X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=examples.tex;h=c1f3df1eb64e416fb00c52f809cdcfc562138324;hp=c87f59bdc792d78d7d5e88e0158510616bf63b56;hb=71b85fe2012cdf733f6e72137038d7d9960ddf08;hpb=7b6c4a0dba2844a0ecc1fd96a951f9c4c021f0a4 diff --git a/examples.tex b/examples.tex index c87f59b..c1f3df1 100644 --- a/examples.tex +++ b/examples.tex @@ -1,10 +1,37 @@ \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 $\Frac{R}$. + \end{section} + \begin{section}{Algorithm} An example of an algorithm (bogosort) environment. @@ -28,20 +55,34 @@ \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 - $\inverse{f}$. + $\inverse{f}$. 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 $\factorial{10}$. - + 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: + % + \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 @@ -51,7 +92,29 @@ 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: $\intersect{A}{B}$, - $\intersectthree{A}{B}{C}$. + $\intersectthree{A}{B}{C}$. 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 $\Rn[1]$, + % + \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} @@ -68,7 +131,14 @@ 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} @@ -81,7 +151,9 @@ 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 + $\pseudoinverse{L}$. 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 @@ -94,10 +166,37 @@ The set of all bounded linear operators from $V$ to $W$ is $\boundedops[W]{V}$. 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, 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 + \end{align*} + % + Its form should also survive in different font sizes... + \Large + \begin{align*} + Z = \directsumperp{V}{W}\\ + \oplus \oplusperp \oplus \oplusperp + \end{align*} + \Huge + \begin{align*} + Z = \directsumperp{V}{W}\\ + \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, @@ -110,6 +209,14 @@ [0 0], [0 0], [1 0], [0 1] ] \end{tcblisting} + + 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}{Miscellaneous} @@ -180,6 +287,10 @@ fox \end{theorem} + \begin{exercise} + jumps + \end{exercise} + \begin{definition} quod \end{definition} @@ -210,6 +321,10 @@ fox \end{theorem*} + \begin{exercise*} + jumps + \end{exercise*} + \begin{definition*} quod \end{definition*} @@ -227,5 +342,14 @@ 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}