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...

In this paper we define pairwise-$\alpha$-paracompact subsets as a generalisation of $\alpha$-paracompact in topological spaces and study their properties its relations with other bitopological spaces. Several theorems are obtained concerning sets. We introduced the definition $p$-paracompact bitoplogical

A new neural network model is introduced in this paper. The aim of the proposed Sierpinski neural networks is to provide a simple and biologically plausible neural network architecture that produces emergent complex spatio-temporal patterns through the activity of the output neurons of the network. Such networks can play an important role in the analysis and understanding of complex dynamic act...

For a modal algebra (B, f), there are two natural ways to extend f to an operation on the MacNeille completion of B. The resulting structures are called the lower and upper MacNeille completions of (B, f). In this paper we consider lower and upper MacNeille completions for various varieties of modal algebras. In particular, we characterize the varieties of closure algebras and diagonalizable al...

The ‘minimal’ payment—a payment method which minimizes the number of coins in a purse—is presented. We focus on a time series of change given back to a shopper repeating the minimal payment. The delay plot shows visually that the set of successive change possesses a fine structure similar to the Sierpinski gasket. We also estimate effectivity of the minimal-payment method by means of the averag...

Sierpinski space Ω is injective in the category Top of topological spaces, but not in any of the larger cartesian closed categories Conv of convergence spaces and Equ of equilogical spaces. We show that this negative result extends to all sub-cccs of Equ and Conv that are closed under subspaces and contain Top. On the other hand, we study the category PrTop of pretopological spaces that lies in...