examples.tex

   1 \documentclass{report}
   2
   3 \usepackage{mjotex}
   4 \usepackage{mathtools}
   5
   6 \begin{document}
   7
   8   \begin{section}{Algorithm}
   9     An example of an algorithm (bogosort) environment.
  10
  11     \begin{algorithm}[H]
  12       \caption{Sort a list of numbers}
  13       \begin{algorithmic}
  14         \Require{A list of numbers $L$}
  15         \Ensure{A new, sorted copy $M$ of the list $L$}
  16
  17         \State{$M \gets L$}
  18
  19         \While{$M$ is not sorted}
  20           \State{Rearrange $M$ randomly}
  21         \EndWhile
  22
  23         \Return{$M$}
  24       \end{algorithmic}
  25     \end{algorithm}
  26   \end{section}
  27
  28   \begin{section}{Arrow}
  29     The identity operator on $V$ is $\identity{V}$. The composition of
  30     $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  31     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  32     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  33   \end{section}
  34
  35   \begin{section}{Common}
  36     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
  37     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  38     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
  39     of things $\pair{1}{2}$ or a triple of them
  40     $\triple{1}{2}{3}$. The Cartesian product of two sets $A$ and $B$
  41     is $\cartprod{A}{B}$; if we take the product with $C$ as well,
  42     then we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$
  43     and $W$ is $\directsum{V}{W}$ and the factorial of the number $10$
  44     is $\factorial{10}$.
  45
  46     Here are a few common tuple spaces that should not have a
  47     superscript when that superscript would be one: $\Nn[1]$,
  48     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
  49     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
  50     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
  51
  52     We also have a few basic set operations, for example the union of
  53     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
  54     course with union comes intersection: $\intersect{A}{B}$,
  55     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
  56     (indexed) union and intersections of things, like
  57     $\unionmany{k=1}{\infty}{A_{k}}$ or
  58     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
  59     is that they do the right thing in a display equation:
  60     %
  61     \begin{equation*}
  62       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
  63     \end{equation*}
  64     %
  65   \end{section}
  66
  67   \begin{section}{Cone}
  68     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
  69     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
  70     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
  71     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
  72     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
  73     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
  74     \ltcone_{K} y$ with respect to a cone $K$.
  75   \end{section}
  76
  77   \begin{section}{Convex}
  78     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
  79     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
  80     then its lineality space is $\linspace{K}$, its lineality is
  81     $\lin{K}$, and its extreme directions are $\Ext{K}$.
  82   \end{section}
  83
  84   \begin{section}{Font}
  85     We can write things like Carathéodory and Güler and $\mathbb{R}$.
  86   \end{section}
  87
  88   \begin{section}{Linear algebra}
  89     The absolute value of $x$ is $\abs{x}$, or its norm is
  90     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
  91     their tensor product is $\tp{x}{y}$. The Kronecker product of
  92     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
  93     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
  94     $\transpose{L}$. Its trace is $\trace{L}$.
  95
  96     The span of a set $X$ is $\spanof{X}$, and its codimension is
  97     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
  98     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
  99     $\Lie{X}$. We can write a column vector $x \coloneqq
 100     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 101     matrix with $\matricize{x}$. To recover the vector, we use
 102     $\vectorize{\matricize{x}}$.
 103
 104     The set of all bounded linear operators from $V$ to $W$ is
 105     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 106     instead.
 107
 108     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 109     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 110     writing $\directsumperp{V}{W}$. That operator should survive a
 111     display equation, too, and the weight of the circle should match
 112     that of the usual direct sum operator.
 113     %
 114     \begin{align*}
 115       Z = \directsumperp{V}{W}\\
 116       \oplus \oplusperp \oplus \oplusperp
 117     \end{align*}
 118     %
 119     Its form should also survive in different font sizes...
 120     \Large
 121     \begin{align*}
 122       Z = \directsumperp{V}{W}\\
 123       \oplus \oplusperp \oplus \oplusperp
 124     \end{align*}
 125     \Huge
 126     \begin{align*}
 127       Z = \directsumperp{V}{W}\\
 128       \oplus \oplusperp \oplus \oplusperp
 129     \end{align*}
 130     \normalsize
 131   \end{section}
 132
 133   \begin{section}{Listing}
 134     Here's an interactive sage prompt:
 135
 136     \begin{tcblisting}{listing only,
 137                        colback=codebg,
 138                        coltext=codefg,
 139                        listing options={language=sage,style=sage}}
 140     sage: K = Cone([ (1,0), (0,1) ])
 141     sage: K.positive_operator_gens()
 142     [
 143     [1 0]  [0 1]  [0 0]  [0 0]
 144     [0 0], [0 0], [1 0], [0 1]
 145     ]
 146     \end{tcblisting}
 147   \end{section}
 148
 149   \begin{section}{Miscellaneous}
 150     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 151     = 3$.
 152   \end{section}
 153
 154   \begin{section}{Proof by cases}
 155
 156     \begin{proposition}
 157       There are two cases in the following proof.
 158
 159       \begin{proof}
 160         The result should be self-evident once we have considered the
 161         following two cases.
 162         \begin{pcases}
 163           \begin{case}[first case]
 164             Nothing happens in the first case.
 165           \end{case}
 166           \begin{case}[second case]
 167             The same thing happens in the second case.
 168           \end{case}
 169         \end{pcases}
 170
 171         You see?
 172       \end{proof}
 173     \end{proposition}
 174
 175     Here's another one.
 176
 177     \renewcommand{\baselinestretch}{2}
 178     \begin{proposition}
 179       Cases should display intelligently even when the document is
 180       double-spaced.
 181
 182       \begin{proof}
 183         Here we go again.
 184
 185         \begin{pcases}
 186           \begin{case}[first case]
 187             Nothing happens in the first case.
 188           \end{case}
 189           \begin{case}[second case]
 190             The same thing happens in the second case.
 191           \end{case}
 192         \end{pcases}
 193
 194         Now it's over.
 195       \end{proof}
 196     \end{proposition}
 197     \renewcommand{\baselinestretch}{1}
 198   \end{section}
 199
 200   \begin{section}{Theorems}
 201     \begin{corollary}
 202       The
 203     \end{corollary}
 204
 205     \begin{lemma}
 206       quick
 207     \end{lemma}
 208
 209     \begin{proposition}
 210       brown
 211     \end{proposition}
 212
 213     \begin{theorem}
 214       fox
 215     \end{theorem}
 216
 217     \begin{definition}
 218       quod
 219     \end{definition}
 220
 221     \begin{example}
 222       erat
 223     \end{example}
 224
 225     \begin{remark}
 226       demonstradum.
 227     \end{remark}
 228   \end{section}
 229
 230   \begin{section}{Theorems (starred)}
 231     \begin{corollary*}
 232       The
 233     \end{corollary*}
 234
 235     \begin{lemma*}
 236       quick
 237     \end{lemma*}
 238
 239     \begin{proposition*}
 240       brown
 241     \end{proposition*}
 242
 243     \begin{theorem*}
 244       fox
 245     \end{theorem*}
 246
 247     \begin{definition*}
 248       quod
 249     \end{definition*}
 250
 251     \begin{example*}
 252       erat
 253     \end{example*}
 254
 255     \begin{remark*}
 256       demonstradum.
 257     \end{remark*}
 258   \end{section}
 259
 260   \begin{section}{Topology}
 261     The interior of a set $X$ is $\interior{X}$. Its closure is
 262     $\closure{X}$ and its boundary is $\boundary{X}$.
 263   \end{section}
 264
 265 \end{document}