\documentclass{report}
+% Setting hypertexnames=false forces hyperref to use a consistent
+% internal counter for proposition/equation references rather than
+% being clever, which doesn't work after we reset those counters.
+\usepackage[hypertexnames=false]{hyperref}
+\hypersetup{
+ colorlinks=true,
+ linkcolor=blue,
+ citecolor=blue
+}
+
+% We have to load this after hyperref, so that links work, but before
+% mjotex so that mjotex knows to define its glossary entries.
+\usepackage[nonumberlist]{glossaries}
+\makenoidxglossaries
+
+% If you want an index, we can do that too. You'll need to define
+% the "INDICES" variable in the GNUmakefile, though.
+\usepackage{makeidx}
+\makeindex
+
\usepackage{mjotex}
\usepackage{mathtools}
\begin{document}
+ \begin{section}{Algebra}
+ If $R$ is a \index{commutative ring}, then $\polyring{R}{X,Y,Z}$
+ is a multivariate polynomial ring with indeterminates $X$, $Y$,
+ and $Z$, and coefficients in $R$. If $R$ is a moreover an integral
+ domain, then its fraction field is $\Frac{R}$. If $x,y,z \in R$,
+ then $\ideal{\set{x,y,z}}$ is the ideal generated by
+ $\set{x,y,z}$, which is defined to be the smallest ideal in $R$
+ containing that set. Likewise, if we are in an algebra
+ $\mathcal{A}$ and if $x,y,z \in \mathcal{A}$, then
+ $\alg{\set{x,y,z}}$ is the smallest subalgebra of $\mathcal{A}$
+ containing the set $\set{x,y,z}$.
+ \end{section}
+
\begin{section}{Algorithm}
An example of an algorithm (bogosort) environment.
\end{section}
\begin{section}{Arrow}
- The identity operator on $V$ is $\identity{V}$. The composition of
- $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
- $\inverse{f}$.
+ The constant function that always returns $a$ is $\const{a}$. The
+ identity operator on $V$ is $\identity{V}$. The composition of $f$
+ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
+ $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
+ domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
+ \end{section}
+
+ \begin{section}{Calculus}
+ The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
+ \Rn \rightarrow \Rn$.
\end{section}
\begin{section}{Common}
The function $f$ applied to $x$ is $f\of{x}$. We can group terms
like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
- set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
- of things $\pair{1}{2}$ or a triple of them
- $\triple{1}{2}{3}$. The Cartesian product of two sets $A$ and $B$
- is $\cartprod{A}{B}$; if we take the product with $C$ as well,
- then we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$
- and $W$ is $\directsum{V}{W}$ and the factorial of the number $10$
- is $\factorial{10}$.
-
+ set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
+ up to seven, for now:
+ %
+ \begin{itemize}
+ \begin{item}
+ Pair: $\pair{1}{2}$,
+ \end{item}
+ \begin{item}
+ Triple: $\triple{1}{2}{3}$,
+ \end{item}
+ \begin{item}
+ Quadruple: $\quadruple{1}{2}{3}{4}$,
+ \end{item}
+ \begin{item}
+ Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
+ \end{item}
+ \begin{item}
+ Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
+ \end{item}
+ \begin{item}
+ Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
+ \end{item}
+ \end{itemize}
+ %
+ The factorial of the number $10$ is $\factorial{10}$.
+
+ The Cartesian product of two sets $A$ and $B$ is
+ $\cartprod{A}{B}$; if we take the product with $C$ as well, then
+ we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
+ is $\directsum{V}{W}$. Or three things,
+ $\directsumthree{U}{V}{W}$. How about more things? Like
+ $\directsummany{k=1}{\infty}{V_{k}} \ne
+ \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
+ cartesian products adapt nicely to display equations:
+ %
+ \begin{equation*}
+ \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
+ \end{equation*}
+ %
Here are a few common tuple spaces that should not have a
superscript when that superscript would be one: $\Nn[1]$,
$\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
\unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
\end{equation*}
%
+ Finally, we have the four standard types of intervals in $\Rn[1]$,
+ %
+ \begin{align*}
+ \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
+ \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
+ \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
+ \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
+ \end{align*}
+ \end{section}
+
+ \begin{section}{Complex}
+ We sometimes want to conjugate complex numbers like
+ $\compconj{a+bi} = a - bi$.
\end{section}
\begin{section}{Cone}
The conic hull of a set $X$ is $\cone{X}$; its affine hull is
$\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
then its lineality space is $\linspace{K}$, its lineality is
- $\lin{K}$, and its extreme directions are $\Ext{K}$.
+ $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
+ $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
+ proper face, then we write $F \properfaceof K$.
+ \end{section}
+
+ \begin{section}{Euclidean Jordan algebras}
+ The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
+ is $\jp{x}{y}$.
\end{section}
\begin{section}{Font}
- We can write things like Carathéodory and Güler and $\mathbb{R}$.
+ We can write things like Carathéodory and Güler and
+ $\mathbb{R}$. The PostScript Zapf Chancery font is also available
+ in both upper- and lower-case:
+ %
+ \begin{itemize}
+ \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
+ \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
+ \end{itemize}
\end{section}
\begin{section}{Linear algebra}
their tensor product is $\tp{x}{y}$. The Kronecker product of
matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
$L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
- $\transpose{L}$. Its trace is $\trace{L}$.
+ $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
+ concept is the Moore-Penrose pseudoinverse of $L$, denoted by
+ $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
+ $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
+ real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
+ respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
+ simplify because the ``$n$'' does not indicate the arity of a
+ Cartesian product in this case.
The span of a set $X$ is $\spanof{X}$, and its codimension is
$\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
$\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
instead.
+ If you want to solve a system of equations, try Cramer's
+ rule~\cite{ehrenborg}.
+
The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
but what if $W = V^{\perp}$? Then we wish to indicate that fact by
writing $\directsumperp{V}{W}$. That operator should survive a
- display equation, too:
+ display equation, too, and the weight of the circle should match
+ that of the usual direct sum operator.
%
\begin{align*}
Z = \directsumperp{V}{W}\\
- \oplus\oplusperp\oplus\oplusperp
+ \oplus \oplusperp \oplus \oplusperp
\end{align*}
%
Its form should also survive in different font sizes...
\Large
\begin{align*}
Z = \directsumperp{V}{W}\\
- \oplus\oplusperp\oplus\oplusperp
+ \oplus \oplusperp \oplus \oplusperp
\end{align*}
\Huge
\begin{align*}
Z = \directsumperp{V}{W}\\
- \oplus\oplusperp\oplus\oplusperp
+ \oplus \oplusperp \oplus \oplusperp
\end{align*}
\normalsize
\end{section}
\begin{section}{Listing}
- Here's an interactive sage prompt:
+ Here's an interactive SageMath prompt:
\begin{tcblisting}{listing only,
colback=codebg,
[0 0], [0 0], [1 0], [0 1]
]
\end{tcblisting}
- \end{section}
- \begin{section}{Miscellaneous}
- The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
- = 3$.
+ However, the smart way to display a SageMath listing is to load it
+ from an external file (under the ``listings'' subdirectory):
+
+ \sagelisting{example}
+
+ Keeping the listings in separate files makes it easy for the build
+ system to test them.
\end{section}
\begin{section}{Proof by cases}
\renewcommand{\baselinestretch}{1}
\end{section}
+ \begin{section}{Set theory}
+ The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
+ = 3$, and its powerset is $\powerset{X}$.
+ \end{section}
+
\begin{section}{Theorems}
\begin{corollary}
The
fox
\end{theorem}
+ \begin{exercise}
+ jumps
+ \end{exercise}
+
\begin{definition}
quod
\end{definition}
fox
\end{theorem*}
+ \begin{exercise*}
+ jumps
+ \end{exercise*}
+
\begin{definition*}
quod
\end{definition*}
The interior of a set $X$ is $\interior{X}$. Its closure is
$\closure{X}$ and its boundary is $\boundary{X}$.
\end{section}
-
+
+ \setlength{\glslistdottedwidth}{.3\linewidth}
+ \setglossarystyle{listdotted}
+ \glsaddall
+ \printnoidxglossaries
+
+ \bibliographystyle{mjo}
+ \bibliography{local-references}
+
+ \printindex
\end{document}
projects / mjotex.git / blobdiff