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