We introduce a duality for Affine Iterated Function Systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recu...

Fractals have been introduced by Mandelbrot (1975) to define geometrical ensembles, or mathematical sets, that have a fractional dimension. He pioneered the study of very irregular mathematical sets, where the methods of classical calculus cannot be applied. Fractals are not smooth nor differentiable; they are characterized by self-similarity. Their geometrical structure has details at all scal...

We consider two models (A and B) which can describe both two dimensional fragmentation and stochastic fractals. Model A exhibits multifractality on a unique support when describing a fragmentation process and on one of infinitely many possible supports when describing stochastic fractals. Model B obeys simple scaling. PACS numbers: 05.20-y,02.50-r

We describe a principle that can be used to generate self-similarity and fractals in almost any nonlinear system in nature that supports solitons, given that some proper nonadiabatic conditions are met. We illustrate our idea on a particular optics example that also theoretically demonstrates fractals in nonlinear optics.

Before rendering 2D or 3D fractals with iterated function systems, it is necessary to calculate the bounding extent of fractals. We develop a new algorithm to compute the bounding box which closely contains the entire attractor of an iterated function system. r 2003 Elsevier Science Ltd. All rights reserved.

We present extensive computer simulations of a modified Eden model on a tetragonal lattice. This model is able to reproduce the crossover from mass fractal to surface fractal structures observed experimentally in the clustering process of partially hydrolysed silica particles. One of the most interesting phenomena in random growth is that systems with only shortrange interactions can form aggre...

In IFS fractals generated by affine transformations with arbitrary coefficients often there is a lot of chaotic noise. In the present paper we study the effect of related trigonometric coefficients on affine transformations in terms of the IFS fractals generated by them. The use of related trigonometric functions as coefficients reduces the randomness of the scatter and generates meaningful sha...

The internal structure of the iterations of Koch curve and Sierpiński gasket—the known fractals [4]—is described in terms of multihypergraphical membrane systems related to membrane structures [13] and whose membranes are hyperedges of multi-hypergraphs used to define gluing patterns for the components of the iterations of the considered fractals.

Background: Early simple, low-cost diagnosis of schizophrenia may accelerate the beginning of the treatment process. Here, utilizing the projective tools, including fractal images, are some of the diagnostic aids. Objectives: This study aimed to compare the preferences, descriptions, and response latency to fractal images between schizophrenic and healthy individuals. Materials & Methods: In ...

‎The concept of self-similarity on subsets of algebraic varieties‎ ‎is defined by considering algebraic endomorphisms of the variety‎ ‎as `similarity' maps‎. ‎Self-similar fractals are subsets of algebraic varieties‎ ‎which can be written as a finite and disjoint union of‎ ‎`similar' copies‎. ‎Fractals provide a framework in which‎, ‎one can‎ ‎unite some results and conjectures in Diophantine g...