The recent development of analysis on fractal spaces is physically motivated by the study of diffusion in disordered media. The natural questions that arise concern the existence and uniqueness of a suitable Laplace operator, and the behaviour of the associated heat semigroup, on a space which is fractal. The classes of fractals for which these questions were ®rst answered were classes of exact...

We introduce two new techniques to the analysis on fractals. One is based on the presentation of the fractal as the boundary of a countable Gromov hyperbolic graph, whereas the other one consists in taking all possible “backward” extensions of the above hyperbolic graph and considering them as the classes of a discrete equivalence relation on an appropriate compact space. Illustrating these tec...

“Fractal” is a word invented by French mathematician B.B. Mandelbrot around 1970s. He claimed that many patterns of Nature, such as clouds, mountains and coastlines are not lines nor circles which are smooth, but are so irregular, fragmented and exhibit an altogether different level of complexity. From this viewpoint, he called the family of those shapes as fractals. Especially, he made special...

We study the interaction problem of a linear polymer chain, floating in fractal containers that belong to the three-dimensional Sierpinski gasket (3D SG) family of fractals, with a surface-adsorbed linear polymer chain. Each member of the 3D SG fractal family has a fractal impenetrable 2D adsorbing surface, which appears to be 2D SG fractal. The two-polymer system is modelled by two mutually cr...

It is now well known that natural Brownian Motions on various disordered or complex structures are anomalously slow, and that convection in a turbulent flow can create anomalously fast diffusion. In this work we try to understand the basic mechanisms of anomalous diffusion using and developing the tools of homogenisation. These mechanisms of the slow diffusion for instance are well understood f...

Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic ...

Using self–avoiding walk model on three–dimensional Sierpinski fractals (3d SF) we have studied critical properties of self–interacting linear polymers in porous environment, via exact real–space renormalization group (RG) method. We have found that RG equations for 3d SF with base b = 4 are much more complicated than for the previously studied b = 2 and b = 3 3d SFs. Numerical analysis of thes...

We study the problem of critical adsorption of piecewise directed random walks on a boundary of fractal lattices that belong to the Sierpinski gasket family. By applying the exact real space renormalization group method, we calculate the crossover exponent φ, associated with the number of adsorbed steps, for the complete fractal family. We demonstrate that our results are very close to the resu...

Fractals, shapes comprised of self-similar parts, are not merely prescribed linear structures. A wide class of fractals can also arise from the rich dynamics inherent to nonlinear optics. I n 1967, Benoit Mandelbrot published a paper that gave birth to the study of fractals, entitled " How long is the coast of Britain? Statistical self similarity and fractional dimension " 1. According to Mande...

This note introduces a new general conjecture correlating the dimensionality dT of an infinite lattice with N nodes to the asymptotic value of its Wiener Index W(N). In the limit of large N the general asymptotic behavior W(N)≈Ns is proposed, where the exponent s and dT are related by the conjectured formula s=2+1/dT allowing a new definition of dimensionality dW=(s-2)-1. Being related to the t...