examples.tex

   1 \documentclass{report}
   2
   3 % We have to load this before mjotex so that mjotex knows to define
   4 % its glossary entries.
   5 \usepackage[nonumberlist]{glossaries}
   6 \makenoidxglossaries
   7
   8 % If you want an index, we can do that too. You'll need to define
   9 % the "INDICES" variable in the GNUmakefile, though.
  10 \usepackage{makeidx}
  11 \makeindex
  12
  13 \usepackage{mjotex}
  14 \usepackage{mathtools}
  15
  16 \begin{document}
  17
  18   \begin{section}{Algebra}
  19     If $R$ is a \index{commutative ring}, then $\polyring{R}{X,Y,Z}$
  20     is a multivariate polynomial ring with indeterminates $X$, $Y$,
  21     and $Z$, and coefficients in $R$. If $R$ is a moreover an integral
  22     domain, then its fraction field is $\Frac{R}$.
  23   \end{section}
  24
  25   \begin{section}{Algorithm}
  26     An example of an algorithm (bogosort) environment.
  27
  28     \begin{algorithm}[H]
  29       \caption{Sort a list of numbers}
  30       \begin{algorithmic}
  31         \Require{A list of numbers $L$}
  32         \Ensure{A new, sorted copy $M$ of the list $L$}
  33
  34         \State{$M \gets L$}
  35
  36         \While{$M$ is not sorted}
  37           \State{Rearrange $M$ randomly}
  38         \EndWhile
  39
  40         \Return{$M$}
  41       \end{algorithmic}
  42     \end{algorithm}
  43   \end{section}
  44
  45   \begin{section}{Arrow}
  46     The identity operator on $V$ is $\identity{V}$. The composition of
  47     $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  48     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  49     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  50   \end{section}
  51
  52   \begin{section}{Calculus}
  53     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  54     \Rn \rightarrow \Rn$.
  55   \end{section}
  56
  57   \begin{section}{Common}
  58     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
  59     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  60     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
  61     of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
  62     and the factorial of the number $10$ is $\factorial{10}$.
  63
  64     The Cartesian product of two sets $A$ and $B$ is
  65     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
  66     we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
  67     is $\directsum{V}{W}$. Or three things,
  68     $\directsumthree{U}{V}{W}$. How about more things? Like
  69     $\directsummany{k=1}{\infty}{V_{k}} \ne
  70     \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
  71     cartesian products adapt nicely to display equations:
  72     %
  73     \begin{equation*}
  74       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
  75     \end{equation*}
  76     Here are a few common tuple spaces that should not have a
  77     superscript when that superscript would be one: $\Nn[1]$,
  78     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
  79     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
  80     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
  81
  82     We also have a few basic set operations, for example the union of
  83     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
  84     course with union comes intersection: $\intersect{A}{B}$,
  85     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
  86     (indexed) union and intersections of things, like
  87     $\unionmany{k=1}{\infty}{A_{k}}$ or
  88     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
  89     is that they do the right thing in a display equation:
  90     %
  91     \begin{equation*}
  92       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
  93     \end{equation*}
  94
  95     Finally, we have the four standard types of intervals in $\Rn[1]$,
  96     %
  97     \begin{align*}
  98       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
  99       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 100       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 101       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 102     \end{align*}
 103   \end{section}
 104
 105   \begin{section}{Complex}
 106     We sometimes want to conjugate complex numbers like
 107     $\compconj{a+bi} = a - bi$.
 108   \end{section}
 109
 110   \begin{section}{Cone}
 111     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 112     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 113     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 114     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 115     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 116     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 117     \ltcone_{K} y$ with respect to a cone $K$.
 118   \end{section}
 119
 120   \begin{section}{Convex}
 121     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 122     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 123     then its lineality space is $\linspace{K}$, its lineality is
 124     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 125     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 126     proper face, then we write $F \properfaceof K$.
 127   \end{section}
 128
 129   \begin{section}{Euclidean Jordan algebras}
 130     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 131     is $\jp{x}{y}$.
 132   \end{section}
 133
 134   \begin{section}{Font}
 135     We can write things like Carathéodory and Güler and $\mathbb{R}$.
 136   \end{section}
 137
 138   \begin{section}{Linear algebra}
 139     The absolute value of $x$ is $\abs{x}$, or its norm is
 140     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 141     their tensor product is $\tp{x}{y}$. The Kronecker product of
 142     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 143     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 144     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
 145     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 146     $\pseudoinverse{L}$.
 147
 148     The span of a set $X$ is $\spanof{X}$, and its codimension is
 149     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 150     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 151     $\Lie{X}$. We can write a column vector $x \coloneqq
 152     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 153     matrix with $\matricize{x}$. To recover the vector, we use
 154     $\vectorize{\matricize{x}}$.
 155
 156     The set of all bounded linear operators from $V$ to $W$ is
 157     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 158     instead.
 159
 160     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 161     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 162     writing $\directsumperp{V}{W}$. That operator should survive a
 163     display equation, too, and the weight of the circle should match
 164     that of the usual direct sum operator.
 165     %
 166     \begin{align*}
 167       Z = \directsumperp{V}{W}\\
 168       \oplus \oplusperp \oplus \oplusperp
 169     \end{align*}
 170     %
 171     Its form should also survive in different font sizes...
 172     \Large
 173     \begin{align*}
 174       Z = \directsumperp{V}{W}\\
 175       \oplus \oplusperp \oplus \oplusperp
 176     \end{align*}
 177     \Huge
 178     \begin{align*}
 179       Z = \directsumperp{V}{W}\\
 180       \oplus \oplusperp \oplus \oplusperp
 181     \end{align*}
 182     \normalsize
 183   \end{section}
 184
 185   \begin{section}{Listing}
 186     Here's an interactive SageMath prompt:
 187
 188     \begin{tcblisting}{listing only,
 189                        colback=codebg,
 190                        coltext=codefg,
 191                        listing options={language=sage,style=sage}}
 192     sage: K = Cone([ (1,0), (0,1) ])
 193     sage: K.positive_operator_gens()
 194     [
 195     [1 0]  [0 1]  [0 0]  [0 0]
 196     [0 0], [0 0], [1 0], [0 1]
 197     ]
 198     \end{tcblisting}
 199
 200     However, the smart way to display a SageMath listing is to load it
 201     from an external file (under the ``listings'' subdirectory):
 202
 203     \sagelisting{example}
 204
 205     Keeping the listings in separate files makes it easy for the build
 206     system to test them.
 207   \end{section}
 208
 209   \begin{section}{Miscellaneous}
 210     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 211     = 3$.
 212   \end{section}
 213
 214   \begin{section}{Proof by cases}
 215
 216     \begin{proposition}
 217       There are two cases in the following proof.
 218
 219       \begin{proof}
 220         The result should be self-evident once we have considered the
 221         following two cases.
 222         \begin{pcases}
 223           \begin{case}[first case]
 224             Nothing happens in the first case.
 225           \end{case}
 226           \begin{case}[second case]
 227             The same thing happens in the second case.
 228           \end{case}
 229         \end{pcases}
 230
 231         You see?
 232       \end{proof}
 233     \end{proposition}
 234
 235     Here's another one.
 236
 237     \renewcommand{\baselinestretch}{2}
 238     \begin{proposition}
 239       Cases should display intelligently even when the document is
 240       double-spaced.
 241
 242       \begin{proof}
 243         Here we go again.
 244
 245         \begin{pcases}
 246           \begin{case}[first case]
 247             Nothing happens in the first case.
 248           \end{case}
 249           \begin{case}[second case]
 250             The same thing happens in the second case.
 251           \end{case}
 252         \end{pcases}
 253
 254         Now it's over.
 255       \end{proof}
 256     \end{proposition}
 257     \renewcommand{\baselinestretch}{1}
 258   \end{section}
 259
 260   \begin{section}{Theorems}
 261     \begin{corollary}
 262       The
 263     \end{corollary}
 264
 265     \begin{lemma}
 266       quick
 267     \end{lemma}
 268
 269     \begin{proposition}
 270       brown
 271     \end{proposition}
 272
 273     \begin{theorem}
 274       fox
 275     \end{theorem}
 276
 277     \begin{exercise}
 278       jumps
 279     \end{exercise}
 280
 281     \begin{definition}
 282       quod
 283     \end{definition}
 284
 285     \begin{example}
 286       erat
 287     \end{example}
 288
 289     \begin{remark}
 290       demonstradum.
 291     \end{remark}
 292   \end{section}
 293
 294   \begin{section}{Theorems (starred)}
 295     \begin{corollary*}
 296       The
 297     \end{corollary*}
 298
 299     \begin{lemma*}
 300       quick
 301     \end{lemma*}
 302
 303     \begin{proposition*}
 304       brown
 305     \end{proposition*}
 306
 307     \begin{theorem*}
 308       fox
 309     \end{theorem*}
 310
 311     \begin{exercise*}
 312       jumps
 313     \end{exercise*}
 314
 315     \begin{definition*}
 316       quod
 317     \end{definition*}
 318
 319     \begin{example*}
 320       erat
 321     \end{example*}
 322
 323     \begin{remark*}
 324       demonstradum.
 325     \end{remark*}
 326   \end{section}
 327
 328   \begin{section}{Topology}
 329     The interior of a set $X$ is $\interior{X}$. Its closure is
 330     $\closure{X}$ and its boundary is $\boundary{X}$.
 331   \end{section}
 332
 333   \setlength{\glslistdottedwidth}{.3\linewidth}
 334   \setglossarystyle{listdotted}
 335   \glsaddall
 336   \printnoidxglossaries
 337
 338   \printindex
 339 \end{document}