Narrow Results

A nondistributive scator algebra in 1 + 2 dimensions is used to map the quadratic iteration. The hyperbolic numbers square bound set reveals a rich structure when taken into the three-dimensional (3D) hyperbolic scator space. Self-similar small copies of the larger set are obtained along the real axis. These self-similar sets are located at the same positions and have equivalent relative sizes as the small M-set copies found between the Myrberg-Feigenbaum (MF) point and -2 in the complex Mandelbrot set. Furthermore, these small copies are self similar 3D copies of the larger 3D bound set. The real roots of the respective polynomials exhibit basins of attraction in a 3D space. Slices of the 3D confined scator set, labeled (s;x,y), are shown at different planes to give an approximate idea of the 3D objects highly complicated boundary.

The difference between the inverse power function and the negative exponential function is significant. The former suggests a complex distribution, while the latter indicates a simple distribution. However, the association of the power-law distribution with the exponential distribution has been seldom researched. This paper is devoted to exploring the relationships between exponential laws and power laws from the angle of view of urban geography. Using mathematical derivation and numerical experiments, I reveal that a power-law distribution can be created through a semi-moving average process of an exponential distribution. For the distributions defined in a one-dimension space (e.g. Zipf's law), the power exponent is 1; while for those defined in a two-dimension space (e.g. Clark's law), the power exponent is 2. The findings of this study are as follows. First, the exponential distributions suggest a hidden scaling, but the scaling exponents suggest a Euclidean dimension. Second, special power-law distributions can be derived from exponential distributions, but they differ from the typical power-law distributions. Third, it is the real power-law distributions that can be related with fractal dimension. This study discloses an inherent link between simplicity and complexity. In practice, maybe the result presented in this paper can be employed to distinguish the real power laws from spurious power laws (e.g. the fake Zipf distribution).

Animal models represent the basis of our current understanding of the pathophysiology of asthma and are of central importance in the preclinical development of drug therapies. The characterization of irregular lung shapes is a major issue in radiological imaging of mice in these models. The aim of this study was to find out whether differences in lung morphology can be described by fractal geometry. Healthy and asthmatic mouse groups, before and after an acute asthma attack induced by methacholine, were studied. In vivo flat-panel-based high-resolution Computed Tomography (CT) was used for mice's thorax imaging. The digital image data of the mice's lungs were segmented from the surrounding tissue. After that, the lungs were divided by image gray-level thresholds into two additional subsets. One subset contained basically the air transporting bronchial system. The other subset corresponds mainly to the blood vessel system. We estimated the fractal dimension of all sets of the different mouse groups using the mass radius relation (mrr). We found that the air transporting subset of the bronchial lung tissue enables a complete and significant differentiation between all four mouse groups (mean D of control mice before methacholine treatment: 2.64 ± 0.06; after treatment: 2.76 ± 0.03; asthma mice before methacholine treatment: 2.37 ± 0.16; after treatment: 2.71 ± 0.03; p < 0.05). We conclude that the concept of fractal geometry allows a well-defined, quantitative numerical and objective differentiation of lung shapes — applicable most likely also in human asthma diagnostics.

The entropy production and the variational functional of a Laplacian diffusional field around the first four fractal iterations of a linear self-similar tree (von Koch curve) is studied analytically and detailed predictions are stated. In a next stage, these predictions are confronted with results from numerical resolution of the Laplace equation by means of Finite Elements computations. After a brief review of the existing results, the range of distances near the geometric irregularity, the so-called "Near Field", a situation never studied in the past, is treated exhaustively. We notice here that in the Near Field, the usual notion of the active zone approximation introduced by Sapoval et al. [M. Filoche and B. Sapoval, Transfer across random versus deterministic fractal interfaces, Phys. Rev. Lett. 84(25) (2000) 5776;¹ B. Sapoval, M. Filoche, K. Karamanos and R. Brizzi, Can one hear the shape of an electrode? I. Numerical study of the active zone in Laplacian transfer, Eur. Phys. J. B. Condens. Matter Complex Syst. 9(4) (1999) 739-753.]² is strictly inapplicable. The basic new result is that the validity of the active-zone approximation based on irreversible thermodynamics is confirmed in this limit, and this implies a new interpretation of this notion for Laplacian diffusional fields.

Let $S$ be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map

In this paper, we introduce the $p$-circle inversion which generalizes the classical inversion with respect to a circle ($p=2$) and the taxicab inversion $(p=1)$. We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the $p$-circle inversion.

Fractal interpolation is a modern technique for fitting of smooth/non-smooth data. Based on only functional values, we develop two types of ${{𝒞}}^1$-rational fractal interpolation surfaces (FISs) on a rectangular grid in the present paper that contain scaling factors in both directions and two types of positive real parameters which are referred as shape parameters. The graphs of these ${{𝒞}}^1$-rational FISs are the attractors of suitable rational iterated function systems (IFSs) in ${ℝ}^3$ which use a collection of rational IFSs in the $x$-direction and $y$-direction and hence these FISs are self-referential in nature. Using upper bounds of the interpolation error of the $x$-direction and $y$-direction fractal interpolants along the grid lines, we study the convergence results of ${{𝒞}}^1$-rational FISs toward the original function. A numerical illustration is provided to explain the visual quality of our rational FISs. An extra feature of these fractal surface schemes is that it allows subsequent interactive alteration of the shape of the surfaces by changing the scaling factors and shape parameters.

The concept of local fractional derivative was introduced in order to be able to study the local scaling behavior of functions. However it has turned out to be much more useful. It was found that simple equations involving these operators naturally incorporate the fractal sets into the equations. Here, the scope of these equations has been extended further by considering different possibilities for the known function. We have also studied a separable local fractional differential equation along with its method of solution.

We investigate bond percolation on the iterated barycentric subdivision of a triangle, the hexa-carpet, and the non-p.c.f. Sierpinski gasket. With the use of known results on the diamond fractal, we are able to bound the critical probability of bond percolation on the non-p.c.f. gasket and the iterated barycentric subdivision of a triangle from above by 0.282. We then show how both the gasket and hexacarpet fractals are related via the iterated barycentric subdivisions of a triangle: the two spaces exhibit duality properties although they are not themselves dual graphs. Finally, we show the existence of a non-trivial phase transition on all three graphs.

The dynamics seismic activity occurred in the Cocos Plate—Mexico is analyzed through the evolution of Hurst exponent and 3D fractal dimension, under the mathematical fractal structure based on seismic activity time series, taking into account the magnitude ($M$) as the main parameter to be estimated. The seismic activity time series and, annual intervals are considered first for finding the Hurst exponent of each year since 1988 (the year in which the database is consistent) until 2012, and then the following years are accumulated describing the cumulative Hurst exponent. The seismic activity description is based on Cocos Plate data information; during a period conform from 1 January 1988 to 31 December 2012. Analyses were performed following methods, mainly considering that the Hurst exponent analysis provides the ability to find the seismicity behavior time–space, described by parameters obtained under the fractal dimension and complex systems.

This paper analyzes the complexity of stock exchanges through fractal theory. Closing price indices of four stock exchanges with different industry sectors are selected. Degree of complexity is assessed through Higuchi’s fractal dimension. Various window sizes are considered in evaluating the fractal dimension. It is inferred that the data considered as a whole represents random walk for all the four indices. Analysis of financial data through windowing procedure exhibits multi-fractality. Attempts to apply moving averages to reduce noise in the data revealed lower estimates of fractal dimension, which was verified using fractional Brownian motion. A change in the normalization factor in Higuchi’s algorithm did improve the results. It is quintessential to focus on rural development to realize a standard and steady growth of economy. Tools must be devised to settle the issues in this regard. Micro level institutions are necessary for the economic growth of a country like India, which would induce a sporadic development in the present global economical scenario.

The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussinesq equation into a nonlinear local fractional ODE. The exact traveling wave solution is also obtained with aid of the non-differentiable graph. The proposed method, involving the fractal special functions, is efficient for finding the exact solutions of the nonlinear PDEs in fractal domains.

The Sierpinski Triangle (ST) is a fractal which has Haussdorf dimension ${log}_{2}3\approx 1.585$ that has been studied extensively. In this paper, we introduce the Sierpinski Triangle Plane (STP), an infinite extension of the ST that spans the entire real plane but is not a vector subspace or a tiling of the plane with a finite set of STs. STP is shown to be a radial fractal with many interesting and surprising properties.

For beta-transformations, we prove that the Lebesgue measure of any measurable scrambled set is zero, and there exists a scrambled set with full Hausdorff dimension.

Let $N$ be an integer greater than or equal to $2$ and let $x_i^{\prime }s$ be numbers with $x_0<x_1<x_2<\cdots <x_N$. Denote that $I$ is the interval $[x_0,x_N]$ and $\mathrm{\Delta }=\{(x_k,{\mu }_k)\in ℝ\times ℝ:k=0,1,\dots ,N\}$ is a set of points. Suppose that $Y_k$ is a random perturbation of ${\mu }_k$ for $k=0,1,\dots ,N$, and we set ${\mathrm{\Delta }}^{\ast }=\{(x_k,Y_k):k=0,1,\dots ,N\}$. Let $f_{\mathrm{\Delta }}$ and $f_{{\mathrm{\Delta }}^{\ast }}$ be linear fractal interpolation functions on $I$ corresponding to the set of points $\mathrm{\Delta }$ and ${\mathrm{\Delta }}^{\ast }$, respectively. The value $f_{{\mathrm{\Delta }}^{\ast }}(x)$ is random for all $x\in I$. In this paper, we show that the expectation of $f_{{\mathrm{\Delta }}^{\ast }}(x)$ is $f_{\mathrm{\Delta }}(x)$. We also establish estimations for the variance of $f_{{\mathrm{\Delta }}^{\ast }}(x)$ and the expectation of $|f_{{\mathrm{\Delta }}^{\ast }}(x)-f_{\mathrm{\Delta }}(x)|$.

In order to research the properties of the fractional order calculus of broken line segments’ fractal interpolation function (FIF) generated by the linear iterated function system (IFS), the concepts of the Riemann–Liouville fractional order calculus and the method of the IFS are used to prove the properties of the fractional calculus of the broken line segments’ FIF generated by the linear IFS. There are two conclusions as follows. First, the fractional order integral of the broken line segments’ FIF formed by the linear IFS is continuous and first-order differentiable on the closed interval $[0,x_N]$. Second, the broken line segments’ FIF formed by the linear IFS exists with fractional order differential, but the differential function is not continuous.

The aim of this work is to quantitatively explore the texture evolution of amphibole aggregation and residual melt with pressure and temperature. The amphibole aggregation growth from a basaltic melt and the residual melt at high pressure (0.6–2.6GPa) and high temperature (860–970${}^{\circ }$C) exhibit statistical self-similarity which made us consider studying such characteristic by fractal analysis. The bi-phase box counting method was applied for fractal analysis of each product to identify the fractal phase and the fractal dimension was estimated. In the experimental products, the residual melt is identified as the fractal and amphibole as the Euclidean except for one experiment. The results show that the residual melt can be quantified by the fractal dimension $(D_B)$ within the range of 1.782–1.848. The temperature has a significant effect on the morphology of amphibole and the fractal dimension of the residual melt. The higher the crystallization temperature is, the more regular the amphibole grains are. At lower temperature (from 860${}^{\circ }$C to 915${}^{\circ }$C), the fractal dimension of the residual melt decreased with the increasing crystallization temperature, but at higher temperature (970${}^{\circ }$C), the fractal phase changed to amphibole and the fractal dimension of amphibole is 1.816. The pressure may be the dominant factor that controls the morphology of the mineral aggregation and the residual melt. The fractal dimension of melt decreased linearly with the increasing pressure and if the linear relationship between the fractal dimension and pressure can be further verified in the future, it can be used as a potential geological barometer.

In this paper we consider the expectation, the autocovariance, and increments of the deviation of a fractal interpolation function $f_{{\mathrm{\Delta }}_Y}$ corresponding to a random dataset ${\mathrm{\Delta }}_Y=\{(x_k,Y_k):k=0,1,\dots ,N\}$. We show that the covariance of $f_{{\mathrm{\Delta }}_Y}(x)$ and $Y_i$ is a fractal interpolation function on $I$ for each fixed $Y_i$, where $I=[x_0,x_N]$. We also prove that, for a fixed $x\in I$, the covariance of $f_{{\mathrm{\Delta }}_Y}(x)$ and $f_{{\mathrm{\Delta }}_Y}(t)$ is a fractal interpolation function on $I$. A special type of increments of the deviation of $f_{{\mathrm{\Delta }}_Y}$ is also investigated.

The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other interpolant, both smooth and nonsmooth, but the important fact is that this technique provides one of the few methods of nondifferentiable interpolation. In this way, it constitutes a functional model for chaotic processes. This paper studies a generalization of an approximation formula proposed by Dunham Jackson, where a wider range of values of an exponent of the basic trigonometric functions is considered. The trigonometric polynomials are then transformed in close fractal functions that, in general, are not smooth. For suitable election of this parameter, one obtains better conditions of convergence than in the classical case: the hypothesis of continuity alone is enough to ensure the convergence when the sampling frequency is increased. Finally, bounds of discrete fractal Jackson operators and their classical counterparts are proposed.

An important problem in analysis on fractals is the existence and the determination of an eigenform on a given finitely ramified fractal. It is known that on every fractal either with three vertices or with connected interior, an eigenform exists for suitable weights on the cells. In this paper, we prove that if the fractal has three vertices and connected interior, the form having all coefficients equal to $1$ is an eigenform for suitable weights on the cells.