\usepackage{color}
\usepackage{bm}
\usepackage{bbm}


\newcommand{\Mset}[1]{\{#1\}_{m=1}^M}
\newcommand{\Nset}[1]{\{#1\}_{m=1}^N}
\newcommand{\Mvec}[1]{\big(#1\big)_{m=1}^M}

\newcommand{\emphc}[1]{{\color{blue}#1}}
\newcommand{\notec}[1]{{\color{red}#1}}
\newcommand{\soutr}[1]{{\color{red}\sout{#1}}}


%\usetikzlibrary{dsp,chains,matrix,arrows,calc,fit,automata,positioning,scopes}


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


\newcommand{\replace}[2]{{\color{red}{#1}\,\Midline{{#2}}}}
\newcommand{\comment}[1]{{\color{blue}\emph{Comment:} #1}}
\newcommand{\insrt}[1]{{\color{red}#1}}

\newcommand{\norm}[1]{\Vert#1\Vert}
\newcommand{\ones}{\mathbf{1}}
\newcommand{\Trace}[1]{\mathrm{tr}\left(#1\right)}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\med}{\mathrm{med}}
\newcommand{\atoms}[1]{\mathrm{atoms}(#1)}

\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{prp}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}

\theoremstyle{definition}
\newtheorem{dfn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{obs}[thm]{Observation}
\newtheorem{prb}[thm]{Problem}
\newtheorem{exm}[thm]{Example}

\def\dim{\mathrm{dim}}
\newcommand{\Real}{\mathbb{R}}
\newcommand{\Comp}{\mathbb{C}}
\newcommand{\Int}{\mathbb{Z}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\Nat}{\mathbb{N}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\V}{\mathcal{V}}
\newcommand{\solS}{\mathcal{S}}
\newcommand{\solH}{\mathcal{H}}
\newcommand{\tcX}{\tilde\X}


\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cLUV}{\mathcal{L}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}

\newcommand{\kf}{\Bbbk}

\newcommand{\transpose}{\mathrm{T}}

\newcommand{\ind}{\mathcal{T}}
\newcommand{\bss}{\mathcal{S}}
\newcommand{\fss}{\mathcal{H}}
\newcommand{\xss}{\mathcal{X}}
\newcommand{\samp}{T}


\newcommand{\supp}{\text{supp}}


%\markboth{Last modified -- \currenttime ~ \today}{Last modified -- \currenttime ~ \today}

%%%%%%%%%%%%%%%
%YMB

\def\Comp{\mathbb{C}}
\def\Real{\mathbb{R}}
\def\timeline{\mathbf{T}}
\def\lmaps{\mathtt{Lin}}
\def\kg{\kappa}
\def\ig{\rho}
\def\ker{\mathtt{Ker}}
\def\ima{\mathtt{Im}}
\def\grass{\mathtt{Gr}}
\def\obs{\mathcal{O}}
\def\Proj{\mathbb{P}}
\def\circle{\mathtt{S}^1}
\def\bx{\mathbf{x}}
\def\br{\mathbf{r}}
\def\tr{\mathtt{tr}}

\def\eps{\epsilon}
\def\ray{\mathbf{k}}
\def\az{\mathtt{Az}}

\def\ex{\mathbb{E}}
\def\var{\mathbb{V}}
\def\prob{\mathbb{P}}
\def\quadr{\mathtt{Q}}

\def\mtree{\mathtt T}

\def\ma{{$\mathbf{M}$}}
\def\fr{{$\mathbf{F}$}}
\def\lw{{\bm{\lambda}}}
\def\rw{{\bm{\rho}}}

\newcommand\Em{\mathbf{e}}
\newcommand\Eb{\mathbf{b}}
\newcommand\Ed{\mathbf{d}}
\newcommand\cov{\mathtt{Cov}}
\newcommand\wi{\mathbf{w}}

\newcommand{\spc}{\mathbb{X}}
\newcommand{\ph}{\mathbf{PH}}
\newcommand{\brm}{\mathbf{B}}
\newcommand{\brmd}[1]{\brm^{#1}}
\newcommand{\mph}{\bm{\mu}}
\newcommand{\arpath}[3]{{#1}\xrightarrow{\displaystyle{#2}}{#3}}
\newcommand\vertical{\mathtt{B}}


\def\smpl{\bm{\sigma}}
\def\smple{{\smpl^\eps}}
\def\smplo{\breve{\smpl}}
\def\lnk{\bm{L}}
\def\lnko{\breve{\lnk}}
\def\ones{\mathbf{1}}
\def\zeros{\mathbf{0}}
\def\conf{\mathtt{Conf}}
\def\xx{\bm{x}}
\def\yy{\bm{y}}
\def\zz{\bm{z}}
\def\mm{\mathbf{m}}
\def\nn{\mathbf{n}}
\def\ideal{\mathcal{I}}
\def\nset{{\bm{N}}}
\def\sset{\mathrm{I}}
\def\clrs{{\mathfrak{C}}}
\def\col{\bm{c}}

\def\one{\bm{1}}
\def\dualone{\one^\circ}
\def\aal{\bm{\alpha}}
\def\prt{\mathtt{Part}}
\def\diag{\bm{\Delta}}

\def\fprt{\prt^\mathtt{F}}

\def\poset{\mathcal{P}}
\def\join{\wedge}

\def\chic{{\chi_c}}

\def\cube{\mathbb{I}}
\def\fc{{{\sigma}}}
\def\Ne{\diag^\eps}
\def\Se{{\fc}_\eps}
\def\bSe{\bar{\fc}_\eps}
\def\SE{{{\fc}^\eps}}

