On the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity

We prove the analyticity of smooth critical points for O’Hara’s knot energies [Formula: see text], with [Formula: see text] and [Formula: see text], subject to a fixed length constraint. This implies, together with the already established regularity results for O’Hara’s knot energies, that bounded energy critical points of [Formula: see text] subject to a fixed length constraint are not only [Formula: see text] but also analytic. Our approach is based on Cauchy’s method of majorants and a decomposition of the gradient that was adapted from the Möbius energy case [Formula: see text].

On the Complex Dynamics of Continued and Discrete Cauchy’s Method

Let p be a complex polynomial of fixed degree n. In this paper we show that Cauchy’s method may fail to find all zeros of p for any initial guess z0 lying in the complex plane and we propose several ways to find all zeros of a given polynomial using scaled Cauchy’s methods.