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
   8 \ifx\mathbb\undefined
   9   \usepackage{amsfonts}
  10 \fi
  11
  12 \ifx\bigtimes\undefined
  13   \usepackage{mathtools}
  14 \fi
  15
  16 % Place the argument in matching left/right parentheses.
  17 \newcommand*{\of}[1]{ \left({#1}\right) }
  18
  19 % Group terms using parentheses.
  20 \newcommand*{\qty}[1]{ \left({#1}\right) }
  21
  22 % Group terms using square brackets.
  23 \newcommand*{\sqty}[1]{ \left[{#1}\right] }
  24
  25 % Create a set from the given elements
  26 \newcommand*{\set}[1]{\left\lbrace{#1}\right\rbrace}
  27
  28 % A set comprehension, where the ``such that...'' bar is added
  29 % automatically. The bar was chosen over a colon to avoid ambiguity
  30 % with the L : V -> V notation. We can't leverage \set here because \middle
  31 % needs \left and \right present.
  32 \newcommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace}
  33
  34 % A pair of things.
  35 \newcommand*{\pair}[2]{ \left({#1},{#2}\right) }
  36
  37 % A triple of things.
  38 \newcommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) }
  39
  40 % A four-tuple of things.
  41 \newcommand*{\quadruple}[4]{ \left({#1},{#2},{#3},{#4}\right) }
  42
  43 % A five-tuple of things.
  44 \newcommand*{\quintuple}[5]{ \left({#1},{#2},{#3},{#4},{#5}\right) }
  45
  46 % A six-tuple of things.
  47 \newcommand*{\sextuple}[6]{ \left({#1},{#2},{#3},{#4},{#5},{#6}\right) }
  48
  49 % A seven-tuple of things.
  50 \newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
  51
  52 % The Cartesian product of two things.
  53 \newcommand*{\cartprod}[2]{ {#1}\times{#2} }
  54
  55 % The Cartesian product of three things.
  56 \newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
  57
  58 % The direct sum of two things.
  59 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
  60
  61 % The direct sum of three things.
  62 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  63
  64 % The factorial operator.
  65 \newcommand*{\factorial}[1]{ {#1}! }
  66
  67 %
  68 % Product spaces
  69 %
  70 % All of the product spaces (for example, R^n) that follow default to
  71 % an exponent of ``n'', but that exponent can be changed by providing
  72 % it as an optional argument. If the exponent given is ``1'', then it
  73 % will be omitted entirely.
  74 %
  75
  76 % The natural n-space, N x N x N x ... x N.
  77 \newcommand*{\Nn}[1][n]{
  78   \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  79 }
  80
  81 \ifdefined\newglossaryentry
  82   \newglossaryentry{N}{
  83     name={\ensuremath{\Nn[1]}},
  84     description={the set of natural numbers},
  85     sort=N
  86   }
  87 \fi
  88
  89 % The integral n-space, Z x Z x Z x ... x Z.
  90 \newcommand*{\Zn}[1][n]{
  91   \mathbb{Z}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  92 }
  93
  94 \ifdefined\newglossaryentry
  95   \newglossaryentry{Z}{
  96     name={\ensuremath{\Zn[1]}},
  97     description={the ring of integers},
  98     sort=Z
  99   }
 100 \fi
 101
 102 % The rational n-space, Q x Q x Q x ... x Q.
 103 \newcommand*{\Qn}[1][n]{
 104   \mathbb{Q}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 105 }
 106
 107 \ifdefined\newglossaryentry
 108   \newglossaryentry{Q}{
 109     name={\ensuremath{\Qn[1]}},
 110     description={the field of rational numbers},
 111     sort=Q
 112   }
 113 \fi
 114
 115 % The real n-space, R x R x R x ... x R.
 116 \newcommand*{\Rn}[1][n]{
 117   \mathbb{R}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 118 }
 119
 120 \ifdefined\newglossaryentry
 121   \newglossaryentry{R}{
 122     name={\ensuremath{\Rn[1]}},
 123     description={the field of real numbers},
 124     sort=R
 125   }
 126 \fi
 127
 128
 129 % The complex n-space, C x C x C x ... x C.
 130 \newcommand*{\Cn}[1][n]{
 131   \mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 132 }
 133
 134 \ifdefined\newglossaryentry
 135   \newglossaryentry{C}{
 136     name={\ensuremath{\Cn[1]}},
 137     description={the field of complex numbers},
 138     sort=C
 139   }
 140 \fi
 141
 142
 143 % The space of real symmetric n-by-n matrices.
 144 \newcommand*{\Sn}[1][n]{
 145   \mathcal{S}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 146 }
 147
 148 \ifdefined\newglossaryentry
 149   \newglossaryentry{Sn}{
 150     name={\ensuremath{\Sn}},
 151     description={the set of $n$-by-$n$ real symmetric matrices},
 152     sort=Sn
 153   }
 154 \fi
 155
 156 % The space of complex Hermitian n-by-n matrices.
 157 \newcommand*{\Hn}[1][n]{
 158   \mathcal{H}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 159 }
 160
 161 \ifdefined\newglossaryentry
 162   \newglossaryentry{Hn}{
 163     name={\ensuremath{\Hn}},
 164     description={the set of $n$-by-$n$ complex Hermitian matrices},
 165     sort=Hn
 166   }
 167 \fi
 168
 169 %
 170 % Basic set operations
 171 %
 172
 173 % The union of its two arguments.
 174 \newcommand*{\union}[2]{ {#1}\cup{#2} }
 175
 176 % A three-argument union.
 177 \newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
 178
 179 % The intersection of its two arguments.
 180 \newcommand*{\intersect}[2]{ {#1}\cap{#2} }
 181
 182 % A three-argument intersection.
 183 \newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
 184
 185 % An indexed arbitrary binary operation such as the union or
 186 % intersection of an infinite number of sets. The first argument is
 187 % the operator symbol to use, such as \cup for a union. The second
 188 % argument is the lower index, for example k=1. The third argument is
 189 % the upper index, such as \infty. Finally the fourth argument should
 190 % contain the things (e.g. indexed sets) to be operated on.
 191 \newcommand*{\binopmany}[4]{
 192   \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} }
 193              { {#1}_{#2}^{#3}{#4} }
 194              { {#1}_{#2}^{#3}{#4} }
 195              { {#1}_{#2}^{#3}{#4} }
 196 }
 197
 198 \newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
 199 \newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
 200 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
 201 \newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
 202
 203
 204 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
 205 % the real line.
 206 \newcommand*{\intervaloo}[2]{ \left({#1},{#2}\right) } % open-open
 207 \newcommand*{\intervaloc}[2]{ \left({#1},{#2}\right] } % open-closed
 208 \newcommand*{\intervalco}[2]{ \left[{#1},{#2}\right) } % closed-open
 209 \newcommand*{\intervalcc}[2]{ \left[{#1},{#2}\right] } % closed-closed
 210
 211
 212 \fi