\documentclass[mathserif]{beamer}
%\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{amsfonts, amscd, amssymb, amsthm, amsmath}
\usepackage{pdfsync}%leaves makers for tex searching
\usepackage{enumerate}


%%% Color  %%%--------------------------------------------------------------
\usepackage{color}
\definecolor{dgreen}{rgb}{0,.7,.2}
\definecolor{dblue}{rgb}{0,0,.8}
\definecolor{lblue}{rgb}{.2,.2,1}
\definecolor{purple}{rgb}{.6,0,.7}
\definecolor{dred}{rgb}{.7, 0, 0.1}
\newcommand{\NOTE}[1]{{\color{blue}#1}}
\newcommand{\MOVED}[1]{{\color{gray}#1}}
\newcommand{\blue}[1]{{\color{blue} #1 }}


%%% Beamer  %%%--------------------------------------------------------------
\usenavigationsymbolstemplate{}

\usefonttheme[%
%  	stillsansserifsmall, 	% will cause 'small' text to be still typeset using sans serif. This refers to
  					% the text in the headline, footline, and sidebars. Using this options is often
  					% advisable since small  text is often easier to read in sans serif.
   	onlymath, % typset math in serif but nothing else
 ]{serif}

\usecolortheme[rgb={0,0,1}]{structure}
\setbeamercolor{alerted text}{fg=blue}
%\beamertemplatenavigationsymbolsempty 



%%%%%% Theorems %%%%% 
\theoremstyle{plain}
	\newtheorem{thm}{Theorem}
	\newtheorem{prop}[thm]{Proposition}
	\newtheorem{lem}[thm]{Lemma}
	\newtheorem*{claim}{Claim}
	\newtheorem*{cor}{Corollary}
\theoremstyle{definition}
	\newtheorem*{defn}{Definition}
	\newtheorem*{ex}{Example}
	\newtheorem*{fex}{For example:}



%%% Index %%%------------------------------------------------------------
%\usepackage{makeidx}
\newcommand{\dfn}[1]{\emph{#1}}
\newcommand{\subdfn}[2]{\emph{#2}}


%%% Pictures %%%--------------------------------------------------------- 
%\usepackage{etex} %fixes the fight that pictex has with every other drawing package
%\usepackage{pictexwd}
\usepackage{tikz}
%\usepackage{graphicx}
%\usepackage{epstopdf}
%\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}
%\usepackage[pdftex,bookmarks]{hyperref}




\usetikzlibrary{scopes}
\usepgflibrary{shapes.geometric}
\def\ShiftX{\pgftransformxshift}
\def\ShiftY{\pgftransformyshift}
\tikzstyle{V}=[draw, fill =black, circle, inner sep=0pt, minimum size=2pt]
\tikzstyle{over}=[draw=white,double=black,line width=2pt, double distance=1pt]
\tikzstyle{bV}=[draw, fill =black, circle, inner sep=0pt, minimum size=4pt, outer sep=1pt]

\def\ShiftX{\pgftransformxshift}

\def\Over[#1,#2][#3,#4]{ %1,2=start position; 3,4=end position
	\draw[over]   (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);}
\def\Under[#1,#2][#3,#4]{ %1,2=start position; 3,4=end position
	\draw  (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);}
\def\Cross[#1,#2][#3,#4]{%Mimic over, under follows
	\Under[#3,#2][#1,#4]\Over[#1,#2][#3,#4]}

\def\Nodes[#1][#2]{
 \foreach \x in {1,...,#2} {
%       \draw (\x,0) circle (0.4cm);
	 \filldraw [black] (\x,#1) circle (2pt);
	 }
	}



\newcommand{\VCenter}[2]{
\begin{tikzpicture}[xscale=\XUNIT, yscale=\YUNIT]
		\draw[white] (0,0.5*#1)--(0,-0.5*#1);
		\node at (0,0) {#2};
\end{tikzpicture}
}

\newcounter{r}
\newcommand\Part[1]{
        \setcounter{r}{1}
	 \foreach \x in {#1}{
 	{\ifnum\value{r}=1
		\draw (0,\value{r}-1)--(\x,\value{r}-1); 
		\fi}
	\draw (0,\value{r}) to (\x,\value{r});
   	\foreach \y in {0, ..., \x} {\draw (\y,\value{r})--(\y,\value{r}-1);}
	\addtocounter{r}{1}
 }}
 \def\PartUNIT{.175}
%Self-contained tikz images for \Part above. 
\newcommand{\PART}[1]{
\begin{matrix}
\begin{tikzpicture}[xscale=\PartUNIT, yscale=-\PartUNIT] 
	\Part{#1}
\end{tikzpicture}
\end{matrix}
}




%%% Alphabets %%%--------------------------------------------------------
\def\cA{\mathcal{A}}\def\cB{\mathcal{B}}\def\cC{\mathcal{C}}\def\cD{\mathcal{D}}\def\cE{\mathcal{E}}\def\cF{\mathcal{F}}\def\cG{\mathcal{G}}\def\cH{\mathcal{H}}\def\cI{\mathcal{I}}\def\cJ{\mathcal{J}}\def\cK{\mathcal{K}}\def\cL{\mathcal{L}}\def\cM{\mathcal{M}}\def\cN{\mathcal{N}}\def\cO{\mathcal{O}}\def\cP{\mathcal{P}}\def\cQ{\mathcal{Q}}\def\cR{\mathcal{R}}\def\cS{\mathcal{S}}\def\cT{\mathcal{T}}\def\cU{\mathcal{U}}\def\cV{\mathcal{V}}\def\cW{\mathcal{W}}\def\cX{\mathcal{X}}\def\cY{\mathcal{Y}}\def\cZ{\mathcal{Z}}

\def\AA{\mathbb{A}} \def\BB{\mathbb{B}} \def\CC{\mathbb{C}} \def\DD{\mathbb{D}} \def\EE{\mathbb{E}} \def\FF{\mathbb{F}} \def\GG{\mathbb{G}} \def\HH{\mathbb{H}} \def\II{\mathbb{I}} \def\JJ{\mathbb{J}} \def\KK{\mathbb{K}} \def\LL{\mathbb{L}} \def\MM{\mathbb{M}} \def\NN{\mathbb{N}} \def\OO{\mathbb{O}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\SS{\mathbb{S}} \def\TT{\mathbb{T}} \def\UU{\mathbb{U}} \def\VV{\mathbb{V}} \def\WW{\mathbb{W}} \def\XX{\mathbb{X}} \def\YY{\mathbb{Y}} \def\ZZ{\mathbb{Z}}  

\def\fa{\mathfrak{a}} \def\fb{\mathfrak{b}} \def\fc{\mathfrak{c}} \def\fd{\mathfrak{d}} \def\fe{\mathfrak{e}} \def\ff{\mathfrak{f}} \def\fg{\mathfrak{g}} \def\fh{\mathfrak{h}} \def\fj{\mathfrak{j}} \def\fk{\mathfrak{k}} \def\fl{\mathfrak{l}} \def\fm{\mathfrak{m}} \def\fn{\mathfrak{n}} \def\fo{\mathfrak{o}} \def\fp{\mathfrak{p}} \def\fq{\mathfrak{q}} \def\fr{\mathfrak{r}} \def\fs{\mathfrak{s}} \def\ft{\mathfrak{t}} \def\fu{\mathfrak{u}} \def\fv{\mathfrak{v}} \def\fw{\mathfrak{w}} \def\fx{\mathfrak{x}} \def\fy{\mathfrak{y}} \def\fz{\mathfrak{z}}
\def\fgl{\mathfrak{gl}}  \def\fsl{\mathfrak{sl}}  \def\fso{\mathfrak{so}}  \def\fsp{\mathfrak{sp}}  \def\fsu{\mathfrak{su}}  \def\fs{\mathfrak{s}}
\def\GL{\mathrm{GL}} \def\SL{\mathrm{SL}}  \def\SO{\mathrm{SO}} \def\Sp{\mathrm{Sp}}  \def\SU{\mathrm{SU}} 

\def\cc{\mathbf{c}}\def\ii{\mathbf{i}}\def\jj{\mathbf{j}}\def\kk{\mathbf{k}}\def\uu{\mathbf{u}}\def\vv{\mathbf{v}}\def\ww{\mathbf{w}}


\def\<{\langle} \def\>{\rangle}
\def\ad{\mathrm{ad}} 
\def\Alg{\mathrm{Alg}}
\def\Aut{\mathrm{Aut}}
\def\ch{\mathrm{ch}}
\def\dim{\mathrm{dim}} 
\def\End{\mathrm{End}} 
\def\ev{\mathrm{ev}} 
\def\half{\hbox{$\frac12$}}
\def\Hom{\mathrm{Hom}} 
\def\id{\mathrm{id}}
\def\img{\mathrm{img}}
\def\Ind{\mathrm{Ind}}
\def\Irr{\mathrm{Irr}} 
\def\ker{\mathrm{ker}}
%\def\one{\mathbbm{1}}
\def\normeq{\unlhd}
\def\proj{\mathrm{proj}}
\def\qtr{\mathrm{qtr}} 
\def\rank{\mathrm{rank}}
\def\rad{\mathrm{rad}} 
\def\Res{\mathrm{Res}} 
\def\Sym{\mathrm{Sym}} 
\def\tr{\mathrm{tr}} 
\def\Tr{\mathrm{Tr}} 
\def\vep{\varepsilon}
\def\wt{\mathrm{wt}}
\def\Lie{\mathrm{Lie}}
\def\Example{\medskip \noindent {\bf Example. }}

