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 commutative ring\index{commutative ring}, then
  30     $\polyring{R}{X,Y,Z}$ is a multivariate polynomial ring with
  31     indeterminates $X$, $Y$, and $Z$, and coefficients in $R$. If $R$
  32     is a moreover an integral domain, then its fraction field is
  33     $\Frac{R}$. If $x,y,z \in R$, then $\ideal{\set{x,y,z}}$ is the
  34     ideal generated by $\set{x,y,z}$, which is defined to be the
  35     smallest ideal in $R$ containing that set. Likewise, if we are in
  36     an algebra $\mathcal{A}$ 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
  40     If $R$ has a multiplicative identity (that is, a unit) element,
  41     then that element is denoted by $\unit{R}$.
  42   \end{section}
  43
  44   \begin{section}{Algorithm}
  45     An example of an algorithm (bogosort) environment.
  46
  47     \begin{algorithm}[H]
  48       \caption{Sort a list of numbers}
  49       \begin{algorithmic}
  50         \Require{A list of numbers $L$}
  51         \Ensure{A new, sorted copy $M$ of the list $L$}
  52
  53         \State{$M \gets L$}
  54
  55         \While{$M$ is not sorted}
  56           \State{Rearrange $M$ randomly}
  57         \EndWhile{}
  58
  59         \Return{$M$}
  60       \end{algorithmic}
  61     \end{algorithm}
  62   \end{section}
  63
  64   \begin{section}{Arrow}
  65     The constant function that always returns $a$ is $\const{a}$. The
  66     identity operator on $V$ is $\identity{V}$. The composition of $f$
  67     and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  68     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  69     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  70   \end{section}
  71
  72   \begin{section}{Calculus}
  73     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  74     \Rn \rightarrow \Rn$.
  75   \end{section}
  76
  77   \begin{section}{Common}
  78     The function $f$ applied to $x$ is $f\of{x}$, and the restriction
  79     of $f$ to a subset $X$ of its domain is $\restrict{f}{X}$. We can
  80     group terms like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c -
  81         d}}$. The tuples go up to seven, for now:
  82     %
  83     \begin{itemize}
  84       \begin{item}
  85         Pair: $\pair{1}{2}$,
  86       \end{item}
  87       \begin{item}
  88         Triple: $\triple{1}{2}{3}$,
  89       \end{item}
  90       \begin{item}
  91         Quadruple: $\quadruple{1}{2}{3}{4}$,
  92       \end{item}
  93       \begin{item}
  94         Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
  95       \end{item}
  96       \begin{item}
  97         Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
  98       \end{item}
  99       \begin{item}
 100         Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
 101       \end{item}
 102     \end{itemize}
 103     %
 104     The factorial of the number $10$ is $\factorial{10}$, and the
 105     least common multiple of $4$ and $6$ is $\lcm{\set{4,6}} =
 106     12$.
 107
 108     The direct sum of $V$ and $W$ is $\directsum{V}{W}$. Or three
 109     things, $\directsumthree{U}{V}{W}$. How about more things? Like
 110     $\directsummany{k=1}{\infty}{V_{k}}$. Those direct sums
 111     adapt nicely to display equations:
 112     %
 113     \begin{equation*}
 114       \directsummany{k=1}{\infty}{V_{k}} \ne \emptyset.
 115     \end{equation*}
 116     %
 117     Here are a few common tuple spaces that should not have a
 118     superscript when that superscript would be one: $\Nn[1]$,
 119     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
 120     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
 121     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. The symbols $\Fn[1]$, $\Fn[2]$,
 122     et cetera, are available for use with a generic field.
 123
 124     Finally, we have the four standard types of intervals in $\Rn[1]$,
 125     %
 126     \begin{align*}
 127       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 128       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 129       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 130       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 131     \end{align*}
 132   \end{section}
 133
 134   \begin{section}{Complex}
 135     We sometimes want to conjugate complex numbers like
 136     $\compconj{a+bi} = a - bi$.
 137   \end{section}
 138
 139   \begin{section}{Cone}
 140     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 141     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 142     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 143     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 144     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 145     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 146     \ltcone_{K} y$ with respect to a cone $K$.
 147   \end{section}
 148
 149   \begin{section}{Convex}
 150     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 151     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 152     then its lineality space is $\linspace{K}$, its lineality is
 153     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 154     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 155     proper face, then we write $F \properfaceof K$.
 156   \end{section}
 157
 158   \begin{section}{Euclidean Jordan algebras}
 159     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 160     is $\jp{x}{y}$.
 161   \end{section}
 162
 163   \begin{section}{Font}
 164     We can write things like Carathéodory and Güler and
 165     $\mathbb{R}$. The PostScript Zapf Chancery font is also available
 166     in both upper- and lower-case:
 167     %
 168     \begin{itemize}
 169       \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
 170       \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
 171     \end{itemize}
 172   \end{section}
 173
 174   \begin{section}{Linear algebra}
 175     The absolute value of $x$ is $\abs{x}$, or its norm is
 176     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 177     their tensor product is $\tp{x}{y}$. The Kronecker product of
 178     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 179     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 180     $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the
 181     set of its eigenvalues---is $\spectrum{L}$. Another
 182     matrix-specific concept is the Moore-Penrose pseudoinverse of $L$,
 183     denoted by $\pseudoinverse{L}$. Finally, the rank of a matrix $L$
 184     is $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
 185     real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
 186     respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
 187     simplify because the ``$n$'' does not indicate the arity of a
 188     Cartesian product in this case. A handy way to represent the
 189     matrix $A \in \Rn[n \times n]$ whose only non-zero entries are on
 190     the diagonal is $\diag{\colvec{A_{11},A_{22},\ldots,A_{nn}}}$.
 191
 192     The span of a set $X$ is $\spanof{X}$, and its codimension is
 193     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 194     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 195     $\Lie{X}$. We can write a column vector $x \coloneqq
 196     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 197     matrix with $\matricize{x}$. To recover the vector, we use
 198     $\vectorize{\matricize{x}}$.
 199
 200     The set of all bounded linear operators from $V$ to $W$ is
 201     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 202     instead.
 203
 204     If you want to solve a system of equations, try Cramer's
 205     rule~\cite{ehrenborg}.
 206
 207     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 208     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 209     writing $\directsumperp{V}{W}$. That operator should survive a
 210     display equation, too, and the weight of the circle should match
 211     that of the usual direct sum operator.
 212     %
 213     \begin{align*}
 214       Z = \directsumperp{V}{W}\\
 215       \oplus \oplusperp \oplus \oplusperp
 216     \end{align*}
 217     %
 218     Its form should also survive in different font sizes\ldots
 219     \Large
 220     \begin{align*}
 221       Z = \directsumperp{V}{W}\\
 222       \oplus \oplusperp \oplus \oplusperp
 223     \end{align*}
 224     \Huge
 225     \begin{align*}
 226       Z = \directsumperp{V}{W}\\
 227       \oplus \oplusperp \oplus \oplusperp
 228     \end{align*}
 229     \normalsize
 230   \end{section}
 231
 232   \begin{section}{Listing}
 233     Here's an interactive SageMath prompt:
 234
 235     \begin{tcblisting}{listing only,
 236                        colback=codebg,
 237                        coltext=codefg,
 238                        listing options={language=sage,style=sage}}
 239     sage: K = Cone([ (1,0), (0,1) ])
 240     sage: K.positive_operator_gens()
 241     [
 242     [1 0]  [0 1]  [0 0]  [0 0]
 243     [0 0], [0 0], [1 0], [0 1]
 244     ]
 245     \end{tcblisting}
 246
 247     However, the smart way to display a SageMath listing is to load it
 248     from an external file (under the ``listings'' subdirectory):
 249
 250     \sagelisting{example}
 251
 252     Keeping the listings in separate files makes it easy for the build
 253     system to test them.
 254   \end{section}
 255
 256   \begin{section}{Proof by cases}
 257
 258     \begin{proposition}
 259       There are two cases in the following proof.
 260
 261       \begin{proof}
 262         The result should be self-evident once we have considered the
 263         following two cases.
 264         \begin{pcases}
 265           \begin{case}[first case]
 266             Nothing happens in the first case.
 267           \end{case}
 268           \begin{case}[second case]
 269             The same thing happens in the second case.
 270           \end{case}
 271         \end{pcases}
 272
 273         You see?
 274       \end{proof}
 275     \end{proposition}
 276
 277     Here's another one.
 278
 279     \renewcommand{\baselinestretch}{2}
 280     \begin{proposition}
 281       Cases should display intelligently even when the document is
 282       double-spaced.
 283
 284       \begin{proof}
 285         Here we go again.
 286
 287         \begin{pcases}
 288           \begin{case}[first case]
 289             Nothing happens in the first case.
 290           \end{case}
 291           \begin{case}[second case]
 292             The same thing happens in the second case.
 293           \end{case}
 294         \end{pcases}
 295
 296         Now it's over.
 297       \end{proof}
 298     \end{proposition}
 299     \renewcommand{\baselinestretch}{1}
 300   \end{section}
 301
 302   \begin{section}{Set theory}
 303     Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The
 304     cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} =
 305     3$, and its powerset is $\powerset{X}$.
 306
 307     We also have a few basic set operations, for example the union of
 308     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
 309     course with union comes intersection: $\intersect{A}{B}$,
 310     $\intersectthree{A}{B}{C}$. The Cartesian product of two sets $A$
 311     and $B$ is there too: $\cartprod{A}{B}$. If we take the product
 312     with $C$ as well, then we obtain $\cartprodthree{A}{B}{C}$.
 313
 314     We can also take an arbitrary (indexed) union, intersection, or
 315     Cartesian product of things, like
 316     $\unionmany{k=1}{\infty}{A_{k}}$,
 317     $\intersectmany{k=1}{\infty}{B_{k}}$, or
 318     $\cartprodmany{k=1}{\infty}{C_{k}}$. The best part about those is
 319     that they do the right thing in a display equation:
 320     %
 321     \begin{equation*}
 322       \unionmany{k=1}{\infty}{A_{k}}
 323       \ne
 324       \intersectmany{k=1}{\infty}{B_{k}}
 325       \ne
 326       \cartprodmany{k=1}{\infty}{C_{k}}.
 327     \end{equation*}
 328     %
 329   \end{section}
 330
 331   \begin{section}{Theorems}
 332     \begin{corollary}
 333       The
 334     \end{corollary}
 335
 336     \begin{lemma}
 337       quick
 338     \end{lemma}
 339
 340     \begin{proposition}
 341       brown
 342     \end{proposition}
 343
 344     \begin{theorem}
 345       fox
 346     \end{theorem}
 347
 348     \begin{exercise}
 349       jumps
 350     \end{exercise}
 351
 352     \begin{definition}
 353       quod
 354     \end{definition}
 355
 356     \begin{example}
 357       erat
 358     \end{example}
 359
 360     \begin{remark}
 361       demonstradum.
 362     \end{remark}
 363   \end{section}
 364
 365   \begin{section}{Theorems (starred)}
 366     \begin{corollary*}
 367       The
 368     \end{corollary*}
 369
 370     \begin{lemma*}
 371       quick
 372     \end{lemma*}
 373
 374     \begin{proposition*}
 375       brown
 376     \end{proposition*}
 377
 378     \begin{theorem*}
 379       fox
 380     \end{theorem*}
 381
 382     \begin{exercise*}
 383       jumps
 384     \end{exercise*}
 385
 386     \begin{definition*}
 387       quod
 388     \end{definition*}
 389
 390     \begin{example*}
 391       erat
 392     \end{example*}
 393
 394     \begin{remark*}
 395       demonstradum.
 396     \end{remark*}
 397   \end{section}
 398
 399   \begin{section}{Topology}
 400     The interior of a set $X$ is $\interior{X}$. Its closure is
 401     $\closure{X}$ and its boundary is $\boundary{X}$.
 402   \end{section}
 403
 404   \setlength{\glslistdottedwidth}{.3\linewidth}
 405   \setglossarystyle{listdotted}
 406   \glsaddall{}
 407   \printnoidxglossaries{}
 408
 409   \bibliographystyle{mjo}
 410   \bibliography{local-references}
 411
 412   \printindex
 413 \end{document}