X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=examples.tex;h=c7959732d44ce74297d43c228dd90ae267dd255a;hp=cdf2359d4fe22b68738d76ccca052d8b8d719b89;hb=a42e99e28d22bd0a313d4bac23cd4278627be1a3;hpb=d4712abb88a2f342f31bbd60426d1a983df05ac2 diff --git a/examples.tex b/examples.tex index cdf2359..c795973 100644 --- a/examples.tex +++ b/examples.tex @@ -1,15 +1,41 @@ \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 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 $R$ 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} @@ -33,8 +59,9 @@ \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 + 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} @@ -45,44 +72,49 @@ \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 $\triple{1}{2}{3}$, - and 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: + The function $f$ applied to $x$ is $f\of{x}$, and the restriction + of $f$ to a subset $X$ of its domain is $\restrict{f}{X}$. We can + group terms like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - + d}}$. 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 direct sum of $V$ and $W$ is $\directsum{V}{W}$. Or three + things, $\directsumthree{U}{V}{W}$. How about more things? Like + $\directsummany{k=1}{\infty}{V_{k}}$. Those direct sums + adapt nicely to display equations: % \begin{equation*} - \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}. + \directsummany{k=1}{\infty}{V_{k}} \ne \emptyset. \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 superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$, - $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. - - We also have a few basic set operations, for example the union of - two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of - course with union comes intersection: $\intersect{A}{B}$, - $\intersectthree{A}{B}{C}$. We can also take an arbitrary - (indexed) union and intersections of things, like - $\unionmany{k=1}{\infty}{A_{k}}$ or - $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those - is that they do the right thing in a display equation: - % - \begin{equation*} - \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}} - \end{equation*} - - Finally, we have the four standard types of intervals in $\Rn[1]$, + $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. 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 },\\ @@ -122,7 +154,14 @@ \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} @@ -133,7 +172,12 @@ $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific concept is the Moore-Penrose pseudoinverse of $L$, denoted by - $\pseudoinverse{L}$. + $\pseudoinverse{L}$. 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 @@ -147,6 +191,9 @@ $\boundedops[W]{V}$. 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 @@ -196,11 +243,6 @@ system to test them. \end{section} - \begin{section}{Miscellaneous} - The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} - = 3$. - \end{section} - \begin{section}{Proof by cases} \begin{proposition} @@ -247,6 +289,35 @@ \renewcommand{\baselinestretch}{1} \end{section} + \begin{section}{Set theory} + Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The + cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} = + 3$, and its powerset is $\powerset{X}$. + + We also have a few basic set operations, for example the union of + two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of + course with union comes intersection: $\intersect{A}{B}$, + $\intersectthree{A}{B}{C}$. The Cartesian product of two sets $A$ + and $B$ is there too: $\cartprod{A}{B}$. If we take the product + with $C$ as well, then we obtain $\cartprodthree{A}{B}{C}$. + + We can also take an arbitrary (indexed) union, intersection, or + Cartesian product of things, like + $\unionmany{k=1}{\infty}{A_{k}}$, + $\intersectmany{k=1}{\infty}{B_{k}}$, or + $\cartprodmany{k=1}{\infty}{C_{k}}$. The best part about those is + that they do the right thing in a display equation: + % + \begin{equation*} + \unionmany{k=1}{\infty}{A_{k}} + \ne + \intersectmany{k=1}{\infty}{B_{k}} + \ne + \cartprodmany{k=1}{\infty}{C_{k}}. + \end{equation*} + % + \end{section} + \begin{section}{Theorems} \begin{corollary} The @@ -320,4 +391,13 @@ $\closure{X}$ and its boundary is $\boundary{X}$. \end{section} + \setlength{\glslistdottedwidth}{.3\linewidth} + \setglossarystyle{listdotted} + \glsaddall + \printnoidxglossaries + + \bibliographystyle{mjo} + \bibliography{local-references} + + \printindex \end{document}