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gitweb.michael.orlitzky.com - mjotex.git/blob - mjo-common.tex
2 % Only the most commonly-used macros. Needed by everything else.
4 \ifx\havemjocommon\undefined
11 % Needed for \bigtimes.
12 \usepackage{mathtools
}
14 % Place the argument in matching left/right parentheses.
15 \newcommand*
{\of}[1]{ \left(
{#1}\right)
}
17 % Group terms using parentheses.
18 \newcommand*
{\qty}[1]{ \left(
{#1}\right)
}
20 % Group terms using square brackets.
21 \newcommand*
{\sqty}[1]{ \left[{#1}\right] }
23 % Create a set from the given elements
24 \newcommand*
{\set}[1]{\left\lbrace{#1}\right\rbrace}
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}
33 \newcommand*
{\pair}[2]{ \left(
{#1},
{#2}\right)
}
36 \newcommand*
{\triple}[3]{ \left(
{#1},
{#2},
{#3}\right)
}
38 % The Cartesian product of two things.
39 \newcommand*
{\cartprod}[2]{ {#1}\times{#2} }
41 % The Cartesian product of three things.
42 \newcommand*
{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
44 % The direct sum of two things.
45 \newcommand*
{\directsum}[2]{ {#1}\oplus{#2} }
47 % The direct sum of three things.
48 \newcommand*
{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
50 % The factorial operator.
51 \newcommand*
{\factorial}[1]{ {#1}!
}
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.
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
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
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
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
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
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
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
98 % Basic set operations
101 % The union of its two arguments.
102 \newcommand*
{\union}[2]{ {#1}\cup{#2} }
104 % A three-argument union.
105 \newcommand*
{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
107 % The intersection of its two arguments.
108 \newcommand*
{\intersect}[2]{ {#1}\cap{#2} }
110 % A three-argument intersection.
111 \newcommand*
{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
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} }
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} }
132 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
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