The diffusion of ideas is often closely connected to the creation and diffusion of knowledge and to the technological evolution of society. Because of this, knowledge creation, exchange and its subsequent transformation into innovations for improved welfare and economic growth is briefly described from a historical point of view. Next, three approaches are discussed for modeling the diffusion o...

Fractal proteins are a new evolvable method of mapping genotype to phenotype through a developmental process, where genes are expressed into proteins comprised of subsets of the Mandelbrot Set. The resulting network of gene and protein interactions can be designed by evolution to produce specific patterns, that in turn can be used to solve problems. Here the use of fractal gene regulatory netwo...

Here we sketch a new derivation of Zipf’s law for word frequencies based on optimal coding. The structure of the derivation is reminiscent of Mandelbrot’s random typing model but it has multiple advantages over random typing: (1) it starts from realistic cognitive pressures (2) it does not require fine tuning of parameters and (3) it sheds light on the origins of other statistical laws of langu...

We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's...

54 Generally, a fractal most often referred to as “a rough shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole.” The father of fractal geometry Benoît Mandelbrot formulated this term in 1975. A fractal has a complicated shape. Certain features characterize a graphical representation of the fractal by mathematical equation: it is self...

One of the major points of contention in studying and modeling nancial returns is whether or not the variance of the returns is nite or in nite (sometimes referred to as the Bachelier-Samuelson Gaussian world versus the Mandelbrot stable world). A di erent formulation of the question asks how heavy the tails of the nancial returns are. The available empirical evidence can be, and has been, inte...

Chains with long-range interactions are considered. The interactions are defined such that each nth particle interacts only with chain particles with the numbers n ± a(m), where m = 1, 2, 3, . . . and a(m) is an integer-valued function. Exponential type functions a(m) = b, where b = 2, 3, . . . , are discussed. The correspondent pseudodifferential equations of chain oscillations are obtained. D...

Recently, a growing interest has emerged for examining the potential of Image Processing tools to assist Art Investigation. Simultaneously, several research works showed the interest of using multifractal analysis for the description of homogeneous textures in images. In this context, the goal of the present contribution is to study the benefits of using the wavelet leader based multifractal fo...

A lot of formal and informal recreational study took place in the fields of Meromorphic Maps, since Mandelbrot popularized the map z ← z + c. An immediate generalization of the Mandelbrot z ← z + c also known as the Multibrot family were also studied. In the current paper, general truncated polynomial maps of the form z ← ∑ p≥2 apx p + c are studied. Two fundamental properties of these polynomi...