It is well known that the discrete Sierpinski triangle can be defined as the nonzero residues modulo 2 of Pascal’s triangle, and that from this definition one can easily construct a tileset with which the discrete Sierpinski triangle self-assembles in Winfree’s tile assembly model. In this paper we introduce an infinite class of discrete self-similar fractals that are defined by the residues mo...

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to products of fractals. While our resu...

The Sierpinski tetrahedron is two-dimensional with respect to fractal dimensions though it is realized in threedimensional space, and it has square projections in three orthogonal directions. We study its generalizations and present two-dimensional fractals with many square projections. One is generated from a hexagonal bipyramid which has square projections not in three but in six directions. ...

We refute the claims made by Riera and Chalub [Phys. Rev. E 58, 4001 (1998)] by demonstrating that they have not provided enough data (requisite in their series expansion method) to draw reliable conclusions about criticality of self-avoiding walks on the Sierpinski gasket family of fractals.

A new mathematical concept of abstract similarity is introduced and illustrated in the space infinite strings on a finite number symbols. The problem chaos presence for Sierpinski fractals, Koch curve, as well Cantor set solved by considering natural map. This accomplished Poincaré, Li-Yorke Devaney chaos, including multi-dimensional cases. Original numerical simula...

With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper proves Sobolev embedding theorems in the general framework of Dirichlet spaces. Under suitable assumptions that are verified variety settings, we obtain whole family Gagliardo-Nirenberg Trudinger-Moser inequalities with optimal exponents. These turn out to depend not only on Hausdorf...

For a class of post–critically finite (p.c.f.) fractals, which includes the Sierpinski gasket (SG), there is a satisfactory theory of analysis due to Kigami, including energy, harmonic functions and Laplacians. In particular, the Laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods. The probabilistic method is also available for non–p...

We explain, in detail, the theory of Fractal Compression introduced by Barnsley and extended by Jacquin. We begin by brieey discussing metric spaces, leading to contrac-tive mappings and their xed points. Then, IFS and Local IFS are introduced. These concepts form the basic underlying theory of Fractal Compression. We then discuss our implementation of the theory and the results of our eeort. O...

Recent years have seen considerable developments in the theory of analysis on certain fractal sets from both probabilistic and analytic viewpoints [1, 10, 19]. In this theory, either a Dirichlet energy form or a diffusion on the fractal is used to construct a weak Laplacian with respect to an appropriate measure, and thereby to define smooth functions. As a result the Laplacian eigenfunctions a...

We investigate Benford's law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained limit of iterative sets, unique measures their components follow a geometric distribution, which is Benford most bases. Building on this intuition, we aim study distribution more complicated fractals. examine Laurent coefficients Riemann mapping Taylor its re...