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gitweb.michael.orlitzky.com - mjotex.git/blob - mjo-cone.tex
4 % The operator families Z(K), LL(K), etc. can technically be defined on
5 % sets other than cones, but nobody cares.
8 \usepackage{amssymb
} % \succcurlyeq and friends
16 % The nonnegative orthant in the given number of dimensions.
17 \newcommand*
{\Rnplus}[1][n
]{ \Rn[#1]_
{+
} }
19 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
20 \newcommand*
{\Lnplus}[1][n
]{ \mathcal{L
}^
{{#1}}_
{+
} }
22 % The PSD cone in a space of symmetric matrices.
23 \newcommand*
{\Snplus}[1][n
]{ \mathcal{S
}^
{{#1}}_
{+
} }
25 % The PSD cone in a space of Hermitian matrices.
26 \newcommand*
{\Hnplus}[1][n
]{ \mathcal{H
}^
{{#1}}_
{+
} }
30 % Some collections of linear operators
33 % The set of all S-operators on its argument.
34 \newcommand*
{\Sof}[1]{ \mathbf{S
} \of{ {#1} } }
36 % The cone of all Z-operators on its argument.
37 \newcommand*
{\Zof}[1]{ \mathbf{Z
} \of{ {#1} } }
39 % The space of Lyapunov-like operators on its argument.
40 \newcommand*
{\LL}[1]{ \mathbf{LL
}\of{ {#1} } }
44 % Cone inequality operators.
47 % Standard cone inequalities.
48 \newcommand*
{\gek}{\succcurlyeq}
49 \newcommand*
{\gtk}{\succ}
50 \newcommand*
{\lek}{\preccurlyeq}
51 \newcommand*
{\ltk}{\prec}
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} }
61 % And negated versions of some of those...
62 \newcommand*
{\ngeqkstar}{ \ineqkstar{\nsucceq} }
63 \newcommand*
{\ngtrkstar}{ \ineqkstar{\nsucc} }