Fractal Fields Research Articles

Fractal Theory and Field Cover Experiments: Implications for the Fractal Characteristics and Radon Diffusion Behavior of Soils and Rocks.

Radon diffusion and transport through different media is a complex process affected by many factors. In this study, the fractal theories and field covering experiments were used to study the fractal characteristics of particle size distribution (PSD) of six kinds of geotechnical materials (e.g., waste rock, sand, laterite, kaolin, mixture of sand and laterite, and mixture of waste rock and laterite) and their effects on radon diffusion. In addition, the radon diffusion coefficient and diffusion length were calculated. Moreover, new formulas for estimating diffusion coefficient and diffusion length functional of fractal dimension d of PSD were proposed. These results demonstrate the following points: (1) the fractal dimension d of the PSD can be used to characterize the property of soils and rocks in the studies of radon diffusion behavior; (2) the diffusion coefficient and diffusion length decrease with increasing fractal dimension of PSD; and (3) the effectiveness of final covers in reducing radon exhalation of uranium tailings impoundments can be evaluated on the basis of the fractal dimension of PSD of materials.

Fractal vector optical fields.

We introduce the concept of a fractal, which provides an alternative approach for flexibly engineering the optical fields and their focal fields. We propose, design, and create a new family of optical fields-fractal vector optical fields, which build a bridge between the fractal and vector optical fields. The fractal vector optical fields have polarization states exhibiting fractal geometry, and may also involve the phase and/or amplitude simultaneously. The results reveal that the focal fields exhibit self-similarity, and the hierarchy of the fractal has the "weeding" role. The fractal can be used to engineer the focal field.

A Complex Story: Universal Preference vs. Individual Differences Shaping Aesthetic Response to Fractals Patterns.

Fractal patterns offer one way to represent the rough complexity of the natural world. Whilst they dominate many of our visual experiences in nature, little large-scale perceptual research has been done to explore how we respond aesthetically to these patterns. Previous research (Taylor et al., 2011) suggests that the fractal patterns with mid-range fractal dimensions (FDs) have universal aesthetic appeal. Perceptual and aesthetic responses to visual complexity have been more varied with findings suggesting both linear (Forsythe et al., 2011) and curvilinear (Berlyne, 1970) relationships. Individual differences have been found to account for many of the differences we see in aesthetic responses but some, such as culture, have received little attention within the fractal and complexity research fields. This two-study article aims to test preference responses to FD and visual complexity, using a large cohort (N = 443) of participants from around the world to allow universality claims to be tested. It explores the extent to which age, culture and gender can predict our preferences for fractally complex patterns. Following exploratory analysis that found strong correlations between FD and visual complexity, a series of linear mixed-effect models were implemented to explore if each of the individual variables could predict preference. The first tested a linear complexity model (likelihood of selecting the more complex image from the pair of images) and the second a mid-range FD model (likelihood of selecting an image within mid-range). Results show that individual differences can reliably predict preferences for complexity across culture, gender and age. However, in fitting with current findings the mid-range models show greater consistency in preference not mediated by gender, age or culture. This article supports the established theory that the mid-range fractal patterns appear to be a universal construct underlying preference but also highlights the fragility of universal claims by demonstrating individual differences in preference for the interrelated concept of visual complexity. This highlights a current stalemate in the field of empirical aesthetics.

Analysis of fractals with combined partition

The space—time properties in the general theory of relativity, as well as the discreteness and non-Archimedean property of space in the quantum theory of gravitation, are discussed. It is emphasized that the properties of bodies in non-Archimedean spaces coincide with the properties of the field of P-adic numbers and fractals. It is suggested that parton showers, used for describing interactions between particles and nuclei at high energies, have a fractal structure. A mechanism of fractal formation with combined partition is considered. The modified SePaC method is offered for the analysis of such fractals. The BC, PaC, and SePaC methods for determining a fractal dimension and other fractal characteristics (numbers of levels and values of a base of forming a fractal) are considered. It is found that the SePaC method has advantages for the analysis of fractals with combined partition.

Use of Computational Fluid Dynamic to Compare the Pressure Loss between a Parallel Flow Field Plate with a Parallel-baffle Flow Field Plate and a Fractal Parallel-baffle Flow Field Plate in a Proton Exchange Membrane Fuel Cell

This research provided CFD tests to verify and understand the reagents fluid dynamic behavior in the channels of the new parallel-baffle flow field plate during a direct ethanol proton exchange membrane fuel cell (DE-PEMFC) operation. The results showed that. The results showed that the fractal plate showed a similar behavior normal plate and a repetition through fractals, the fluid dynamic behavior.

Computational Fractal Logic for Quantum Physics and Cosmology

In the spirit of the work of Ramanujan’s infinite Cesaro sum and the formalization of his work by G. Hardy, we propose that quantum physics and cosmology are governed by fractal logic rather than ordinary classical logic. Various general considerations as well as specific examples are given to illustrate and support our thesis leading to a revision of the standard model of elementary particles as well as relativity. These examples range from the practical aspect to almost esoteric considerations but at the end, everything converges towards a unity of theory and computation presented in the form of modern fractal logic and transfinite quantum field theory in a Cantorian spacetime. We conclude that the universe is akin to a transfinite Turing machine, or a Transfinite Infinite capacity of 'tHooft Automaton.

High Energy Physics and Cosmology as Computation

The present paper is basically written as a non-apologetic strong defence of the thesis that computation is part and parcel of a physical theory and by no means a mere numerical evaluation of the prediction of a theory which comes towards the end. Various general considerations as well as specific examples are given to illustrate and support our arguments. These examples range from the practical aspect to almost esoteric considerations but at the end, everything converges towards a unity of theory and computation presented in the form of modern fractal logic and transfinite quantum field theory in a Cantorian spacetime. It is true that all our examples are taken from physics but our discussion is applicable in equal measure to a much wider aspect of life.

Local Fractional Homotopy Perturbation Method for Solving Non-Homogeneous Heat Conduction Equations in Fractal Domains

In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media.

Electromagnetic waves in non-integer dimensional spaces and fractals

Electromagnetic waves in non-integer dimensional spaces are considered in the framework of continuous models of fractal media and fields. Using the recently suggested vector calculus for non-integer dimensional space, we consider electromagnetic fields in isotropic case. This D-dimensional calculus allows us to describe fractal properties by continuous models with non-integer dimensional spaces. We prove that the wave equation for non-integer dimensional space is similar to equation of waves in non-fractal medium with heterogeneity of power-law type. The speed of electromagnetic waves and the effective refractive index of non-integer dimensional spaces and fractals are discussed.

Spectral dimension and Bohr's formula for Schrödinger operators on unbounded fractal spaces

We establish an asymptotic formula for the eigenvalue counting function of the Schrödinger operator for some unbounded potentials V on several types of unbounded fractal spaces. We give sufficient conditions for Bohr’s formula to hold on metric measure spaces which admit a cellular decomposition, and then verify these conditions for fractafolds and fractal fields based on nested fractals. In particular, we partially answer a question of Fan, Khandker, and Strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow-up of a Sierpinski gasket.

The fractal nature materials microstructure influence on electrochemical energy sources

With increasing of the world energy crisis, research for new, renewable and alternative energy sources are in growth. The focus is on research areas, sometimes of minor importance and applications, where the different synthesis methods and microstructure properties optimization, performed significant improvement of output materials? and components? electro-physical properties, which is important for higher energy efficiency and in the electricity production (batteries and battery systems, fuel cells and hydrogen energy) contribution. Also, the storage tanks capacity improvement, for the energy produced on such way, which is one of the most important development issues in the energy sphere, represents a very promising research and application area. Having in mind, the results achieved in the electrochemical energy sources field, especially electrolyte development, these energy sources, materials fractal nature optimization analysis contribution, have been investigated. Based on materials fractal structure research field, particularly electronic materials, we have performed microstructure influence parameters research in electrochemistry area. We have investigated the Ho2O3 concentration influence (from 0.01wt% to 1wt%) and sintering temperature (from 1320?C to 1380?C), as consolidation parameters, and thus, also open the electrochemical function fractalization door and in the basic thermodynamic parameters the fractal correction introduced. The fractal dimension dependence on additive concentration is also investigated.

English

The main concept of Cross Field Antenna is getting phase shift between electric fields of the two cylinders of the antenna equals to 90 degrees. This is done by connecting a coil to the antenna circuit. The objective of doing this is to get rid of near field region and make far field region starting from antenna surface. Fractal shape antennas have many characteristics that not find in classical antennas. Among them is, multiband and wide band behaviour, antenna size is not depend on the operating frequency and easy matching of the antenna circuit. Here, combination of cross field antenna and fractal antenna is done; to get fractal shape cross field antenna. The proposed fractal shape is Sierpinski Carpet. Fractal shape cross field antenna has attracted wide band behaviour and self-tuning.

Ghost quintessence in fractal gravity

In this study, using the time-like fractal theory of gravity, we mainly focus on the ghost dark energy model which was recently suggested to explain the present acceleration of the cosmic expansion. Next, we establish a connection between the quintessence scalar field and fractal ghost dark energy density. This correspondence allows us to reconstruct the potential and the dynamics of a fractal canonical scalar field (the fractal quintessence) according to the evolution of ghost dark energy density.

A Derivation of Proca Equations on Cantor Sets: A Local Fractional Approach

In a recent paper published at Advances in High Energy Physics (AHEP) journal, Yang Zhao et al. derived Maxwell equations on Cantor sets from the local fractional vector calculus. It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Using the same approach, elsewhere Yang, Baleanu & Tenreiro Machado derived systems of Navier-Stokes equations on Cantor sets. However, so far there is no derivation of Proca equations on Cantor sets. Therefore, in this paper we present for the first time a derivation of Proca equations and GravitoElectroMagnetic (GEM) Proca-type equations on Cantor sets. Considering that Proca equations may be used to explain electromagnetic effects in superconductor, We suggest that Proca equations on Cantor sets can describe electromagnetic of fractal superconductors; besides GEM Proca-type equations on Cantor sets may be used to explain some gravitoelectromagnetic effects of superconductor for fractal media. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.

Fraktálna analýza osteoporózy: prístup podielom vierohodnosti

Based on the traditional fractal theory and on the paper of Stehlík, (2009) the range of fractal dimension of osteoporosis vertebras is analysed. First we give an insight into the field of fractals and the usa­ge of fractals in medicine. After this we show how the analytical tool of Stehlík, (2009) may be applied to the osteoporosis vertebras. It turns out that the used method can be applied very well and that it could help with medical diagnosis. Real data example illustrates the methods discussed.

Emergence of complex and spinor wave functions in scale relativity. II. Lorentz invariance and bi-spinors

Owing to the non-differentiable nature of the theory of Scale Relativity, the emergence of complex wave functions, then of spinors and bi-spinors occurs naturally in its framework. The wave function is here a manifestation of the velocity field of geodesics of a continuous and non-differentiable (therefore fractal) space-time. In a first paper (Paper I), we have presented the general argument which leads to this result using an elaborate and more detailed derivation than previously displayed. We have therefore been able to show how the complex wave function emerges naturally from the doubling of the velocity field and to revisit the derivation of the non-relativistic Schrödinger equation of motion. In the present paper (Paper II), we deal with relativistic motion and detail the natural emergence of the bi-spinors from such first principles of the theory. Moreover, while Lorentz invariance has been up to now inferred from mathematical results obtained in stochastic mechanics, we display here a new and detailed derivation of the way one can obtain a Lorentz invariant expression for the expectation value of the product of two independent fractal fluctuation fields in the sole framework of the theory of Scale Relativity. These new results allow us to enhance the robustness of our derivation of the two main equations of motion of relativistic quantum mechanics (the Klein-Gordon and Dirac equations) which we revisit here at length.

A Central Limit Result in the Wavelet Domain for Minimum Contrast Estimation of Fractal Random Fields

A minimum contrast criterion for consistently estimating the spatial fractal dimension in the wavelet domain is presented in [M. D. Ruiz--Medina and R. M. Crujeiras, Comm. Statist. Theory Methods., 40 (2011), pp. 3599--3613] for a class of semiparametric fractal Gaussian random fields, which includes fractional Brownian motion and Riesz--Bessel motion as particular cases. In this work, asymptotic normality of this minimum contrast estimator is obtained using central limit results for multiple stochastic integrals (see [N. Nualart and G. Peccati, Ann. Probab., 33 (2005), pp. 91--115]).

Human behavior recognition based on stratified fractal conditional random field

In view of real-time issue of the Hidden Conditional Random Field(HCRF) and marked deviation problem of the Latent-Dynamic Conditional Random Field(LDCRF) during behavior transforming,this article proposed a kind of behavior recognition algorithm based on Stratified Fractal Conditional Random Field(SFCRF).The proposed algorithm improved LDCRF and put forward the concept of score mark,which made the integrity and direction of human behavior specific.The experimental results show that the proposed algorithm can obtain better recognition effect than Conditional Random Field(CRF),HCRF and LDCRF.

Complete High Dimensional Inverse Characterization of Fractal Surfaces and Volumes

In the present paper, we are describing a methodology for the determination of the complete set of parameters associated with the Weierstrass-Mandelbrot (W-M) function that can describe a fractal scalar field distribution defined by measured or computed data distributed on a surface or in a volume. Our effort is motivated not only by the need for accurate fractal surface and volume reconstruction but also by the need to be able to describe analytically a scalar field quantity distribution on a surface or in a volume that corresponds to various material properties distributions for engineering and science applications. Our method involves utilizing a refactoring of the W-M function that permits defining the characterization problem as a high dimensional inverse problem solved by singular value decomposition for the so-called phases of the function. Coupled with this process is a second level exhaustive search that enables the determination of the density of the frequencies involved in defining the trigonometric functions participating in the definition of the W-M function. Numerical applications of the proposed method on both synthetic and actual surface and volume data, validate the efficiency and the accuracy of the proposed approach. This approach constitutes a radical departure from the traditional fractal dimension characterization studies and opens the road for a very large number of applications.

Use of PCM Boards for Solar Cell Cooling

This contribution explains the use of fractal theory to describe thermal properties of materials. The basic idea is rooted in the theory of fractal fields defined in E-dimensional Euclidian space. Generic equations describing heat distribution are then specialized to describe the changes in heat transfer as a response to step-wise increases in the amount of heat added to the system. This model was then applied to the study of properties of a systems consisting of solar cells attached on a phase-change material (PCM) back sheet board. The aim of this study is to evaluate the ability of PCM boards to decrease the working temperature of solar cells and thus to increase the efficiency of the cells. Regression of experimental data was used to obtain model parameters. The parameters obtained this way were the thermal diffusivity, thermal conductivity, and specific heat, as well as parameters of the heat source and parameters related to the heat losses of the system. The method was then verified against parameters of the system based on poly-methyl-methacrylate and then applied to a PCM with a phase-change temperature of 25 °C. The values of the thermal parameters were determined at temperatures where both components of the PCM composite (Micronal® and gypsum wall) were solid and below the phase-change temperature and then again at temperatures where one of the components (gypsum) was still solid, while the other one was already liquid (wax). The attempt to determine the parameters during the phase change was not made due the physicochemical processes taking place which would alter the measured data.