examples.tex

   1 \documentclass{report}
   2
   3 % We have to load this before mjotex so that mjotex knows to define
   4 % its glossary entries.
   5 \usepackage[nonumberlist]{glossaries}
   6 \makenoidxglossaries
   7
   8 \usepackage{mjotex}
   9 \usepackage{mathtools}
  10
  11 \begin{document}
  12
  13   \begin{section}{Algebra}
  14     If $R$ is a commutative ring, then $\polyring{R}{X,Y,Z}$ is a
  15     multivariate polynomial ring with indeterminates $X$, $Y$, and
  16     $Z$, and coefficients in $R$. If $R$ is a moreover an integral
  17     domain, then its fraction field is $\Frac{R}$.
  18   \end{section}
  19
  20   \begin{section}{Algorithm}
  21     An example of an algorithm (bogosort) environment.
  22
  23     \begin{algorithm}[H]
  24       \caption{Sort a list of numbers}
  25       \begin{algorithmic}
  26         \Require{A list of numbers $L$}
  27         \Ensure{A new, sorted copy $M$ of the list $L$}
  28
  29         \State{$M \gets L$}
  30
  31         \While{$M$ is not sorted}
  32           \State{Rearrange $M$ randomly}
  33         \EndWhile
  34
  35         \Return{$M$}
  36       \end{algorithmic}
  37     \end{algorithm}
  38   \end{section}
  39
  40   \begin{section}{Arrow}
  41     The identity operator on $V$ is $\identity{V}$. The composition of
  42     $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  43     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  44     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  45   \end{section}
  46
  47   \begin{section}{Calculus}
  48     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  49     \Rn \rightarrow \Rn$.
  50   \end{section}
  51
  52   \begin{section}{Common}
  53     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
  54     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  55     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
  56     of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
  57     and the factorial of the number $10$ is $\factorial{10}$.
  58
  59     The Cartesian product of two sets $A$ and $B$ is
  60     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
  61     we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
  62     is $\directsum{V}{W}$. Or three things,
  63     $\directsumthree{U}{V}{W}$. How about more things? Like
  64     $\directsummany{k=1}{\infty}{V_{k}} \ne
  65     \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
  66     cartesian products adapt nicely to display equations:
  67     %
  68     \begin{equation*}
  69       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
  70     \end{equation*}
  71     Here are a few common tuple spaces that should not have a
  72     superscript when that superscript would be one: $\Nn[1]$,
  73     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
  74     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
  75     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
  76
  77     We also have a few basic set operations, for example the union of
  78     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
  79     course with union comes intersection: $\intersect{A}{B}$,
  80     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
  81     (indexed) union and intersections of things, like
  82     $\unionmany{k=1}{\infty}{A_{k}}$ or
  83     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
  84     is that they do the right thing in a display equation:
  85     %
  86     \begin{equation*}
  87       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
  88     \end{equation*}
  89
  90     Finally, we have the four standard types of intervals in $\Rn[1]$,
  91     %
  92     \begin{align*}
  93       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
  94       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
  95       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
  96       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
  97     \end{align*}
  98   \end{section}
  99
 100   \begin{section}{Complex}
 101     We sometimes want to conjugate complex numbers like
 102     $\compconj{a+bi} = a - bi$.
 103   \end{section}
 104
 105   \begin{section}{Cone}
 106     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 107     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 108     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 109     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 110     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 111     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 112     \ltcone_{K} y$ with respect to a cone $K$.
 113   \end{section}
 114
 115   \begin{section}{Convex}
 116     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 117     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 118     then its lineality space is $\linspace{K}$, its lineality is
 119     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 120     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 121     proper face, then we write $F \properfaceof K$.
 122   \end{section}
 123
 124   \begin{section}{Euclidean Jordan algebras}
 125     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 126     is $\jp{x}{y}$.
 127   \end{section}
 128
 129   \begin{section}{Font}
 130     We can write things like Carathéodory and Güler and $\mathbb{R}$.
 131   \end{section}
 132
 133   \begin{section}{Linear algebra}
 134     The absolute value of $x$ is $\abs{x}$, or its norm is
 135     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 136     their tensor product is $\tp{x}{y}$. The Kronecker product of
 137     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 138     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 139     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
 140     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 141     $\pseudoinverse{L}$.
 142
 143     The span of a set $X$ is $\spanof{X}$, and its codimension is
 144     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 145     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 146     $\Lie{X}$. We can write a column vector $x \coloneqq
 147     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 148     matrix with $\matricize{x}$. To recover the vector, we use
 149     $\vectorize{\matricize{x}}$.
 150
 151     The set of all bounded linear operators from $V$ to $W$ is
 152     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 153     instead.
 154
 155     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 156     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 157     writing $\directsumperp{V}{W}$. That operator should survive a
 158     display equation, too, and the weight of the circle should match
 159     that of the usual direct sum operator.
 160     %
 161     \begin{align*}
 162       Z = \directsumperp{V}{W}\\
 163       \oplus \oplusperp \oplus \oplusperp
 164     \end{align*}
 165     %
 166     Its form should also survive in different font sizes...
 167     \Large
 168     \begin{align*}
 169       Z = \directsumperp{V}{W}\\
 170       \oplus \oplusperp \oplus \oplusperp
 171     \end{align*}
 172     \Huge
 173     \begin{align*}
 174       Z = \directsumperp{V}{W}\\
 175       \oplus \oplusperp \oplus \oplusperp
 176     \end{align*}
 177     \normalsize
 178   \end{section}
 179
 180   \begin{section}{Listing}
 181     Here's an interactive SageMath prompt:
 182
 183     \begin{tcblisting}{listing only,
 184                        colback=codebg,
 185                        coltext=codefg,
 186                        listing options={language=sage,style=sage}}
 187     sage: K = Cone([ (1,0), (0,1) ])
 188     sage: K.positive_operator_gens()
 189     [
 190     [1 0]  [0 1]  [0 0]  [0 0]
 191     [0 0], [0 0], [1 0], [0 1]
 192     ]
 193     \end{tcblisting}
 194
 195     However, the smart way to display a SageMath listing is to load it
 196     from an external file (under the ``listings'' subdirectory):
 197
 198     \sagelisting{example}
 199
 200     Keeping the listings in separate files makes it easy for the build
 201     system to test them.
 202   \end{section}
 203
 204   \begin{section}{Miscellaneous}
 205     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 206     = 3$.
 207   \end{section}
 208
 209   \begin{section}{Proof by cases}
 210
 211     \begin{proposition}
 212       There are two cases in the following proof.
 213
 214       \begin{proof}
 215         The result should be self-evident once we have considered the
 216         following two cases.
 217         \begin{pcases}
 218           \begin{case}[first case]
 219             Nothing happens in the first case.
 220           \end{case}
 221           \begin{case}[second case]
 222             The same thing happens in the second case.
 223           \end{case}
 224         \end{pcases}
 225
 226         You see?
 227       \end{proof}
 228     \end{proposition}
 229
 230     Here's another one.
 231
 232     \renewcommand{\baselinestretch}{2}
 233     \begin{proposition}
 234       Cases should display intelligently even when the document is
 235       double-spaced.
 236
 237       \begin{proof}
 238         Here we go again.
 239
 240         \begin{pcases}
 241           \begin{case}[first case]
 242             Nothing happens in the first case.
 243           \end{case}
 244           \begin{case}[second case]
 245             The same thing happens in the second case.
 246           \end{case}
 247         \end{pcases}
 248
 249         Now it's over.
 250       \end{proof}
 251     \end{proposition}
 252     \renewcommand{\baselinestretch}{1}
 253   \end{section}
 254
 255   \begin{section}{Theorems}
 256     \begin{corollary}
 257       The
 258     \end{corollary}
 259
 260     \begin{lemma}
 261       quick
 262     \end{lemma}
 263
 264     \begin{proposition}
 265       brown
 266     \end{proposition}
 267
 268     \begin{theorem}
 269       fox
 270     \end{theorem}
 271
 272     \begin{exercise}
 273       jumps
 274     \end{exercise}
 275
 276     \begin{definition}
 277       quod
 278     \end{definition}
 279
 280     \begin{example}
 281       erat
 282     \end{example}
 283
 284     \begin{remark}
 285       demonstradum.
 286     \end{remark}
 287   \end{section}
 288
 289   \begin{section}{Theorems (starred)}
 290     \begin{corollary*}
 291       The
 292     \end{corollary*}
 293
 294     \begin{lemma*}
 295       quick
 296     \end{lemma*}
 297
 298     \begin{proposition*}
 299       brown
 300     \end{proposition*}
 301
 302     \begin{theorem*}
 303       fox
 304     \end{theorem*}
 305
 306     \begin{exercise*}
 307       jumps
 308     \end{exercise*}
 309
 310     \begin{definition*}
 311       quod
 312     \end{definition*}
 313
 314     \begin{example*}
 315       erat
 316     \end{example*}
 317
 318     \begin{remark*}
 319       demonstradum.
 320     \end{remark*}
 321   \end{section}
 322
 323   \begin{section}{Topology}
 324     The interior of a set $X$ is $\interior{X}$. Its closure is
 325     $\closure{X}$ and its boundary is $\boundary{X}$.
 326   \end{section}
 327
 328   \setlength{\glslistdottedwidth}{.3\linewidth}
 329   \setglossarystyle{listdotted}
 330   \glsaddall
 331   \printnoidxglossaries
 332 \end{document}