A direct construction of a Laplacian on the Sierpinski gasket as a limit of difference quotients was given by Kigami [Ki1], who later extended the method to a class of self-similar fractals called post critically finite (p.c.f.) [Ki2, Ki3]. At about the same time, Kusuoka and Zhou [KZ] introduced what we will call the method of averages for defining a Laplacian on the Sierpinski carpet, a fract...

The present paper focuses on the order-disorder transition of an Ising model on a self-similar lattice. We present a numerical study, based on the Monte Carlo method in conjunction with the finite size scaling method, of the critical properties of the Ising model on a two dimensional deterministic fractal lattice of Hausdorff dimension dH = ln 8/ ln 3 = 1.89278926.... We give evidence of the ex...

We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets to their boundaries, where boundaries mean the triangles and rectangles which confine gaskets and carpets. As an application, we construct diffusion processes on ...

We have discovered a new fractal pattern, coffee exhibits on a horizontal surface, with a fractal dimension 06 . 0 88 . 1 ± , when a heavy coffee droplet interacts with the surface of lighter milk. Parts of coffee pattern on the surface disappear due to Rayleigh–Taylor instability, and the annihilative behavior produces the fractal pattern. Coffee fractals and the Sierpinski carpet have common ...

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 product of fractals. While our resul...