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 $\zero{R}$. The stabilizer (or isotropy)
  43     subgroup of $G$ that fixes $x$ is $\Stab{G}{x}$.
  44
  45     If $I$ is an ideal, then $\variety{I}$ is the variety that
  46     corresponds to it.
  47   \end{section}
  48
  49   \begin{section}{Algorithm}
  50     An example of an algorithm (bogosort) environment.
  51
  52     \begin{algorithm}[H]
  53       \caption{Sort a list of numbers}
  54       \begin{algorithmic}
  55         \Require{A list of numbers $L$}
  56         \Ensure{A new, sorted copy $M$ of the list $L$}
  57
  58         \State{$M \gets L$}
  59
  60         \While{$M$ is not sorted}
  61           \State{Rearrange $M$ randomly}
  62         \EndWhile{}
  63
  64         \State{\Return{$M$}}
  65       \end{algorithmic}
  66     \end{algorithm}
  67   \end{section}
  68
  69   \begin{section}{Arrow}
  70     The constant function that always returns $a$ is $\const{a}$. The
  71     identity operator on $V$ is $\identity{V}$. The composition of $f$
  72     and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  73     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  74     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  75   \end{section}
  76
  77   \begin{section}{Calculus}
  78     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  79     \Rn \rightarrow \Rn$.
  80   \end{section}
  81
  82   \begin{section}{Common}
  83     The function $f$ applied to $x$ is $f\of{x}$, and the restriction
  84     of $f$ to a subset $X$ of its domain is $\restrict{f}{X}$. We can
  85     group terms like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c -
  86         d}}$. The tuples go up to seven, for now, and then we give up
  87     and use the general construct:
  88     %
  89     \begin{itemize}
  90       \begin{item}
  91         Pair: $\pair{1}{2}$,
  92       \end{item}
  93       \begin{item}
  94         Triple: $\triple{1}{2}{3}$,
  95       \end{item}
  96       \begin{item}
  97         Quadruple: $\quadruple{1}{2}{3}{4}$,
  98       \end{item}
  99       \begin{item}
 100         Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
 101       \end{item}
 102       \begin{item}
 103         Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
 104       \end{item}
 105       \begin{item}
 106         Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
 107       \end{item}
 108       \begin{item}
 109         Tuple: $\tuple{1,2,\ldots,8675309}$.
 110       \end{item}
 111     \end{itemize}
 112     %
 113     The factorial of the number $10$ is $\factorial{10}$, and the
 114     least common multiple of $4$ and $6$ is $\lcm{\set{4,6}} =
 115     12$.
 116
 117     The direct sum of $V$ and $W$ is $\directsum{V}{W}$. Or three
 118     things, $\directsumthree{U}{V}{W}$. How about more things? Like
 119     $\directsummany{k=1}{\infty}{V_{k}}$. Those direct sums
 120     adapt nicely to display equations:
 121     %
 122     \begin{equation*}
 123       \directsummany{k=1}{\infty}{V_{k}} \ne \emptyset.
 124     \end{equation*}
 125     %
 126     Here are a few common tuple spaces that should not have a
 127     superscript when that superscript would be one: $\Nn[1]$,
 128     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
 129     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
 130     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. The symbols $\Fn[1]$, $\Fn[2]$,
 131     et cetera, are available for use with a generic field.
 132
 133     Finally, we have the four standard types of intervals in $\Rn[1]$,
 134     %
 135     \begin{align*}
 136       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 137       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 138       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 139       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 140     \end{align*}
 141   \end{section}
 142
 143   \begin{section}{Complex}
 144     We sometimes want to conjugate complex numbers like
 145     $\compconj{a+bi} = a - bi$.
 146   \end{section}
 147
 148   \begin{section}{Cone}
 149     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 150     are $\Rnplus$, $\Rnplusplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.
 151     If cones $K_{1}$ and $K_{2}$ are given, we can define
 152     $\posops{K_{1}}$, $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$,
 153     $\Zof{K_{1}}$, $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can
 154     also define $x \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K}
 155     y$, and $x \ltcone_{K} y$ with respect to a cone $K$.
 156   \end{section}
 157
 158   \begin{section}{Convex}
 159     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 160     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 161     then its lineality space is $\linspace{K}$, its lineality is
 162     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 163     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 164     proper face, then we write $F \properfaceof K$.
 165   \end{section}
 166
 167   \begin{section}{Euclidean Jordan algebras}
 168     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 169     $V$ is $\jp{x}{y}$. The Jordan-automorphism group of $V$ is
 170     $\JAut{V}$. Two popular operators in an EJA are its quadratic
 171     representation and ``left multiplication by'' operator. For a
 172     given $x$, they are, respectively, $\quadrepr{x}$ and
 173     $\leftmult{x}$.
 174   \end{section}
 175
 176   \begin{section}{Font}
 177     We can write things like Carathéodory and Güler and
 178     $\mathbb{R}$. The PostScript Zapf Chancery font is also available
 179     in both upper- and lower-case:
 180     %
 181     \begin{itemize}
 182       \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
 183       \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
 184     \end{itemize}
 185   \end{section}
 186
 187   \begin{section}{Hurwitz}
 188     Here lies the Hurwitz algebras, like the quaternions
 189     $\quaternions$ and octonions $\octonions$.
 190   \end{section}
 191
 192   \begin{section}{Linear algebra}
 193     The absolute value of $x$ is $\abs{x}$, or its norm is
 194     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 195     their tensor product is $\tp{x}{y}$. The Kronecker product of
 196     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 197     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 198     $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the
 199     set of its eigenvalues---is $\spectrum{L}$. Another
 200     matrix-specific concept is the Moore-Penrose pseudoinverse of $L$,
 201     denoted by $\pseudoinverse{L}$. Finally, the rank of a matrix $L$
 202     is $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
 203     real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
 204     respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
 205     simplify because the ``$n$'' does not indicate the arity of a
 206     Cartesian product in this case. A handy way to represent the
 207     matrix $A \in \Rn[n \times n]$ whose only non-zero entries are on
 208     the diagonal is $\diag{\colvec{A_{11},A_{22},\ldots,A_{nn}}}$.
 209
 210     The span of a set $X$ is $\spanof{X}$, and its codimension is
 211     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 212     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 213     $\Lie{X}$. We can write a column vector $x \coloneqq
 214     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 215     matrix with $\matricize{x}$. To recover the vector, we use
 216     $\vectorize{\matricize{x}}$.
 217
 218     The set of all bounded linear operators from $V$ to $W$ is
 219     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 220     instead. If you have matrices instead, then the general linear
 221     group of $n$-by-$n$ matrices with entries in $\mathbb{F}$ is
 222     $\GL{n}{\mathbb{F}}$.
 223
 224     If you want to solve a system of equations, try Cramer's
 225     rule~\cite{ehrenborg}. Or at least the reduced row-echelon form of
 226     the matrix, $\rref{A}$.
 227
 228     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 229     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 230     writing $\directsumperp{V}{W}$. That operator should survive a
 231     display equation, too, and the weight of the circle should match
 232     that of the usual direct sum operator.
 233     %
 234     \begin{align*}
 235       Z = \directsumperp{V}{W}\\
 236       \oplus \oplusperp \oplus \oplusperp
 237     \end{align*}
 238     %
 239     Its form should also survive in different font sizes\ldots
 240     \Large
 241     \begin{align*}
 242       Z = \directsumperp{V}{W}\\
 243       \oplus \oplusperp \oplus \oplusperp
 244     \end{align*}
 245     \Huge
 246     \begin{align*}
 247       Z = \directsumperp{V}{W}\\
 248       \oplus \oplusperp \oplus \oplusperp
 249     \end{align*}
 250     \normalsize
 251   \end{section}
 252
 253   \begin{section}{Listing}
 254     Here's an interactive SageMath prompt:
 255
 256     \begin{tcblisting}{listing only,
 257                        colback=codebg,
 258                        coltext=codefg,
 259                        listing options={language=sage,style=sage}}
 260     sage: K = Cone([ (1,0), (0,1) ])
 261     sage: K.positive_operator_gens()
 262     [
 263     [1 0]  [0 1]  [0 0]  [0 0]
 264     [0 0], [0 0], [1 0], [0 1]
 265     ]
 266     \end{tcblisting}
 267
 268     However, the smart way to display a SageMath listing is to load it
 269     from an external file (under the ``listings'' subdirectory):
 270
 271     \sagelisting{example}
 272
 273     Keeping the listings in separate files makes it easy for the build
 274     system to test them.
 275   \end{section}
 276
 277   \begin{section}{Proof by cases}
 278
 279     \begin{proposition}
 280       There are two cases in the following proof.
 281
 282       \begin{proof}
 283         The result should be self-evident once we have considered the
 284         following two cases.
 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         You see?
 295       \end{proof}
 296     \end{proposition}
 297
 298     Here's another one.
 299
 300     \renewcommand{\baselinestretch}{2}
 301     \begin{proposition}
 302       Cases should display intelligently even when the document is
 303       double-spaced.
 304
 305       \begin{proof}
 306         Here we go again.
 307
 308         \begin{pcases}
 309           \begin{case}[first case]
 310             Nothing happens in the first case.
 311           \end{case}
 312           \begin{case}[second case]
 313             The same thing happens in the second case.
 314           \end{case}
 315         \end{pcases}
 316
 317         Now it's over.
 318       \end{proof}
 319     \end{proposition}
 320     \renewcommand{\baselinestretch}{1}
 321   \end{section}
 322
 323   \begin{section}{Set theory}
 324     Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The
 325     cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} =
 326     3$, and its powerset is $\powerset{X}$.
 327
 328     We also have a few basic set operations, for example the union of
 329     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
 330     course with union comes intersection: $\intersect{A}{B}$,
 331     $\intersectthree{A}{B}{C}$. The Cartesian product of two sets $A$
 332     and $B$ is there too: $\cartprod{A}{B}$. If we take the product
 333     with $C$ as well, then we obtain $\cartprodthree{A}{B}{C}$.
 334
 335     We can also take an arbitrary (indexed) union, intersection, or
 336     Cartesian product of things, like
 337     $\unionmany{k=1}{\infty}{A_{k}}$,
 338     $\intersectmany{k=1}{\infty}{B_{k}}$, or
 339     $\cartprodmany{k=1}{\infty}{C_{k}}$. The best part about those is
 340     that they do the right thing in a display equation:
 341     %
 342     \begin{equation*}
 343       \unionmany{k=1}{\infty}{A_{k}}
 344       \ne
 345       \intersectmany{k=1}{\infty}{B_{k}}
 346       \ne
 347       \cartprodmany{k=1}{\infty}{C_{k}}.
 348     \end{equation*}
 349     %
 350   \end{section}
 351
 352   \begin{section}{Theorems}
 353     \begin{corollary}
 354       The
 355     \end{corollary}
 356
 357     \begin{lemma}
 358       quick
 359     \end{lemma}
 360
 361     \begin{proposition}
 362       brown
 363     \end{proposition}
 364
 365     \begin{theorem}
 366       fox
 367     \end{theorem}
 368
 369     \begin{exercise}
 370       jumps
 371     \end{exercise}
 372
 373     \begin{definition}
 374       quod
 375     \end{definition}
 376
 377     \begin{example}
 378       erat
 379     \end{example}
 380
 381     \begin{remark}
 382       demonstradum.
 383     \end{remark}
 384   \end{section}
 385
 386   \begin{section}{Theorems (starred)}
 387     \begin{corollary*}
 388       The
 389     \end{corollary*}
 390
 391     \begin{lemma*}
 392       quick
 393     \end{lemma*}
 394
 395     \begin{proposition*}
 396       brown
 397     \end{proposition*}
 398
 399     \begin{theorem*}
 400       fox
 401     \end{theorem*}
 402
 403     \begin{exercise*}
 404       jumps
 405     \end{exercise*}
 406
 407     \begin{definition*}
 408       quod
 409     \end{definition*}
 410
 411     \begin{example*}
 412       erat
 413     \end{example*}
 414
 415     \begin{remark*}
 416       demonstradum.
 417     \end{remark*}
 418   \end{section}
 419
 420   \begin{section}{Topology}
 421     The interior of a set $X$ is $\interior{X}$. Its closure is
 422     $\closure{X}$ and its boundary is $\boundary{X}$.
 423   \end{section}
 424
 425   \setlength{\glslistdottedwidth}{.3\linewidth}
 426   \setglossarystyle{listdotted}
 427   \glsaddall{}
 428   \printnoidxglossaries{}
 429
 430   \bibliographystyle{mjo}
 431   \bibliography{local-references}
 432
 433   \printindex
 434 \end{document}