Uniform Harnack inequalities for harmonic functions on the preand graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the construction of Brownian motion on Sierpinski carpets embedded in Md , d > 3, estimates on the fundamental solution of the heat equation, and Sobolev and Poincaré inequalities. The Sierpinski carpets (S...

In this paper, we firstly devise a new and general p-ary subdivision scheme based on normal vectors with multi-parameters to generate fractals. Rich and colorful fractals including some known fractals and a lot of unknown ones can be generated directly and conveniently by using it uniformly. The method is easy to use and effective in generating fractals since the values of the parameters and th...

Connections between the analysis on fractals and the iteration of rational functions were discovered in the earliest publications on diffusion processes on certain self-similar sets, such as the Sierpiński gasket (see, for instance [3, 38]). The connection stems from the fact that time on the successive approximations of the fractal is modelled by a branching process. The relation of branching ...

The ‘analysis on fractals’ and ‘analysis on metric spaces’ communities have tended to work independently. Metric spaces such as the Sierpinski carpet fail to satisfy some of the properties which are generally assumed for metric spaces. This survey discusses analysis on the Sierpinski carpet, with particular emphasis on the properties of the heat kernel. 1. Background and history Percolation was...

The Sierpinski gasket admits a locally isometric ramified self-covering. A semifinite spectral triple is constructed on the resulting solenoidal space, and its main geometrical features are discussed.

We study the dynamic structure factor S(k,t) of proteins at large wave numbers k, kR(g)≫1, where R(g) is the gyration radius. At this regime measurements are sensitive to internal dynamics, and we focus on vibrational dynamics of folded proteins. Exploiting the analogy between proteins and fractals, we perform a general analytic calculation of the displacement two-point correlation functions, <...

We apply a recently obtained three-critical-point theorem of B. Ricceri to prove the existence of at least three solutions of certain two-parameter Dirichlet problems defined on the Sierpinski gasket. We also show the existence of at least three nonzero solutions of certain perturbed two-parameter Dirichlet problems on the Sierpinski gasket, using both the mountain pass theorem of Ambrosetti an...

Recently, Bennett arononled that he proved a conjecture of Sierpinski on triangular numbers. In this paper, we firstly modified the mistakes in reference [7] of Bennett and [8] of Chen and Fang, and then using Störmer’s theorem of the solutions of Pell equation, and a deep result of primitive divisor of Bilu, Hanrot and Voutier, we proved that there do not exist four distinct triangular numbers...

If a torsion-free hyperbolic group G has 1-dimensional boundary ∂∞G, then ∂∞G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When ∂∞G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3-dimensional hyperbolic Poincaré duality group. We also construct a “topologically rigid” hyperbolic group G: any homeomorphism of ∂∞G is induced by an ...

1 Presented to the Society, December 27, 1939. The author is indebted to Professor W. A. Hurwitz for his indispensable advice in the preparation of this paper. 2 W. Sierpinski, Leçons sur les Nombres Transfinis, Paris, 1928, p. 202; hereafter referred to as Sierpinski (I). The theorem was first stated for ordinals of the first and second ordinal class by G. Cantor, Beitrdge zur Begründung der t...