No abstract

We investigate percolation phenomena in multifractal objects that are built in a simple way. In these objects the multifractality comes directly from the geometric tiling. We identify some differences between percolation in the proposed multifractals and in a regular lattice. There are basically two sources of these differences. The first is related to the coordination number, which changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability. Depending on a parameter characterizing the multifractal and the lattice size, the histogram can have two peaks. We observe that the percolation threshold for the multifractal is lower than that for the square lattice. We compute the fractal dimension of the percolating cluster and the critical exponent beta. Despite the topological differences, we find that the percolation in a multifractal support is in the same universality class as standard percolation.

Recently a multifractal object, Q mf , was proposed to allow the study of percolation properties in a multifractal support. The area and the number of neighbors of the blocks of Q mf show a non-trivial behavior. The value of the probability of occupation at the percolation threshold, pc, is a function of ρ, a parameter of Q mf which is related to its anisotropy. We investigate the relation between pc and the average number of neighbors of the blocks as well as the anisotropy of Q mf . I IntroductionDue to the work of many physicists, and particularly to the contributions of Dietrich Stauffer, percolation theory has became a powerful tool in Science describing phenomena in many areas as geology, biology, magnetism, or social phenomena [1,2]. Despite the enormous success of percolation it has been a theory studied in a support (lattice) that has a single dimension. The only references relating percolation and multifractality concern to the multifractal properties of some quantities of the spanning cluster at the percolation threshold [3,4].Recently, a model to study percolation in a multifractal was proposed [5] in the literature. In fact, the authors have created an original multifractal object, Q mf , and an efficient way to estimate its percolation properties. In this work we study in detail the method to estimate p c for this multifractal and discuss the relation between p c , some topologic characteristics, and the anisotropy of Q mf .The multifractal object we develop, Q mf , is an intuitive generalization of the square lattice [5]. Suppose that in the construction of the square lattice we use the following algorithm: take a square of size 1 and cut it symmetrically with vertical and horizontal lines. Repeat this process n-times; at the n th step we have a regular square lattice with 2 n × 2 n cells. The setup algorithm of Q mf is quite similar, the main difference is that we do not cut the square in a symmetric way. In section 2 we explain in detail this algorithm.The development of Q mf has a twofold motivation. Firstly, there are systems like oil reservoirs that show multifractal properties [6] and are good candidates to be modeled by such object. Secondly, there is indeed a much more general scope: we want to study percolation phenomena in lattices that are not regular, but that are multifractal in the geometrical sense. It is important to know how site percolation transition happens in lattices in which the cells vary in size and also in the number of neighbors.In this work we analyse some geometric and topologic properties of the percolation cluster generated on Q mf at the percolation threshold. The paper is organized as follows: in Section II we present the process of construction of Q mf and the algorithm for the estimation of p c , in Section III we show the numerical simulations concerning the percolation threshold p c and the topologic properties of Q mf ; and finally in Section IV we present our final remarks and comments. II The modelIn this section we show the process of building the multifrac...

No abstract

Language disorders that affect children with autism spectrum disorder can bring significant damage to their overall development. Research shows that the best treatment practices for this population are those that start early, have active parental collaboration and are held in natural milieu. In Brazil, early intervention programs are typically held in clinical environments and rarely involve parents as intervention agents. With autism becoming a high-incidence disability, diagnosed early in life, it is paramount that alternative forms of treatment be introduced in the country. The general purpose of this study was to investigate the feasibility of using a parent-guided intervention model with Brazilian families. The participants included two mothers and their 3 year old sons, who were diagnosed with autism. Outcome measures included estimates of caregiver responsive language input and children's communication turns evaluated during home interactions. The results indicated that both dyads increased their levels of responsiveness in, at least, one routine following treatment. Implications of the findings are discussed considering cultural variables.