This volume is a sequel to the books Fractals: Theory and Applications in Engineering (Springer-Verlag, 1999) and Fractals in Engineering. From Theory to Industrial Applications (Springer-Verlag, 1997), presenting some of the most recent advances in the ?eld. It is a fascinating exercise to follow the progress of knowledge in this interdisciplinary area, as witnessed by these three volumes. First,con?rmingprevioustrendsobservedin1997and1999,appliedma- ematical research on fractals has now reached a mature level, where beautiful theories are developed in direct contact with engineering concerns. The four papers in the Mathematical Aspects section constitute valuable additions to the set of tools needed by the engineer: Synthetic pictures modelling and rendering in computer graphics (Theory and Applications of Fractal Tops, by Michael Barnsley), curve approximation and ¿fractal B-splines¿ (Splines, Fractal Functions, and Besov and Triebel-Lizorkin Spaces, by Peter Mas- pust), deep understanding of the Hol ¿ derian properties of certain stochastic processes useful in a large number of applications (H¿ olderian random fu- tions, by Antoine Ayache et al. ), and study of the invariant measure of a coupled discrete dynamical system (Fractal Stationary Density in Coupled Maps,byJürgen Jost et al. ). The second section of the book describes novel physical applications as well as recent progress on more classical ones. The paper A Network of Fr- tal Force Chains and Their E?ect in Granular Materials under Compression by Luis E. Vallejo et al.

Part 1: Mathematical Aspects.- Theory and Applications of Fractal Tops.- Splines, Fractal Functions, and Besov and Triebel-Lizorkin Spaces.- Hoelderian Random Functions.- Fractal Stationary Density in Coupled Maps.- Part 2: Physics.- A Network of Fractal Force Chains and Their Effect in Granular Materials under Compression.- Percolation and Permeability of Three Dimensional Fracture Networks with a Power Law Size Distribution.- Acoustic Diffraction Patterns from Fractal to Urban Structures: Applications to the Sierpinski Triangle and to a Neoclassical Urban Facade.- Turbulent k - epsilon model of flute-like musical instrument sound production.- Part 3: Chemical Engineering.- A Simple Discrete Stochastic Model for Laser-induced Jet-chemical Etching.- Invariant Structures and Multifractal Measures in 2d Mixing Systems.- Part 4: Finance.- Long Range Dependence in Financial Markets.- Financial Modelling by Multiscale Fractional Brownian Motion.- Part 5: Internet Traffic.- Limiting Fractal Random Processes in Heavy-tailed Systems.- A Non-parametric Test for Self-similarity and Stationarity in Network Traffic.- Part 6: Image Processing.- Continuous Evolution of Functions and Measures Toward Fixed Points of Contraction Mappings.- Various Mathematical Approaches to Extract Information from Textures of Increasing Complexities.- Fractal Inverse Problem: Approximation, Formulation and Differential Methods.