mjo-common.tex

   1 %
   2 % Only the most commonly-used macros. Needed by everything else.
   3 %
   4 \ifx\havemjocommon\undefined
   5 \def\havemjocommon{1}
   6
   7 \ifx\mathbb\undefined
   8   \usepackage{amsfonts}
   9 \fi
  10
  11 \ifx\bigtimes\undefined
  12   \usepackage{mathtools}
  13 \fi
  14
  15 % Place the argument in matching left/right parentheses.
  16 \newcommand*{\of}[1]{ \left({#1}\right) }
  17
  18 % Group terms using parentheses.
  19 \newcommand*{\qty}[1]{ \left({#1}\right) }
  20
  21 % Group terms using square brackets.
  22 \newcommand*{\sqty}[1]{ \left[{#1}\right] }
  23
  24 % Create a set from the given elements
  25 \newcommand*{\set}[1]{\left\lbrace{#1}\right\rbrace}
  26
  27 % A set comprehension, where the ``such that...'' bar is added
  28 % automatically. The bar was chosen over a colon to avoid ambiguity
  29 % with the L : V -> V notation. We can't leverage \set here because \middle
  30 % needs \left and \right present.
  31 \newcommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace}
  32
  33 % A pair of things.
  34 \newcommand*{\pair}[2]{ \left({#1},{#2}\right) }
  35
  36 % A triple of things.
  37 \newcommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) }
  38
  39 % A four-tuple of things.
  40 \newcommand*{\quadruple}[4]{ \left({#1},{#2},{#3},{#4}\right) }
  41
  42 % A five-tuple of things.
  43 \newcommand*{\quintuple}[5]{ \left({#1},{#2},{#3},{#4},{#5}\right) }
  44
  45 % A six-tuple of things.
  46 \newcommand*{\sextuple}[6]{ \left({#1},{#2},{#3},{#4},{#5},{#6}\right) }
  47
  48 % A seven-tuple of things.
  49 \newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
  50
  51 % The Cartesian product of two things.
  52 \newcommand*{\cartprod}[2]{ {#1}\times{#2} }
  53
  54 % The Cartesian product of three things.
  55 \newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
  56
  57 % The direct sum of two things.
  58 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
  59
  60 % The direct sum of three things.
  61 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  62
  63 % The factorial operator.
  64 \newcommand*{\factorial}[1]{ {#1}! }
  65
  66 %
  67 % Product spaces
  68 %
  69 % All of the product spaces (for example, R^n) that follow default to
  70 % an exponent of ``n'', but that exponent can be changed by providing
  71 % it as an optional argument. If the exponent given is ``1'', then it
  72 % will be omitted entirely.
  73 %
  74
  75 % The natural n-space, N x N x N x ... x N.
  76 \newcommand*{\Nn}[1][n]{
  77   \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  78 }
  79
  80 \ifdefined\newglossaryentry
  81   \newglossaryentry{N}{
  82     name={\ensuremath{\Nn[1]}},
  83     description={the set of natural numbers},
  84     sort=N
  85   }
  86 \fi
  87
  88 % The integral n-space, Z x Z x Z x ... x Z.
  89 \newcommand*{\Zn}[1][n]{
  90   \mathbb{Z}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  91 }
  92
  93 \ifdefined\newglossaryentry
  94   \newglossaryentry{Z}{
  95     name={\ensuremath{\Zn[1]}},
  96     description={the ring of integers},
  97     sort=Z
  98   }
  99 \fi
 100
 101 % The rational n-space, Q x Q x Q x ... x Q.
 102 \newcommand*{\Qn}[1][n]{
 103   \mathbb{Q}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 104 }
 105
 106 \ifdefined\newglossaryentry
 107   \newglossaryentry{Q}{
 108     name={\ensuremath{\Qn[1]}},
 109     description={the field of rational numbers},
 110     sort=Q
 111   }
 112 \fi
 113
 114 % The real n-space, R x R x R x ... x R.
 115 \newcommand*{\Rn}[1][n]{
 116   \mathbb{R}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 117 }
 118
 119 \ifdefined\newglossaryentry
 120   \newglossaryentry{R}{
 121     name={\ensuremath{\Rn[1]}},
 122     description={the field of real numbers},
 123     sort=R
 124   }
 125 \fi
 126
 127
 128 % The complex n-space, C x C x C x ... x C.
 129 \newcommand*{\Cn}[1][n]{
 130   \mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 131 }
 132
 133 \ifdefined\newglossaryentry
 134   \newglossaryentry{C}{
 135     name={\ensuremath{\Cn[1]}},
 136     description={the field of complex numbers},
 137     sort=C
 138   }
 139 \fi
 140
 141
 142 %
 143 % Basic set operations
 144 %
 145
 146 % The union of its two arguments.
 147 \newcommand*{\union}[2]{ {#1}\cup{#2} }
 148
 149 % A three-argument union.
 150 \newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
 151
 152 % The intersection of its two arguments.
 153 \newcommand*{\intersect}[2]{ {#1}\cap{#2} }
 154
 155 % A three-argument intersection.
 156 \newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
 157
 158 % An indexed arbitrary binary operation such as the union or
 159 % intersection of an infinite number of sets. The first argument is
 160 % the operator symbol to use, such as \cup for a union. The second
 161 % argument is the lower index, for example k=1. The third argument is
 162 % the upper index, such as \infty. Finally the fourth argument should
 163 % contain the things (e.g. indexed sets) to be operated on.
 164 \newcommand*{\binopmany}[4]{
 165   \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} }
 166              { {#1}_{#2}^{#3}{#4} }
 167              { {#1}_{#2}^{#3}{#4} }
 168              { {#1}_{#2}^{#3}{#4} }
 169 }
 170
 171 \newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
 172 \newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
 173 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
 174 \newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
 175
 176 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
 177 % the real line.
 178 \newcommand*{\intervaloo}[2]{ \left({#1},{#2}\right) } % open-open
 179 \newcommand*{\intervaloc}[2]{ \left({#1},{#2}\right] } % open-closed
 180 \newcommand*{\intervalco}[2]{ \left[{#1},{#2}\right) } % closed-open
 181 \newcommand*{\intervalcc}[2]{ \left[{#1},{#2}\right] } % closed-closed
 182
 183
 184 \fi