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 \input{mjo-common}
   9
  10 % The set of all S-operators on its argument.
  11 \newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
  12
  13 % The cone of all Z-operators on its argument.
  14 \newcommand*{\Zof}[1]{ \mathbf{Z} \of{ {#1} } }
  15
  16 % The space of Lyapunov-like operators on its argument.
  17 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
  18
  19 % Display a ``Discrete Complementarity Set'' (DCS). The first argument
  20 % is the name of the cone, the second argument is a generating set for
  21 % that cone, and the third argument is a generating set for its dual.
  22 \newcommand*{\DCS}[3]{ C\of{{#1}} \cap \qty{ {#2} \times {#3} } }
  23
  24 % Cone inequality operators.
  25 \newcommand*{\gek}{ \succcurlyeq }
  26 \newcommand*{\gtk}{ \succ }
  27 \newcommand*{\lek}{ \preccurlyeq }
  28 \newcommand*{\ltk}{ \prec }
  29
  30 % Starred versions of the cone inequality operators.
  31 \newcommand*{\ineqkstar}[1]{ \mathrel{ \overset{ _{\ast} }{ #1 } } }
  32 \newcommand*{\gekstar}{ \ineqkstar{\gek} }
  33 \newcommand*{\gtkstar}{ \ineqkstar{\gtk} }
  34 \newcommand*{\lekstar}{ \ineqkstar{\lek} }
  35 \newcommand*{\ltkstar}{ \ineqkstar{\ltk} }
  36
  37 % And negated versions of some of those...
  38 \newcommand*{\ngeqkstar}{ \ineqkstar{\nsucceq} }
  39 \newcommand*{\ngtrkstar}{ \ineqkstar{\nsucc} }