Some problems related to the transition density u(t, x) of the diffusion on the Sierpinski gasket are considerd, based on recent rigorous results and detailed numerical calculations. The main contents are an extension of Flory’s formula for the end-to-end distance exponent of self-avoiding walks on the fractal spaces, and an evidence of the oscillatory behavior of u(t, x) on the Sierpinski gask...

We consider a random version of the McMullen–Bedford general Sierpinski carpet which is constructed by randomly choosing patterns in each step instead of a single pattern in its original form. Their Hausdorff, packing and box-counting dimensions are determined. A sufficient condition and a necessary condition for the Hausdorff measures in their dimensions to be positive are given. As an applica...

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...

Self-affinity and self-similarity are fundamental concepts in fractal geometry. In this paper, they are related to collage grammars syntactic devices based on hyperedge replacement that generate sets of collages. Essentially, a collage is a picture consisting of geometric parts like line segments, circles, polygons, polyhedra, etc. The overlay of all collages in a collage language yields a frac...

In this paper we studied the translation surfaces according to a new frame called q-frame in three dimensional Euclidean space. The curvatures of the translation surface are obtained in terms of q-frame curvatures. Finally some special cases are investigated for these surfaces.

We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the C(ℵ)-fibration property. We give and prove the relationship between the C(ℵ)-fibration property and an approximate fibration property. Furthermore, we study the pullback maps for C(ℵ)-fibrations.

In this paper, we first present a new type of the concept of open sets by expressing some properties of arbitrary mappings on a power set. With the generalization of the closure spaces in categorical topology, we introduce the generalized topological spaces and the concept of generalized continuity and become familiar with weak and strong structures for generalized topological spaces. Then, int...