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
   8 \usepackage{amssymb} % \succcurlyeq and friends
   9
  10 \input{mjo-common}
  11
  12 %
  13 % Common cones.
  14 %
  15
  16 % The nonnegative orthant in the given number of dimensions.
  17 \newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
  18
  19 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
  20 \newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
  21
  22 % The PSD cone in a space of symmetric matrices.
  23 \newcommand*{\Snplus}[1][n]{ \mathcal{S}^{{#1}}_{+} }
  24
  25 % The PSD cone in a space of Hermitian matrices.
  26 \newcommand*{\Hnplus}[1][n]{ \mathcal{H}^{{#1}}_{+} }
  27
  28
  29 %
  30 % Some collections of linear operators
  31 %
  32
  33 % The set of all S-operators on its argument.
  34 \newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
  35
  36 % The cone of all Z-operators on its argument.
  37 \newcommand*{\Zof}[1]{ \mathbf{Z} \of{ {#1} } }
  38
  39 % The space of Lyapunov-like operators on its argument.
  40 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
  41
  42
  43 %
  44 % Cone inequality operators.
  45 %
  46
  47 % Standard cone inequalities.
  48 \newcommand*{\gek}{ \succcurlyeq }
  49 \newcommand*{\gtk}{ \succ }
  50 \newcommand*{\lek}{ \preccurlyeq }
  51 \newcommand*{\ltk}{ \prec }
  52
  53
  54 % Starred versions of the cone inequality operators.
  55 \newcommand*{\ineqkstar}[1]{ \mathrel{ \overset{ _{\ast} }{ #1 } } }
  56 \newcommand*{\gekstar}{ \ineqkstar{\gek} }
  57 \newcommand*{\gtkstar}{ \ineqkstar{\gtk} }
  58 \newcommand*{\lekstar}{ \ineqkstar{\lek} }
  59 \newcommand*{\ltkstar}{ \ineqkstar{\ltk} }
  60
  61 % And negated versions of some of those...
  62 \newcommand*{\ngeqkstar}{ \ineqkstar{\nsucceq} }
  63 \newcommand*{\ngtrkstar}{ \ineqkstar{\nsucc} }