X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo-cone.tex;h=b53e23918d36c644d9ead02e35574d9adbdcc2a6;hb=21d18c2a88952e968b02c838abc5cb61dff960c9;hp=16d16546f3f0294e9a50febd00a07d4ab936b4c7;hpb=b5df7cd8456ef72fd2e61a2bb0c47b89b0ffa07e;p=mjotex.git

diff --git a/mjo-cone.tex b/mjo-cone.tex
index 16d1654..b53e239 100644
--- a/mjo-cone.tex
+++ b/mjo-cone.tex
@@ -9,6 +9,47 @@
 
 \input{mjo-common}
 
+% The dual of a subset of an inner-product space; always a closed
+% convex cone.
+\newcommand*{\dual}[1]{ #1^{*} }
+
+%
+% Common cones.
+%
+
+% The nonnegative orthant in the given number of dimensions.
+\newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
+
+% The Lorentz ``ice-cream'' cone in the given number of dimensions.
+\newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
+
+% The PSD cone in a space of symmetric matrices.
+\newcommand*{\Snplus}[1][n]{ \mathcal{S}^{{#1}}_{+} }
+
+% The PSD cone in a space of Hermitian matrices.
+\newcommand*{\Hnplus}[1][n]{ \mathcal{H}^{{#1}}_{+} }
+
+
+%
+% 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} } }
 
@@ -18,11 +59,17 @@
 % The space of Lyapunov-like operators on its argument.
 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
 
+
+%
 % Cone inequality operators.
-\newcommand*{\gek}{ \succcurlyeq }
-\newcommand*{\gtk}{ \succ }
-\newcommand*{\lek}{ \preccurlyeq }
-\newcommand*{\ltk}{ \prec }
+%
+
+% 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 } } }