% Needed for \mathbb.
\usepackage{amsfonts}
+% Needed for \bigtimes.
+\usepackage{mathtools}
+
% Place the argument in matching left/right parntheses.
\providecommand*{\of}[1]{ \left({#1}\right) }
% The direct sum of two things.
\providecommand*{\directsum}[2]{ {#1}\oplus{#2} }
+% The direct sum of three things.
+\providecommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
+
% The factorial operator.
\providecommand*{\factorial}[1]{ {#1}! }
\mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
}
+% The space of symmetric n-by-n matrices.
+\providecommand*{\Sn}[1][n]{
+ \mathcal{S}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
+}
%
% Basic set operations
%
% The union of its two arguments.
-\providecommand*{\union}[2]{ {#1} \cup {#2} }
+\providecommand*{\union}[2]{ {#1}\cup{#2} }
% A three-argument union.
\providecommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
% The intersection of its two arguments.
-\providecommand*{\intersect}[2]{ {#1} \cap {#2} }
+\providecommand*{\intersect}[2]{ {#1}\cap{#2} }
% A three-argument intersection.
\providecommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
% the upper index, such as \infty. Finally the fourth argument should
% contain the things (e.g. indexed sets) to be operated on.
\providecommand*{\binopmany}[4]{
- \mathchoice
- { \underset{#2}{\overset{#3}{#1}}{#4} }
- { {#1}_{#2}^{#3}{#4} }
- { {#1}_{#2}^{#3}{#4} }
- { {#1}_{#2}^{#3}{#4} }
+ \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} }
+ { {#1}_{#2}^{#3}{#4} }
+ { {#1}_{#2}^{#3}{#4} }
+ { {#1}_{#2}^{#3}{#4} }
}
-\providecommand*{\unionmany}[3]{ \binopmany{\cup}{#1}{#2}{#3} }
-\providecommand*{\intersectmany}[3]{ \binopmany{\cap}{#1}{#2}{#3} }
+\providecommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
+\providecommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
+\providecommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
+\providecommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }