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% BB shortcuts (for number spaces, etc)
\newcommand{\C}{\mathbb{C}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\I}{\mathbb{I}}
\newcommand{\J}{\mathbb{J}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\V}{\mathbb{V}}
\newcommand{\Z}{\mathbb{Z}}

% Caligraphic
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\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}}

% other useful shortcuts
\newcommand{\8}{\infty}
\newcommand{\dx}{\text{ d}}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
\newcommand{\hastoeq}{\overset{!}{=}}
\newcommand{\Leftrightarrowequiv}{\overset{\cdot}{\Leftrightarrow}}
\newcommand{\Longleftrightarrowequiv}{\overset{\cdot}{\Longleftrightarrow}}
\newcommand{\Rightarrowequiv}{\overset{\cdot}{\Rightarrow}}
\newcommand{\Leftarrowequiv}{\overset{\cdot}{\Leftarrow}}
\newcommand{\defAs}{\overset{\text{def}}{=}}
\newcommand{\der}[1]{\frac{\mbox{d}}{\mbox{d}#1}}
\newcommand{\dern}[2]{\frac{\mbox{d}^{#2}}{\mbox{d}^{#2}#1}}
\newcommand{\parder}[1]{\frac{\partial}{\partial #1}}
\newcommand{\pardern}[2]{\frac{\partial^{#2}}{\partial^{#2} #1}}
\newcommand{\defEquiv}{\overset{\text{def}}{\Leftrightarrow}}
\newcommand{\defImplies}{\overset{\text{def}}{\Rightarrow}}
\newcommand{\divides}{\hspace{0.25em}|\hspace{0.25em}}
\newcommand{\mbeq}{\overset{!}{=}}
\newcommand{\lcm}{\text{lcm}}
\newcommand{\limit}[2]{\lim_{#1 \rightarrow #2}}
\newcommand{\seq}[1]{(#1_n)_{n \geq 1}}
\newcommand{\ser}[2]{\sum_{k = 1}^{#2} #1_k}
\newcommand{\limni}{\limit{n}{\infty}}
\newcommand{\limnz}{\limit{n}{0}}
\newcommand{\limxz}{\limit{x_0}{0}}
\newcommand{\limhz}{\limit{h}{0}}
\newcommand{\liminfni}{\liminf_{n \rightarrow \infty}}
\newcommand{\limsupni}{\limsup_{n \rightarrow \infty}}
\newcommand{\elementstack}[2]{\genfrac{}{}{0pt}{}{#1}{#2}}
\DeclareMathOperator\arctanh{arctanh}
\DeclareMathOperator\arccosh{arccosh}
\DeclareMathOperator\arcsinh{arcsinh}