We describe relation between analysis on fractals and the theory of self-similar groups. In particular, we focus on the construction of the Laplacian on limit sets of such groups in several concrete examples, and in the general p.c.f. case. We pose a number of open questions.

Riesz potentials and Laplacian of fractal measures in metric spaces are introduced. They deene self{adjoint operators in the Hilbert space L 2 () and the former are shown to be compact. In the euclidean case the corresponding spectral asymptotics are derived by Besov space methods. The inverses of the Riesz potentials are fractal pseudo-diierential operators. For the Laplace operator the spectr...

A novel type of fractals (i.e., Zhang fractals) is yielded via solving time-varying or static nonlinear equations in complex domain by discrete-time complex-valued Zhang dynamics (DTCVZD). The DTCVZD model that uses different types of activation functions can generate various Zhang fractals. These fractals are different from the conventional Newton fractals discovered 30 years ago (since 1983) ...

There is a well developed theory (see [5, 9]) of analysis on certain types of fractal sets, of which the Sierpinski Gasket (SG) is the simplest non-trivial example. In this theory the fractals are viewed as limits of graphs, and notions analogous to the Dirichlet energy and the Laplacian are constructed as renormalized limits of the corresponding objects on the approximating graphs. The nature ...

The present paper analyses and documents the merits of incorporating fractal design in microstrip antenna intended to be mounted on and integrated into the design of smart vehicles. A novel design is proposed for a compact tri-band hexagonal microstrip antenna to be integrated with the body of a smart vehicle for short range communication purpose in an Intelligent Transport System (ITS). This a...

Deterministic and random fractals, within the framework of Iterated Function Systems, have been used to model and study a wide range of phenomena across many areas of science and technology. However, for many applications deterministic fractals are locally too similar near distinct points while standard random fractals have too little local correlation. Random fractals are also slow and difficu...

An exact enumeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold. We use this approach to calculate exactly the first low-density expansion of LBB@) for arbitrary system dimensionality d, where L B B @ ) is the mean number of backbone bonds and p is the bond occupation probability. Standard series extrapolation methods provide es...

A scale-invariant theory is derived for the linear optical properties of nontrivial fractals (fractal clusters) in the collective region, i.e., near the center of the absorption band. A model of dilute fractals is introduced and is used to find the shape of the wings of the absorption line in the binary approximation. Strong fluctuations of the local fields in nontrivial fractals are described....

We present the numbers of spanning forests on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket SGd,b(n) with d = 2 and b = 3, 4 are obtained. We also derive the upper bounds of the asymptotic growth constants for both SGd and SG2,b.