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A note on intermediate differentiability of Lipschitz functions

Luděk Zajíček (1999)

Commentationes Mathematicae Universitatis Carolinae

Let f be a Lipschitz function on a superreflexive Banach space X . We prove that then the set of points of X at which f has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces X ), but it is even σ -porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer.

A note on n-ary Poisson brackets

Michor, Peter W., Vaisman, Izu (2000)

Proceedings of the 19th Winter School "Geometry and Physics"

An n -ary Poisson bracket (or generalized Poisson bracket) on the manifold M is a skew-symmetric n -linear bracket { , ⋯ , } of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order n , i.e., ∑ σ ∈ S 2 n - 1 ( sign σ ) { { f σ 1 , ⋯ , f σ n } , f σ n + 1 , ⋯ , f σ 2 n - 1 } = 0 , S 2 n ...

A Note on the Alexander Theorem on the Complex Plane

Sylwester Zając (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We investigate the Banach manifold consisting of complex r functions on the unit disc having boundary values in a given one-dimensional submanifold of the plane. We show that ∂/∂λ̅ restricted to that submanifold is a Fredholm mapping. Moreover, for any such function we obtain a relation between its homotopy class and the Fredholm index.

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