X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=e11815ef7856b47f0dd5cb0e775c3272caeec427;hb=71dc84ac39e0dc87daf64fe61d7523e33711dc34;hp=98ddfd550b08ea4fd78301eb47c0aea19cb8de7a;hpb=5c4f67545f0988d065f1d52f90eed9233562c9fc;p=mjotex.git diff --git a/examples.tex b/examples.tex index 98ddfd5..e11815e 100644 --- a/examples.tex +++ b/examples.tex @@ -1,10 +1,43 @@ \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. @@ -26,22 +59,60 @@ \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 @@ -61,6 +132,19 @@ \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} @@ -77,11 +161,25 @@ 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} @@ -90,7 +188,14 @@ 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 @@ -104,6 +209,9 @@ $\boundedops[W]{V}$. 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 @@ -130,7 +238,7 @@ \end{section} \begin{section}{Listing} - Here's an interactive sage prompt: + Here's an interactive SageMath prompt: \begin{tcblisting}{listing only, colback=codebg, @@ -143,11 +251,14 @@ [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} @@ -196,6 +307,11 @@ \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 @@ -213,6 +329,10 @@ fox \end{theorem} + \begin{exercise} + jumps + \end{exercise} + \begin{definition} quod \end{definition} @@ -243,6 +363,10 @@ fox \end{theorem*} + \begin{exercise*} + jumps + \end{exercise*} + \begin{definition*} quod \end{definition*} @@ -260,5 +384,14 @@ 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}