Differentiable retracts and a modified inverse function theorem

Lp-INVERSE THEOREM FOR MODIFIED BETA OPERATORS

and B(v+1,n) being the beta function (see, e.g., [3]). It is easily verified that the operators Bn are linear positive operators. Also, Bn(1,x) = 1. It turns out that the order of approximation for the operators (1.1) is at best O(n−1) howsoever smooth the function may be. With the aim of improving the order of approximation, we have to slack the positive condition of these operators for which ...

On the Inverse Function Theorem

A Rough Differentiable Function

A real-valued continuously differentiable function f on the unit interval is constructed such that ∞ ∑ k=1 βf (x, 2 −k) = ∞ holds for every x ∈ [0, 1]. Here βf (x, 2−k) measures the distance of f to the best approximating linear function at scale 2−k around x.

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue’s dominated convergence theorem and Ekeland’s variational principle. As a consequence, the assumptions are substantially weakened: th...

On Converses of Napoleon’s Theorem and a Modified Shape Function

The (negative) Torricelli triangle T1(ABC) of a nondegenerate (positively oriented) triangle ABC is defined to be the triangle A1B1C1, where ABC1, BCA1, and CAB1 are the equilateral triangles drawn outwardly on the sides of ABC. It is known that not every triangle is the Torricelli triangle of some initial triangle, and triangles that are not Torricelli triangles are characterized in [28]. In t...