Restricting uniformly open surjections

Injections, Surjections and More

where  is the natural logarithmic base [4]. In counting all injections, we treat extensions as distinct; for example, the function  : {1 2}→ {1 2} with () =  is not the same as the function  : {1 2} → {1 2 3} with () = , nor is it the same as the function  : {1 2 3}→ {1 2 3} with () = . Let  denote the set of all surjections {1     } → {1    } where  ≥ . ...

A Strong Open Mapping Theorem for Surjections from Cones onto Banach Spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael’s Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse o...

On partitions, surjections, and Stirling numbers

Hereditary Invertible Linear Surjections and Splitting Problems for Selections

Let A + B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h : X → A + B splits into a sum h = f + g of continuous mappings f : X → A and g : X → B? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear ...

Restricting range restricts conclusions

THE EXPERTISE APPROACH AND SKILL ACQUISITION Research on expertise is by definition focused on a restricted sample of individuals. Experts are people who consistently produce outstanding performance in their domains (Ericsson, 2006) and as such are without exception located on the positive side of the skill distribution. The usual approach in the study of expertise is to compare the extreme gro...