Variation and Minkowski dimension of fractal interpolation surface

The fractal interpolation surface on the rectangular domain is discussed in this paper. We study the properties of the oscillation and the variation of bivariate continuous functions. Then we discuss the special properties of bivariate fractal interpolation function, and estimate the value of its variation. Using the relation between the Minkowski dimension of the graph of continuous function and its variation, we obtain the exact value of the Minkowski dimension of the fractal interpolation surface.     

Estimation of fractal dimension and fractal curvatures from digital images

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic, etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behavior, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.     

Backbone fractal dimension and fractal hybrid orbital of protein structure

Fractal geometry analysis provides a useful and desirable tool to characterize the configuration and structure of proteins. In this paper we examined the fractal properties of 750 folded proteins from four different structural classes, namely (1) the α-class (dominated by α-helices), (2) the β-class (dominated by β-pleated sheets), (3) the (α/β)-class (α-helices and β-sheets alternately mixed) and (4) the (α + β)-class (α-helices and β-sheets largely segregated) by using two fractal dimension methods, i.e. “the local fractal dimension” and “the backbone fractal dimension” (a new and useful quantitative parameter). The results showed that the protein molecules exhibit a fractal behavior in the range of 1 ? N ? 15 (N is the number of the interval between two adjacent amino acid residues), and the value of backbone fractal dimension is distinctly greater than that of local fractal dimension for the same protein. The average value of two fractal dimensions decreased in order of α > α/β > α + β > β. Moreover, the mathematical formula for the hybrid orbital model of protein based on the concept of backbone fractal dimension is in good coincidence with that of the similarity dimension. So it is a very accurate and simple method to analyze the hybrid orbital model of protein by using the backbone fractal dimension.     

Affective property of image and fractal dimension

Affective information processing is an advanced research direction in the AI world. Affective Information of image was taken as the objective of research in this paper. The influence of color vision properties’ histograms of image on human emotions was analyzed. Then based on fractal theory, the fractal aspect of different kinds of images was analyzed in keeping with the space domain. After that, psychological testing method of semantic difference was applied to verify the uniformity of the objective and subjective evaluations. At last, a conclusion was drawn that image having different affective properties could be classified by their fractal dimensions.     

THE SMOOTHNESS AND DIMENSION OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, we investigate the smoothness of non-equidistant fractal interpolation functions We obtain the Holder exponents of such fractal interpolation functions by using the technique of operator approximation. At last, We discuss the series expressiong of these functions and give a Box-counting dimension estimation of “critical” fractal interpohltion functions by using our smoothness results.     

Box dimension and fractional integral of linear fractal interpolation functions

In this paper, we present a new method to calculate the box dimension of a graph of continuous functions. Using this method, we obtain the box dimension formula for linear fractal interpolation functions (FIFs). Furthermore we prove that the fractional integral of a linear FIF is also a linear FIF and in some cases, there exists a linear relationship between the order of fractional integral and box dimension of two linear FIFs.     

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

We consider Farey series of rational numbers in terms of fractal sets labeled by the Hausdorff dimension with values defined in the interval 1<h<2 and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of dual topological quantum numbers, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that the universality classes of the quantum Hall transitions are classified in terms of h. The connection between Number Theory and Physics appears naturally in this context.     

The Minkowski dimension and critical effects in fractal evolution of defects

Creation of the programs for determination of the Minkowski dimension and other corresponding fractal characteristics are the main result of this paper. The programs were created also for the one-side fractal dimension. The stable numerical algorithm of multiplicative differentiate is given in this paper. One confirms that the critical effect occurs and it is corresponding to creation of the macroscopic crack through the breaking of the early formed microdefects. The singularity occurs at the moment at which the percolation is simultaneously observed. A character of the singularity is well visible – the jump for tens of orders occurs. The critical effect was observed in the computer simulations of the fatigue process.     

On the connection between the order of the fractional derivative and the Hausdorff dimension of a fractal function

This paper investigates the fractional derivative of a fractal function. It has been proven that there exists certain linear connection between the order of the Weyl-Marchaud fractional derivatives(WMFD) and the Hausdorff dimension of a fractal function. Graphs and numerical results further show this linear relationship.     

The concept of physical and fractal dimension I. The projective dimensions

The projective dimensional analysis based on the projective extension of scaling group and projective dimensional function is presented. We study the notion of self-similarity based on the physical quantities and not on topology. The projective analog of the classical theorem-π has been formulated in terms of (projective) numerical invariants. The additivity dimensions replacing fractal dimension are briefly discussed. Intuitive examples illustrate presented ideas.     

Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension

Recurrent bivariate fractal interpolation surfaces (RBFISs) generalise the notion of affine fractal interpolation surfaces (FISs) in that the iterated system of transformations used to construct such a surface is non-affine. The resulting limit surface is therefore no longer self-affine nor self-similar. Exact values for the box-counting dimension of the RBFISs are obtained. Finally, a methodology to approximate any natural surface using RBFISs is outlined.     

Robust to noise and outliers estimator of correlation dimension

The estimation of correlation dimension of continuous and discreet deterministic chaotic processes corrupted by an additive noise and outliers observations is investigated. In this paper we propose a new estimator of correlation dimension based on similarity between the evolution of Gaussian kernel correlation sum (Gkcs) and that of modified Boltzmann sigmoidal function (mBsf), this estimator is given by the maximum value of the first derivative of logarithmic transform of Gkcs against logarithmic transform of bandwidth, so the proposed estimator is independent of the choice of regression region like other regression estimators of correlation dimension. Simulation study indicates the robustness of proposed estimator to the presence of different types of noise such us independent Gaussian noise, non independent Gaussian noise and uniform noise for high noise level, moreover, this estimator is also robust to presence of 60% of outliers observations. Application of this new estimator with determination of their confidence interval using the moving block bootstrap method to adjusted closed price of S&P500 index daily time series revels the stochastic behavior of such financial time series.     

Discrepancies between metric dimension and partition dimension of a connected graph

Let and be graphs where the set of vertices is the set of points of the integer lattice and the set of edges consists of all pairs of vertices whose city block and chessboard distances, respectively, are 1.In this paper it is shown that the partition dimensions of these graphs are 3 and 4, respectively, while their metric dimensions are not finite. Also, for every n3 there exists an induced subgraph of of order 3n-1 with metric dimension n and partition dimension 3. These examples will answer a question raised by Chartrand, Salehi and Zhang. Furthermore, graphs of order n9 having partition dimension n-2 are characterized, thus completing the characterization of graphs of order n having partition dimension 2, n, or n-1 given by Chartrand, Salehi and Zhang. The list of these graphs includes 23 members.     

A set of formulae on fractal dimension relations and its application to urban form

The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary dimension of cities is derived from the idea of box-counting measurement and the principle of dimensional consistency in this paper. Thus, several practical results are obtained as follows. First, I derive the hyperbolic relation between the boundary dimension and form dimension of cities. Using the relation, we can estimate the form dimension through the boundary dimension and vice versa. Second, I derive the proper scales of fractal dimension: the form dimension comes between 1.5 and 2, and the boundary dimension comes between 1 and 1.5. Third, I derive three form dimension values with special geometric meanings. The first is 4/3, the second is 3/2, and the third is 1 + 2^1/2/2 ≈ 1.7071. The fractal dimension relation formulae are applied to China’s cities and the cities of the United Kingdom, and the computations are consistent with the theoretical expectation. The formulae are useful in the fractal dimension estimation of urban form, and the findings about the fractal parameters are revealing for future city planning and the spatial optimization of cities.     

The concept of physical and fractal dimension II. The differential calculus in dimensional spaces

The projective dimensional analysis based on the projective extension of scaling group and projective dimensional function is studied. The differential calculus corresponding to geometry of dimensional spaces is constructed and examined. At the next step we explore the projective extension of dimensional derivatives. Simple fractal models of various processes with changing fractal dimension illustrate the proposed methods.     

Discrete sources analysis of the correlation between surface plasmon resonance and extraordinary optical transmission

The discrete sources method is adapted to analyze the scattered energy distribution in the entire space in the problem of diffraction by an inhomogeneity in a film. The relation between extraordinary optical transmission through a nanometric film and surface plasmon resonance is studied numerically. It is shown that these phenomena are closely correlated.     

Effects of GSM microwaves, pulsed magnetic field, and temperature on fractal dimension of brain tumors

Fractal dimension of a two-dimensional C-6 rat glioma tumors growing in the microwave field generated by signal simulation of the Global System for Mobile communications (GSM) with frequency 960 MHz was found significantly enhanced as compared with field free tumors growing at different temperatures and on the other hand a strong pulsed magnetic field lowered fractal dimension of tumors.     

Estimates for the fractal dimension and the number of determining modes for invariant sets of dynamical systems

Majorants of the fractal dimension and of the number of determining modes for unbounded sets, invariant with respect to operators of semigroups of classes 1 and 2, are obtained. They are computed for the Navier-Stokes equations (two- and three-dimensional) under the first boundary condition and under periodicity conditions in the spaces and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 105–129, 1987.