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Holomorphic extension maps for spaces of Whitney jets.

Jean Schmets, Manuel Valdivia (2001)

RACSAM

The key result (Theorem 1) provides the existence of a holomorphic approximation map for some space of C∞-functions on an open subset of Rn. This leads to results about the existence of a continuous linear extension map from the space of the Whitney jets on a closed subset F of Rn into a space of holomorphic functions on an open subset D of Cn such that D ∩ Rn = RnF.

How to define "convex functions" on differentiable manifolds

Stefan Rolewicz (2009)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In the paper a class of families (M) of functions defined on differentiable manifolds M with the following properties: 1 . if M is a linear manifold, then (M) contains convex functions, 2 . (·) is invariant under diffeomorphisms, 3 . each f ∈ (M) is differentiable on a dense G δ -set, is investigated.

Hukuhara's differentiable iteration semigroups of linear set-valued functions

Andrzej Smajdor (2004)

Annales Polonici Mathematici

Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. A family F t : t 0 of continuous linear set-valued functions F t : K c c ( K ) is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function Φ ( t , x ) = F t ( x ) is a solution of the problem D t Φ ( t , x ) = Φ ( t , G ( x ) ) : = Φ ( t , y ) : y G ( x ) , Φ(0,x) = x, for x ∈ K and t ≥ 0, where D t Φ ( t , x ) denotes the Hukuhara derivative of Φ(t,x) with respect to t and G ( x ) : = l i m s 0 + ( F s ( x ) - x ) / s for x ∈ K.

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