An intensity-expansion method to treat non-stationary time series: an application to the distance between prime numbers

We study the fractal properties of the distances between consecutive primes. The distance sequence is found to be well described by a non-stationary exponential probability distribution. We propose an intensity-expansion method to treat this non-stationarity and we find that the statistics underlying the distance between consecutive primes is Gaussian and that, by transforming the distance sequence into a stationary one, the range of Gaussian randomness of the sequence increases.Comment: 11 pages, 7 figures, in press on 'Chaos, Solitons & Fractals

Random walk through fractal environments

We analyze random walk through fractal environments, embedded in 3-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e. of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D of the fractal is less than 2, there is though always a finite rate of unaffected escape. Random walks through fractal sets with D less or equal 2 can thus be considered as defective Levy walks. The distribution of jump increments for D > 2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk-increments. It is shown that the particles undergo anomalous, enhanced diffusion for D_F < 2, the diffusion is dominated by the finite escape rate. Diffusion for D_F > 2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality (SOC) models give rise to enhanced diffusion. The analytical results are illustrated by Monte-Carlo simulations.Comment: 22 pages, 16 figures; in press at Phys. Rev. E, 200

A multiple exp-function method for nonlinear differential equations and its application

A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure which generalizes Hirota's perturbation scheme. With help of Maple, an application of the approach to the $3+1$ dimensional potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure

Study of the island morphology at the early stages of Fe/Mo(110) MBE growth

We present theoretical study of morphology of Fe islands grown at Mo(110) surface in sub-monolayer MBE mode. We utilize atomistic SOS model with bond counting, and interactions of Fe adatom up to third nearest neighbors. We performed KMC simulations for different values of adatom interactions and varying temperatures. We have found that, while for the low temperature islands are fat fractals, for the temperature 500K islands have faceted rhombic-like shape. For the higher temperature, islands acquire a rounded shape. In order to evaluated qualitatively morphological changes, we measured averaged aspect ration of islands. We calculated dependence of the average aspect ratio on the temperature, and on the strength of interactions of an adatom with neighbors.Comment: 6 pages, 6 figures. Proceedings of 11-th Symposium on Surface Physics, Prague 200

Application of Fractal Growth Patterns in Housing Layout Design

In early phases of design, during the process of form-exploration, architects -- knowingly or unknowingly -- have used mathematics as their guiding tool to evolve a formal methodology of design. Fundamental compositional principles such as symmetry, rhythm and proportion are based on specific mathematical underpinnings. However, very often the designer comes across a situation where these underlying mathematical principles need to be overlapped or interfaced. Applying fractal concepts to the order can accommodate this complex diversity. Fractals allow us to provide a combination of order and surprise in a rhythmic composition using a specific mathematical geometry. Fractals are typically unit-based and, can thus allow exploration in architectural designs which have a ‘unit’ as a fundamental issue or necessity. The design of housing layout stands out prominently among such architectural problems and, can thus be one such instance in which fractals may be used as a design tool. Commonly seen organisational patterns in housing layout designs create rigidity and monotony, while others like clustered groups are too inconsistent and can create disorder. The research tries applying fractal ordering principles to strike a balance between these extremes by creating an orderly arrangement of houses with an underlying variation in the pattern. The traditional processes of creating housing layouts is quite cumbersome. With the mathematical power of computers, fractal ordering principles are used as Iterative functions to generate multiple design options. The research investigates the potential of the emergent patterns of fractals as an organisational principle in designing housing layouts, while limiting it based on site constraints, size and the transforming rules. In doing so, the objective is to explore the computational and mathematical basis of repetitive patterns in architectural order and compositions. The study also aims at developing a computer application, based on algorithms using fractals, which offers capabilities as a conceptual and organisational tool for a housing layout. The application is implemented, tested and its results are demonstrated using a live terrain data. Search Keywords for This Page Fractals in architecture and design, Fractal geometry in architecture, House patterns designs, Fractal geometry in architecture and design, Fractals in architecture, Fractal houses, Housing layout design, Fractals architecture, Fractal architecture buildin