examples.tex

   1 \documentclass{report}
   2
   3 % Setting hypertexnames=false forces hyperref to use a consistent
   4 % internal counter for proposition/equation references rather than
   5 % being clever, which doesn't work after we reset those counters.
   6 \usepackage[hypertexnames=false]{hyperref}
   7 \hypersetup{
   8   colorlinks=true,
   9   linkcolor=blue,
  10   citecolor=blue
  11 }
  12
  13 % We have to load this after hyperref, so that links work, but before
  14 % mjotex so that mjotex knows to define its glossary entries.
  15 \usepackage[nonumberlist]{glossaries}
  16 \makenoidxglossaries
  17
  18 % If you want an index, we can do that too. You'll need to define
  19 % the "INDICES" variable in the GNUmakefile, though.
  20 \usepackage{makeidx}
  21 \makeindex
  22
  23 \usepackage{mjotex}
  24 \usepackage{mathtools}
  25
  26 \begin{document}
  27
  28   \begin{section}{Algebra}
  29     If $R$ is a \index{commutative ring}, then $\polyring{R}{X,Y,Z}$
  30     is a multivariate polynomial ring with indeterminates $X$, $Y$,
  31     and $Z$, and coefficients in $R$. If $R$ is a moreover an integral
  32     domain, then its fraction field is $\Frac{R}$.
  33   \end{section}
  34
  35   \begin{section}{Algorithm}
  36     An example of an algorithm (bogosort) environment.
  37
  38     \begin{algorithm}[H]
  39       \caption{Sort a list of numbers}
  40       \begin{algorithmic}
  41         \Require{A list of numbers $L$}
  42         \Ensure{A new, sorted copy $M$ of the list $L$}
  43
  44         \State{$M \gets L$}
  45
  46         \While{$M$ is not sorted}
  47           \State{Rearrange $M$ randomly}
  48         \EndWhile
  49
  50         \Return{$M$}
  51       \end{algorithmic}
  52     \end{algorithm}
  53   \end{section}
  54
  55   \begin{section}{Arrow}
  56     The constant function that always returns $a$ is $\const{a}$. The
  57     identity operator on $V$ is $\identity{V}$. The composition of $f$
  58     and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  59     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  60     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  61   \end{section}
  62
  63   \begin{section}{Calculus}
  64     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  65     \Rn \rightarrow \Rn$.
  66   \end{section}
  67
  68   \begin{section}{Common}
  69     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
  70     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  71     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
  72     of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
  73     and the factorial of the number $10$ is $\factorial{10}$.
  74
  75     The Cartesian product of two sets $A$ and $B$ is
  76     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
  77     we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
  78     is $\directsum{V}{W}$. Or three things,
  79     $\directsumthree{U}{V}{W}$. How about more things? Like
  80     $\directsummany{k=1}{\infty}{V_{k}} \ne
  81     \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
  82     cartesian products adapt nicely to display equations:
  83     %
  84     \begin{equation*}
  85       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
  86     \end{equation*}
  87     Here are a few common tuple spaces that should not have a
  88     superscript when that superscript would be one: $\Nn[1]$,
  89     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
  90     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
  91     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
  92
  93     We also have a few basic set operations, for example the union of
  94     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
  95     course with union comes intersection: $\intersect{A}{B}$,
  96     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
  97     (indexed) union and intersections of things, like
  98     $\unionmany{k=1}{\infty}{A_{k}}$ or
  99     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
 100     is that they do the right thing in a display equation:
 101     %
 102     \begin{equation*}
 103       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
 104     \end{equation*}
 105
 106     Finally, we have the four standard types of intervals in $\Rn[1]$,
 107     %
 108     \begin{align*}
 109       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 110       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 111       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 112       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 113     \end{align*}
 114   \end{section}
 115
 116   \begin{section}{Complex}
 117     We sometimes want to conjugate complex numbers like
 118     $\compconj{a+bi} = a - bi$.
 119   \end{section}
 120
 121   \begin{section}{Cone}
 122     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 123     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 124     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 125     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 126     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 127     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 128     \ltcone_{K} y$ with respect to a cone $K$.
 129   \end{section}
 130
 131   \begin{section}{Convex}
 132     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 133     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 134     then its lineality space is $\linspace{K}$, its lineality is
 135     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 136     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 137     proper face, then we write $F \properfaceof K$.
 138   \end{section}
 139
 140   \begin{section}{Euclidean Jordan algebras}
 141     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 142     is $\jp{x}{y}$.
 143   \end{section}
 144
 145   \begin{section}{Font}
 146     We can write things like Carathéodory and Güler and $\mathbb{R}$.
 147   \end{section}
 148
 149   \begin{section}{Linear algebra}
 150     The absolute value of $x$ is $\abs{x}$, or its norm is
 151     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 152     their tensor product is $\tp{x}{y}$. The Kronecker product of
 153     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 154     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 155     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
 156     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 157     $\pseudoinverse{L}$.
 158
 159     The span of a set $X$ is $\spanof{X}$, and its codimension is
 160     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 161     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 162     $\Lie{X}$. We can write a column vector $x \coloneqq
 163     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 164     matrix with $\matricize{x}$. To recover the vector, we use
 165     $\vectorize{\matricize{x}}$.
 166
 167     The set of all bounded linear operators from $V$ to $W$ is
 168     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 169     instead.
 170
 171     If you want to solve a system of equations, try Cramer's
 172     rule~\cite{ehrenborg}.
 173
 174     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 175     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 176     writing $\directsumperp{V}{W}$. That operator should survive a
 177     display equation, too, and the weight of the circle should match
 178     that of the usual direct sum operator.
 179     %
 180     \begin{align*}
 181       Z = \directsumperp{V}{W}\\
 182       \oplus \oplusperp \oplus \oplusperp
 183     \end{align*}
 184     %
 185     Its form should also survive in different font sizes...
 186     \Large
 187     \begin{align*}
 188       Z = \directsumperp{V}{W}\\
 189       \oplus \oplusperp \oplus \oplusperp
 190     \end{align*}
 191     \Huge
 192     \begin{align*}
 193       Z = \directsumperp{V}{W}\\
 194       \oplus \oplusperp \oplus \oplusperp
 195     \end{align*}
 196     \normalsize
 197   \end{section}
 198
 199   \begin{section}{Listing}
 200     Here's an interactive SageMath prompt:
 201
 202     \begin{tcblisting}{listing only,
 203                        colback=codebg,
 204                        coltext=codefg,
 205                        listing options={language=sage,style=sage}}
 206     sage: K = Cone([ (1,0), (0,1) ])
 207     sage: K.positive_operator_gens()
 208     [
 209     [1 0]  [0 1]  [0 0]  [0 0]
 210     [0 0], [0 0], [1 0], [0 1]
 211     ]
 212     \end{tcblisting}
 213
 214     However, the smart way to display a SageMath listing is to load it
 215     from an external file (under the ``listings'' subdirectory):
 216
 217     \sagelisting{example}
 218
 219     Keeping the listings in separate files makes it easy for the build
 220     system to test them.
 221   \end{section}
 222
 223   \begin{section}{Miscellaneous}
 224     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 225     = 3$.
 226   \end{section}
 227
 228   \begin{section}{Proof by cases}
 229
 230     \begin{proposition}
 231       There are two cases in the following proof.
 232
 233       \begin{proof}
 234         The result should be self-evident once we have considered the
 235         following two cases.
 236         \begin{pcases}
 237           \begin{case}[first case]
 238             Nothing happens in the first case.
 239           \end{case}
 240           \begin{case}[second case]
 241             The same thing happens in the second case.
 242           \end{case}
 243         \end{pcases}
 244
 245         You see?
 246       \end{proof}
 247     \end{proposition}
 248
 249     Here's another one.
 250
 251     \renewcommand{\baselinestretch}{2}
 252     \begin{proposition}
 253       Cases should display intelligently even when the document is
 254       double-spaced.
 255
 256       \begin{proof}
 257         Here we go again.
 258
 259         \begin{pcases}
 260           \begin{case}[first case]
 261             Nothing happens in the first case.
 262           \end{case}
 263           \begin{case}[second case]
 264             The same thing happens in the second case.
 265           \end{case}
 266         \end{pcases}
 267
 268         Now it's over.
 269       \end{proof}
 270     \end{proposition}
 271     \renewcommand{\baselinestretch}{1}
 272   \end{section}
 273
 274   \begin{section}{Theorems}
 275     \begin{corollary}
 276       The
 277     \end{corollary}
 278
 279     \begin{lemma}
 280       quick
 281     \end{lemma}
 282
 283     \begin{proposition}
 284       brown
 285     \end{proposition}
 286
 287     \begin{theorem}
 288       fox
 289     \end{theorem}
 290
 291     \begin{exercise}
 292       jumps
 293     \end{exercise}
 294
 295     \begin{definition}
 296       quod
 297     \end{definition}
 298
 299     \begin{example}
 300       erat
 301     \end{example}
 302
 303     \begin{remark}
 304       demonstradum.
 305     \end{remark}
 306   \end{section}
 307
 308   \begin{section}{Theorems (starred)}
 309     \begin{corollary*}
 310       The
 311     \end{corollary*}
 312
 313     \begin{lemma*}
 314       quick
 315     \end{lemma*}
 316
 317     \begin{proposition*}
 318       brown
 319     \end{proposition*}
 320
 321     \begin{theorem*}
 322       fox
 323     \end{theorem*}
 324
 325     \begin{exercise*}
 326       jumps
 327     \end{exercise*}
 328
 329     \begin{definition*}
 330       quod
 331     \end{definition*}
 332
 333     \begin{example*}
 334       erat
 335     \end{example*}
 336
 337     \begin{remark*}
 338       demonstradum.
 339     \end{remark*}
 340   \end{section}
 341
 342   \begin{section}{Topology}
 343     The interior of a set $X$ is $\interior{X}$. Its closure is
 344     $\closure{X}$ and its boundary is $\boundary{X}$.
 345   \end{section}
 346
 347   \setlength{\glslistdottedwidth}{.3\linewidth}
 348   \setglossarystyle{listdotted}
 349   \glsaddall
 350   \printnoidxglossaries
 351
 352   \bibliographystyle{mjo}
 353   \bibliography{local-references}
 354
 355   \printindex
 356 \end{document}