mjo-cone.tex

   1 %
   2 % Cone stuff.
   3 %
   4 % The operator families Z(K), LL(K), etc. can technically be defined on
   5 % sets other than cones, but nobody cares.
   6 %
   7 \ifx\havemjocone\undefined
   8 \def\havemjocone{1}
   9
  10
  11 \ifx\succcurlyeq\undefined
  12   \usepackage{amssymb} % \succcurlyeq, \preccurlyeq
  13 \fi
  14
  15 \input{mjo-common}
  16
  17 % The dual of a subset of an inner-product space; always a closed
  18 % convex cone.
  19 \newcommand*{\dual}[1]{ #1^{*} }
  20
  21 %
  22 % Common cones.
  23 %
  24
  25 % The nonnegative orthant in the given number of dimensions.
  26 \newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
  27
  28 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
  29 \newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
  30
  31 % The PSD cone in a space of symmetric matrices.
  32 \newcommand*{\Snplus}[1][n]{ \Sn[#1]_{+} }
  33
  34 % The PSD cone in a space of Hermitian matrices.
  35 \newcommand*{\Hnplus}[1][n]{ \Hn[#1]_{+} }
  36
  37
  38 %
  39 % Some collections of linear operators.
  40 %
  41
  42 % The set of all positive operators on its argument. This uses the
  43 % same magic as \boundedops to accept either one or two arguments. If
  44 % one argument is given, the domain and codomain are equal and the
  45 % positive operators fix a subset of that space. When two arguments
  46 % are given, the positive operators send the first argument to a
  47 % subset of the second.
  48 \newcommand*{\posops}[2][]{
  49   \pi\of{ {#2}
  50     \if\relax\detokenize{#1}\relax
  51       {}%
  52     \else
  53       {,{#1}}%
  54     \fi
  55   }
  56 }
  57
  58 % The set of all S-operators on its argument.
  59 \newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
  60
  61 % The cone of all Z-operators on its argument.
  62 \newcommand*{\Zof}[1]{ \mathbf{Z} \of{ {#1} } }
  63
  64 % The space of Lyapunov-like operators on its argument.
  65 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
  66
  67 % The Lyapunov rank of the given cone.
  68 \newcommand*{\lyapunovrank}[1]{ \beta\of{ {#1} } }
  69
  70 % Cone inequality operators.
  71 \newcommand*{\gecone}{\succcurlyeq}
  72 \newcommand*{\gtcone}{\succ}
  73 \newcommand*{\lecone}{\preccurlyeq}
  74 \newcommand*{\ltcone}{\prec}
  75
  76
  77 \fi