A mapping is called - -convex (or - -concave) with respect to in second argument if, for any nonempty finite subset,,,, with and, there exists such that.
(1 For any, is open; (2) is closed; (3) is lsc; (4 for all and, ; (5)for all, is -concave with respect to in second argument; (6 for all, is lsc; (7) is usc, where for all.
If is a - -convex mapping (or a - -concave mapping) with respect to in second argument, then for any, is a -convex mapping (or a -concave mapping) with respect to.
Let be a -Lipschitz continuous random operator and be -Lipschitz continuous in the first argument, -Lipschitz continuous in the second argument and -Lipschitz continuous in the third argument, respectively.
Let be -strongly monotone and -Lipschitz continuous, and let be a -Lipschitz continuous random operator, and let be -strongly accretive with respect to and -Lipschitz continuous in the first argument, and -Lipschitz continuous in the second argument, -Lipschitz continuous in the third argument, respectively.
Clearly, F is g.h.c. in the first argument and l.s.c. in the second argument.
They are linear in the first argument and antilinear in the second argument.
The maps and are Carathéodory functions, which means that they are measurable in the first argument and continuous in the second one.
For σ ≥ 0 and x′ ≪ 1, the fixation probability G0 x0, σ) is in good approximation linear in its first argument.
Predicates have a fixed finite arity in FOL, and nothing precludes binding at once a variable in the first argument of one predicate and in the second argument of another predicate.
