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 $\triple{1}{2}{3}$,
  40     and the factorial of the number $10$ is $\factorial{10}$.
  41
  42     The Cartesian product of two sets $A$ and $B$ is
  43     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
  44     we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
  45     is $\directsum{V}{W}$. Or three things,
  46     $\directsumthree{U}{V}{W}$. How about more things? Like
  47     $\directsummany{k=1}{\infty}{V_{k}} \ne
  48     \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
  49     cartesian products adapt nicely to display equations:
  50     %
  51     \begin{equation*}
  52       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
  53     \end{equation*}
  54     Here are a few common tuple spaces that should not have a
  55     superscript when that superscript would be one: $\Nn[1]$,
  56     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
  57     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
  58     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
  59
  60     We also have a few basic set operations, for example the union of
  61     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
  62     course with union comes intersection: $\intersect{A}{B}$,
  63     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
  64     (indexed) union and intersections of things, like
  65     $\unionmany{k=1}{\infty}{A_{k}}$ or
  66     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
  67     is that they do the right thing in a display equation:
  68     %
  69     \begin{equation*}
  70       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
  71     \end{equation*}
  72   \end{section}
  73
  74   \begin{section}{Cone}
  75     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
  76     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
  77     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
  78     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
  79     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
  80     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
  81     \ltcone_{K} y$ with respect to a cone $K$.
  82   \end{section}
  83
  84   \begin{section}{Convex}
  85     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
  86     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
  87     then its lineality space is $\linspace{K}$, its lineality is
  88     $\lin{K}$, and its extreme directions are $\Ext{K}$.
  89   \end{section}
  90
  91   \begin{section}{Font}
  92     We can write things like Carathéodory and Güler and $\mathbb{R}$.
  93   \end{section}
  94
  95   \begin{section}{Linear algebra}
  96     The absolute value of $x$ is $\abs{x}$, or its norm is
  97     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
  98     their tensor product is $\tp{x}{y}$. The Kronecker product of
  99     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 100     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 101     $\transpose{L}$. Its trace is $\trace{L}$.
 102
 103     The span of a set $X$ is $\spanof{X}$, and its codimension is
 104     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 105     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 106     $\Lie{X}$. We can write a column vector $x \coloneqq
 107     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 108     matrix with $\matricize{x}$. To recover the vector, we use
 109     $\vectorize{\matricize{x}}$.
 110
 111     The set of all bounded linear operators from $V$ to $W$ is
 112     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 113     instead.
 114
 115     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 116     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 117     writing $\directsumperp{V}{W}$. That operator should survive a
 118     display equation, too, and the weight of the circle should match
 119     that of the usual direct sum operator.
 120     %
 121     \begin{align*}
 122       Z = \directsumperp{V}{W}\\
 123       \oplus \oplusperp \oplus \oplusperp
 124     \end{align*}
 125     %
 126     Its form should also survive in different font sizes...
 127     \Large
 128     \begin{align*}
 129       Z = \directsumperp{V}{W}\\
 130       \oplus \oplusperp \oplus \oplusperp
 131     \end{align*}
 132     \Huge
 133     \begin{align*}
 134       Z = \directsumperp{V}{W}\\
 135       \oplus \oplusperp \oplus \oplusperp
 136     \end{align*}
 137     \normalsize
 138   \end{section}
 139
 140   \begin{section}{Listing}
 141     Here's an interactive sage prompt:
 142
 143     \begin{tcblisting}{listing only,
 144                        colback=codebg,
 145                        coltext=codefg,
 146                        listing options={language=sage,style=sage}}
 147     sage: K = Cone([ (1,0), (0,1) ])
 148     sage: K.positive_operator_gens()
 149     [
 150     [1 0]  [0 1]  [0 0]  [0 0]
 151     [0 0], [0 0], [1 0], [0 1]
 152     ]
 153     \end{tcblisting}
 154   \end{section}
 155
 156   \begin{section}{Miscellaneous}
 157     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 158     = 3$.
 159   \end{section}
 160
 161   \begin{section}{Proof by cases}
 162
 163     \begin{proposition}
 164       There are two cases in the following proof.
 165
 166       \begin{proof}
 167         The result should be self-evident once we have considered the
 168         following two cases.
 169         \begin{pcases}
 170           \begin{case}[first case]
 171             Nothing happens in the first case.
 172           \end{case}
 173           \begin{case}[second case]
 174             The same thing happens in the second case.
 175           \end{case}
 176         \end{pcases}
 177
 178         You see?
 179       \end{proof}
 180     \end{proposition}
 181
 182     Here's another one.
 183
 184     \renewcommand{\baselinestretch}{2}
 185     \begin{proposition}
 186       Cases should display intelligently even when the document is
 187       double-spaced.
 188
 189       \begin{proof}
 190         Here we go again.
 191
 192         \begin{pcases}
 193           \begin{case}[first case]
 194             Nothing happens in the first case.
 195           \end{case}
 196           \begin{case}[second case]
 197             The same thing happens in the second case.
 198           \end{case}
 199         \end{pcases}
 200
 201         Now it's over.
 202       \end{proof}
 203     \end{proposition}
 204     \renewcommand{\baselinestretch}{1}
 205   \end{section}
 206
 207   \begin{section}{Theorems}
 208     \begin{corollary}
 209       The
 210     \end{corollary}
 211
 212     \begin{lemma}
 213       quick
 214     \end{lemma}
 215
 216     \begin{proposition}
 217       brown
 218     \end{proposition}
 219
 220     \begin{theorem}
 221       fox
 222     \end{theorem}
 223
 224     \begin{definition}
 225       quod
 226     \end{definition}
 227
 228     \begin{example}
 229       erat
 230     \end{example}
 231
 232     \begin{remark}
 233       demonstradum.
 234     \end{remark}
 235   \end{section}
 236
 237   \begin{section}{Theorems (starred)}
 238     \begin{corollary*}
 239       The
 240     \end{corollary*}
 241
 242     \begin{lemma*}
 243       quick
 244     \end{lemma*}
 245
 246     \begin{proposition*}
 247       brown
 248     \end{proposition*}
 249
 250     \begin{theorem*}
 251       fox
 252     \end{theorem*}
 253
 254     \begin{definition*}
 255       quod
 256     \end{definition*}
 257
 258     \begin{example*}
 259       erat
 260     \end{example*}
 261
 262     \begin{remark*}
 263       demonstradum.
 264     \end{remark*}
 265   \end{section}
 266
 267   \begin{section}{Topology}
 268     The interior of a set $X$ is $\interior{X}$. Its closure is
 269     $\closure{X}$ and its boundary is $\boundary{X}$.
 270   \end{section}
 271
 272 \end{document}