mjo-common.tex

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