
\usepackage{amsmath}
%\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{stmaryrd}

\newcommand {\br}[1]{\left(#1\right)}
\newcommand {\brbig}[1]{\bigl( #1 \bigr)}
\newcommand {\brBig}[1]{\Bigl( #1 \Bigr)}
\newcommand {\brbigg}[1]{\biggl( #1 \biggr)}
\newcommand {\brBigg}[1]{\Bigg( #1 \Biggr)}
\newcommand {\tbr}[1]{\left(\!#1 \!\right)}
\newcommand {\ttbr}[1]{\left(\!\!#1 \!\!\right)}
\newcommand {\tttbr}[1]{\left(\!\!\! #1 \!\!\!\right)}
\newcommand {\sqb}[1]{\left[#1\right]}
\newcommand {\sqbP}[1]{\left[\: #1 \:\right]}
\newcommand {\sqbbig}[1]{\bigl[ #1 \bigr]}
\newcommand {\sqbBig}[1]{\Bigl[ #1 \Bigr]}
\newcommand {\sqbbigg}[1]{\biggl[ #1 \biggr]}
\newcommand {\sqbBigg}[1]{\Bigg[ #1 \Biggr]}
\newcommand {\cbr}[1]{\left\{#1 \right\}}
\newcommand {\cbrP}[1]{\left\{\: #1\: \right\}}
\newcommand {\nm}[1]{\left\Arrowvert\, #1 \,\right\Arrowvert}
\newcommand {\nmi}[1]{\left\Arrowvert\, #1 \,\right\Arrowvert_\infty}
\newcommand {\inp}[1]{\left\langle #1 \right\rangle}
\newcommand {\ninp}[1]{\langle #1 \rangle}
\newcommand {\bs}[1]{\vert #1 \vert}
\newcommand {\abs}[1]{\left\vert\, #1 \,\right\vert}
\newcommand {\absbig}[1]{\bigl\vert #1 \bigr\vert}
\newcommand {\absBig}[1]{\Bigl\vert #1 \Bigr\vert}
\newcommand {\absbigg}[1]{\biggl\vert #1 \biggr\vert}
\newcommand {\absBigg}[1]{\Biggl\vert #1 \Biggr\vert}
\newcommand {\series}[1]{\!\!\ll\!\!{#1}\!\!\gg\!\!}
\newcommand {\bbracket}[1]{\llbracket{#1}\rrbracket}

\newcommand {\short}[2]{{\delimitershortfall100pt \delimiterfactor600 {#1{#2}}} 
\delimiterfactor1000}

\newcommand {\m}[1]{$ #1 $}
\newcommand {\V}[1]{\mbox{\boldmath$#1$}}
\newcommand {\h}[1]{\widehat{#1}}
\newcommand {\te}[1]{\widetilde{#1}}
%\newcommand {\ob}[1]{\overline{#1}}
\newcommand {\ul}[1]{\underline{#1}}
\newcommand {\wbar}[1]{\overline{#1}}

\newcommand {\fr}[2]{\frac{#1}{#2}}
\newcommand {\ovr}[1]{\frac{1}{#1}}
\newcommand {\tovr}[1]{{\textstyle \frac{1}{#1}}}

\newcommand {\dd}{\partial}
\newcommand {\pD}[2]{\frac{d #1}{d #2}}
\newcommand {\pd}[2]{\frac{\dd #1}{\dd #2}}
\newcommand {\pfd}[2]{\frac{\delta #1}{\delta #2}}

\newcommand {\ms}{\! - \!}
\newcommand {\ps}{\! +\!}
\newcommand {\es}{\! =\!}
\newcommand {\tes}{\!\! = \!\!}
\newcommand {\tsetminus}{\! \setminus \!}
\newcommand {\tin}{\! \in \!}
\newcommand {\tcdot}{\!\cdot\!}
\newcommand {\tsubset}{\!\subset\!}
\newcommand {\tto}{\!\to\!}
\newcommand {\tmapsto}{\!\mapsto\!} 
\newcommand {\quadd}{\quad\enspace}
\newcommand {\qqquad}{\qquad\quad}
\newcommand {\pull}{\!\!\!\!}
\newcommand {\pad}[1]{\, #1 \,}
%\newcommand {\ggf}{\hspace{12pt}}
\newcommand {\<}[1]{\hspace{-0.1 em}#1\hspace{-0.1 em}}
\newcommand {\ts}{\hspace{.11111em}}
\newcommand {\tts}{\hspace{.055555em}}

\newcommand {\bi}[1]{#1#1}
\newcommand {\tri}[1]{#1#1#1}
\newcommand {\qd}[1]{#1#1#1#1}
\newcommand {\oct}[1]{#1#1#1#1#1#1#1#1}

\newcommand {\sgn}{\mathrm{sgn}}
\newcommand {\vol}{\,\mathrm{vol}\,}


\newcommand {\ignore}[1]{}
\newcommand {\mc}[1]{\mathcal{#1}}

\newcommand {\bra}[1]{\left\langle  #1 \right \vert}
\newcommand {\ket}[1]{\left \vert #1 \right \rangle}
\newcommand {\braket}[2]{\left\langle #1 \vert #2 \right \rangle}

\newcommand {\etext}[1]{\ \text{#1}\ }
\newcommand {\dtext}[1]{\ \ \text{#1}\ \ }
\newcommand {\qtext}[1]{\quad\text{#1}\quad}

\newcommand {\Real}{\mathbb{R}}
\newcommand {\Natural}{\mathbb{N}}
\newcommand {\Input}{\mathcal{X}}
\newcommand {\Output}{\mathcal{Y}}
\newcommand {\Hilbert}{\mathcal{H}}

\newcommand {\Xcal}{\mathcal{X}}
\newcommand {\Ycal}{\mathcal{Y}}
\newcommand {\Zcal}{\mathcal{Z}}
\newcommand {\Hcal}{\mathcal{H}}
\newcommand {\Fcal}{\mathcal{F}}
\newcommand {\Ccal}{\mathcal{C}}
\newcommand {\Lcal}{\mathcal{L}}
\newcommand {\Mcal}{\mathcal{M}}
\newcommand {\FF}{\mathbb{F}}
\newcommand {\RR}{\mathbb{R}}
\newcommand {\NN}{\mathbb{N}}
\newcommand {\ZZ}{\mathbb{Z}}
\newcommand {\CC}{\mathbb{C}}

\newcommand {\EE}{\mathbb{E}}
\newcommand {\PP}{\mathbb{P}}

\newcommand {\eps}{\ensuremath{\epsilon}}

\newcommand{\rf}[1]{(\ref{#1})}
