Catálogo de publicaciones - libros

We consider an iterated function system (IFS) of one-to-one contractive maps on a compact metric space. We define the of an IFS; define an associated symbolic dynamical system; present and explain a fast algorithm for computing the top; describe an example in one dimension with a rich history going back to work of A. Rényi [, Acta Math. Acad. Sci. Hung., (1957), pp. 477–493]; and we show how tops may be used to help to model and render synthetic pictures in applications in computer graphics.

In this paper, some of the relationships between splines and fractal functions are considered. In particular, it is shown that fractal analogs of B-splines can be defined. Moreover, conditions on fractal functions are derived that guarantee their inclusion in approximation spaces, such as Besov and Triebel-Lizorkin spaces. These conditions generalize earlier results obtained in [8].

Hölder regularity which plays a key rôle in fractal geometry raises an increasing interest in probability and statistics. In this paper we discuss various aspects of local and global regularity for stochastic processes and random fields. As a main result we show the invariability of the pointwise Hölder exponent of a continuous and nowhere differentiable random field which has stationary increments and satisfies a zero-one law. We also survey some recent uses of Hölder spaces in limit theorems for stochastic processes and statistics.

We study the invariant measure or the stationary density of a coupled discrete dynamical system as a function of the coupling parameter (0 < < 1/4). The dynamical system considered is chaotic and unsynchronized for this range of parameter values. We find that the stationary density, restricted on the synchronization manifold, is a fractal function. We find the lower bound on the fractal dimension of the graph of this function and show that it changes continuously with the coupling parameter.

Granular materials forming part of civil engineering structures such as rockfill dams and the granular base in pavement systems are subjected to large compressive stresses resulting from gravity and traffic loads respectively. As a result of these compressive stresses, the granular materials break into pieces of different sizes. The size distribution of the broken granular material has been found to be fractal in nature. However, there is no explanation to date about the mechanisms that cause the granular materials to develop a fractal size distribution. In the present study, a compression test designed to crush granular materials is presented. The tests used 5 mm glass beads and a plexiglass cylinder having an internal diameter equal to 5 cm. As a result of compression in the cylinder, the glass beads broke into pieces that had a fractal size distribution. The compression test was numerically simulated using the Discrete Element Method (DEM). The DEM simulation indicated that the particles developed a network of force chains in order to resist the compressive stress. These force chains did not have a uniform intensity but was found to vary widely through out the sample. Also, the distribution of the force chains in the sample did not involve all the grains but only a selective number of them. Thus, the force chains did not cover the whole sample. Using the box method, it was determined that the distribution of the force chains in the sample was fractal in nature. Also, the intensity of the force chains in the sample was found to be fractal in nature. Thus, the fractal nature of the intensity of the force chains and their distribution were found to be the main reason why granular material develop fractal fragments as a result of compression.

The percolation and permeability of fracture networks is investigated numerically by using a three dimensional model of plane polygons randomly located and oriented in space with sizes following a power law distribution. The influence of the range and exponent of the size distribution, of the fracture shapes and of the exponent of the individual fracture conductivity is examined. A dimensionless fracture density, which involves a shape factor, proves to be an adequate percolation parameter. In these terms, the critical density is nearly invariant, over a wide range of shape and size distribution parameters. The permeability is determined by solving the flow equations after triangulating the fracture networks. Eventually, a general expression is proposed, which is the product of the volumetric surface area, weighted by the individual fracture conductivities, and of a fairly universal function of the dimensionless network density, which accounts for the influences of the fracture shape and size distributions. Two analytical formulas are proposed which successfully fit the numerical data.

The concept of fractal geometry, introduced by B. Mandelbrot has been explored in diverse areas of science, including acoustics [7]. The first part of this work relates the properties of far-field Fraunhofer diffraction region in wave acoustics for characterizing reflection on a self-similar structure. Therefore, the computation of the spatial Fourier transform of the Sierpinski triangle leads to its scattering intensity distribution, which describes its acoustical interference behavior. As a major application of this method, an urban facade scatter densitometry will be compared to acoustic measurements of the first reflections of its surface. The good agreement between computation and measurement allows to validate the spatial Fourier transform of the facade as an indicator of acoustic scattering.

The sound production of flute-like musical instruments like the transvers flute is governed by the coupling between the mouth or embouchure hole in which the flute player blows and the flute tube. Here the flute tubes eigenfrequencies forces the self-sustained oscillation of the generator region at the blowing hole into the tubes resonance frequencies. This paper supposes an explanation for this behaviour. Experiments show a very small amount of energy supplied by the players blowing actually getting into the tube of about 3.5%. So most of the air flow is blown into the room outside the flute. The modelling of the flutes presented here shows a turbulent description of the process as consistent with the experimental findings. The flute tube, which forces the flow in its direction leads to a large directional change of the flow, which results in a strong turbulent viscous damping of the system. So there is strong evidence, that in the nonlinear coupled flute system of blowing and tube the tube forces the blowing system in the tubes eigenfrequencies because the tubes air column is much less damped than the generator region at the soundhole.

Recently developed processes based on laser-induced liquid jet-chemical etching provide efficient methods for high resolution microstructuring of metals. Like in other abrasive techniques (water-jet cutting, laser cutting, ion sputtering etc.) a spontaneous formation of ripples in the surface morphology has been observed depending upon the choice of system parameters. In this paper we present a discrete stochastic model describing the joint action of removal of material by chemical etching and thermally activated diffusion initiated by a moving laser leading to structure formation of a surface. Depending on scan speed and laser power different surface morphologies are observed ranging from rough surface structures to the formation of ripples. The continuum equation associated to the discrete model is shown to be a modified Kuramoto-Sivashinsky equation in a frame comoving with the laser beam. Fourier and wavelet techniques as well as large deviation spectra are used for a characterization of the surfaces.

This article analyzes the relationship between geometric invariant structures in two-dimensional mixing systems and measure-theoretical properties associated with the spatial distribution of stable/unstable manifolds of periodic points. Specifically, a connection is established between the Bowen measure associated with the spatial distribution of periodic points and the -measures characterizing the distribution of stable/unstable leaves throughout the mixing space. This result is made possible through the introduction of the concept of of two measures.