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