X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=mjo-cone.tex;h=78e87415f587dddb5832215cf91aad5079d04fd3;hp=e35b5eb4babb13b16b202f1c52acd50a1076b3d2;hb=HEAD;hpb=c547f718395b105994c5559eea84d9aa2eac4173

diff --git a/mjo-cone.tex b/mjo-cone.tex
index e35b5eb..00a1309 100644
--- a/mjo-cone.tex
+++ b/mjo-cone.tex
@@ -4,8 +4,59 @@
 % 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}
 
-\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 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]{ \Sn[#1]_{+} }
+
+% The PSD cone in a space of Hermitian matrices.
+\newcommand*{\Hnplus}[1][n]{ \Hn[#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} } }
@@ -16,24 +67,14 @@
 % The space of Lyapunov-like operators on its argument.
 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
 
-% Display a ``Discrete Complementarity Set'' (DCS). The first argument
-% is the name of the cone, the second argument is a generating set for
-% that cone, and the third argument is a generating set for its dual.
-\newcommand*{\DCS}[3]{ C\of{{#1}} \cap \qty{ {#2} \times {#3} } }
+% The Lyapunov rank of the given cone.
+\newcommand*{\lyapunovrank}[1]{ \beta\of{ {#1} } }
 
 % Cone inequality operators.
-\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}
+
+
+\fi