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 % Needed for \mathbb.
   9 \usepackage{amsfonts}
  10
  11 % Needed for \bigtimes.
  12 \usepackage{mathtools}
  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 % The Cartesian product of two things.
  39 \newcommand*{\cartprod}[2]{ {#1}\times{#2} }
  40
  41 % The Cartesian product of three things.
  42 \newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
  43
  44 % The direct sum of two things.
  45 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
  46
  47 % The direct sum of three things.
  48 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  49
  50 % The factorial operator.
  51 \newcommand*{\factorial}[1]{ {#1}! }
  52
  53 %
  54 % Product spaces
  55 %
  56 % All of the product spaces (for example, R^n) that follow default to
  57 % an exponent of ``n'', but that exponent can be changed by providing
  58 % it as an optional argument. If the exponent given is ``1'', then it
  59 % will be omitted entirely.
  60 %
  61
  62 % The natural n-space, N x N x N x ... x N.
  63 \newcommand*{\Nn}[1][n]{
  64   \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  65 }
  66
  67 % The integral n-space, Z x Z x Z x ... x Z.
  68 \newcommand*{\Zn}[1][n]{
  69   \mathbb{Z}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  70 }
  71
  72 % The rational n-space, Q x Q x Q x ... x Q.
  73 \newcommand*{\Qn}[1][n]{
  74   \mathbb{Q}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  75 }
  76
  77 % The real n-space, R x R x R x ... x R.
  78 \newcommand*{\Rn}[1][n]{
  79   \mathbb{R}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  80 }
  81
  82 % The complex n-space, C x C x C x ... x C.
  83 \newcommand*{\Cn}[1][n]{
  84   \mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  85 }
  86
  87 % The space of real symmetric n-by-n matrices.
  88 \newcommand*{\Sn}[1][n]{
  89   \mathcal{S}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  90 }
  91
  92 % The space of complex Hermitian n-by-n matrices.
  93 \newcommand*{\Hn}[1][n]{
  94   \mathcal{H}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  95 }
  96
  97 %
  98 % Basic set operations
  99 %
 100
 101 % The union of its two arguments.
 102 \newcommand*{\union}[2]{ {#1}\cup{#2} }
 103
 104 % A three-argument union.
 105 \newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
 106
 107 % The intersection of its two arguments.
 108 \newcommand*{\intersect}[2]{ {#1}\cap{#2} }
 109
 110 % A three-argument intersection.
 111 \newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
 112
 113 % An indexed arbitrary binary operation such as the union or
 114 % intersection of an infinite number of sets. The first argument is
 115 % the operator symbol to use, such as \cup for a union. The second
 116 % argument is the lower index, for example k=1. The third argument is
 117 % the upper index, such as \infty. Finally the fourth argument should
 118 % contain the things (e.g. indexed sets) to be operated on.
 119 \newcommand*{\binopmany}[4]{
 120   \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} }
 121              { {#1}_{#2}^{#3}{#4} }
 122              { {#1}_{#2}^{#3}{#4} }
 123              { {#1}_{#2}^{#3}{#4} }
 124 }
 125
 126 \newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
 127 \newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
 128 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
 129 \newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
 130
 131
 132 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
 133 % the real line.
 134 \newcommand*{\intervaloo}[2]{ \left({#1},{#2}\right) } % open-open
 135 \newcommand*{\intervaloc}[2]{ \left({#1},{#2}\right] } % open-closed
 136 \newcommand*{\intervalco}[2]{ \left[{#1},{#2}\right) } % closed-open
 137 \newcommand*{\intervalcc}[2]{ \left[{#1},{#2}\right] } % closed-closed
 138
 139
 140 \fi