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     $\inverse{f}$.
  32   \end{section}
  33
  34   \begin{section}{Common}
  35     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
  36     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  37     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
  38     of things $\pair{1}{2}$ or a triple of them
  39     $\triple{1}{2}{3}$. The Cartesian product of two sets $A$ and $B$
  40     is $\cartprod{A}{B}$; if we take the product with $C$ as well,
  41     then we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$
  42     and $W$ is $\directsum{V}{W}$ and the factorial of the number $10$
  43     is $\factorial{10}$.
  44
  45     Here are a few common tuple spaces that should not have a
  46     superscript when that superscript would be one: $\Nn[1]$,
  47     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
  48     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
  49     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
  50
  51     We also have a few basic set operations, for example the union of
  52     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
  53     course with union comes intersection: $\intersect{A}{B}$,
  54     $\intersectthree{A}{B}{C}$.
  55   \end{section}
  56
  57   \begin{section}{Cone}
  58     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
  59     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
  60     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
  61     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
  62     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
  63     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
  64     \ltcone_{K} y$ with respect to a cone $K$.
  65   \end{section}
  66
  67   \begin{section}{Convex}
  68     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
  69     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
  70     then its lineality space is $\linspace{K}$, its lineality is
  71     $\lin{K}$, and its extreme directions are $\Ext{K}$.
  72   \end{section}
  73
  74   \begin{section}{Font}
  75     We can write things like Carathéodory and Güler and $\mathbb{R}$.
  76   \end{section}
  77
  78   \begin{section}{Linear algebra}
  79     The absolute value of $x$ is $\abs{x}$, or its norm is
  80     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
  81     their tensor product is $\tp{x}{y}$. The Kronecker product of
  82     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
  83     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
  84     $\transpose{L}$. Its trace is $\trace{L}$.
  85
  86     The span of a set $X$ is $\spanof{X}$, and its codimension is
  87     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
  88     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
  89     $\Lie{X}$. We can write a column vector $x \coloneqq
  90     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
  91     matrix with $\matricize{x}$. To recover the vector, we use
  92     $\vectorize{\matricize{x}}$.
  93
  94     The set of all bounded linear operators from $V$ to $W$ is
  95     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
  96     instead.
  97   \end{section}
  98
  99   \begin{section}{Listing}
 100     Here's an interactive sage prompt:
 101
 102     \begin{tcblisting}{listing only,
 103                        colback=codebg,
 104                        coltext=codefg,
 105                        listing options={language=sage,style=sage}}
 106     sage: K = Cone([ (1,0), (0,1) ])
 107     sage: K.positive_operator_gens()
 108     [
 109     [1 0]  [0 1]  [0 0]  [0 0]
 110     [0 0], [0 0], [1 0], [0 1]
 111     ]
 112     \end{tcblisting}
 113   \end{section}
 114
 115   \begin{section}{Miscellaneous}
 116     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 117     = 3$.
 118   \end{section}
 119
 120   \begin{section}{Proof by cases}
 121
 122     \begin{proposition}
 123       There are two cases in the following proof.
 124
 125       \begin{proof}
 126         The result should be self-evident once we have considered the
 127         following two cases.
 128         \begin{pcases}
 129           \begin{case}[first case]
 130             Nothing happens in the first case.
 131           \end{case}
 132           \begin{case}[second case]
 133             The same thing happens in the second case.
 134           \end{case}
 135         \end{pcases}
 136
 137         You see?
 138       \end{proof}
 139     \end{proposition}
 140
 141     Here's another one.
 142
 143     \renewcommand{\baselinestretch}{2}
 144     \begin{proposition}
 145       Cases should display intelligently even when the document is
 146       double-spaced.
 147
 148       \begin{proof}
 149         Here we go again.
 150
 151         \begin{pcases}
 152           \begin{case}[first case]
 153             Nothing happens in the first case.
 154           \end{case}
 155           \begin{case}[second case]
 156             The same thing happens in the second case.
 157           \end{case}
 158         \end{pcases}
 159
 160         Now it's over.
 161       \end{proof}
 162     \end{proposition}
 163     \renewcommand{\baselinestretch}{1}
 164   \end{section}
 165
 166   \begin{section}{Theorems}
 167     \begin{corollary}
 168       The
 169     \end{corollary}
 170
 171     \begin{lemma}
 172       quick
 173     \end{lemma}
 174
 175     \begin{proposition}
 176       brown
 177     \end{proposition}
 178
 179     \begin{theorem}
 180       fox
 181     \end{theorem}
 182
 183     \begin{definition}
 184       quod
 185     \end{definition}
 186
 187     \begin{example}
 188       erat
 189     \end{example}
 190
 191     \begin{remark}
 192       demonstradum.
 193     \end{remark}
 194   \end{section}
 195
 196   \begin{section}{Theorems (starred)}
 197     \begin{corollary*}
 198       The
 199     \end{corollary*}
 200
 201     \begin{lemma*}
 202       quick
 203     \end{lemma*}
 204
 205     \begin{proposition*}
 206       brown
 207     \end{proposition*}
 208
 209     \begin{theorem*}
 210       fox
 211     \end{theorem*}
 212
 213     \begin{definition*}
 214       quod
 215     \end{definition*}
 216
 217     \begin{example*}
 218       erat
 219     \end{example*}
 220
 221     \begin{remark*}
 222       demonstradum.
 223     \end{remark*}
 224   \end{section}
 225
 226   \begin{section}{Topology}
 227     The interior of a set $X$ is $\interior{X}$. Its closure is
 228     $\closure{X}$ and its boundary is $\boundary{X}$.
 229   \end{section}
 230
 231 \end{document}