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} % for \of, \Rn, etc.
  16 \input{mjo-linear_algebra} % \Sn and \Hn
  17
  18 % The dual of a subset of an inner-product space; always a closed
  19 % convex cone.
  20 \newcommand*{\dual}[1]{ #1^{*} }
  21
  22 %
  23 % Common cones.
  24 %
  25
  26 % The nonnegative and strictly positive orthants in the given number
  27 % of dimensions.
  28 \newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
  29 \newcommand*{\Rnplusplus}[1][n]{ \Rn[#1]_{++} }
  30
  31 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
  32 \newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
  33
  34 % The PSD cone in a space of symmetric matrices.
  35 \newcommand*{\Snplus}[1][n]{ \Sn[#1]_{+} }
  36
  37 % The PSD cone in a space of Hermitian matrices.
  38 \newcommand*{\Hnplus}[1][n]{ \Hn[#1]_{+} }
  39
  40
  41 %
  42 % Some collections of linear operators.
  43 %
  44
  45 % The set of all positive operators on its argument. This uses the
  46 % same magic as \boundedops to accept either one or two arguments. If
  47 % one argument is given, the domain and codomain are equal and the
  48 % positive operators fix a subset of that space. When two arguments
  49 % are given, the positive operators send the first argument to a
  50 % subset of the second.
  51 \newcommand*{\posops}[2][]{
  52   \pi\of{ {#2}
  53     \if\relax\detokenize{#1}\relax
  54       {}%
  55     \else
  56       {,{#1}}%
  57     \fi
  58   }
  59 }
  60
  61 % The set of all S-operators on its argument.
  62 \newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
  63
  64 % The cone of all Z-operators on its argument.
  65 \newcommand*{\Zof}[1]{ \mathbf{Z} \of{ {#1} } }
  66
  67 % The space of Lyapunov-like operators on its argument.
  68 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
  69
  70 % The Lyapunov rank of the given cone.
  71 \newcommand*{\lyapunovrank}[1]{ \beta\of{ {#1} } }
  72
  73 % Cone inequality operators.
  74 \newcommand*{\gecone}{\succcurlyeq}
  75 \newcommand*{\gtcone}{\succ}
  76 \newcommand*{\lecone}{\preccurlyeq}
  77 \newcommand*{\ltcone}{\prec}
  78
  79
  80 \fi