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 $f\of{x}$, and the restriction
  76     of $f$ to a subset $X$ of its domain is $\restrict{f}{X}$. We can
  77     group terms like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c -
  78         d}}$. The tuples go 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 direct sum of $V$ and $W$ is $\directsum{V}{W}$. Or three
 104     things, $\directsumthree{U}{V}{W}$. How about more things? Like
 105     $\directsummany{k=1}{\infty}{V_{k}}$. Those direct sums
 106     adapt nicely to display equations:
 107     %
 108     \begin{equation*}
 109       \directsummany{k=1}{\infty}{V_{k}} \ne \emptyset.
 110     \end{equation*}
 111     %
 112     Here are a few common tuple spaces that should not have a
 113     superscript when that superscript would be one: $\Nn[1]$,
 114     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
 115     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
 116     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. Finally, we have the four standard
 117     types of intervals in $\Rn[1]$,
 118     %
 119     \begin{align*}
 120       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 121       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 122       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 123       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 124     \end{align*}
 125   \end{section}
 126
 127   \begin{section}{Complex}
 128     We sometimes want to conjugate complex numbers like
 129     $\compconj{a+bi} = a - bi$.
 130   \end{section}
 131
 132   \begin{section}{Cone}
 133     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 134     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 135     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 136     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 137     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 138     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 139     \ltcone_{K} y$ with respect to a cone $K$.
 140   \end{section}
 141
 142   \begin{section}{Convex}
 143     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 144     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 145     then its lineality space is $\linspace{K}$, its lineality is
 146     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 147     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 148     proper face, then we write $F \properfaceof K$.
 149   \end{section}
 150
 151   \begin{section}{Euclidean Jordan algebras}
 152     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 153     is $\jp{x}{y}$.
 154   \end{section}
 155
 156   \begin{section}{Font}
 157     We can write things like Carathéodory and Güler and
 158     $\mathbb{R}$. The PostScript Zapf Chancery font is also available
 159     in both upper- and lower-case:
 160     %
 161     \begin{itemize}
 162       \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
 163       \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
 164     \end{itemize}
 165   \end{section}
 166
 167   \begin{section}{Linear algebra}
 168     The absolute value of $x$ is $\abs{x}$, or its norm is
 169     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 170     their tensor product is $\tp{x}{y}$. The Kronecker product of
 171     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 172     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 173     $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the
 174     set of its eigenvalues---is $\spectrum{L}$. Another matrix-specific
 175     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 176     $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
 177     $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
 178     real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
 179     respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
 180     simplify because the ``$n$'' does not indicate the arity of a
 181     Cartesian product in this case.
 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}{Proof by cases}
 248
 249     \begin{proposition}
 250       There are two cases in the following proof.
 251
 252       \begin{proof}
 253         The result should be self-evident once we have considered the
 254         following two cases.
 255         \begin{pcases}
 256           \begin{case}[first case]
 257             Nothing happens in the first case.
 258           \end{case}
 259           \begin{case}[second case]
 260             The same thing happens in the second case.
 261           \end{case}
 262         \end{pcases}
 263
 264         You see?
 265       \end{proof}
 266     \end{proposition}
 267
 268     Here's another one.
 269
 270     \renewcommand{\baselinestretch}{2}
 271     \begin{proposition}
 272       Cases should display intelligently even when the document is
 273       double-spaced.
 274
 275       \begin{proof}
 276         Here we go again.
 277
 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         Now it's over.
 288       \end{proof}
 289     \end{proposition}
 290     \renewcommand{\baselinestretch}{1}
 291   \end{section}
 292
 293   \begin{section}{Set theory}
 294     Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The
 295     cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} =
 296     3$, and its powerset is $\powerset{X}$.
 297
 298     We also have a few basic set operations, for example the union of
 299     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
 300     course with union comes intersection: $\intersect{A}{B}$,
 301     $\intersectthree{A}{B}{C}$. The Cartesian product of two sets $A$
 302     and $B$ is there too: $\cartprod{A}{B}$. If we take the product
 303     with $C$ as well, then we obtain $\cartprodthree{A}{B}{C}$.
 304
 305     We can also take an arbitrary (indexed) union, intersection, or
 306     Cartesian product of things, like
 307     $\unionmany{k=1}{\infty}{A_{k}}$,
 308     $\intersectmany{k=1}{\infty}{B_{k}}$, or
 309     $\cartprodmany{k=1}{\infty}{C_{k}}$. The best part about those is
 310     that they do the right thing in a display equation:
 311     %
 312     \begin{equation*}
 313       \unionmany{k=1}{\infty}{A_{k}}
 314       \ne
 315       \intersectmany{k=1}{\infty}{B_{k}}
 316       \ne
 317       \cartprodmany{k=1}{\infty}{C_{k}}.
 318     \end{equation*}
 319     %
 320   \end{section}
 321
 322   \begin{section}{Theorems}
 323     \begin{corollary}
 324       The
 325     \end{corollary}
 326
 327     \begin{lemma}
 328       quick
 329     \end{lemma}
 330
 331     \begin{proposition}
 332       brown
 333     \end{proposition}
 334
 335     \begin{theorem}
 336       fox
 337     \end{theorem}
 338
 339     \begin{exercise}
 340       jumps
 341     \end{exercise}
 342
 343     \begin{definition}
 344       quod
 345     \end{definition}
 346
 347     \begin{example}
 348       erat
 349     \end{example}
 350
 351     \begin{remark}
 352       demonstradum.
 353     \end{remark}
 354   \end{section}
 355
 356   \begin{section}{Theorems (starred)}
 357     \begin{corollary*}
 358       The
 359     \end{corollary*}
 360
 361     \begin{lemma*}
 362       quick
 363     \end{lemma*}
 364
 365     \begin{proposition*}
 366       brown
 367     \end{proposition*}
 368
 369     \begin{theorem*}
 370       fox
 371     \end{theorem*}
 372
 373     \begin{exercise*}
 374       jumps
 375     \end{exercise*}
 376
 377     \begin{definition*}
 378       quod
 379     \end{definition*}
 380
 381     \begin{example*}
 382       erat
 383     \end{example*}
 384
 385     \begin{remark*}
 386       demonstradum.
 387     \end{remark*}
 388   \end{section}
 389
 390   \begin{section}{Topology}
 391     The interior of a set $X$ is $\interior{X}$. Its closure is
 392     $\closure{X}$ and its boundary is $\boundary{X}$.
 393   \end{section}
 394
 395   \setlength{\glslistdottedwidth}{.3\linewidth}
 396   \setglossarystyle{listdotted}
 397   \glsaddall
 398   \printnoidxglossaries
 399
 400   \bibliographystyle{mjo}
 401   \bibliography{local-references}
 402
 403   \printindex
 404 \end{document}