usenames]{color}


\usepackage{wasysym}


% these were deprecated by the centos tex installation, but this is no longer a problem:
\usepackage[yyyymmdd,hhmmss]{datetime}   
\usepackage{framed}  % needed for \begin{shaded}
 \usepackage{mdframed}
 \usepackage{wrapfig}
 \usepackage{yfonts}  
%\input epsf




\def\figblock#1#2#3#4{
\vskip.04in
\hskip-.24in
\parbox{#1\textwidth}{#3
}
\parfig{#2}{#4}
}


\def\ketbra#1#2{ | #1 \rangle\hskip-2pt\langle #2|}


\def\TT{{\bf T}}


\def\bn{{\bf n}}
\def\nn{{\bf n}}
\def\bb{{\bf b}}

\newcommand{\arxivlink}[1]{\htmladdnormallink{#1}{http://arxiv.org/abs/#1}}




%%%%%%%%%%%  new as of 2017-02-15





\def\futureNote#1{
{\footnotesize
\color{DVcolor}[#1]
}}



\def\sfE{\mathsf{E}}


\newmdenv[%
%    backgroundcolor=red!8,
        backgroundcolor=lightgray,
    linecolor=black,
%    outerlinewidth=1pt,
    outerlinewidth=2pt,
%    roundcorner=5mm,
%    skipabove=\baselineskip,
%    skipbelow=\baselineskip,
%  bottomline=false,
%    leftline=false,rightline=false,
%    frametitle=\mbox{},
]{boxedandshaded}

\def\End{\text{End}}
\def\Hom{\text{Hom}}


\def\thump{\hfill$\blacksquare$}




%%%%%%%%%%% new as of 2016-03-16

\def\parsnip{-.22in}

\def\parfig#1#2{
\parbox{#1\textwidth}
{\includegraphics[width=#1\textwidth]{#2}}
}


%%%%%%%%%%%  new as of 2015-07-08


\def\angstrom{\buildrel{\circ} \over {\mathrm{A}}}

\def\mathmap#1#2#3#4{
#1 & \to &  #2 \cr 
#3 & \mapsto & #4 
}



% from diptarka
\def\WIDERULE{\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}}



%%%%%%%%%%%%

\def\partials#1#2#3{\( {\partial {#1} \over \partial {#2}} \)_{#3} }
%\def\subsubsubsection#1{{\bf\large #1}}
\def\subsubsubsection#1{{\bf #1}}
\def\eqbm{equilibrium}

\def\breakS{\vfill\eject}
% for posted version, you can remove pagebreaks by 
% \def\breakS{}



%%%%%%% making revtex unnecessary:
%
%\textheight 22cm\textwidth 15.5cm
%\oddsidemargin 14pt\evensidemargin 0pt\topmargin -40pt
%\topmargin-20pt
%\def\baselinestretch{1.1}
%\addtolength{\parskip}{1ex}
%\jot=.5ex
%
%\setlength\arraycolsep{2pt}
%
%
%\newcommand{\preprint}[1]{\begin{table}[t]  %%
%             \begin{flushright}               %%
%             {#1}                             %%
%             \end{flushright}                 %%
%             \end{table}}                     %%
%\renewcommand{\title}[1]{\vbox{\center\LARGE{#1}}\vspace{5mm}}
%\renewcommand{\author}[1]{\vbox{\center#1}\vspace{5mm}}
%\newcommand{\address}[1]{\vbox{\center\em#1}}
%\newcommand{\email}[1]{\vbox{\center\tt#1}\vspace{5mm}}
%
%\renewcommand{\date}[1]{\vbox{\center#1}}
%
%\newcommand{\pra}{Phys.~Rev.~A}
%\newcommand{\prb}{Phys.~Rev.~B}
%\newcommand{\prc}{Phys.~Rev.~C}
%\newcommand{\prd}{Phys.~Rev.~D}
%\newcommand{\pre}{Phys.~Rev.~E}
%\newcommand{\prl}{Phys.~Rev.~Lett.}

%%%%%%%% things above here need to be adjusted in the template.

%%%%%%%%%%%% EQUATION NUMBERING  (must renew in appendix)
\numberwithin{equation}{section}

\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}




% the Arovas phantom dagger
\def\nd{{ \vphantom{\dagger}}}




\newcommand{\vev}[1]{\langle #1 \rangle}
\def\mapright#1{\!\!\!\smash{
\mathop{\longrightarrow}\limits^{#1}}\!\!\!}
\newcommand{\bigoint}{\displaystyle \oint}
%\newcommand{\ltimes}{\times \hspace{-6pt}
%\raisebox{.2ex}{\mbox{$\scriptscriptstyle |$}} \;}
\newlength{\extraspace}
\setlength{\extraspace}{2mm}
\newlength{\extraspaces}
\setlength{\extraspaces}{2.5mm}
\newcounter{dummy}
\addtolength{\abovedisplayskip}{\extraspaces}
\addtolength{\belowdisplayskip}{\extraspaces}
\addtolength{\abovedisplayshortskip}{\extraspace}
\addtolength{\belowdisplayshortskip}{\extraspace}
%\newcommand{\ee}{\end{equation}}
%\newcommand{\figuurnum}{\arabic{fignum}}
\newcommand{\figuur}[3]{
%\addtocounter{fignum}{1}
%\addcontentsline{lof}{figure}{\protect
%\numberline{\arabic{section}.\arabic{fignum}}{#3}}
\hspace{-3mm}{\it }\ %\figuurnum.
\begin{figure}[t]\begin{center}
\leavevmode\hbox{\epsfxsize=#2 \epsffile{#1.eps}}\\[3mm]
\medskip
\parbox{15cm}{%\small \bf Fig.\ %\figuurnum: 
#3}
\end{center} \end{figure}\hspace{-3.5mm}}
\newcommand{\figuurb}[3]{
%\addtocounter{fignum}{1}
%\addcontentsline{lof}{figure}{\protect
%\numberline{\arabic{section}.\arabic{fignum}}{#3}}
\hspace{-3mm}{\it }\ %\figuurnum.
\begin{figure}[ht]\begin{center}
\leavevmode\hbox{\epsfxsize=#2 \epsffile{#1.eps}}\\[3mm]
\parbox{11cm}{%\small \bf Fig.\ %\figuurnum: 
#3}
\end{center} \end{figure}\hspace{-3.5mm}}
\newcommand{\fig}{{\it}}% fig.}\ }




%\usepackage{textcomp}
%\usepackage{fontenc}
%\usepackage{bm}
%\usepackage{amsmath}
%\usepackage{yfonts}  % this is the one SS use
%\usepackage{mathbbol}
%\usepackage{arev}
%\usepackage{mbboard}
%\usepackage{devnagri}
%\usepackage{fc}


%\textwidth = 6.5 in
%\textheight = 9 in
%\oddsidemargin = 0.0 in
%\evensidemargin = 0.0 in
%\topmargin = 0.0 in
%\headheight = 0.0 in
%\headsep = 0.0 in
%\parskip = 0.2in
%\parindent = 0.1in


\def\be{\begin{equation}}
\def\ee{\end{equation}}



\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}









\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}

\def\dbar{{\mathchar'26\mkern-12mu \dd}} 
\def\bdelta{ \slashed{\delta}}
\def\CU{{\cal U}}
%
%%MATH MACROS
%Greek letters and their bars
\def\ep{\epsilon}
\def\bep{\bar\epsilon}
\def\blam{\bar\lambda}
\def\bsig{\bar\sigma}
\def\bpsi{\bar\psi}
\def\bphi{\bar\phi}
\def\bp{\bar\partial}
\def\bthet{\overline \theta}
\def\Dbar{\overline D}
\def\bgam{\bar\gamma}
\def\p{\partial}
\def\tilp{\tilde\partial}
\def\eps{\epsilon}
\def\half{{1\over 2}}
\def\Tr{{{\rm Tr~ }}}
\def\tr{{\rm tr}}
\def\Re{{\rm Re\hskip0.1em}}
\def\Im{{\rm Im\hskip0.1em}}
\def\even{{\rm even}}
\def\odd{{\rm odd}}
\def\lcm{{\rm lcm}}
\def\diag{{\rm diag}}
\def\bra#1{\left\langle#1\right|}
\def\ket#1{\left|#1\right\rangle}
\def\bbra#1{{\langle\langle}#1|}
\def\kket#1{|#1\rangle\rangle}
\def\vev#1{\left\langle{#1}\right\rangle}
\def\Dslash{\rlap{\hskip0.2em/}D}
\def\dslash{\rlap{{\hskip0.08em/}}\partial}

\def\CA{{\cal A}}
\def\CB{{\cal B}}
\def\CC{{\cal C}}
\def\CD{{\cal D}}
\def\CE{{\cal E}}
\def\CF{{\cal F}}
\def\CG{{\cal G}}
\def\CH{{\cal H}}
\def\CI{{\cal I}}
\def\CJ{{\cal J}}
\def\CK{{\cal K}}
\def\CL{{\cal L}}
\def\CM{{\cal M}}
\def\CN{{\cal N}}
\def\CO{{\cal O}}%AEL
\def\CP{{\cal P}}
\def\CQ{{\cal Q}}
\def\CR{{\cal R}}
\def\CS{{\cal S}}
\def\CT{{\cal T}}
\def\CU{{\cal U}}
\def\CV{{\cal V}}
\def\CW{{\cal W}}
\def\CX{{\cal X}}%AEL
\def\CY{{\cal Y}}
\def\CZ{{\cal Z}}

%\def\ad#1#2{{\delta\over\delta\sigma^{#1}(#2)}}
\def\ppt{{\partial\over\partial t}}
\def\comment#1{[#1]}
\def\nl{\hfill\break}
\def\a{\alpha}
\def\ta{\tilde\alpha}
\def\ap{\alpha'}
\def\sqap{\sqrt{\alpha'}}
\def\bA{\bar A}
\def\floor#1{{#1}}
\def\sgn{{\rm sgn\ }}
\def\I{{\rm I}}
\def\IA{{\rm IA}}
\def\II{\relax{I\kern-.10em I}}
\def\IIa{{\II}a}
\def\IIb{{\II}b}
\def\TeV{{\rm TeV}}
\def\hk{hyperk\"ahler\  }
\def\Hk{Hyperk\"ahler\  }
\def\cascade{{\cal A}}
%
%{\buildrel b \to i \over \longrightarrow}
\def\imp{$\Rightarrow$}
%\def\IZ{\relax\ifmmode\mathchoice
%{\hbox{\cmss Z\kern-.4em Z}}{\hbox{\cmss Z\kern-.4em Z}}
%{\lower.9pt\hbox{\cmsss Z\kern-.4em Z}}
%{\lower1.2pt\hbox{\cmsss Z\kern-.4em Z}}
%\else{\cmss Z\kern-.4emZ}\fi}
%\def\IZ{\relax{\rm Z\kern-.34em Z}}
\def\IZ{\mathbb{Z}}
\def\IB{\relax{\rm I\kern-.18em B}}
%\def\IC{{\relax\hbox{$\inbar\kern-.3em{\rm C}$}}}
\def\IC{\mathbb{C}}
\def\ID{\relax{\rm I\kern-.18em D}}
\def\IE{\relax{\rm I\kern-.18em E}}
\def\IF{\relax{\rm I\kern-.18em F}}
\def\IG{\relax\hbox{$\inbar\kern-.3em{\rm G}$}}
\def\IGa{\relax\hbox{${\rm I}\kern-.18em\Gamma$}}
\def\IH{\relax{\rm I\kern-.18em H}}
\def\II{\relax{\rm I\kern-.18em I}}
\def\IK{\relax{\rm I\kern-.18em K}}
%\def\IP{\relax{\rm I\kern-.18em P}}
\def\IP{\mathbb{P}}
%\def\IX{\relax{\rm X\kern-.01em X}}
%this doesn't work
\def\IX{{\bf X}}
%
\def\barom{\overline{\Omega}}
\def\barA{\bar{A}}
\def\jb{{\bar \jmath}}
\def\Hom{{\rm Hom}}
\def\inbar{\,\vrule height1.5ex width.4pt depth0pt}
\def\mod{{\rm\; mod\;}}
\def\ndt{\noindent}
\def\p{\partial}
\def\pab{\pb_{\bar A} }
\def\pb{{\bar \p}}
\def\pgp{\pb g g^{-1}}
\font\cmss=cmss10 \font\msss=cmss10 at 7pt
%\def\IR{\relax{\rm I\kern-.18em R}}
\def\IR{\mathbb{R}}
\def\pbar{\bar{\p}}
\def\qmvw{\CM_{\vec \zeta}(\vec v, \vec w) }
%i made these.
\def\sdtimes{\mathbin{\hbox{\hskip2pt\vrule
height 4.1pt depth -.3pt width .30pt\hskip-2pt$\times$}}}
%\def\simgt{\hskip0.05in\relax{ > 
%{\lower5.0pt\hbox{\kern-1.05em $\sim$}} } \hskip0.05in}
%\def\simlt{\hskip0.05in\relax{ <
%{\lower5.0pt\hbox{\kern-1.05em $\sim$}} } \hskip0.05in}
%\def\simlt{\hskip0.05in\relax{ 
%\raise3.0pt\hbox{ $<$
%{\lower5.0pt\hbox{\kern-1.05em $\sim$}} } \hskip0.05in}
\def\simgt{\hskip0.05in\relax{ 
\raise3.0pt\hbox{ $>$
{\lower5.0pt\hbox{\kern-1.05em $\sim$}} }} \hskip0.05in}
\def\ker{{\rm ker\ }}
\def\cok{{\rm cok\ }}
\def\im{{\rm im\ }}
\def\ind{{\rm ind\ }}
\def\ub{{\bar{u}}}
\def\vol{{\rm vol}}
\def\Vol{{\rm Vol}}
\def\wb{{\bar{w}}}
\def\zb {{\bar{z}}}
\def\one {{\bf 1}}
\def\dpr{^{\prime\dagger}}
%
\def\BR{\IR}
\def\BZ{Z} % for now
\def\BP{\IP}
\def\BR{\IR}
\def\BC{\IC}
\def\BM{\QM}
\def\BH{\QH}
\def\BX{\QX}
%
\def\ls{l_s}
\def\ms{m_s}
\def\gs{g_s}
\def\lp10{\ell_p^{10}}
\def\lp11{\ell_p^{11}}
\def\R11{R_{11}}
\def\mb{{m_{\rm brane}}}
\def\vb{{v_{\rm brane}}}
%
\def\brs{Q_{{\rm B}}}
\def\brsb{\bar{Q}_{{\rm B}}}
\def\dels{\delta_{{\rm S}}}
\def\delb{\delta_{{\rm B}}}
\def\zb{\bar{z}}
\def\frac#1#2{{#1 \over #2}}
\def\epb{\bar{\epsilon}}
\def\tts{{\rm tt}{}^\ast}
\def\ch{{\rm ch}}
\def\td{{\rm Td}}
\def\ahat{\hat{A}}
\def\cft#1{{\rm CFT_{#1}}}
\def\dS{\partial \Sigma}
%\def\ni{\noindent}
\def\imt{{\rm Im} \tau}
\def\mfl{ (-1)^{F_L}}
\def\mfr{ (-1)^{F_R}}

\def\up{\uparrow}
\def\down{\downarrow}

\def\norm#1{\vrule\hskip-1.2pt \vrule \hskip1.2pt #1  \hskip1.2pt \vrule\hskip-1.2pt \vrule}
\def\norm#1{|\hskip-1.5pt | \hskip1.5pt #1  \hskip1.5pt |\hskip-1.5pt |}
%$ \norm{A}$

%identity operator from doyon-fonseca
\def\Ione{\hbox{$1\hskip -1.2pt\vrule depth 0pt height 1.53ex width 0.7pt
                  \vrule depth 0pt height 0.3pt width 0.12em$}}


\def\norm#1{\vrule\hskip-1.2pt \vrule \hskip1.2pt #1  \hskip1.2pt \vrule\hskip-1.2pt \vrule}
\def\norm#1{|\hskip-1.5pt | \hskip1.5pt #1  \hskip1.5pt |\hskip-1.5pt |}
%$ \norm{A}$


\def\thetafunction#1#2#3#4{
\vartheta\left[\matrix{ #1 \cr #2} \right] (#3 |#4)}



%% from the topological vertex paper


%%                              TABLEAUX.TEX
%%      This  macro file is for producing a ``Young Tableau'' which is
%%      an array of little squares sometimes used in mathematical physics.
%%      For instance, the command $\tableau{6 3 2}$ will produce a tableau
%%      with 6 squares in the top row, 3 in the next, and 2 in the last.
%%                                  OOOOOO
%%      This tableau will look like OOO    but made of squares instead of O's.
%%                                  OO
%%      Any number of rows may be present, each having a nonzero number of
%%      squares.
%%
%%      A tableau is math mode material, so use $ or $$ to enclose it.
%%
%%      The size and line-thickness of the little boxes are controlled by the
%%      dimension parameters --
%%              \tableauside=1.0ex              %(size)
%%              \tableaurule=0.4pt              %(line-thickness)
%%      Change them if you want.
%%
%%                                                      -- Doug Eardley 9/19/8%%
%%
\newdimen\tableauside\tableauside=1.0ex
\newdimen\tableaurule\tableaurule=0.4pt
\newdimen\tableaustep
\def\phantomhrule#1{\hbox{\vbox to0pt{\hrule height\tableaurule width#1\vss}}}
\def\phantomvrule#1{\vbox{\hbox to0pt{\vrule width\tableaurule height#1\hss}}}
\def\sqr{\vbox{%
  \phantomhrule\tableaustep
  \hbox{\phantomvrule\tableaustep\kern\tableaustep\phantomvrule\tableaustep}%
  \hbox{\vbox{\phantomhrule\tableauside}\kern-\tableaurule}}}
\def\squares#1{\hbox{\count0=#1\noindent\loop\sqr
  \advance\count0 by-1 \ifnum\count0>0\repeat}}
\def\tableau#1{\vcenter{\offinterlineskip
  \tableaustep=\tableauside\advance\tableaustep by-\tableaurule
  \kern\normallineskip\hbox
    {\kern\normallineskip\vbox
      {\gettableau#1 0 }%
     \kern\normallineskip\kern\tableaurule}%
  \kern\normallineskip\kern\tableaurule}}
\def\gettableau#1 {\ifnum#1=0\let\next=\null\else
  \squares{#1}\let\next=\gettableau\fi\next}

\tableauside=1.0ex
\tableaurule=0.4pt


%% from shiraz


%\font\smallrm=cmr8
% \def\app#1#2{\global\meqno=1\global\subsecno=0\xdef\secsym{\hbox{#1.}}
% \bigbreak\bigskip\noindent{\bf Appendix.}\message{(#1. #2)}
% \writetoca{Appendix {#1.} {#2}}\par\nobreak\medskip\nobreak}
% %
% %       \eqn\label{a+b=c}       gives displayed equation, numbered
% %                               consecutively within sections.
%%     \eqnn and \eqna define labels in advance (of eqalign?)
% %
% \def\eqnn#1{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
% \global\advance\meqno by1\wrlabeL#1}
% \def\eqna#1{\xdef #1##1{\hbox{$(\secsym\the\meqno##1)$}}
% \writedef{#1\numbersign1\leftbracket#1{\numbersign1}}%
% \global\advance\meqno by1\wrlabeL{#1$\{\}$}}
% \def\eqn#1#2{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
% \global\advance\meqno by1$$#2\eqno#1\eqlabeL#1$$}
%
%\def\IZ{\relax\ifmmode\mathchoice
% {\hbox{\cmss Z\kern-.4em Z}}{\hbox{\cmss Z\kern-.4em Z}}
% {\lower.9pt\hbox{\cmsss Z\kern-.4em Z}}
% {\lower1.2pt\hbox{\cmsss Z\kern-.4em Z}}\else{\cmss Z\kern-.4em Z}\fi}



%\global\newcount\itemno \global\itemno=0
%\def\itemized{\global\itemno=0\bigbreak\bigskip\noindent}
%\def\itemaut#1{\global\advance\itemno by1\noindent\item{\the\itemno.}#1}

%\itemized
%\itemaut{First this.}
%\itemaut{Then that.}


\def\bs{\backslash}
\def\bfe{{\bf e}}

\def\del{\partial}

\def\({\left(}
\def\){\right)}

\def\sidebarb{\color{darkblue}
\vskip.2in
\hrule 
}

\def\sidebare{
\vskip.2in
\hrule 
\cob
}


\def\ppi{\boldsymbol{\pi}}
\def\pphi{\boldsymbol{\phi}}
%\def\pphi{\phi}
\def\grad{\vec \nabla}
\def\vB{\vec B}
\def\vE{\vec E}
\def\vA{\vec A}
\def\vAA{ \vec{\bf A}}
\def\vEE{{{\vec {\bf E}}}}
\def\vBB{{\vec {\bf B}}}

\def\cc{{\bf c}}

\def\ii{{\bf i}}


\def\CK{{\cal K}}
\def\xx{{\bf x}}
\def\yy{{\bf y}}

\def\dd{\text{d}}

\def\pp{{\bf p}}
\def\aa{{\bf a}}
\def\qq{{\bf q}}

\def\AA{{\bf A}}
\def\BB{{\bf B}}
\def\EE{{\bf E}}
\def\HH{{\bf H}}
\def\KK{{\bf K}}
\def\JJ{{\bf J}}
\def\MM{{\bf M}}
\def\NN{{\bf N}}
\def\PP{{\bf P}}
\def\QQ{{\bf Q}}
\def\RR{{\bf R}}
\def\SS{{\bf S}}
\def\uu{{\bf u}}
\def\ww{{\bf w}}


\def\UU{{\bf U}}
\def\VV{{\bf V}}
\def\WW{{\bf W}}
\def\rrho{{\boldsymbol\rho}}
\def\ssigma{{\boldsymbol\sigma}}
\def\ttau{{\boldsymbol\tau}}
\def\SSigma{{\boldsymbol\Sigma}}
\def\Prob{\text{{\bf Prob}}}

 \def\XX{{\bf X}}
 \def\ZZ{{\bf Z}}
\def\YY{{\bf Y}}
\def\cnot{\CC_\text{not}}

\def\DDD{\boldsymbol{\Delta}}

%Approximately less than operators:
\def\lsim{\mathrel{\mathstrut\smash{\ooalign{\raise2.5pt\hbox{$<$}\cr\lower2.5pt\hbox{$\sim$}}}}}
\def\gsim{\mathrel{\mathstrut\smash{\ooalign{\raise2.5pt\hbox{$>$}\cr\lower2.5pt\hbox{$\sim$}}}}}

\def\MSbar{\overline{\text{MS}}}


%A leftright arrow which acts like \vec:
\def\overleftrightarrow#1{\vbox{\ialign{##\crcr
     $\leftrightarrow$\crcr\noalign{\kern-0pt\nointerlineskip}
     $\hfil\displaystyle{#1}\hfil$\crcr}}}
     
     \def\overleftarrow#1{\vbox{\ialign{##\crcr
     $\leftarrow$\crcr\noalign{\kern-0pt\nointerlineskip}
     $\hfil\displaystyle{#1}\hfil$\crcr}}}


%%ENGLISH MACROS
\def\eg{{\it e.g.}}
\def\ie{{\it i.e.}}
\def\cf{{\it c.f.}}
\def\etal{{\it et. al.}}
\hyphenation{Di-men-sion-al}

%%% additions new to 239a


\def\hh{{\bf h}}

\def\gSO{\textsf{SO}}
\def\gG{\textsf{G}}

\def\gO{\textsf{O}}
\def\gSU{\textsf{SU}}
\def\gU{\textsf{U}}
\def\gS{\textsf{S}}


\def\gs{\text{gs}}


%%% further innovations

\def\eqnfig#1#2{\parbox{#1}{
\includegraphics[height=#1]{#2}
}}



\newif{\ifeq}           % defines a new condition @eq tested by the conditional \ifeq
\eqtrue                 % if uncommented, declares @eq to be true
%\eqfalse               % if uncommented, declares @eq to be false
                                %
                                % to use this, wrap text with the conditional, eg:
                                %
                                % \ifeq
                                % SHOW THIS IFF \eqtrue HAS BEEN DECLARED
                                % \fi

\newcounter{lecturecounter}
% \addtocounter{lecturecounter}{19}


%\newcounter{lecturecounter}
%%[counter]
\newcommand{\Lec}{
\stepcounter{lecturecounter}
{\color{blue}
\hfill[End of Lecture \arabic{lecturecounter}]
\cob}
% \addtocounter{lecturecounter}{1}
}


%\newcounter{bar}
%\newcommand{\Lec}{%
%        \stepcounter{bar}%
%        \thebar}


%% SKIP TO HERE

%\renewcommand{\thesubsection}{(\alph{subsection})}
%\renewcommand{\thesubsection}{(\roman{subsection})}
%\newpage
%\setcounter{page}{1}
%\def\chapternumber{1}
%\renewcommand{\thesection}{\chapternumber}
%\renewcommand{\thepage}{\chapternumber-\arabic{page}}

%\title{Lecture Notes}
%% Lecture Notes}
%%Chapter \chapternumber: }
%\author{Lecturer: McGreevy
%%\footnote
%%{}
%}
%\date{{\tiny Last updated: \today, \currenttime}
%}
%\begin{document}
%\maketitle
%
%%\vfill
%
%\renewcommand{\contentsname}{}
%%\vskip-.5in
%%\breakS
%
%\tableofcontents
%
%\breakS
%
%\section{}
%\end{document}
