X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=mjo-cone.tex;h=b770d753ec3ad8855a0ab91081550ee106737056;hp=a7507d60778bf4a01b560cb1b41c97e36a77ff9b;hb=eaa1f615a11b7010088cb848ce909c47beb3fb4d;hpb=38147a40c34416d5dbd87f44d8786e3dd73a132f

diff --git a/mjo-cone.tex b/mjo-cone.tex
index a7507d6..b770d75 100644
--- a/mjo-cone.tex
+++ b/mjo-cone.tex
@@ -9,6 +9,10 @@
 
 \input{mjo-common}
 
+% The dual of a subset of an inner-product space; always a closed
+% convex cone.
+\newcommand*{\dual}[1]{ #1^{*} }
+
 %
 % Common cones.
 %
@@ -27,9 +31,25 @@
 
 
 %
-% Some collections of linear operators
+% Some collections of linear operators.
 %
 
+% The set of all positive operators on its argument. This uses the
+% same magic as \boundedops to accept either one or two arguments. If
+% one argument is given, the domain and codomain are equal and the
+% positive operators fix a subset of that space. When two arguments
+% are given, the positive operators send the first argument to a
+% subset of the second.
+\newcommand*{\posops}[2][]{
+  \pi\of{ {#2}
+    \if\relax\detokenize{#1}\relax
+      {}%
+    \else
+      {,{#1}}%
+    \fi
+  }
+}
+
 % The set of all S-operators on its argument.
 \newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
 
@@ -39,25 +59,11 @@
 % The space of Lyapunov-like operators on its argument.
 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
 
+% The Lyapunov rank of the given cone.
+\newcommand*{\lyapunovrank}[1]{ \beta\of{ {#1} } }
 
-%
 % Cone inequality operators.
-%
-
-% Standard cone inequalities.
-\newcommand*{\gek}{\succcurlyeq}
-\newcommand*{\gtk}{\succ}
-\newcommand*{\lek}{\preccurlyeq}
-\newcommand*{\ltk}{\prec}
-
-
-% Starred versions of the cone inequality operators.
-\newcommand*{\ineqkstar}[1]{ \mathrel{ \overset{ _{\ast} }{ #1 } } }
-\newcommand*{\gekstar}{ \ineqkstar{\gek} }
-\newcommand*{\gtkstar}{ \ineqkstar{\gtk} }
-\newcommand*{\lekstar}{ \ineqkstar{\lek} }
-\newcommand*{\ltkstar}{ \ineqkstar{\ltk} }
-
-% And negated versions of some of those...
-\newcommand*{\ngeqkstar}{ \ineqkstar{\nsucceq} }
-\newcommand*{\ngtrkstar}{ \ineqkstar{\nsucc} }
+\newcommand*{\gecone}{\succcurlyeq}
+\newcommand*{\gtcone}{\succ}
+\newcommand*{\lecone}{\preccurlyeq}
+\newcommand*{\ltcone}{\prec}