Biological systems are by nature complex and this complexity has been shown to be important in maintaining homeostasis. The plant microtubule cytoskeleton is a highly complex system, with contributing factors through interactions with microtubule-associated proteins (MAPs), expression of multiple tubulin isoforms, and post-translational modification of tubulin and MAPs. Some of this complexity ...

Fractals with different levels of self-similarity and magnification are defined as reduced fractals. It is shown that spectra these fractals can be constructed used to describe complexity natural phenomena. Specific applications biological systems, such green algae, performed, it suggested the obtained classify considered algae by identifying associated them. The ranges for determined their ext...

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...

Riesz potentials of fractal measures μ in metric spaces and their inverses are introduced . They define self–adjoint operators in the Hilbert space L2(μ) and the former are shown to be compact. In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operato...

In undergraduate classrooms, while teaching chaos and fractals, it is taught as if there no relation between these two. By using some non linear oscillators we demonstrate that a connection fractals. plotting the phase space diagrams of four nonlinear box counting method finding fractal dimension established chaotic nature oscillators. The awareness all systems are good fractals will add more i...

This paper discusses one method of producing fractals, namely that of iterated function systems. We first establish the tools of Hausdorff measure and Hausdorff dimension to analyze fractals, as well as some concepts in the theory of metric spaces. The latter allows us to prove the existence and uniqueness of fractals as fixed points of iterated function systems. We discuss the connection betwe...

V-variable fractals, where V is a positive integer, are intuitively fractals with at most V different “forms” or “shapes” at all levels of magnification. In this paper we describe how V-variable fractals can be used for the purpose of image compression.