On McMullen-like mappings

We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d<1$ and the simple critical values of $f_{\lambda}$ belong to the trap door. We generalize this behavior defining a McMullen-like mapping as a rational map $f$ associated to a hyperbolic postcritically finite polynomial $P$ and a pole data $\mathcal{D}$ where we encode, basically, the location of every pole of $f$ and the local degree at each pole. In the McMullen family, the polynomial $P$ is $z\mapsto z^n$ and the pole data $\mathcal{D}$ is the pole located at the origin that maps to infinity with local degree $d$. As in the McMullen family $f_{\lambda}$, we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial $P$ and the pole data $\mathcal{D}$. We prove that the arithmetic condition is necessary using the theory of Thurston's obstructions, and sufficient by quasiconformal surgery.Comment: 21 pages, 2 figure

The resolving number of a graph

We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.Comment: 13 pages, 3 figure

Resolving sets for breaking symmetries of graphs

This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent study on other functions related to these sets. Thus, we obtain lower and upper bounds on all these functions by means of very diverse tools. Among them are some adequate constructions of graphs, a variant of a classical result in graph domination and a polynomial time algorithm that produces both distinguishing sets and determining sets. Further, we consider specific families of graphs where the restrictions of these functions can be computed. To this end, we utilize two well-known objects in graph theory: $k$-dominating sets and matchings.Comment: 24 pages, 12 figure

Enseñanza basada en contenidos: una experiencia para el desarrollo de competencias del EEES en grados en Ingeniería Informática

En esta ponencia describimos una estrategia docente diseñada para potenciar la implicación del alumno en el desarrollo de las clases. Aprovechando la creación de grupos con docencia en inglés en los grados en Ingeniería Informática, hemos utilizado en nuestra asignatura una metodología activa basada en la participación del alumno durante las clases. En estos grupos, un objetivo adicional es que el alumno progrese en su dominio del inglés, que claramente se alcanzará si el alumno practica activamente el idioma. Siguiendo esta idea, hemos enfocado nuestra asignatura de forma que el alumno sea el protagonista de las clases, como se indica en el Espacio Europeo de Educación Superior (EEES), realizando diversas actividades durante el curso. Con este enfoque hemos obtenido buenos resultados finales, a nivel de calificaciones y de dominio del idioma, además de desarrollar otras competencias del EEES (capacidad de exposición en público, trabajo en grupo, actitud crítica, etc).SUMMARY -- This paper describes a teaching approach designed to promote the involvement of students in class. Due to the creation of English-teaching groups in Computer Engineering Degrees, we have used an active methodology based on the participation of students in class. In these groups, an additional goal is the progress of students in their fluency in English, an aspect that will be clearly achieved if they actively practice this language. Following this idea, we have oriented our subject so that the student plays an active role in class, as indicated in the European Higher Education Area (EHEA), by means of different activities during the semester. With this approach, we have obtained good final results in terms of qualifications and English level, and other skills indicated by the EHEA (such as public presentation skills, cooperative work, critical attitude,...).Peer Reviewe

Topological properties of the immediate basins of attraction for the secant method

We study the discrete dynamical system defined on a subset of $R^2$ given by the iterates of the secant method applied to a real polynomial $p$. Each simple real root $\alpha$ of $p$ has associated its basin of attraction $\mathcal A(\alpha)$ formed by the set of points converging towards the fixed point $(\alpha,\alpha)$ of $S$. We denote by $\mathcal A^*(\alpha)$ its immediate basin of attraction, that is, the connected component of $\mathcal A(\alpha)$ which contains $(\alpha,\alpha)$. We focus on some topological properties of $\mathcal A^*(\alpha)$, when $\alpha$ is an internal real root of $p$. More precisely, we show the existence of a 4-cycle in $\partial \mathcal A^*(\alpha)$ and we give conditions on $p$ to guarantee the simple connectivity of $\mathcal A^*(\alpha)$.Comment: 24 pages, 19 figure

Simultaneous bifurcation of limit cycles from two nests of periodic orbits

AbstractLet z˙=f(z) be an holomorphic differential equation having a center at p, and consider the following perturbation z˙=f(z)+εR(z,z¯). We give an integral expression, similar to an Abelian integral, whose zeroes control the limit cycles that bifurcate from the periodic orbits of the period annulus of p. This expression is given in terms of the linearizing map of z˙=f(z) at p. The result is applied to control the simultaneous bifurcation of limit cycles from the two period annuli of z˙=iz+z2, after a polynomial perturbation