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 $\unit{R}$. Its additive identity
  42     element is $\zero{R}$.
  43   \end{section}
  44
  45   \begin{section}{Algorithm}
  46     An example of an algorithm (bogosort) environment.
  47
  48     \begin{algorithm}[H]
  49       \caption{Sort a list of numbers}
  50       \begin{algorithmic}
  51         \Require{A list of numbers $L$}
  52         \Ensure{A new, sorted copy $M$ of the list $L$}
  53
  54         \State{$M \gets L$}
  55
  56         \While{$M$ is not sorted}
  57           \State{Rearrange $M$ randomly}
  58         \EndWhile{}
  59
  60         \State{\Return{$M$}}
  61       \end{algorithmic}
  62     \end{algorithm}
  63   \end{section}
  64
  65   \begin{section}{Arrow}
  66     The constant function that always returns $a$ is $\const{a}$. The
  67     identity operator on $V$ is $\identity{V}$. The composition of $f$
  68     and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  69     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  70     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  71   \end{section}
  72
  73   \begin{section}{Calculus}
  74     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  75     \Rn \rightarrow \Rn$.
  76   \end{section}
  77
  78   \begin{section}{Common}
  79     The function $f$ applied to $x$ is $f\of{x}$, and the restriction
  80     of $f$ to a subset $X$ of its domain is $\restrict{f}{X}$. We can
  81     group terms like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c -
  82         d}}$. The tuples go up to seven, for now, and then we give up
  83     and use the general construct:
  84     %
  85     \begin{itemize}
  86       \begin{item}
  87         Pair: $\pair{1}{2}$,
  88       \end{item}
  89       \begin{item}
  90         Triple: $\triple{1}{2}{3}$,
  91       \end{item}
  92       \begin{item}
  93         Quadruple: $\quadruple{1}{2}{3}{4}$,
  94       \end{item}
  95       \begin{item}
  96         Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
  97       \end{item}
  98       \begin{item}
  99         Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
 100       \end{item}
 101       \begin{item}
 102         Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
 103       \end{item}
 104       \begin{item}
 105         Tuple: $\tuple{1,2,\ldots,8675309}$.
 106       \end{item}
 107     \end{itemize}
 108     %
 109     The factorial of the number $10$ is $\factorial{10}$, and the
 110     least common multiple of $4$ and $6$ is $\lcm{\set{4,6}} =
 111     12$.
 112
 113     The direct sum of $V$ and $W$ is $\directsum{V}{W}$. Or three
 114     things, $\directsumthree{U}{V}{W}$. How about more things? Like
 115     $\directsummany{k=1}{\infty}{V_{k}}$. Those direct sums
 116     adapt nicely to display equations:
 117     %
 118     \begin{equation*}
 119       \directsummany{k=1}{\infty}{V_{k}} \ne \emptyset.
 120     \end{equation*}
 121     %
 122     Here are a few common tuple spaces that should not have a
 123     superscript when that superscript would be one: $\Nn[1]$,
 124     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
 125     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
 126     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. The symbols $\Fn[1]$, $\Fn[2]$,
 127     et cetera, are available for use with a generic field.
 128
 129     Finally, we have the four standard types of intervals in $\Rn[1]$,
 130     %
 131     \begin{align*}
 132       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 133       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 134       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 135       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 136     \end{align*}
 137   \end{section}
 138
 139   \begin{section}{Complex}
 140     We sometimes want to conjugate complex numbers like
 141     $\compconj{a+bi} = a - bi$.
 142   \end{section}
 143
 144   \begin{section}{Cone}
 145     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 146     are $\Rnplus$, $\Rnplusplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.
 147     If cones $K_{1}$ and $K_{2}$ are given, we can define
 148     $\posops{K_{1}}$, $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$,
 149     $\Zof{K_{1}}$, $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can
 150     also define $x \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K}
 151     y$, and $x \ltcone_{K} y$ with respect to a cone $K$.
 152   \end{section}
 153
 154   \begin{section}{Convex}
 155     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 156     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 157     then its lineality space is $\linspace{K}$, its lineality is
 158     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 159     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 160     proper face, then we write $F \properfaceof K$.
 161   \end{section}
 162
 163   \begin{section}{Euclidean Jordan algebras}
 164     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 165     is $\jp{x}{y}$.
 166   \end{section}
 167
 168   \begin{section}{Font}
 169     We can write things like Carathéodory and Güler and
 170     $\mathbb{R}$. The PostScript Zapf Chancery font is also available
 171     in both upper- and lower-case:
 172     %
 173     \begin{itemize}
 174       \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
 175       \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
 176     \end{itemize}
 177   \end{section}
 178
 179   \begin{section}{Hurwitz}
 180     Here lies the Hurwitz algebras, like the quaternions
 181     $\quaternions$ and octonions $\octonions$.
 182   \end{section}
 183
 184   \begin{section}{Linear algebra}
 185     The absolute value of $x$ is $\abs{x}$, or its norm is
 186     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 187     their tensor product is $\tp{x}{y}$. The Kronecker product of
 188     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 189     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 190     $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the
 191     set of its eigenvalues---is $\spectrum{L}$. Another
 192     matrix-specific concept is the Moore-Penrose pseudoinverse of $L$,
 193     denoted by $\pseudoinverse{L}$. Finally, the rank of a matrix $L$
 194     is $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
 195     real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
 196     respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
 197     simplify because the ``$n$'' does not indicate the arity of a
 198     Cartesian product in this case. A handy way to represent the
 199     matrix $A \in \Rn[n \times n]$ whose only non-zero entries are on
 200     the diagonal is $\diag{\colvec{A_{11},A_{22},\ldots,A_{nn}}}$.
 201
 202     The span of a set $X$ is $\spanof{X}$, and its codimension is
 203     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 204     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 205     $\Lie{X}$. We can write a column vector $x \coloneqq
 206     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 207     matrix with $\matricize{x}$. To recover the vector, we use
 208     $\vectorize{\matricize{x}}$.
 209
 210     The set of all bounded linear operators from $V$ to $W$ is
 211     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 212     instead.
 213
 214     If you want to solve a system of equations, try Cramer's
 215     rule~\cite{ehrenborg}. Or at least the reduced row-echelon form of
 216     the matrix, $\rref{A}$.
 217
 218     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 219     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 220     writing $\directsumperp{V}{W}$. That operator should survive a
 221     display equation, too, and the weight of the circle should match
 222     that of the usual direct sum operator.
 223     %
 224     \begin{align*}
 225       Z = \directsumperp{V}{W}\\
 226       \oplus \oplusperp \oplus \oplusperp
 227     \end{align*}
 228     %
 229     Its form should also survive in different font sizes\ldots
 230     \Large
 231     \begin{align*}
 232       Z = \directsumperp{V}{W}\\
 233       \oplus \oplusperp \oplus \oplusperp
 234     \end{align*}
 235     \Huge
 236     \begin{align*}
 237       Z = \directsumperp{V}{W}\\
 238       \oplus \oplusperp \oplus \oplusperp
 239     \end{align*}
 240     \normalsize
 241   \end{section}
 242
 243   \begin{section}{Listing}
 244     Here's an interactive SageMath prompt:
 245
 246     \begin{tcblisting}{listing only,
 247                        colback=codebg,
 248                        coltext=codefg,
 249                        listing options={language=sage,style=sage}}
 250     sage: K = Cone([ (1,0), (0,1) ])
 251     sage: K.positive_operator_gens()
 252     [
 253     [1 0]  [0 1]  [0 0]  [0 0]
 254     [0 0], [0 0], [1 0], [0 1]
 255     ]
 256     \end{tcblisting}
 257
 258     However, the smart way to display a SageMath listing is to load it
 259     from an external file (under the ``listings'' subdirectory):
 260
 261     \sagelisting{example}
 262
 263     Keeping the listings in separate files makes it easy for the build
 264     system to test them.
 265   \end{section}
 266
 267   \begin{section}{Proof by cases}
 268
 269     \begin{proposition}
 270       There are two cases in the following proof.
 271
 272       \begin{proof}
 273         The result should be self-evident once we have considered the
 274         following two cases.
 275         \begin{pcases}
 276           \begin{case}[first case]
 277             Nothing happens in the first case.
 278           \end{case}
 279           \begin{case}[second case]
 280             The same thing happens in the second case.
 281           \end{case}
 282         \end{pcases}
 283
 284         You see?
 285       \end{proof}
 286     \end{proposition}
 287
 288     Here's another one.
 289
 290     \renewcommand{\baselinestretch}{2}
 291     \begin{proposition}
 292       Cases should display intelligently even when the document is
 293       double-spaced.
 294
 295       \begin{proof}
 296         Here we go again.
 297
 298         \begin{pcases}
 299           \begin{case}[first case]
 300             Nothing happens in the first case.
 301           \end{case}
 302           \begin{case}[second case]
 303             The same thing happens in the second case.
 304           \end{case}
 305         \end{pcases}
 306
 307         Now it's over.
 308       \end{proof}
 309     \end{proposition}
 310     \renewcommand{\baselinestretch}{1}
 311   \end{section}
 312
 313   \begin{section}{Set theory}
 314     Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The
 315     cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} =
 316     3$, and its powerset is $\powerset{X}$.
 317
 318     We also have a few basic set operations, for example the union of
 319     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
 320     course with union comes intersection: $\intersect{A}{B}$,
 321     $\intersectthree{A}{B}{C}$. The Cartesian product of two sets $A$
 322     and $B$ is there too: $\cartprod{A}{B}$. If we take the product
 323     with $C$ as well, then we obtain $\cartprodthree{A}{B}{C}$.
 324
 325     We can also take an arbitrary (indexed) union, intersection, or
 326     Cartesian product of things, like
 327     $\unionmany{k=1}{\infty}{A_{k}}$,
 328     $\intersectmany{k=1}{\infty}{B_{k}}$, or
 329     $\cartprodmany{k=1}{\infty}{C_{k}}$. The best part about those is
 330     that they do the right thing in a display equation:
 331     %
 332     \begin{equation*}
 333       \unionmany{k=1}{\infty}{A_{k}}
 334       \ne
 335       \intersectmany{k=1}{\infty}{B_{k}}
 336       \ne
 337       \cartprodmany{k=1}{\infty}{C_{k}}.
 338     \end{equation*}
 339     %
 340   \end{section}
 341
 342   \begin{section}{Theorems}
 343     \begin{corollary}
 344       The
 345     \end{corollary}
 346
 347     \begin{lemma}
 348       quick
 349     \end{lemma}
 350
 351     \begin{proposition}
 352       brown
 353     \end{proposition}
 354
 355     \begin{theorem}
 356       fox
 357     \end{theorem}
 358
 359     \begin{exercise}
 360       jumps
 361     \end{exercise}
 362
 363     \begin{definition}
 364       quod
 365     \end{definition}
 366
 367     \begin{example}
 368       erat
 369     \end{example}
 370
 371     \begin{remark}
 372       demonstradum.
 373     \end{remark}
 374   \end{section}
 375
 376   \begin{section}{Theorems (starred)}
 377     \begin{corollary*}
 378       The
 379     \end{corollary*}
 380
 381     \begin{lemma*}
 382       quick
 383     \end{lemma*}
 384
 385     \begin{proposition*}
 386       brown
 387     \end{proposition*}
 388
 389     \begin{theorem*}
 390       fox
 391     \end{theorem*}
 392
 393     \begin{exercise*}
 394       jumps
 395     \end{exercise*}
 396
 397     \begin{definition*}
 398       quod
 399     \end{definition*}
 400
 401     \begin{example*}
 402       erat
 403     \end{example*}
 404
 405     \begin{remark*}
 406       demonstradum.
 407     \end{remark*}
 408   \end{section}
 409
 410   \begin{section}{Topology}
 411     The interior of a set $X$ is $\interior{X}$. Its closure is
 412     $\closure{X}$ and its boundary is $\boundary{X}$.
 413   \end{section}
 414
 415   \setlength{\glslistdottedwidth}{.3\linewidth}
 416   \setglossarystyle{listdotted}
 417   \glsaddall{}
 418   \printnoidxglossaries{}
 419
 420   \bibliographystyle{mjo}
 421   \bibliography{local-references}
 422
 423   \printindex
 424 \end{document}