X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=mjo-cone.tex;h=78e87415f587dddb5832215cf91aad5079d04fd3;hp=1199d4f55be54130d2660f7814f9f857d8590853;hb=HEAD;hpb=ad5dafa4ee98c4729b33bebc1817c09cae1b881a

diff --git a/mjo-cone.tex b/mjo-cone.tex
index 1199d4f..00a1309 100644
--- a/mjo-cone.tex
+++ b/mjo-cone.tex
@@ -4,32 +4,60 @@
 % The operator families Z(K), LL(K), etc. can technically be defined on
 % sets other than cones, but nobody cares.
 %
+\ifx\havemjocone\undefined
+\def\havemjocone{1}
 
-\usepackage{amssymb} % \succcurlyeq and friends
 
-\input{mjo-common}
+\ifx\succcurlyeq\undefined
+  \usepackage{amssymb} % \succcurlyeq, \preccurlyeq
+\fi
+
+\input{mjo-common} % for \of, \Rn, etc.
+\input{mjo-linear_algebra} % \Sn and \Hn
+
+% 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]{ \mathbb{R}^{{#1}}_{+} }
+% The nonnegative and strictly positive orthants in the given number
+% of dimensions.
+\newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
+\newcommand*{\Rnplusplus}[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}}_{+} }
+\newcommand*{\Snplus}[1][n]{ \Sn[#1]_{+} }
 
 % The PSD cone in a space of Hermitian matrices.
-\newcommand*{\Hnplus}[1][n]{ \mathcal{H}^{{#1}}_{+} }
+\newcommand*{\Hnplus}[1][n]{ \Hn[#1]_{+} }
 
 
 %
-% 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 +67,14 @@
 % 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 }
-
+\newcommand*{\gecone}{\succcurlyeq}
+\newcommand*{\gtcone}{\succ}
+\newcommand*{\lecone}{\preccurlyeq}
+\newcommand*{\ltcone}{\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} }
+\fi