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     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. Finally, we have the four standard
 122     types of intervals in $\Rn[1]$,
 123     %
 124     \begin{align*}
 125       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 126       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 127       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 128       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 129     \end{align*}
 130   \end{section}
 131
 132   \begin{section}{Complex}
 133     We sometimes want to conjugate complex numbers like
 134     $\compconj{a+bi} = a - bi$.
 135   \end{section}
 136
 137   \begin{section}{Cone}
 138     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 139     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 140     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 141     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 142     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 143     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 144     \ltcone_{K} y$ with respect to a cone $K$.
 145   \end{section}
 146
 147   \begin{section}{Convex}
 148     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 149     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 150     then its lineality space is $\linspace{K}$, its lineality is
 151     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 152     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 153     proper face, then we write $F \properfaceof K$.
 154   \end{section}
 155
 156   \begin{section}{Euclidean Jordan algebras}
 157     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 158     is $\jp{x}{y}$.
 159   \end{section}
 160
 161   \begin{section}{Font}
 162     We can write things like Carathéodory and Güler and
 163     $\mathbb{R}$. The PostScript Zapf Chancery font is also available
 164     in both upper- and lower-case:
 165     %
 166     \begin{itemize}
 167       \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
 168       \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
 169     \end{itemize}
 170   \end{section}
 171
 172   \begin{section}{Linear algebra}
 173     The absolute value of $x$ is $\abs{x}$, or its norm is
 174     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 175     their tensor product is $\tp{x}{y}$. The Kronecker product of
 176     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 177     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 178     $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the
 179     set of its eigenvalues---is $\spectrum{L}$. Another
 180     matrix-specific concept is the Moore-Penrose pseudoinverse of $L$,
 181     denoted by $\pseudoinverse{L}$. Finally, the rank of a matrix $L$
 182     is $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
 183     real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
 184     respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
 185     simplify because the ``$n$'' does not indicate the arity of a
 186     Cartesian product in this case. A handy way to represent the
 187     matrix $A \in \Rn[n \times n]$ whose only non-zero entries are on
 188     the diagonal is $\diag{\colvec{A_{11},A_{22},\ldots,A_{nn}}}$.
 189
 190     The span of a set $X$ is $\spanof{X}$, and its codimension is
 191     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 192     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 193     $\Lie{X}$. We can write a column vector $x \coloneqq
 194     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 195     matrix with $\matricize{x}$. To recover the vector, we use
 196     $\vectorize{\matricize{x}}$.
 197
 198     The set of all bounded linear operators from $V$ to $W$ is
 199     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 200     instead.
 201
 202     If you want to solve a system of equations, try Cramer's
 203     rule~\cite{ehrenborg}.
 204
 205     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 206     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 207     writing $\directsumperp{V}{W}$. That operator should survive a
 208     display equation, too, and the weight of the circle should match
 209     that of the usual direct sum operator.
 210     %
 211     \begin{align*}
 212       Z = \directsumperp{V}{W}\\
 213       \oplus \oplusperp \oplus \oplusperp
 214     \end{align*}
 215     %
 216     Its form should also survive in different font sizes\ldots
 217     \Large
 218     \begin{align*}
 219       Z = \directsumperp{V}{W}\\
 220       \oplus \oplusperp \oplus \oplusperp
 221     \end{align*}
 222     \Huge
 223     \begin{align*}
 224       Z = \directsumperp{V}{W}\\
 225       \oplus \oplusperp \oplus \oplusperp
 226     \end{align*}
 227     \normalsize
 228   \end{section}
 229
 230   \begin{section}{Listing}
 231     Here's an interactive SageMath prompt:
 232
 233     \begin{tcblisting}{listing only,
 234                        colback=codebg,
 235                        coltext=codefg,
 236                        listing options={language=sage,style=sage}}
 237     sage: K = Cone([ (1,0), (0,1) ])
 238     sage: K.positive_operator_gens()
 239     [
 240     [1 0]  [0 1]  [0 0]  [0 0]
 241     [0 0], [0 0], [1 0], [0 1]
 242     ]
 243     \end{tcblisting}
 244
 245     However, the smart way to display a SageMath listing is to load it
 246     from an external file (under the ``listings'' subdirectory):
 247
 248     \sagelisting{example}
 249
 250     Keeping the listings in separate files makes it easy for the build
 251     system to test them.
 252   \end{section}
 253
 254   \begin{section}{Proof by cases}
 255
 256     \begin{proposition}
 257       There are two cases in the following proof.
 258
 259       \begin{proof}
 260         The result should be self-evident once we have considered the
 261         following two cases.
 262         \begin{pcases}
 263           \begin{case}[first case]
 264             Nothing happens in the first case.
 265           \end{case}
 266           \begin{case}[second case]
 267             The same thing happens in the second case.
 268           \end{case}
 269         \end{pcases}
 270
 271         You see?
 272       \end{proof}
 273     \end{proposition}
 274
 275     Here's another one.
 276
 277     \renewcommand{\baselinestretch}{2}
 278     \begin{proposition}
 279       Cases should display intelligently even when the document is
 280       double-spaced.
 281
 282       \begin{proof}
 283         Here we go again.
 284
 285         \begin{pcases}
 286           \begin{case}[first case]
 287             Nothing happens in the first case.
 288           \end{case}
 289           \begin{case}[second case]
 290             The same thing happens in the second case.
 291           \end{case}
 292         \end{pcases}
 293
 294         Now it's over.
 295       \end{proof}
 296     \end{proposition}
 297     \renewcommand{\baselinestretch}{1}
 298   \end{section}
 299
 300   \begin{section}{Set theory}
 301     Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The
 302     cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} =
 303     3$, and its powerset is $\powerset{X}$.
 304
 305     We also have a few basic set operations, for example the union of
 306     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
 307     course with union comes intersection: $\intersect{A}{B}$,
 308     $\intersectthree{A}{B}{C}$. The Cartesian product of two sets $A$
 309     and $B$ is there too: $\cartprod{A}{B}$. If we take the product
 310     with $C$ as well, then we obtain $\cartprodthree{A}{B}{C}$.
 311
 312     We can also take an arbitrary (indexed) union, intersection, or
 313     Cartesian product of things, like
 314     $\unionmany{k=1}{\infty}{A_{k}}$,
 315     $\intersectmany{k=1}{\infty}{B_{k}}$, or
 316     $\cartprodmany{k=1}{\infty}{C_{k}}$. The best part about those is
 317     that they do the right thing in a display equation:
 318     %
 319     \begin{equation*}
 320       \unionmany{k=1}{\infty}{A_{k}}
 321       \ne
 322       \intersectmany{k=1}{\infty}{B_{k}}
 323       \ne
 324       \cartprodmany{k=1}{\infty}{C_{k}}.
 325     \end{equation*}
 326     %
 327   \end{section}
 328
 329   \begin{section}{Theorems}
 330     \begin{corollary}
 331       The
 332     \end{corollary}
 333
 334     \begin{lemma}
 335       quick
 336     \end{lemma}
 337
 338     \begin{proposition}
 339       brown
 340     \end{proposition}
 341
 342     \begin{theorem}
 343       fox
 344     \end{theorem}
 345
 346     \begin{exercise}
 347       jumps
 348     \end{exercise}
 349
 350     \begin{definition}
 351       quod
 352     \end{definition}
 353
 354     \begin{example}
 355       erat
 356     \end{example}
 357
 358     \begin{remark}
 359       demonstradum.
 360     \end{remark}
 361   \end{section}
 362
 363   \begin{section}{Theorems (starred)}
 364     \begin{corollary*}
 365       The
 366     \end{corollary*}
 367
 368     \begin{lemma*}
 369       quick
 370     \end{lemma*}
 371
 372     \begin{proposition*}
 373       brown
 374     \end{proposition*}
 375
 376     \begin{theorem*}
 377       fox
 378     \end{theorem*}
 379
 380     \begin{exercise*}
 381       jumps
 382     \end{exercise*}
 383
 384     \begin{definition*}
 385       quod
 386     \end{definition*}
 387
 388     \begin{example*}
 389       erat
 390     \end{example*}
 391
 392     \begin{remark*}
 393       demonstradum.
 394     \end{remark*}
 395   \end{section}
 396
 397   \begin{section}{Topology}
 398     The interior of a set $X$ is $\interior{X}$. Its closure is
 399     $\closure{X}$ and its boundary is $\boundary{X}$.
 400   \end{section}
 401
 402   \setlength{\glslistdottedwidth}{.3\linewidth}
 403   \setglossarystyle{listdotted}
 404   \glsaddall{}
 405   \printnoidxglossaries{}
 406
 407   \bibliographystyle{mjo}
 408   \bibliography{local-references}
 409
 410   \printindex
 411 \end{document}