% --------- DEFINITIONS ----------------------
\def\bz{\mathbf{0}}
\def\bB{\mathbf{B}}
\def\ba{\boldsymbol{\alpha}}
\def\bb{\boldsymbol{\beta}}
\def\bPsi{\boldsymbol{\Psi}}
\def\invPsi{\boldsymbol{\Psi}^{-1}}
\def\brho{\boldsymbol{\rho}}
\def\bxi{\boldsymbol{\xi}}
\def\bphi{\boldsymbol{\phi}}
\def\bmu{\boldsymbol{\mu}}
\def\bGa{\boldsymbol{\Gamma}}
\def\bga{\boldsymbol{\gamma}}
\def\bLa{\boldsymbol{\Lambda}}
\def\bOm{\boldsymbol{\Omega}}
\def\bSi{\boldsymbol{\Sigma}}
\def\bve{\boldsymbol{\varepsilon}}
\def\b0{\mathbf{0}}
\def\bI{\mathbf{I}}
\def\bJ{\mathbf{J}}
\def\I_Vg{\mathbf{I}_{V_g}}
\def\J_Vg{\mathbf{J}_{V_g}}
\def\bX{\mathbf{X}}
\def\br{\mathbf{r}}
\def\Wg{\mathbf{W}_g}
\def\Wgs{\mathbf{W}_{gs}}
\def\bM{\mathbf{M}}
\def\Mgs{\mathbf{M}_{gs}}
\def\tr{\text{tr}}
\def\tV{\text{V}}
\def\V{\text{Var}}

\def\bA{\mathbf{A}}
\def\bC{\mathbf{C}}
\def\bc{\mathbf{c}}
\def\bT{\mathbf{T}}
\def\bW{\mathbf{W}}
\def\bX{\mathbf{X}}
\def\bY{\mathbf{Y}} %%%
\def\bZ{\mathbf{Z}}
\def\bU{\mathbf{U}}
\def\bV{\mathbf{V}}
\def\b1{\mathbf{1}}

\def\rN{{\rm N}} % roman N

\def\etal{{\it et.al.\/}}

% --------- Greek Letters ----------------------
\def\al{\alpha}
\def\be{\beta}
\def\De{\Delta}
\def\de{\delta}
\def\ga{\gamma}
\def\Ga{\Gamma}
\def\ka{\kappa}
\def\la{\lambda}
\def\La{\Lambda}
\def\om{\omega}
\def\Om{\Omega}
\def\Th{\Theta}
\def\th{\theta}
\def\si{\sigma}
\def\Si{\Sigma}
\def\vp{\varphi}
\def\be{\beta}
\def\e{\epsilon}
\def\ve{\varepsilon}
\def\bve{\boldsymbol{\varepsilon}}

\def\beqa{\begin{eqnarray*}}
\def\eeqa{\end{eqnarray*}}

\def\tg{{\tilde{g}}}
\def\cD{{\cal D}}   % calligraphic letters
\def\cO{{\cal O}}
\def\cF{{\cal F}}
\def\cA{{\cal A}}
\def\cB{{\cal B}}
\def\cN{{\cal N}}
\def\cC{{\cal C}}
\def\cS{{\cal S}}

\def\md{{m_D}}
\def\p{\partial}


%\def\Mp{\mathbf{M}^{\mathbf{p}} % use in math mode
\def\IN{\!\in\!}
\def\Q1{\bar{Q}_1}
\def\hvp{\Hat{\vp}}
\def\rd{\bR^d}
\def\Lr{L^2(\bR)}
\def\Lrd{L^2(\bR^d)}
\def\zd{\bZ^d}
\def\lzd{\ell^2(\zd)}
\def\intrd{\int_{\rd}}
\def\ra{\rightarrow}
\def\Ra{\Rightarrow}
\def\lra{\longrightarrow}
\def\Lra{\Longrightarrow}
\def\Llra{\Longleftrightarrow}
\def\lms{\longmapsto}
\def\sset{\subset}
\def\sseq{\subseteq}
\def\hf{\widehat{f}}
\def\<{\left<}
\def\>{\right>}
\def\bks{\backslash}

\newcommand{\cx}[1]{{\color{red} #1}}       % from Jose
\newcommand{\cy}[1]{{\color{green} #1}}
\newcommand{\cz}[1]{{\color{brown} #1}}
\newcommand{\cw}[1]{{\color{blue} #1}}
\newcommand{\cv}[1]{{\color{magenta} #1}}
\newcommand{\cq}[1]{{\color{emerald} #1}}
\newcommand{\cs}[1]{{\color{bblue} #1}}
\newcommand{\ct}[1]{{\color{mblue} #1}}
\newcommand{\cn}[1]{{\color{rblue} #1}}
\newcommand{\mB}[1]{{\color{Blue} #1}} % my blue, darker than \cw

\newcommand{\co}[1]{{\color{orange} #1}}
\newcommand{\cyl}[1]{{\color{yellow} #1}}
\newcommand{\cp}[1]{{\color{purple} #1}}