Skip to main content\( {\labelitemi}{$\diamond$}
\def\arraystretch{1.5}
\newcommand{\contentsfinish}{}
\newcommand{\separator}{\begin{center}\rule{\columnwidth}{\arrayrulewidth}\end{center}}
\newcommand{\tosay}[1]{\begin{center}\text{\fbox{\scriptsize{#1}}}\end{center}}
\renewcommand{\cftsecfont}{}
\renewcommand{\cftsecpagefont}{}
\renewcommand{\descriptionlabel}[1]{\hspace{\labelsep}\smallcaps{#1}}
\def\oldequation{\equation}
\def\endoldequation{\endequation}
\newcommand{\nl}{
}
\newcommand{\runin}[1]{\textls[50]{\otherscshape #1}}
\renewcommand{\sectionmark}[1]{}
\renewcommand{\subsectionmark}[1]{}
\renewcommand{\sectionmark}[1]{\markboth{{\thesection}.\ \smallcaps{#1}}{\thesection.\ \smallcaps{#1}}}
\renewcommand{\subsectionmark}[1]{}
\renewcommand{\sectionmark}[1]{\markboth{{\scriptsize\thesection}.\ \smallcaps{#1}}{}}
\renewcommand{\subsectionmark}[1]{}
\newcommand{\makedefaultsection}[2][true]{
}
\newcommand{\timestamp}{{\color{red}Last updated: {\currenttime\ (UTC), \today}}}
\renewcommand{\le}{\leqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\ge}{\geqslant}
\newcommand{\ideal}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\subgp}[1]{\left\langle\, #1 \,\right\rangle}
\def\p{\varphi}
\def\isomorphic{\cong}
\def\imp{\to}
\def\iff{\leftrightarrow}
\def\Gal{\text{Gal}}
\def\im{\text{im}}
\def\ran{\text{ran}}
\def\normal{\vartriangleleft}
\renewcommand{\qedsymbol}{$\checkmark$}
\def\lcm{{\text{lcm}\,}}
\newcommand{\q}[1]{{\color{red} {#1}}}
\newcommand{\startimportant}[1]{\end{[{Hint:} #1]\end}}
\renewcommand{\textcircled}[1]{\tikz[baseline=(char.base)]{\node[shape=circle,draw,inner sep=2pt,color=red] (char) {#1};}}
\newcommand{\crossout}[1]{\tikz[baseline=(char.base)]{\node[mynode, cross out,draw] (char) {#1};}}
\newcommand{\set}[1]{\left\{ {#1} \right\}}
\newcommand{\setof}[2]{{\left\{#1\,|\,#2\right\}}}
\def\C{{\mathbb C}}
\def\Z{{\mathbb Z}}
\def\bF{{\mathbb F}}
\def\Q{{\mathbb Q}}
\def\N{{\mathbb N}}
\def\U{{\mathcal U}}
\def\pow{{\mathcal P}}
\def\R{{\mathbb R}}
\newcommand{\h}[1]{{\textbf{#1}}}
\def\presnotes{}
\renewcommand{\d}{\displaystyle}
\newcommand{\N}{\mathbb N}
\newcommand{\B}{\mathbf B}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\U}{\mathcal U}
\newcommand{\pow}{\mathcal P}
\newcommand{\inv}{^{-1}}
\newcommand{\st}{:}
\renewcommand{\iff}{\leftrightarrow}
\newcommand{\Iff}{\Leftrightarrow}
\newcommand{\imp}{\rightarrow}
\newcommand{\Imp}{\Rightarrow}
\newcommand{\isom}{\cong}
\renewcommand{\bar}{\overline}
\newcommand{\card}[1]{\left| #1 \right|}
\newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}}
\newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}}
\newcommand{\va}[1]{\vtx{above}{#1}}
\newcommand{\vb}[1]{\vtx{below}{#1}}
\newcommand{\vr}[1]{\vtx{right}{#1}}
\newcommand{\vl}[1]{\vtx{left}{#1}}
\renewcommand{\v}{\vtx{above}{}}
\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\,}$}}}
\)
