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 \input{mjo-font} % amsfonts and \mathpzc
   9
  10 \ifx\bigtimes\undefined
  11   \usepackage{mathtools}
  12 \fi
  13
  14 % Place the argument in matching left/right parentheses.
  15 \newcommand*{\of}[1]{ \left({#1}\right) }
  16
  17 % Group terms using parentheses.
  18 \newcommand*{\qty}[1]{ \left({#1}\right) }
  19
  20 % Group terms using square brackets.
  21 \newcommand*{\sqty}[1]{ \left[{#1}\right] }
  22
  23 % Create a set from the given elements
  24 \newcommand*{\set}[1]{\left\lbrace{#1}\right\rbrace}
  25
  26 % A set comprehension, where the ``such that...'' bar is added
  27 % automatically. The bar was chosen over a colon to avoid ambiguity
  28 % with the L : V -> V notation. We can't leverage \set here because \middle
  29 % needs \left and \right present.
  30 \newcommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace}
  31
  32 % A pair of things.
  33 \newcommand*{\pair}[2]{ \left({#1},{#2}\right) }
  34
  35 % A triple of things.
  36 \newcommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) }
  37
  38 % A four-tuple of things.
  39 \newcommand*{\quadruple}[4]{ \left({#1},{#2},{#3},{#4}\right) }
  40
  41 % A five-tuple of things.
  42 \newcommand*{\quintuple}[5]{ \left({#1},{#2},{#3},{#4},{#5}\right) }
  43
  44 % A six-tuple of things.
  45 \newcommand*{\sextuple}[6]{ \left({#1},{#2},{#3},{#4},{#5},{#6}\right) }
  46
  47 % A seven-tuple of things.
  48 \newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
  49
  50 % The Cartesian product of two things.
  51 \newcommand*{\cartprod}[2]{ {#1}\times{#2} }
  52
  53 % The Cartesian product of three things.
  54 \newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
  55
  56 % The direct sum of two things.
  57 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
  58
  59 % The direct sum of three things.
  60 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  61
  62 % The factorial operator.
  63 \newcommand*{\factorial}[1]{ {#1}! }
  64
  65 %
  66 % Product spaces
  67 %
  68 % All of the product spaces (for example, R^n) that follow default to
  69 % an exponent of ``n'', but that exponent can be changed by providing
  70 % it as an optional argument. If the exponent given is ``1'', then it
  71 % will be omitted entirely.
  72 %
  73
  74 % The natural n-space, N x N x N x ... x N.
  75 \newcommand*{\Nn}[1][n]{
  76   \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  77 }
  78
  79 \ifdefined\newglossaryentry
  80   \newglossaryentry{N}{
  81     name={\ensuremath{\Nn[1]}},
  82     description={the set of natural numbers},
  83     sort=N
  84   }
  85 \fi
  86
  87 % The integral n-space, Z x Z x Z x ... x Z.
  88 \newcommand*{\Zn}[1][n]{
  89   \mathbb{Z}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  90 }
  91
  92 \ifdefined\newglossaryentry
  93   \newglossaryentry{Z}{
  94     name={\ensuremath{\Zn[1]}},
  95     description={the ring of integers},
  96     sort=Z
  97   }
  98 \fi
  99
 100 % The rational n-space, Q x Q x Q x ... x Q.
 101 \newcommand*{\Qn}[1][n]{
 102   \mathbb{Q}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 103 }
 104
 105 \ifdefined\newglossaryentry
 106   \newglossaryentry{Q}{
 107     name={\ensuremath{\Qn[1]}},
 108     description={the field of rational numbers},
 109     sort=Q
 110   }
 111 \fi
 112
 113 % The real n-space, R x R x R x ... x R.
 114 \newcommand*{\Rn}[1][n]{
 115   \mathbb{R}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 116 }
 117
 118 \ifdefined\newglossaryentry
 119   \newglossaryentry{R}{
 120     name={\ensuremath{\Rn[1]}},
 121     description={the field of real numbers},
 122     sort=R
 123   }
 124 \fi
 125
 126
 127 % The complex n-space, C x C x C x ... x C.
 128 \newcommand*{\Cn}[1][n]{
 129   \mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 130 }
 131
 132 \ifdefined\newglossaryentry
 133   \newglossaryentry{C}{
 134     name={\ensuremath{\Cn[1]}},
 135     description={the field of complex numbers},
 136     sort=C
 137   }
 138 \fi
 139
 140
 141 % The space of real symmetric n-by-n matrices. Does not reduce to
 142 % merely "S" when n=1 since S^{n} does not mean an n-fold cartesian
 143 % product of S^{1}.
 144 \newcommand*{\Sn}[1][n]{ \mathcal{S}^{#1} }
 145 \ifdefined\newglossaryentry
 146   \newglossaryentry{Sn}{
 147     name={\ensuremath{\Sn}},
 148     description={the set of $n$-by-$n$ real symmetric matrices},
 149     sort=Sn
 150   }
 151 \fi
 152
 153 % The space of complex Hermitian n-by-n matrices. Does not reduce to
 154 % merely "H" when n=1 since H^{n} does not mean an n-fold cartesian
 155 % product of H^{1}.
 156 \newcommand*{\Hn}[1][n]{ \mathcal{H}^{#1} }
 157 \ifdefined\newglossaryentry
 158   \newglossaryentry{Hn}{
 159     name={\ensuremath{\Hn}},
 160     description={the set of $n$-by-$n$ complex Hermitian matrices},
 161     sort=Hn
 162   }
 163 \fi
 164
 165
 166 %
 167 % Basic set operations
 168 %
 169
 170 % The union of its two arguments.
 171 \newcommand*{\union}[2]{ {#1}\cup{#2} }
 172
 173 % A three-argument union.
 174 \newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
 175
 176 % The intersection of its two arguments.
 177 \newcommand*{\intersect}[2]{ {#1}\cap{#2} }
 178
 179 % A three-argument intersection.
 180 \newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
 181
 182 % An indexed arbitrary binary operation such as the union or
 183 % intersection of an infinite number of sets. The first argument is
 184 % the operator symbol to use, such as \cup for a union. The second
 185 % argument is the lower index, for example k=1. The third argument is
 186 % the upper index, such as \infty. Finally the fourth argument should
 187 % contain the things (e.g. indexed sets) to be operated on.
 188 \newcommand*{\binopmany}[4]{
 189   \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} }
 190              { {#1}_{#2}^{#3}{#4} }
 191              { {#1}_{#2}^{#3}{#4} }
 192              { {#1}_{#2}^{#3}{#4} }
 193 }
 194
 195 \newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
 196 \newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
 197 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
 198 \newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
 199
 200 \newcommand*{\powerset}[1]{\mathpzc{P}\of{{#1}}}
 201 \ifdefined\newglossaryentry
 202   \newglossaryentry{powerset}{
 203     name={\ensuremath{\powerset{X}}},
 204     description={the ``powerset,'' or set of all subsets of $X$},
 205     sort=p
 206   }
 207 \fi
 208
 209 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
 210 % the real line.
 211 \newcommand*{\intervaloo}[2]{ \left({#1},{#2}\right) } % open-open
 212 \newcommand*{\intervaloc}[2]{ \left({#1},{#2}\right] } % open-closed
 213 \newcommand*{\intervalco}[2]{ \left[{#1},{#2}\right) } % closed-open
 214 \newcommand*{\intervalcc}[2]{ \left[{#1},{#2}\right] } % closed-closed
 215
 216
 217 \fi