Inverse functions of polynomials and its applications to initialize the search of solutions of polynomials and polynomial systems

Abstract

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinitedimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold.

An infinite family of one-step iterators for solving nonlinear equations to increase the order of convergence and a new algorithm of global convergence

“…Some recent results published by the author in [13] are resumed, that will be needed throughout this writing. Note 1.…”

“…Furthermore, we are obtaining hopeful results to generalize this method in order to locate and solve all the real roots of nonlinear systems, by improving and generalizing the ideas expressed in [13] for polynomial systems.…”

Explicit Construction of the Inverse of an Analytic Real Function: Some Applications

“…In this section, we review some outcomes, previously published by the authors, (see [1,2]), which will be used in the following sections. Such a summary has been written in detail for the sake of clarity and the self-developed reading of these lines.…”

“…where B n,k ( f (1) , f (2) , ..., f (n) ) are the partial Bell polynomials, with k = 1, 2, ..., n. Substituting B n,k , according to (34), we arrive at:…”

“…In this paper, we have taken as a basis the previous works [1,2], in which the inverse of a polynomial function is constructed, with the aim of generalizing the methods developed there to any analytic function. That is to say: given an analytic function around the point x 0 :…”