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}{Calculus}
  36     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  37     \Rn \rightarrow \Rn$.
  38   \end{section}
  39
  40   \begin{section}{Common}
  41     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
  42     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  43     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
  44     of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
  45     and the factorial of the number $10$ is $\factorial{10}$.
  46
  47     The Cartesian product of two sets $A$ and $B$ is
  48     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
  49     we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
  50     is $\directsum{V}{W}$. Or three things,
  51     $\directsumthree{U}{V}{W}$. How about more things? Like
  52     $\directsummany{k=1}{\infty}{V_{k}} \ne
  53     \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
  54     cartesian products adapt nicely to display equations:
  55     %
  56     \begin{equation*}
  57       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
  58     \end{equation*}
  59     Here are a few common tuple spaces that should not have a
  60     superscript when that superscript would be one: $\Nn[1]$,
  61     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
  62     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
  63     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
  64
  65     We also have a few basic set operations, for example the union of
  66     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
  67     course with union comes intersection: $\intersect{A}{B}$,
  68     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
  69     (indexed) union and intersections of things, like
  70     $\unionmany{k=1}{\infty}{A_{k}}$ or
  71     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
  72     is that they do the right thing in a display equation:
  73     %
  74     \begin{equation*}
  75       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
  76     \end{equation*}
  77
  78     Finally, we have the four standard types of intervals in $\Rn[1]$,
  79     %
  80     \begin{align*}
  81       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
  82       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
  83       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
  84       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
  85     \end{align*}
  86   \end{section}
  87
  88   \begin{section}{Complex}
  89     We sometimes want to conjugate complex numbers like
  90     $\compconj{a+bi} = a - bi$.
  91   \end{section}
  92
  93   \begin{section}{Cone}
  94     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
  95     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
  96     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
  97     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
  98     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
  99     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 100     \ltcone_{K} y$ with respect to a cone $K$.
 101   \end{section}
 102
 103   \begin{section}{Convex}
 104     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 105     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 106     then its lineality space is $\linspace{K}$, its lineality is
 107     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 108     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 109     proper face, then we write $F \properfaceof K$.
 110   \end{section}
 111
 112   \begin{section}{Font}
 113     We can write things like Carathéodory and Güler and $\mathbb{R}$.
 114   \end{section}
 115
 116   \begin{section}{Linear algebra}
 117     The absolute value of $x$ is $\abs{x}$, or its norm is
 118     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 119     their tensor product is $\tp{x}{y}$. The Kronecker product of
 120     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 121     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 122     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
 123     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 124     $\pseudoinverse{L}$.
 125
 126     The span of a set $X$ is $\spanof{X}$, and its codimension is
 127     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 128     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 129     $\Lie{X}$. We can write a column vector $x \coloneqq
 130     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 131     matrix with $\matricize{x}$. To recover the vector, we use
 132     $\vectorize{\matricize{x}}$.
 133
 134     The set of all bounded linear operators from $V$ to $W$ is
 135     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 136     instead.
 137
 138     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 139     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 140     writing $\directsumperp{V}{W}$. That operator should survive a
 141     display equation, too, and the weight of the circle should match
 142     that of the usual direct sum operator.
 143     %
 144     \begin{align*}
 145       Z = \directsumperp{V}{W}\\
 146       \oplus \oplusperp \oplus \oplusperp
 147     \end{align*}
 148     %
 149     Its form should also survive in different font sizes...
 150     \Large
 151     \begin{align*}
 152       Z = \directsumperp{V}{W}\\
 153       \oplus \oplusperp \oplus \oplusperp
 154     \end{align*}
 155     \Huge
 156     \begin{align*}
 157       Z = \directsumperp{V}{W}\\
 158       \oplus \oplusperp \oplus \oplusperp
 159     \end{align*}
 160     \normalsize
 161   \end{section}
 162
 163   \begin{section}{Listing}
 164     Here's an interactive sage prompt:
 165
 166     \begin{tcblisting}{listing only,
 167                        colback=codebg,
 168                        coltext=codefg,
 169                        listing options={language=sage,style=sage}}
 170     sage: K = Cone([ (1,0), (0,1) ])
 171     sage: K.positive_operator_gens()
 172     [
 173     [1 0]  [0 1]  [0 0]  [0 0]
 174     [0 0], [0 0], [1 0], [0 1]
 175     ]
 176     \end{tcblisting}
 177   \end{section}
 178
 179   \begin{section}{Miscellaneous}
 180     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 181     = 3$.
 182   \end{section}
 183
 184   \begin{section}{Proof by cases}
 185
 186     \begin{proposition}
 187       There are two cases in the following proof.
 188
 189       \begin{proof}
 190         The result should be self-evident once we have considered the
 191         following two cases.
 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         You see?
 202       \end{proof}
 203     \end{proposition}
 204
 205     Here's another one.
 206
 207     \renewcommand{\baselinestretch}{2}
 208     \begin{proposition}
 209       Cases should display intelligently even when the document is
 210       double-spaced.
 211
 212       \begin{proof}
 213         Here we go again.
 214
 215         \begin{pcases}
 216           \begin{case}[first case]
 217             Nothing happens in the first case.
 218           \end{case}
 219           \begin{case}[second case]
 220             The same thing happens in the second case.
 221           \end{case}
 222         \end{pcases}
 223
 224         Now it's over.
 225       \end{proof}
 226     \end{proposition}
 227     \renewcommand{\baselinestretch}{1}
 228   \end{section}
 229
 230   \begin{section}{Theorems}
 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{exercise}
 248       jumps
 249     \end{exercise}
 250
 251     \begin{definition}
 252       quod
 253     \end{definition}
 254
 255     \begin{example}
 256       erat
 257     \end{example}
 258
 259     \begin{remark}
 260       demonstradum.
 261     \end{remark}
 262   \end{section}
 263
 264   \begin{section}{Theorems (starred)}
 265     \begin{corollary*}
 266       The
 267     \end{corollary*}
 268
 269     \begin{lemma*}
 270       quick
 271     \end{lemma*}
 272
 273     \begin{proposition*}
 274       brown
 275     \end{proposition*}
 276
 277     \begin{theorem*}
 278       fox
 279     \end{theorem*}
 280
 281     \begin{exercise*}
 282       jumps
 283     \end{exercise*}
 284
 285     \begin{definition*}
 286       quod
 287     \end{definition*}
 288
 289     \begin{example*}
 290       erat
 291     \end{example*}
 292
 293     \begin{remark*}
 294       demonstradum.
 295     \end{remark*}
 296   \end{section}
 297
 298   \begin{section}{Topology}
 299     The interior of a set $X$ is $\interior{X}$. Its closure is
 300     $\closure{X}$ and its boundary is $\boundary{X}$.
 301   \end{section}
 302
 303 \end{document}