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