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     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
  55     (indexed) union and intersections of things, like
  56     $\unionmany{k=1}{\infty}{A_{k}}$ or
  57     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
  58     is that they do the right thing in a display equation:
  59     %
  60     \begin{equation*}
  61       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
  62     \end{equation*}
  63     %
  64   \end{section}
  65
  66   \begin{section}{Cone}
  67     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
  68     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
  69     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
  70     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
  71     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
  72     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
  73     \ltcone_{K} y$ with respect to a cone $K$.
  74   \end{section}
  75
  76   \begin{section}{Convex}
  77     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
  78     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
  79     then its lineality space is $\linspace{K}$, its lineality is
  80     $\lin{K}$, and its extreme directions are $\Ext{K}$.
  81   \end{section}
  82
  83   \begin{section}{Font}
  84     We can write things like Carathéodory and Güler and $\mathbb{R}$.
  85   \end{section}
  86
  87   \begin{section}{Linear algebra}
  88     The absolute value of $x$ is $\abs{x}$, or its norm is
  89     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
  90     their tensor product is $\tp{x}{y}$. The Kronecker product of
  91     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
  92     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
  93     $\transpose{L}$. Its trace is $\trace{L}$.
  94
  95     The span of a set $X$ is $\spanof{X}$, and its codimension is
  96     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
  97     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
  98     $\Lie{X}$. We can write a column vector $x \coloneqq
  99     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 100     matrix with $\matricize{x}$. To recover the vector, we use
 101     $\vectorize{\matricize{x}}$.
 102
 103     The set of all bounded linear operators from $V$ to $W$ is
 104     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 105     instead.
 106   \end{section}
 107
 108   \begin{section}{Listing}
 109     Here's an interactive sage prompt:
 110
 111     \begin{tcblisting}{listing only,
 112                        colback=codebg,
 113                        coltext=codefg,
 114                        listing options={language=sage,style=sage}}
 115     sage: K = Cone([ (1,0), (0,1) ])
 116     sage: K.positive_operator_gens()
 117     [
 118     [1 0]  [0 1]  [0 0]  [0 0]
 119     [0 0], [0 0], [1 0], [0 1]
 120     ]
 121     \end{tcblisting}
 122   \end{section}
 123
 124   \begin{section}{Miscellaneous}
 125     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 126     = 3$.
 127   \end{section}
 128
 129   \begin{section}{Proof by cases}
 130
 131     \begin{proposition}
 132       There are two cases in the following proof.
 133
 134       \begin{proof}
 135         The result should be self-evident once we have considered the
 136         following two cases.
 137         \begin{pcases}
 138           \begin{case}[first case]
 139             Nothing happens in the first case.
 140           \end{case}
 141           \begin{case}[second case]
 142             The same thing happens in the second case.
 143           \end{case}
 144         \end{pcases}
 145
 146         You see?
 147       \end{proof}
 148     \end{proposition}
 149
 150     Here's another one.
 151
 152     \renewcommand{\baselinestretch}{2}
 153     \begin{proposition}
 154       Cases should display intelligently even when the document is
 155       double-spaced.
 156
 157       \begin{proof}
 158         Here we go again.
 159
 160         \begin{pcases}
 161           \begin{case}[first case]
 162             Nothing happens in the first case.
 163           \end{case}
 164           \begin{case}[second case]
 165             The same thing happens in the second case.
 166           \end{case}
 167         \end{pcases}
 168
 169         Now it's over.
 170       \end{proof}
 171     \end{proposition}
 172     \renewcommand{\baselinestretch}{1}
 173   \end{section}
 174
 175   \begin{section}{Theorems}
 176     \begin{corollary}
 177       The
 178     \end{corollary}
 179
 180     \begin{lemma}
 181       quick
 182     \end{lemma}
 183
 184     \begin{proposition}
 185       brown
 186     \end{proposition}
 187
 188     \begin{theorem}
 189       fox
 190     \end{theorem}
 191
 192     \begin{definition}
 193       quod
 194     \end{definition}
 195
 196     \begin{example}
 197       erat
 198     \end{example}
 199
 200     \begin{remark}
 201       demonstradum.
 202     \end{remark}
 203   \end{section}
 204
 205   \begin{section}{Theorems (starred)}
 206     \begin{corollary*}
 207       The
 208     \end{corollary*}
 209
 210     \begin{lemma*}
 211       quick
 212     \end{lemma*}
 213
 214     \begin{proposition*}
 215       brown
 216     \end{proposition*}
 217
 218     \begin{theorem*}
 219       fox
 220     \end{theorem*}
 221
 222     \begin{definition*}
 223       quod
 224     \end{definition*}
 225
 226     \begin{example*}
 227       erat
 228     \end{example*}
 229
 230     \begin{remark*}
 231       demonstradum.
 232     \end{remark*}
 233   \end{section}
 234
 235   \begin{section}{Topology}
 236     The interior of a set $X$ is $\interior{X}$. Its closure is
 237     $\closure{X}$ and its boundary is $\boundary{X}$.
 238   \end{section}
 239
 240 \end{document}