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 % The Cartesian product of two things.
  41 \newcommand*{\cartprod}[2]{ {#1}\times{#2} }
  42
  43 % The Cartesian product of three things.
  44 \newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
  45
  46 % The direct sum of two things.
  47 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
  48
  49 % The direct sum of three things.
  50 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  51
  52 % The factorial operator.
  53 \newcommand*{\factorial}[1]{ {#1}! }
  54
  55 %
  56 % Product spaces
  57 %
  58 % All of the product spaces (for example, R^n) that follow default to
  59 % an exponent of ``n'', but that exponent can be changed by providing
  60 % it as an optional argument. If the exponent given is ``1'', then it
  61 % will be omitted entirely.
  62 %
  63
  64 % The natural n-space, N x N x N x ... x N.
  65 \newcommand*{\Nn}[1][n]{
  66   \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  67 }
  68
  69 \ifdefined\newglossaryentry
  70   \newglossaryentry{N}{
  71     name={\ensuremath{\Nn[1]}},
  72     description={the set of natural numbers},
  73     sort=N
  74   }
  75 \fi
  76
  77 % The integral n-space, Z x Z x Z x ... x Z.
  78 \newcommand*{\Zn}[1][n]{
  79   \mathbb{Z}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  80 }
  81
  82 \ifdefined\newglossaryentry
  83   \newglossaryentry{Z}{
  84     name={\ensuremath{\Zn[1]}},
  85     description={the ring of integers},
  86     sort=Z
  87   }
  88 \fi
  89
  90 % The rational n-space, Q x Q x Q x ... x Q.
  91 \newcommand*{\Qn}[1][n]{
  92   \mathbb{Q}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  93 }
  94
  95 \ifdefined\newglossaryentry
  96   \newglossaryentry{Q}{
  97     name={\ensuremath{\Qn[1]}},
  98     description={the field of rational numbers},
  99     sort=Q
 100   }
 101 \fi
 102
 103 % The real n-space, R x R x R x ... x R.
 104 \newcommand*{\Rn}[1][n]{
 105   \mathbb{R}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 106 }
 107
 108 \ifdefined\newglossaryentry
 109   \newglossaryentry{R}{
 110     name={\ensuremath{\Rn[1]}},
 111     description={the field of real numbers},
 112     sort=R
 113   }
 114 \fi
 115
 116
 117 % The complex n-space, C x C x C x ... x C.
 118 \newcommand*{\Cn}[1][n]{
 119   \mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 120 }
 121
 122 \ifdefined\newglossaryentry
 123   \newglossaryentry{C}{
 124     name={\ensuremath{\Cn[1]}},
 125     description={the field of complex numbers},
 126     sort=C
 127   }
 128 \fi
 129
 130
 131 % The space of real symmetric n-by-n matrices.
 132 \newcommand*{\Sn}[1][n]{
 133   \mathcal{S}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 134 }
 135
 136 \ifdefined\newglossaryentry
 137   \newglossaryentry{Sn}{
 138     name={\ensuremath{\Sn}},
 139     description={the set of $n$-by-$n$ real symmetric matrices},
 140     sort=Sn
 141   }
 142 \fi
 143
 144 % The space of complex Hermitian n-by-n matrices.
 145 \newcommand*{\Hn}[1][n]{
 146   \mathcal{H}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 147 }
 148
 149 \ifdefined\newglossaryentry
 150   \newglossaryentry{Hn}{
 151     name={\ensuremath{\Hn}},
 152     description={the set of $n$-by-$n$ complex Hermitian matrices},
 153     sort=Hn
 154   }
 155 \fi
 156
 157 %
 158 % Basic set operations
 159 %
 160
 161 % The union of its two arguments.
 162 \newcommand*{\union}[2]{ {#1}\cup{#2} }
 163
 164 % A three-argument union.
 165 \newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
 166
 167 % The intersection of its two arguments.
 168 \newcommand*{\intersect}[2]{ {#1}\cap{#2} }
 169
 170 % A three-argument intersection.
 171 \newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
 172
 173 % An indexed arbitrary binary operation such as the union or
 174 % intersection of an infinite number of sets. The first argument is
 175 % the operator symbol to use, such as \cup for a union. The second
 176 % argument is the lower index, for example k=1. The third argument is
 177 % the upper index, such as \infty. Finally the fourth argument should
 178 % contain the things (e.g. indexed sets) to be operated on.
 179 \newcommand*{\binopmany}[4]{
 180   \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} }
 181              { {#1}_{#2}^{#3}{#4} }
 182              { {#1}_{#2}^{#3}{#4} }
 183              { {#1}_{#2}^{#3}{#4} }
 184 }
 185
 186 \newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
 187 \newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
 188 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
 189 \newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
 190
 191
 192 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
 193 % the real line.
 194 \newcommand*{\intervaloo}[2]{ \left({#1},{#2}\right) } % open-open
 195 \newcommand*{\intervaloc}[2]{ \left({#1},{#2}\right] } % open-closed
 196 \newcommand*{\intervalco}[2]{ \left[{#1},{#2}\right) } % closed-open
 197 \newcommand*{\intervalcc}[2]{ \left[{#1},{#2}\right] } % closed-closed
 198
 199
 200 \fi