X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=examples.tex;h=2d05431e2ceb2fcdcdf3314980f3348929a69ea8;hp=e0ffcb97a11ce5a97a2e837e7c994fd19714e1ae;hb=HEAD;hpb=987368c596bfe23dbcbaf5260a0d7011c9d3fc1e diff --git a/examples.tex b/examples.tex index e0ffcb9..383cef2 100644 --- a/examples.tex +++ b/examples.tex @@ -1,10 +1,51 @@ \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\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 $\set{x,y,z}$. + + If $R$ has a multiplicative identity (that is, a unit) element, + then that element is denoted by $\unit{R}$. Its additive identity + element is $\zero{R}$. The stabilizer (or isotropy) + subgroup of $G$ that fixes $x$ is $\Stab{G}{x}$. + + If $I$ is an ideal, then $\variety{I}$ is the variety that + corresponds to it. + \end{section} + \begin{section}{Algorithm} An example of an algorithm (bogosort) environment. @@ -18,71 +59,134 @@ \While{$M$ is not sorted} \State{Rearrange $M$ randomly} - \EndWhile + \EndWhile{} - \Return{$M$} + \State{\Return{$M$}} \end{algorithmic} \end{algorithm} \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} - \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 $\factorial{10}$. + \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 $f\of{x}$, 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, and then we give up + and use the general construct: + % + \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} + \begin{item} + Tuple: $\tuple{1,2,\ldots,8675309}$. + \end{item} + \end{itemize} + % + The factorial of the number $10$ is $\factorial{10}$, and the + least common multiple of $4$ and $6$ is $\lcm{\set{4,6}} = + 12$. + + 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 \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: $\Nn[2]$, $\Zn[2]$, - $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. + $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. The symbols $\Fn[1]$, $\Fn[2]$, + et cetera, are available for use with a generic field. - 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 $\Rn[1]$, % + \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 dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones - are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$. If cones - $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$, - $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$, - $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x - \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x - \ltcone_{K} y$ with respect to a cone $K$. + are $\Rnplus$, $\Rnplusplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$. + If cones $K_{1}$ and $K_{2}$ are given, we can define + $\posops{K_{1}}$, $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, + $\Zof{K_{1}}$, $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can + also define $x \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} + y$, and $x \ltcone_{K} y$ with respect to a cone $K$. \end{section} \begin{section}{Convex} 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 + $V$ is $\jp{x}{y}$. The Jordan-automorphism group of $V$ is + $\JAut{V}$. Two popular operators in an EJA are its quadratic + representation and ``left multiplication by'' operator. For a + given $x$, they are, respectively, $\quadrepr{x}$ and + $\leftmult{x}$. \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}{Hurwitz} + Here lies the Hurwitz algebras, like the quaternions + $\quaternions$ and octonions $\octonions$. \end{section} \begin{section}{Linear algebra} @@ -91,7 +195,17 @@ 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 $\trace{L}$, and its spectrum---the + set of its eigenvalues---is $\spectrum{L}$. 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. A handy way to represent the + matrix $A \in \Rn[n \times n]$ whose only non-zero entries are on + the diagonal is $\diag{\colvec{A_{11},A_{22},\ldots,A_{nn}}}$. 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 @@ -103,7 +217,13 @@ The set of all bounded linear operators from $V$ to $W$ is $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$ - instead. + instead. If you have matrices instead, then the general linear + group of $n$-by-$n$ matrices with entries in $\mathbb{F}$ is + $\GL{n}{\mathbb{F}}$. + + If you want to solve a system of equations, try Cramer's + rule~\cite{ehrenborg}. Or at least the reduced row-echelon form of + the matrix, $\rref{A}$. 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 @@ -116,7 +236,7 @@ \oplus \oplusperp \oplus \oplusperp \end{align*} % - Its form should also survive in different font sizes... + Its form should also survive in different font sizes\ldots \Large \begin{align*} Z = \directsumperp{V}{W}\\ @@ -131,7 +251,7 @@ \end{section} \begin{section}{Listing} - Here's an interactive sage prompt: + Here's an interactive SageMath prompt: \begin{tcblisting}{listing only, colback=codebg, @@ -144,11 +264,14 @@ [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} @@ -197,6 +320,35 @@ \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 @@ -214,6 +366,10 @@ fox \end{theorem} + \begin{exercise} + jumps + \end{exercise} + \begin{definition} quod \end{definition} @@ -244,6 +400,10 @@ fox \end{theorem*} + \begin{exercise*} + jumps + \end{exercise*} + \begin{definition*} quod \end{definition*} @@ -261,5 +421,14 @@ 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}