%%%%%%%%%%%%% Luke's standard style
%%%%%%%%%%%%% 10/28/08 version

\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}

\usepackage{booktabs}



\renewcommand{\ln}{\log}
\newcommand{\mzero}{\ensuremath{\mathbf{0}}}
\newcommand{\mx}{\ensuremath{\mathbf{x}}}
\newcommand{\eqquestion}{\ensuremath{\stackrel{\text{\tiny ?}}{=}}}


% Shortcuts for "blackboard-style" symbols for reals
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
\newcommand{\Q}{\ensuremath{\mathbb{Q}}}

% Shortcut for Lagrangian symbol
\newcommand{\Lag}{\ensuremath{\mathcal{L}}}

% Preferred emptyset symbol
\renewcommand{\emptyset}{\ensuremath{\varnothing}}

% Preferred epsilon symbol
\renewcommand{\epsilon}{\varepsilon}

% Shortcuts for non-italicized operators
%\DeclareMathOperator{\E}{E}			% To get regular, non-italicized E
\newcommand{\E}{\ensuremath{\mathbb{E}}}	% To get blackboard E
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\DeclareMathOperator{\rank}{rank}

% New math operators: argmax and argmin
\DeclareMathOperator*{\argmax}{argmax}
\DeclareMathOperator*{\argmin}{argmin}

% \pref as a shortcut for preference (and ensure math mode)
\newcommand{\pref}{\ensuremath{\succsim}}

% \spref as a shortcut for strict preference (and ensure math mode)
\newcommand{\spref}{\ensuremath{\succ}}

% Sometimes forget to use \nabla for gradient
\newcommand{\grad}{\ensuremath{\nabla}}

% Define abs and norm per AMS recommendations
\providecommand{\abs}[1]{\lvert#1\rvert}
\providecommand{\norm}[1]{\lVert#1\rVert}

% Slanted fractions (from Clayton Featherstone)
\def\slantfrac#1#2{\,^{#1}\!/_{\! #2}}