Skip to main content\(\usepackage{mathtools}
\DeclareMathOperator{\fixedset}{fix}
\DeclareMathOperator{\support}{supp}
\DeclareMathOperator{\lcm}{lcm}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\upto}[1]{\set{0,1,\dotsc,#1-1}}
\newcommand{\seq}[1]{\left(#1\right)}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
\newcommand{\after}{\circ}
\newcommand{\sym}[1]{\mathcal{S}_{#1}}
\newcommand{\fix}[1]{\fixedset{#1}}
\newcommand{\supp}[1]{\support{#1}}
\newcommand{\xto}[1]{\xrightarrow{#1}}
\newcommand{\arr}[1]{\overrightarrow{#1}}
\newcommand{\rem}{\%}
\newcommand{\divides}{\mid}
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Zp}{\Ints^+}
\newcommand{\Rats}{\mathbb{Q}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Comps}{\mathbb{C}}
\newcommand{\Mats}[1]{\mathbb{M}_{#1}}
\renewcommand{\vec}[1]{\mathbf{#1}}
\newcommand{\cvec}[1]{\begin{bmatrix}#1\end{bmatrix}}
\newcommand{\vv}{\vec{v}}
\newcommand{\CV}[1]{\Comps^{#1}}
\newcommand{\entry}[2]{\left[#1\right]_{#2}}
\newcommand{\hip}[2]{\left\langle#1,#2\right\rangle}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\dist}[3]{d_{#1}\left(#2,#3\right)}
\newcommand{\norm}[1]{\left\|#1\right\|}
\DeclareMathOperator{\spn}{span}
\newcommand{\spanset}[1]{\spn #1}
\DeclareMathOperator{\Aut}{Aut}
\newcommand{\rswap}[2]{R_{#1}\leftrightarrow R_{#2}}
\newcommand{\rscale}[2]{\left(#1\right) R_{#2}}
\newcommand{\relim}[3]{\left(#1\right) R_{#2}+R_{#3}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
