\usepackage{amssymb,amsmath}
\usepackage{stmaryrd,paralist}
\usepackage{bbm}
\usepackage{graphics,graphicx,tikz-cd}

%\usepackage[latin1]{inputenc}
\usepackage{subfigure}
\usepackage{hyperref,color}
%\usepackage[dvipsnames]{xcolor}
\graphicspath{{Figures/}} 
\graphicspath{{./Figures/}}

\usepackage{url}




\newcommand{\OB}[1]{\textcolor{blue}{OB: #1}}
\newcommand{\ob}[1]{\textcolor{blue}{#1}}
\newcommand{\theod}[1]{\textcolor{olive}{#1}}

\newtheorem{theo}{Theorem}
\numberwithin{theo}{section}
\newtheorem{thm}[theo]{Theorem}
\newtheorem{lem}[theo]{Lemma}
\newtheorem{lemma}[theo]{Lemma}
\newtheorem{prop}[theo]{Proposition}
\newtheorem{proposition}[theo]{Proposition}
\newtheorem{claim}[theo]{Claim}
%\newtheorem{fact}[theo]{Fact}
\newtheorem{Def}[theo]{Definition}
\newtheorem{definition}[theo]{Definition}
\newtheorem{notation}[theo]{Notation}
\newtheorem{cor}[theo]{Corollary}
\theoremstyle{remark}
\newtheorem{remark}[theo]{Remark}
\newtheorem{example}[theo]{Example}
\newtheorem{rk}[theo]{Remark}


\newenvironment{Remark}{
\refstepcounter{theo}
\smallbreak\noindent%
\textbf{Remark~\thetheo.}}%
{\medbreak}

\newcommand{\ds}{\displaystyle}
\def\ni{\noindent}




\newcommand{\bitem}{\begin{compactitem}}
\newcommand{\eitem}{\end{compactitem}}



\def\ZZ{\mathbb{Z}}
\def\NN{\mathbb{N}}
\def\QQ{\mathbb{Q}}
\def\RR{\mathbb{R}}

\def\bz{\mathbb{Z}}
\def\bn{\mathbb{N}}
\def\br{\mathbb{R}}
\def\rp{\mathbb{R}^+}


\def\cA{\mathcal{A}}
\def\cB{\mathcal{B}}
\def\cM{\mathcal{M}}
\def\cF{\mathcal{F}}
\def\cG{\mathcal{G}}
\def\cO{\mathcal{O}}
\def\cR{\mathcal{R}}
\def\cS{\mathcal{S}}
\def\cT{\mathcal{T}}
 
 
\def\mA{\mathcal{A}}
\def\mB{\mathcal{B}}
\def\mC{\mathcal{C}}
\def\mD{\mathcal{D}}
\def\mE{\mathcal{E}}
\def\mF{\mathcal{F}}
\def\mG{\mathcal{G}}
\def\mL{\mathcal{L}}
\def\mM{\mathcal{M}}
\def\mO{\mathcal{O}}
\def\mP{\mathcal{P}}
\def\mR{\mathcal{R}}
\def\mS{\mathcal{S}}
\def\mT{\mathcal{T}}
  
\def\cW{\mathcal{W}}
\def\cL{\mathcal{L}}

\def\cX{\mathcal{X}}


\def\bA{\mathbf{A}}
\def\bE{\mathbf{E}}
\def\bL{\mathbf{L}}
\def\bM{\mathbf{M}}
\def\bO{\mathbf{O}}
\def\bW{\mathbf{W}}
\def\bX{\mathbf{X}}
\def\bAL{\mathbf{AL}}
\def\bT{\mathbf{T}}


\def\mShi{\mathcal{S}hi}

%\def\hH{\widehat{H}}
\newcommand{\Id}{\textrm{Id}}



\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\de}{\delta}
\newcommand{\om}{\omega}
\newcommand{\si}{\sigma}
\newcommand{\eps}{\epsilon}
\renewcommand{\th}{\theta}


\newcommand{\Om}{\Omega}
\newcommand{\Ga}{\Gamma}
\newcommand{\De}{\Delta}
\newcommand{\Th}{\Theta}
\newcommand{\Ups}{\Upsilon}





\newcommand{\ov}[1]{\overline{#1}}
\newcommand{\wt}[1]{\widetilde{#1}}

\newcommand{\bal}{\ov{\al}}
\newcommand{\bbe}{\ov{\be}}
\newcommand{\bga}{\ov{\ga}}
\newcommand{\bde}{\ov{\de}}
\newcommand{\bth}{\ov{\th}}
\newcommand{\bPhi}{\ov{\Phi}}
\newcommand{\bPsi}{\ov{\Psi}}


\newcommand{\fig}[3]{\begin{figure}[h!]\begin{center}\includegraphics[#1]{#2.pdf}\end{center}\caption{#3}\label{fig:#2}\end{figure}}

\newcommand{\dd}{\textbf{d}}
\newcommand{\ee}{\textbf{e}}
\newcommand{\ff}{\textbf{f}}
\newcommand{\mm}{\textbf{m}}
\newcommand{\vv}{\textbf{v}}
\newcommand{\xx}{\textbf{x}}
\newcommand{\yy}{\textbf{y}}

\newcommand{\fS}{\mathfrak{S}} % Symmetric group


\DeclareMathOperator{\cadet}{\!\textrm{-cadet}}
\DeclareMathOperator{\child}{\!\textrm{-child}}
\renewcommand{\path}{\textrm{path}}
\DeclareMathOperator{\drift}{\textrm{drift}}
\newcommand{\pathT}{\path_T}
\DeclareMathOperator{\driftT}{\drift_T}
\DeclareMathOperator{\blocks}{\textrm{blocks}}
\DeclareMathOperator{\shadow}{\textrm{shadow}}
\DeclareMathOperator{\start}{\textrm{start}}
\DeclareMathOperator{\eend}{\textrm{end}}
\DeclareMathOperator{\comp}{\textrm{comp}}


\DeclareMathOperator{\tr}{\textrm{Triple}}
\DeclareMathOperator{\trmn}{\tr_m^n}

\newcommand{\wT}{{\wt{T}}}
\newcommand{\vD}{{\vec{\mD}}}
\newcommand{\oV}{{\ov{V}}}
\newcommand{\oD}{{\ov{D}}}
\newcommand{\oE}{{\ov{E}}}
\newcommand{\oG}{{\ov{\mG}}}


\DeclareMathOperator{\Tmn}{\mT_m^n}
\DeclareMathOperator{\Tmd}{\mT_m^d}
\DeclareMathOperator{\Tmnd}{\mT_{mn}^d}
\DeclareMathOperator{\oT}{\ov{\mT}}
\DeclareMathOperator{\oTmn}{\ov{\mT}_m^n}
\DeclareMathOperator{\wA}{\wt{\mA}}
\DeclareMathOperator{\wTmn}{\wt{\mT}_m^n}



%\DeclareMathOperator{\precT}{\overset{T}{\prec}}
%\DeclareMathOperator{\preceqT}{\overset{T}{\preceq}}
%\DeclareMathOperator{\precL}{\overset{L}{\prec}}
\DeclareMathOperator{\precT}{\prec_T}
\DeclareMathOperator{\precwT}{\prec_{\wt T}}
\DeclareMathOperator{\preceqT}{\preceq_T}
%\DeclareMathOperator{\succT}{\succ_T}
\DeclareMathOperator{\succeqT}{\succeq_T}
\DeclareMathOperator{\succeqwT}{\succeq_{\wt T}}
\DeclareMathOperator{\precL}{\prec_L}
\DeclareMathOperator{\preceqL}{\preceq_L}


\newcommand{\musim}{\overset{\mu}{\sim}}
\newcommand{\nmusim}{\overset{\mu}{\nsim}}
\newcommand{\Lsim}{\overset{L}{\sim}}
\newcommand{\nLsim}{\overset{L}{\nsim}}


