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 $f\of{x}$. We can group terms
  76     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  77     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
  78     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 Cartesian product of two sets $A$ and $B$ is
 104     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
 105     we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
 106     is $\directsum{V}{W}$. Or three things,
 107     $\directsumthree{U}{V}{W}$. How about more things? Like
 108     $\directsummany{k=1}{\infty}{V_{k}} \ne
 109     \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
 110     cartesian products adapt nicely to display equations:
 111     %
 112     \begin{equation*}
 113       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
 114     \end{equation*}
 115     Here are a few common tuple spaces that should not have a
 116     superscript when that superscript would be one: $\Nn[1]$,
 117     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
 118     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
 119     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
 120
 121     We also have a few basic set operations, for example the union of
 122     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
 123     course with union comes intersection: $\intersect{A}{B}$,
 124     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
 125     (indexed) union and intersections of things, like
 126     $\unionmany{k=1}{\infty}{A_{k}}$ or
 127     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
 128     is that they do the right thing in a display equation:
 129     %
 130     \begin{equation*}
 131       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
 132     \end{equation*}
 133
 134     Finally, we have the four standard types of intervals in $\Rn[1]$,
 135     %
 136     \begin{align*}
 137       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 138       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 139       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 140       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 141     \end{align*}
 142   \end{section}
 143
 144   \begin{section}{Complex}
 145     We sometimes want to conjugate complex numbers like
 146     $\compconj{a+bi} = a - bi$.
 147   \end{section}
 148
 149   \begin{section}{Cone}
 150     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 151     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 152     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 153     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 154     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 155     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 156     \ltcone_{K} y$ with respect to a cone $K$.
 157   \end{section}
 158
 159   \begin{section}{Convex}
 160     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 161     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 162     then its lineality space is $\linspace{K}$, its lineality is
 163     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 164     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 165     proper face, then we write $F \properfaceof K$.
 166   \end{section}
 167
 168   \begin{section}{Euclidean Jordan algebras}
 169     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 170     is $\jp{x}{y}$.
 171   \end{section}
 172
 173   \begin{section}{Font}
 174     We can write things like Carathéodory and Güler and $\mathbb{R}$.
 175   \end{section}
 176
 177   \begin{section}{Linear algebra}
 178     The absolute value of $x$ is $\abs{x}$, or its norm is
 179     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 180     their tensor product is $\tp{x}{y}$. The Kronecker product of
 181     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 182     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 183     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
 184     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 185     $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
 186     $\rank{L}$.
 187
 188     The span of a set $X$ is $\spanof{X}$, and its codimension is
 189     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 190     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 191     $\Lie{X}$. We can write a column vector $x \coloneqq
 192     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 193     matrix with $\matricize{x}$. To recover the vector, we use
 194     $\vectorize{\matricize{x}}$.
 195
 196     The set of all bounded linear operators from $V$ to $W$ is
 197     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 198     instead.
 199
 200     If you want to solve a system of equations, try Cramer's
 201     rule~\cite{ehrenborg}.
 202
 203     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 204     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 205     writing $\directsumperp{V}{W}$. That operator should survive a
 206     display equation, too, and the weight of the circle should match
 207     that of the usual direct sum operator.
 208     %
 209     \begin{align*}
 210       Z = \directsumperp{V}{W}\\
 211       \oplus \oplusperp \oplus \oplusperp
 212     \end{align*}
 213     %
 214     Its form should also survive in different font sizes...
 215     \Large
 216     \begin{align*}
 217       Z = \directsumperp{V}{W}\\
 218       \oplus \oplusperp \oplus \oplusperp
 219     \end{align*}
 220     \Huge
 221     \begin{align*}
 222       Z = \directsumperp{V}{W}\\
 223       \oplus \oplusperp \oplus \oplusperp
 224     \end{align*}
 225     \normalsize
 226   \end{section}
 227
 228   \begin{section}{Listing}
 229     Here's an interactive SageMath prompt:
 230
 231     \begin{tcblisting}{listing only,
 232                        colback=codebg,
 233                        coltext=codefg,
 234                        listing options={language=sage,style=sage}}
 235     sage: K = Cone([ (1,0), (0,1) ])
 236     sage: K.positive_operator_gens()
 237     [
 238     [1 0]  [0 1]  [0 0]  [0 0]
 239     [0 0], [0 0], [1 0], [0 1]
 240     ]
 241     \end{tcblisting}
 242
 243     However, the smart way to display a SageMath listing is to load it
 244     from an external file (under the ``listings'' subdirectory):
 245
 246     \sagelisting{example}
 247
 248     Keeping the listings in separate files makes it easy for the build
 249     system to test them.
 250   \end{section}
 251
 252   \begin{section}{Miscellaneous}
 253     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 254     = 3$.
 255   \end{section}
 256
 257   \begin{section}{Proof by cases}
 258
 259     \begin{proposition}
 260       There are two cases in the following proof.
 261
 262       \begin{proof}
 263         The result should be self-evident once we have considered the
 264         following two cases.
 265         \begin{pcases}
 266           \begin{case}[first case]
 267             Nothing happens in the first case.
 268           \end{case}
 269           \begin{case}[second case]
 270             The same thing happens in the second case.
 271           \end{case}
 272         \end{pcases}
 273
 274         You see?
 275       \end{proof}
 276     \end{proposition}
 277
 278     Here's another one.
 279
 280     \renewcommand{\baselinestretch}{2}
 281     \begin{proposition}
 282       Cases should display intelligently even when the document is
 283       double-spaced.
 284
 285       \begin{proof}
 286         Here we go again.
 287
 288         \begin{pcases}
 289           \begin{case}[first case]
 290             Nothing happens in the first case.
 291           \end{case}
 292           \begin{case}[second case]
 293             The same thing happens in the second case.
 294           \end{case}
 295         \end{pcases}
 296
 297         Now it's over.
 298       \end{proof}
 299     \end{proposition}
 300     \renewcommand{\baselinestretch}{1}
 301   \end{section}
 302
 303   \begin{section}{Theorems}
 304     \begin{corollary}
 305       The
 306     \end{corollary}
 307
 308     \begin{lemma}
 309       quick
 310     \end{lemma}
 311
 312     \begin{proposition}
 313       brown
 314     \end{proposition}
 315
 316     \begin{theorem}
 317       fox
 318     \end{theorem}
 319
 320     \begin{exercise}
 321       jumps
 322     \end{exercise}
 323
 324     \begin{definition}
 325       quod
 326     \end{definition}
 327
 328     \begin{example}
 329       erat
 330     \end{example}
 331
 332     \begin{remark}
 333       demonstradum.
 334     \end{remark}
 335   \end{section}
 336
 337   \begin{section}{Theorems (starred)}
 338     \begin{corollary*}
 339       The
 340     \end{corollary*}
 341
 342     \begin{lemma*}
 343       quick
 344     \end{lemma*}
 345
 346     \begin{proposition*}
 347       brown
 348     \end{proposition*}
 349
 350     \begin{theorem*}
 351       fox
 352     \end{theorem*}
 353
 354     \begin{exercise*}
 355       jumps
 356     \end{exercise*}
 357
 358     \begin{definition*}
 359       quod
 360     \end{definition*}
 361
 362     \begin{example*}
 363       erat
 364     \end{example*}
 365
 366     \begin{remark*}
 367       demonstradum.
 368     \end{remark*}
 369   \end{section}
 370
 371   \begin{section}{Topology}
 372     The interior of a set $X$ is $\interior{X}$. Its closure is
 373     $\closure{X}$ and its boundary is $\boundary{X}$.
 374   \end{section}
 375
 376   \setlength{\glslistdottedwidth}{.3\linewidth}
 377   \setglossarystyle{listdotted}
 378   \glsaddall
 379   \printnoidxglossaries
 380
 381   \bibliographystyle{mjo}
 382   \bibliography{local-references}
 383
 384   \printindex
 385 \end{document}