On the inverse function theorem: A counterexample

On the Inverse Function Theorem

A counterexample to the a-‘theorem’

We exhibit a renormalization group flow for a four-dimensional gauge theory along which the conformal central charge a increases. The flow connects the maximally superconformal point of an N = 2 gauge theory with gauge group SU(N+1) and Nf = 2N flavors in the ultraviolet, to a strongly-coupled superconformal point of the SU(N) gauge theory with Nf = 2N massless flavors in the infrared. Our exam...

A Counterexample on a Theorem by Khojasteh, Goodarzi, and Razani

In the paper by Khojasteh et al. 2010 , the authors tried to generalize Branciari’s theorem, introducing the new integral type contraction mappings. In this note we give a counterexample on their main statement Theorem 2.9 . Also we give a comment explaining what the mistake in the proof is, and suggesting what conditions might be appropriate in generalizing fixed point results to cone spaces, ...

A Counterexample to a Theorem of Chase

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...