Simple permutations with order 4n+24n + 2. Part I

The problem of genealogy of permutations has been solved partially by Stefan (odd order) and Acosta-Hum\'anez & Bernhardt (power of two). It is well known that Sharkovskii's theorem shows the relationship between the cardinal of the set of periodic points of a continuous map, but simple permutations will show the behavior of those periodic points. This paper studies the structure of permutations of mixed order $4n+2$, its properties and a way to describe its genealogy by using Pasting and Reversing.Comment: 17 page

Non-integrability of the Armbruster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory

We show the non-integrability of the three-parameter Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory, with the exception of the three already known integrable cases. We use Poincar\'e sections to illustrate the breakdown of regular motion for some parameter values.Comment: Accepted for publication in SIAM Journal on Applied Dynamical Systems. Adapted version for arxiv with 19 pages and 11 figure

Algunas propiedades de Z_n[x] siendo n no necesariamente primo

In this paper we present some originals and elementary results related with some properties of monic polynomials with coefficients belonging to Zn, where n is not prime. In particular we introduce a function to compute the number of roots of such polynomials. This paper is based on the BS thesis ”Grupos Diedros y del Tipo ( p, q)”([2]), written by the author under the supervision of Jairo Charris Castan ̃eda and Jesu ́s Hernando Pérez (Pelusa).En este artículo se presentan algunos resultados originales y elementales relacionados con algunas propiedades de poli- nomios mónicos con coeficientes en Zn, siendo n no necesariamente primo. En particular se introduce una función para calcular el número de ráıces de tales polinomios. Este art ículo está basado en la tesis de grado ”Grupos Diedros y del Tipo (p, q)”([2]), presentada por el autor bajo la dirección de Jairo Charris Castañeda y Jesús Hernando Pérez (Pelusa)

Pasting and Reversing Approach to Matrix Theory

The aim of this paper is to study some aspects of matrix theory through Pasting and Reversing. We start giving a summary of previous results concerning to Pasting and Reversing over vectors and matrices, after we rewrite such properties of Pasting and Reversing in matrix theory using linear mappings to finish with new properties and new sets in matrix theory involving Pasting and Reversing. In particular we introduce new linear mappings: Palindromicing and Antipalindromicing mappings, which allow us to obtain palindromic and antipalindromic vectors and matrices.Comment: 19 page

Non-autonomous hamiltonian systems and Morales-Ramis theory

In this paper we present an approach towards the comprehensive analysis of the non-integrability of differential equations in the form $\ddot x=f(x,t)$ which is analogous to Hamiltonian systems with $1+1/2$ degree of freedom. In particular, we analyze the non-integrability of some important families of differential equations such as Painlevé II, Sitnikov and Hill-Schrödinger equation. We emphasize in Painlevé II, showing its non-integrability through three different Hamiltonian systems, and also in Sitnikov in which two different version including numerical results are shown. The main tool to study the non-integrability of these kind of Hamiltonian systems is Morales-Ramis theory. This paper is a very slight improvement of the talk with the same title delivered by the author in SIAM Conference on Applications of Dynamical Systems 2007