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