example.tex

\documentclass[a4paper, 12pt]{report}

\newcommand{\ispdfversion}{true}

\input{preamble.tex}

\begin{document}
	
	\chapter{An Example Chapter} % (fold)
	\label{cha:an_example_chapter}

		In this chapter we demonstrate the features of the preamble.tex file. Section~\ref{sec:example_environment_and_mathematics_layout} covers the use of the example environment and what mathematics looks like, while Section~\ref{sec:algorithm_and_a_quotation} shows the Algorithm environment and a quotation.

		\section{Example environment and mathematics layout} % (fold)
		\label{sec:example_environment_and_mathematics_layout}

			\lipsum[1]

			\begin{example}
				Let $f=yx-x^2+x$ and $g=2y+x+1 \; \in \Q[x,y]$. We use degree lexicographic order with $y>x$. Dividing $f$ by $g$ gives us:
				\[
					\setlength{\arraycolsep}{3pt}
					\renewcommand{\arraystretch}{1.3}
					\begin{array}{r r r r r r r r r r r r}
						&&&&& yx &&& - & x^2 & + & x \\
						\cline{2-12}
						\multicolumn{1}{r|}{2y+x+1}%
						 & 2y^2x & - &  yx^2 & + & 3yx \\
					\end{array}
				\]

				In reduction notation this division looks as follows:

				\[
					f \myrightarrow{g} -2yx^2+2yx \myrightarrow{g} 2yx+x^3+x^2 \myrightarrow{g} x^3-x.
				\]
			\end{example}

			Math numbers are set $0123456789$, normal numbers are set 0123456789.
		
		% section example_environment_and_mathematics_layout (end)

		\section{Algorithm and a quotation} % (fold)
		\label{sec:algorithm_and_a_quotation}

			\begin{algorithm}[Multivariate Division Algorithm]
			\label{alg:multi}
			Input: $f,f_1,\ldots,f_t \in \F[x_1,\ldots,x_n]$ with $f_i\neq 0 \; (1\leq i\leq t)$.
			
			Output: $u_1,\ldots,u_t,r \in \F[x_1,\ldots,x_n]$ such that $f = u_1f_1 + \ldots + u_tf_t + r$, $r$ is reduced with respect to $\{f_1,\ldots,f_t\}$ and $\mathrm{lp}(f) = \max(\mathrm{lp}(f_i),\mathrm{lp}(u_i),\mathrm{lp}(r))$.
			
			Initialisation: $u_i:=0$, $r:=0$, $h=f$.
			
			While $h\neq 0$ Do
			
			Loop
		\end{algorithm}

		\begin{shadequote}
			Alice was beginning to get very tired of sitting by her sister on the
bank, and of having nothing to do: once or twice she had peeped into the
book her sister was reading, but it had no pictures or conversations in
it, `and what is the use of a book,' thought Alice `without pictures or
conversation?'
		\end{shadequote}
		
		% section algorithm_and_a_quotation (end)
	
	% chapter an_example_chapter (end)

\end{document}