$$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\SO}{\operatorname{SO}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\length}{\operatorname{length}}
\newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}}
\newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}}
\newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}}
\newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}}
\newcommand{\rsum}[1]{{\textstyle\sum_{#1}}}
\newcommand{\partitions}[1]{\mathcal{P}_{#1}}
\newcommand{\erf}{\operatorname{erf}}
\newcommand{\ihat}{\hat{\imath}}
\newcommand{\jhat}{\hat{\jmath}}
\newcommand{\khat}{\hat{k}}
\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}
\newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}
$$