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
  73     Finally, we have the four standard types of intervals in $\Rn[1]$,
  74     %
  75     \begin{align*}
  76       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
  77       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
  78       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
  79       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
  80     \end{align*}
  81   \end{section}
  82
  83   \begin{section}{Complex}
  84     We sometimes want to conjugate complex numbers like
  85     $\compconj{a+bi} = a - bi$.
  86   \end{section}
  87
  88   \begin{section}{Cone}
  89     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
  90     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
  91     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
  92     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
  93     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
  94     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
  95     \ltcone_{K} y$ with respect to a cone $K$.
  96   \end{section}
  97
  98   \begin{section}{Convex}
  99     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 100     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 101     then its lineality space is $\linspace{K}$, its lineality is
 102     $\lin{K}$, and its extreme directions are $\Ext{K}$.
 103   \end{section}
 104
 105   \begin{section}{Font}
 106     We can write things like Carathéodory and Güler and $\mathbb{R}$.
 107   \end{section}
 108
 109   \begin{section}{Linear algebra}
 110     The absolute value of $x$ is $\abs{x}$, or its norm is
 111     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 112     their tensor product is $\tp{x}{y}$. The Kronecker product of
 113     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 114     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 115     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
 116     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 117     $\pseudoinverse{L}$.
 118
 119     The span of a set $X$ is $\spanof{X}$, and its codimension is
 120     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 121     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 122     $\Lie{X}$. We can write a column vector $x \coloneqq
 123     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 124     matrix with $\matricize{x}$. To recover the vector, we use
 125     $\vectorize{\matricize{x}}$.
 126
 127     The set of all bounded linear operators from $V$ to $W$ is
 128     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 129     instead.
 130
 131     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 132     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 133     writing $\directsumperp{V}{W}$. That operator should survive a
 134     display equation, too, and the weight of the circle should match
 135     that of the usual direct sum operator.
 136     %
 137     \begin{align*}
 138       Z = \directsumperp{V}{W}\\
 139       \oplus \oplusperp \oplus \oplusperp
 140     \end{align*}
 141     %
 142     Its form should also survive in different font sizes...
 143     \Large
 144     \begin{align*}
 145       Z = \directsumperp{V}{W}\\
 146       \oplus \oplusperp \oplus \oplusperp
 147     \end{align*}
 148     \Huge
 149     \begin{align*}
 150       Z = \directsumperp{V}{W}\\
 151       \oplus \oplusperp \oplus \oplusperp
 152     \end{align*}
 153     \normalsize
 154   \end{section}
 155
 156   \begin{section}{Listing}
 157     Here's an interactive sage prompt:
 158
 159     \begin{tcblisting}{listing only,
 160                        colback=codebg,
 161                        coltext=codefg,
 162                        listing options={language=sage,style=sage}}
 163     sage: K = Cone([ (1,0), (0,1) ])
 164     sage: K.positive_operator_gens()
 165     [
 166     [1 0]  [0 1]  [0 0]  [0 0]
 167     [0 0], [0 0], [1 0], [0 1]
 168     ]
 169     \end{tcblisting}
 170   \end{section}
 171
 172   \begin{section}{Miscellaneous}
 173     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 174     = 3$.
 175   \end{section}
 176
 177   \begin{section}{Proof by cases}
 178
 179     \begin{proposition}
 180       There are two cases in the following proof.
 181
 182       \begin{proof}
 183         The result should be self-evident once we have considered the
 184         following two cases.
 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         You see?
 195       \end{proof}
 196     \end{proposition}
 197
 198     Here's another one.
 199
 200     \renewcommand{\baselinestretch}{2}
 201     \begin{proposition}
 202       Cases should display intelligently even when the document is
 203       double-spaced.
 204
 205       \begin{proof}
 206         Here we go again.
 207
 208         \begin{pcases}
 209           \begin{case}[first case]
 210             Nothing happens in the first case.
 211           \end{case}
 212           \begin{case}[second case]
 213             The same thing happens in the second case.
 214           \end{case}
 215         \end{pcases}
 216
 217         Now it's over.
 218       \end{proof}
 219     \end{proposition}
 220     \renewcommand{\baselinestretch}{1}
 221   \end{section}
 222
 223   \begin{section}{Theorems}
 224     \begin{corollary}
 225       The
 226     \end{corollary}
 227
 228     \begin{lemma}
 229       quick
 230     \end{lemma}
 231
 232     \begin{proposition}
 233       brown
 234     \end{proposition}
 235
 236     \begin{theorem}
 237       fox
 238     \end{theorem}
 239
 240     \begin{exercise}
 241       jumps
 242     \end{exercise}
 243
 244     \begin{definition}
 245       quod
 246     \end{definition}
 247
 248     \begin{example}
 249       erat
 250     \end{example}
 251
 252     \begin{remark}
 253       demonstradum.
 254     \end{remark}
 255   \end{section}
 256
 257   \begin{section}{Theorems (starred)}
 258     \begin{corollary*}
 259       The
 260     \end{corollary*}
 261
 262     \begin{lemma*}
 263       quick
 264     \end{lemma*}
 265
 266     \begin{proposition*}
 267       brown
 268     \end{proposition*}
 269
 270     \begin{theorem*}
 271       fox
 272     \end{theorem*}
 273
 274     \begin{exercise*}
 275       jumps
 276     \end{exercise*}
 277
 278     \begin{definition*}
 279       quod
 280     \end{definition*}
 281
 282     \begin{example*}
 283       erat
 284     \end{example*}
 285
 286     \begin{remark*}
 287       demonstradum.
 288     \end{remark*}
 289   \end{section}
 290
 291   \begin{section}{Topology}
 292     The interior of a set $X$ is $\interior{X}$. Its closure is
 293     $\closure{X}$ and its boundary is $\boundary{X}$.
 294   \end{section}
 295
 296 \end{document}