examples.tex

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