X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=mjo-common.tex;h=6b357abdbec9b5cdd53dd4c037d2d2e67375432b;hp=27c4c7b3f3d8ee8e81f03bab8f5df3afa6102e3d;hb=HEAD;hpb=37b342a8a6fada8b0fbe828bdf97f346b538f5f4

diff --git a/mjo-common.tex b/mjo-common.tex
index 27c4c7b..ccb22da 100644
--- a/mjo-common.tex
+++ b/mjo-common.tex
@@ -4,13 +4,12 @@
 \ifx\havemjocommon\undefined
 \def\havemjocommon{1}
 
-
 \ifx\mathbb\undefined
   \usepackage{amsfonts}
 \fi
 
-\ifx\bigtimes\undefined
-  \usepackage{mathtools}
+\ifx\restriction\undefined
+  \usepackage{amssymb}
 \fi
 
 % Place the argument in matching left/right parentheses.
@@ -22,15 +21,6 @@
 % Group terms using square brackets.
 \newcommand*{\sqty}[1]{ \left[{#1}\right] }
 
-% Create a set from the given elements
-\newcommand*{\set}[1]{\left\lbrace{#1}\right\rbrace}
-
-% A set comprehension, where the ``such that...'' bar is added
-% automatically. The bar was chosen over a colon to avoid ambiguity
-% with the L : V -> V notation. We can't leverage \set here because \middle
-% needs \left and \right present.
-\newcommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace}
-
 % A pair of things.
 \newcommand*{\pair}[2]{ \left({#1},{#2}\right) }
 
@@ -49,21 +39,40 @@
 % A seven-tuple of things.
 \newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
 
-% The Cartesian product of two things.
-\newcommand*{\cartprod}[2]{ {#1}\times{#2} }
-
-% The Cartesian product of three things.
-\newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
+% A free-form tuple of things. Useful for when the exact number is not
+% known, such as when \ldots will be stuck in the middle of the list,
+% and when you don't want to think in column-vector terms, e.g. with
+% elements of an abstract Cartesian product space.
+\newcommand*{\tuple}[1]{ \left({#1}\right) }
 
-% The direct sum of two things.
-\newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
-
-% The direct sum of three things.
-\newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
+% The "least common multiple of" function. Takes a nonempty set of
+% things that can be multiplied and ordered as its argument. Name
+% chosen for synergy with \gcd, which *does* exist already.
+\newcommand*{\lcm}[1]{ \operatorname{lcm}\of{{#1}} }
+\ifdefined\newglossaryentry
+  \newglossaryentry{lcm}{
+    name={\ensuremath{\lcm{X}}},
+    description={the least common multiple of the elements of $X$},
+    sort=l
+  }
+\fi
 
 % The factorial operator.
 \newcommand*{\factorial}[1]{ {#1}! }
 
+% Restrict the first argument (a function) to the second argument (a
+% subset of that functions domain). Abused for polynomials to specify
+% an associated function with a particular domain (also its codomain,
+% in the case of univariate polynomials).
+\newcommand*{\restrict}[2]{{#1}{\restriction}_{#2}}
+\ifdefined\newglossaryentry
+  \newglossaryentry{restriction}{
+    name={\ensuremath{\restrict{f}{X}}},
+    description={the restriction of $f$ to $X$},
+    sort=r
+  }
+\fi
+
 %
 % Product spaces
 %
@@ -139,48 +148,19 @@
   }
 \fi
 
-
-% The space of real symmetric n-by-n matrices.
-\newcommand*{\Sn}[1][n]{
-  \mathcal{S}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
-}
-
-\ifdefined\newglossaryentry
-  \newglossaryentry{Sn}{
-    name={\ensuremath{\Sn}},
-    description={the set of $n$-by-$n$ real symmetric matrices},
-    sort=Sn
-  }
-\fi
-
-% The space of complex Hermitian n-by-n matrices.
-\newcommand*{\Hn}[1][n]{
-  \mathcal{H}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
+% The n-dimensional product space of a generic field F.
+\newcommand*{\Fn}[1][n]{
+  \mathbb{F}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
 }
 
 \ifdefined\newglossaryentry
-  \newglossaryentry{Hn}{
-    name={\ensuremath{\Hn}},
-    description={the set of $n$-by-$n$ complex Hermitian matrices},
-    sort=Hn
+  \newglossaryentry{F}{
+    name={\ensuremath{\Fn[1]}},
+    description={a generic field},
+    sort=F
   }
 \fi
 
-%
-% Basic set operations
-%
-
-% The union of its two arguments.
-\newcommand*{\union}[2]{ {#1}\cup{#2} }
-
-% A three-argument union.
-\newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
-
-% The intersection of its two arguments.
-\newcommand*{\intersect}[2]{ {#1}\cap{#2} }
-
-% A three-argument intersection.
-\newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
 
 % An indexed arbitrary binary operation such as the union or
 % intersection of an infinite number of sets. The first argument is
@@ -195,11 +175,6 @@
              { {#1}_{#2}^{#3}{#4} }
 }
 
-\newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
-\newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
-\newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
-\newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
-
 
 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
 % the real line.