examples.tex

   1 \documentclass{report}
   2
   3 % Setting hypertexnames=false forces hyperref to use a consistent
   4 % internal counter for proposition/equation references rather than
   5 % being clever, which doesn't work after we reset those counters.
   6 \usepackage[hypertexnames=false]{hyperref}
   7 \hypersetup{
   8   colorlinks=true,
   9   linkcolor=blue,
  10   citecolor=blue
  11 }
  12
  13 % We have to load this after hyperref, so that links work, but before
  14 % mjotex so that mjotex knows to define its glossary entries.
  15 \usepackage[nonumberlist]{glossaries}
  16 \makenoidxglossaries
  17
  18 % If you want an index, we can do that too. You'll need to define
  19 % the "INDICES" variable in the GNUmakefile, though.
  20 \usepackage{makeidx}
  21 \makeindex
  22
  23 \usepackage{mjotex}
  24 \usepackage{mathtools}
  25
  26 \begin{document}
  27
  28   \begin{section}{Algebra}
  29     If $R$ is a \index{commutative ring}, then $\polyring{R}{X,Y,Z}$
  30     is a multivariate polynomial ring with indeterminates $X$, $Y$,
  31     and $Z$, and coefficients in $R$. If $R$ is a moreover an integral
  32     domain, then its fraction field is $\Frac{R}$. If $x,y,z \in R$,
  33     then $\ideal{\set{x,y,z}}$ is the ideal generated by
  34     $\set{x,y,z}$, which is defined to be the smallest ideal in $R$
  35     containing that set. Likewise, if we are in an algebra
  36     $\mathcal{A}$ and if $x,y,z \in \mathcal{A}$, then
  37     $\alg{\set{x,y,z}}$ is the smallest subalgebra of $\mathcal{A}$
  38     containing the set $\set{x,y,z}$.
  39   \end{section}
  40
  41   \begin{section}{Algorithm}
  42     An example of an algorithm (bogosort) environment.
  43
  44     \begin{algorithm}[H]
  45       \caption{Sort a list of numbers}
  46       \begin{algorithmic}
  47         \Require{A list of numbers $L$}
  48         \Ensure{A new, sorted copy $M$ of the list $L$}
  49
  50         \State{$M \gets L$}
  51
  52         \While{$M$ is not sorted}
  53           \State{Rearrange $M$ randomly}
  54         \EndWhile
  55
  56         \Return{$M$}
  57       \end{algorithmic}
  58     \end{algorithm}
  59   \end{section}
  60
  61   \begin{section}{Arrow}
  62     The constant function that always returns $a$ is $\const{a}$. The
  63     identity operator on $V$ is $\identity{V}$. The composition of $f$
  64     and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  65     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  66     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  67   \end{section}
  68
  69   \begin{section}{Calculus}
  70     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  71     \Rn \rightarrow \Rn$.
  72   \end{section}
  73
  74   \begin{section}{Common}
  75     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
  76     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  77     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
  78     up to seven, for now:
  79     %
  80     \begin{itemize}
  81       \begin{item}
  82         Pair: $\pair{1}{2}$,
  83       \end{item}
  84       \begin{item}
  85         Triple: $\triple{1}{2}{3}$,
  86       \end{item}
  87       \begin{item}
  88         Quadruple: $\quadruple{1}{2}{3}{4}$,
  89       \end{item}
  90       \begin{item}
  91         Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
  92       \end{item}
  93       \begin{item}
  94         Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
  95       \end{item}
  96       \begin{item}
  97         Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
  98       \end{item}
  99     \end{itemize}
 100     %
 101     The factorial of the number $10$ is $\factorial{10}$.
 102
 103     The Cartesian product of two sets $A$ and $B$ is
 104     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
 105     we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
 106     is $\directsum{V}{W}$. Or three things,
 107     $\directsumthree{U}{V}{W}$. How about more things? Like
 108     $\directsummany{k=1}{\infty}{V_{k}} \ne
 109     \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
 110     cartesian products adapt nicely to display equations:
 111     %
 112     \begin{equation*}
 113       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
 114     \end{equation*}
 115     %
 116     Here are a few common tuple spaces that should not have a
 117     superscript when that superscript would be one: $\Nn[1]$,
 118     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
 119     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
 120     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. Likewise we have the $n$-by-$n$
 121     real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
 122     respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
 123     simplify because the ``$n$'' does not indicate the arity of a
 124     Cartesian product in this case.
 125
 126     We also have a few basic set operations, for example the union of
 127     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
 128     course with union comes intersection: $\intersect{A}{B}$,
 129     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
 130     (indexed) union and intersections of things, like
 131     $\unionmany{k=1}{\infty}{A_{k}}$ or
 132     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
 133     is that they do the right thing in a display equation:
 134     %
 135     \begin{equation*}
 136       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
 137     \end{equation*}
 138     %
 139     The powerset of $X$ displays nicely, as $\powerset{X}$. Finally,
 140     we have the four standard types of intervals in $\Rn[1]$,
 141     %
 142     \begin{align*}
 143       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 144       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 145       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 146       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 147     \end{align*}
 148   \end{section}
 149
 150   \begin{section}{Complex}
 151     We sometimes want to conjugate complex numbers like
 152     $\compconj{a+bi} = a - bi$.
 153   \end{section}
 154
 155   \begin{section}{Cone}
 156     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 157     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 158     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 159     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 160     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 161     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 162     \ltcone_{K} y$ with respect to a cone $K$.
 163   \end{section}
 164
 165   \begin{section}{Convex}
 166     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 167     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 168     then its lineality space is $\linspace{K}$, its lineality is
 169     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 170     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 171     proper face, then we write $F \properfaceof K$.
 172   \end{section}
 173
 174   \begin{section}{Euclidean Jordan algebras}
 175     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 176     is $\jp{x}{y}$.
 177   \end{section}
 178
 179   \begin{section}{Font}
 180     We can write things like Carathéodory and Güler and
 181     $\mathbb{R}$. The PostScript Zapf Chancery font is also available
 182     in both upper- and lower-case:
 183     %
 184     \begin{itemize}
 185       \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
 186       \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
 187     \end{itemize}
 188   \end{section}
 189
 190   \begin{section}{Linear algebra}
 191     The absolute value of $x$ is $\abs{x}$, or its norm is
 192     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 193     their tensor product is $\tp{x}{y}$. The Kronecker product of
 194     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 195     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 196     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
 197     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 198     $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
 199     $\rank{L}$.
 200
 201     The span of a set $X$ is $\spanof{X}$, and its codimension is
 202     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 203     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 204     $\Lie{X}$. We can write a column vector $x \coloneqq
 205     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 206     matrix with $\matricize{x}$. To recover the vector, we use
 207     $\vectorize{\matricize{x}}$.
 208
 209     The set of all bounded linear operators from $V$ to $W$ is
 210     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 211     instead.
 212
 213     If you want to solve a system of equations, try Cramer's
 214     rule~\cite{ehrenborg}.
 215
 216     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 217     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 218     writing $\directsumperp{V}{W}$. That operator should survive a
 219     display equation, too, and the weight of the circle should match
 220     that of the usual direct sum operator.
 221     %
 222     \begin{align*}
 223       Z = \directsumperp{V}{W}\\
 224       \oplus \oplusperp \oplus \oplusperp
 225     \end{align*}
 226     %
 227     Its form should also survive in different font sizes...
 228     \Large
 229     \begin{align*}
 230       Z = \directsumperp{V}{W}\\
 231       \oplus \oplusperp \oplus \oplusperp
 232     \end{align*}
 233     \Huge
 234     \begin{align*}
 235       Z = \directsumperp{V}{W}\\
 236       \oplus \oplusperp \oplus \oplusperp
 237     \end{align*}
 238     \normalsize
 239   \end{section}
 240
 241   \begin{section}{Listing}
 242     Here's an interactive SageMath prompt:
 243
 244     \begin{tcblisting}{listing only,
 245                        colback=codebg,
 246                        coltext=codefg,
 247                        listing options={language=sage,style=sage}}
 248     sage: K = Cone([ (1,0), (0,1) ])
 249     sage: K.positive_operator_gens()
 250     [
 251     [1 0]  [0 1]  [0 0]  [0 0]
 252     [0 0], [0 0], [1 0], [0 1]
 253     ]
 254     \end{tcblisting}
 255
 256     However, the smart way to display a SageMath listing is to load it
 257     from an external file (under the ``listings'' subdirectory):
 258
 259     \sagelisting{example}
 260
 261     Keeping the listings in separate files makes it easy for the build
 262     system to test them.
 263   \end{section}
 264
 265   \begin{section}{Miscellaneous}
 266     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 267     = 3$.
 268   \end{section}
 269
 270   \begin{section}{Proof by cases}
 271
 272     \begin{proposition}
 273       There are two cases in the following proof.
 274
 275       \begin{proof}
 276         The result should be self-evident once we have considered the
 277         following two cases.
 278         \begin{pcases}
 279           \begin{case}[first case]
 280             Nothing happens in the first case.
 281           \end{case}
 282           \begin{case}[second case]
 283             The same thing happens in the second case.
 284           \end{case}
 285         \end{pcases}
 286
 287         You see?
 288       \end{proof}
 289     \end{proposition}
 290
 291     Here's another one.
 292
 293     \renewcommand{\baselinestretch}{2}
 294     \begin{proposition}
 295       Cases should display intelligently even when the document is
 296       double-spaced.
 297
 298       \begin{proof}
 299         Here we go again.
 300
 301         \begin{pcases}
 302           \begin{case}[first case]
 303             Nothing happens in the first case.
 304           \end{case}
 305           \begin{case}[second case]
 306             The same thing happens in the second case.
 307           \end{case}
 308         \end{pcases}
 309
 310         Now it's over.
 311       \end{proof}
 312     \end{proposition}
 313     \renewcommand{\baselinestretch}{1}
 314   \end{section}
 315
 316   \begin{section}{Theorems}
 317     \begin{corollary}
 318       The
 319     \end{corollary}
 320
 321     \begin{lemma}
 322       quick
 323     \end{lemma}
 324
 325     \begin{proposition}
 326       brown
 327     \end{proposition}
 328
 329     \begin{theorem}
 330       fox
 331     \end{theorem}
 332
 333     \begin{exercise}
 334       jumps
 335     \end{exercise}
 336
 337     \begin{definition}
 338       quod
 339     \end{definition}
 340
 341     \begin{example}
 342       erat
 343     \end{example}
 344
 345     \begin{remark}
 346       demonstradum.
 347     \end{remark}
 348   \end{section}
 349
 350   \begin{section}{Theorems (starred)}
 351     \begin{corollary*}
 352       The
 353     \end{corollary*}
 354
 355     \begin{lemma*}
 356       quick
 357     \end{lemma*}
 358
 359     \begin{proposition*}
 360       brown
 361     \end{proposition*}
 362
 363     \begin{theorem*}
 364       fox
 365     \end{theorem*}
 366
 367     \begin{exercise*}
 368       jumps
 369     \end{exercise*}
 370
 371     \begin{definition*}
 372       quod
 373     \end{definition*}
 374
 375     \begin{example*}
 376       erat
 377     \end{example*}
 378
 379     \begin{remark*}
 380       demonstradum.
 381     \end{remark*}
 382   \end{section}
 383
 384   \begin{section}{Topology}
 385     The interior of a set $X$ is $\interior{X}$. Its closure is
 386     $\closure{X}$ and its boundary is $\boundary{X}$.
 387   \end{section}
 388
 389   \setlength{\glslistdottedwidth}{.3\linewidth}
 390   \setglossarystyle{listdotted}
 391   \glsaddall
 392   \printnoidxglossaries
 393
 394   \bibliographystyle{mjo}
 395   \bibliography{local-references}
 396
 397   \printindex
 398 \end{document}