Multifractal characterization of a dental restorative composite after air-polishing

Roughness is of critical importance for the surface of dental materials. Air-polishing is a procedure commonly used on dental surfaces to remove the biofilm, however it can also damage the material surface. As a result its roughness is increased, and the possible fractal dimension, if any, may change. This study reviews atomic force microscope images of a reference dental restorative composite, treated with abrasive powders of either sodium bicarbonate or glycine, for times of 5, 10 or 30 s, and from distances of 2 or 7 mm. To fully characterize the structural complexity of surface damage, the images were analyzed according to both statistical parameters and multifractal approach. The singularity spectrum f(α) provided quantitative values data to characterize the local scale properties of 3D surface geometry at nanometer scale. The lowest roughening of the surfaces was obtained by air-polishing with glycine for 5 s, independent of the used distance. This observation of least damage and thus best treatment was confirmed by the multifractal analysis. Multifractal analysis provides quantitative information complementary to that offered by traditional surface statistical parameters, with great potential for use also in the field of examination of quality of dental material surfaces.     

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.     

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.     

Surface investigation by electrochemical methods and application of chaos theory and fractal geometry

The present study has considered the application of the noise analysis and fractal geometry as a promising dynamic method for exploiting the corrosion mechanism of the stainless steel 304 that is immersed in different concentrations of FeCl₃. The fractal dimension calculated from the electrochemical noise technique has a good correlation with the surface fractal dimension obtained by electrochemical impedance spectroscopy and scanning electron microscopy results. The complexity of system increases by divergence of Electrochemical Potential noise fractal dimension from 1.5 value and also the roughness of surface increases by an increase in surface fractal dimension. As the concentration of FeCl₃ increases (0.001 M, 0.01 M and 0.1 M) the value of Electrochemical Potential noise fractal dimension diverges from 1.5 value (1.57, 1.33 and 1.01 respectively) and the value of surface fractal dimension increases (2.107, 2.425 and 2.756 for impedance results and 2.073, 2.425 and 2.672 for scanning electron microscopy images). These results show that the complexity of system and roughness of the surface increases by an increase in concentration of FeCl₃. The present study has shown that chaos and noise analysis are effective methods for the study of the mechanism of the corrosion process.     

Multifractality in relativistic charged particles produced at SPS energies

The multifractal analysis of relativistic shower particles produced in ³²S–emulsion interactions at 200 AGeV has been investigated using the method of modified multifractal moments, G_q, in pseudo-rapidity space. The anomalous fractal dimension, d_q, and generalized fractal dimensions, D_q, are determined for the present data for different order of moment. The experimental data reflects multifractal geometry in a multipion production process. The downward concave shape of the multifractal spectral function, f(α_q), gives an evidence for self-similar cascade mechanism. The multifractal specific heat has also been evaluated for the present data using the generalized fractal dimensions, D_q. We compared our experimental results with those obtained from simulated events of the Lund Monte Carlo Code FRITIOF and uncorrelated Monte Carlo events, (MC-RAND) generated randomly in pseudorapidity space based on the assumption of independent emission of particles. The experimental data on multifractality has been found to exhibit a remarkable proximity to the analogous data obtained from the FRITIOF code and the uncorrelated Monte Carlo events.     

Electrostatics in fractal geometry: Fractional calculus approach

An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume V_d ～ r^d with the fractal dimension d and a complementary host volume V_h = V₃ ? V_d. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation.     

Characterizations of PG(n − 1, q)\PG(k − 1, q) by Numerical and Polynomial Invariants

We show that the simple matroid PG(n − 1, q)\PG(k − 1, q), for n ≥ 4 and 1 ≤ k ≤ n − 2, is characterized by a variety of numerical and polynomial invariants. In particular, any matroid that has the same Tutte polynomial as PG(n − 1, q)\PG(k − 1, q) is isomorphic to PG(n − 1, q)\PG(k − 1, q).     

Pattern Frequency Sequences and Internal Zeros

Let q be a pattern and let S_n, q(c) be the number of n-permutations having exactly c copies of q. We investigate when the sequence (S_n, q(c))_c ≥ 0 has internal zeros. If q is a monotone pattern it turns out that, except for q = 12 or 21, the nontrivial sequences (those where n is at least the length of q) always have internal zeros. For the pattern q = 1(l + 1)l…2 there are infinitely many sequences which contain internal zeros and when l = 2 there are also infinitely many which do not. In the latter case, the only possible places for internal zeros are the next-to-last or the second-to-last positions. Note that by symmetry this completely determines the existence of internal zeros for all patterns of length at most 3.     

The second law of thermodynamics and multifractal distribution functions: Bin counting,pair correlations,and the Kaplan–Yorke conjecture

We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equilibrium the distribution is a surprisingly isotropic multifractal strange attractor, with the various fractal dimensionalities in the range 1 < D < 4. The attractor is relatively homogeneous, with projected two-dimensional information and correlation dimensions which are nearly independent of direction. Our data indicate that the Kaplan–Yorke conjecture (for the information dimension) fails in the full four-dimensional phase space. We also find no plausible extension of this conjecture to the projected fractal dimensions of the oscillator. The projected growth rate associated with the largest Lyapunov exponent is negative in the one-dimensional coordinate space.     

Formation patterns at the air-grain interfaces in spinning granular films at high rotation rates

Experimental and numerical studies are described in which a thin film of air-immersed grains is spun in vertical and tilted containers about their axis. At high rotation rates a steep depression appears around the axis of rotation. Interesting fractal type patterns with dimension D = 1.7 ± 0.05 are observed at the air-grain interfaces in the depression. By utilizing computer simulations, it is shown that the fractal-like patterns may be associated with a sharp deformation of the volume occupied by the particles within the depression hole due to turbulent diffusion.     

Equivalent Blocks of Finite General Linear Groups in Non-describing Characteristic

J. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GL_n(q) with defect group C_p^α × C_p^α, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GL_n(q) when p = 3, α = 1, in a different way.     

Numerical methods for a class of jump–diffusion systems with random magnitudes

Stochastic differential delay equations with Poisson driven jumps of random magnitude are popular as models in mathematical finance. In this paper, we shall deal with convergence of the semi-implicit Euler method for nonlinear stochastic differential delay equations with random jump magnitudes and show that the approximate solutions strongly converge to the exact solutions with the order 1 − 1/q (q > 1). This result is more general than what they deal with the jump of deterministic magnitude.     

Special Functions on the Sphere with Applications to Minimal Surfaces

A function which is homogeneous in x, y, z of degree n and satisfies V_xx + V_yy + V_zz = 0 is called a spherical harmonic. In polar coordinates, the spherical harmonics take the form rⁿf_n, where f_n is a spherical surface harmonic of degree n. On a sphere, f_n satisfies ▵ f_n + n(n + 1)f_n = 0, where ▵ is the spherical Laplacian. Bounded spherical surface harmonics are well studied, but in certain instances, unbounded spherical surface harmonics may be of interest. For example, if X is a parameterization of a minimal surface and n is the corresponding unit normal, it is known that the support function, w = X · n, satisfies ▵w + 2w = 0 on a branched covering of a sphere with some points removed. While simple in form, the boundary value problem for the support function has a very rich solution set. We illustrate this by using spherical harmonics of degree one to construct a number of classical genus-zero minimal surfaces such as the catenoid, the helicoid, Enneper's surface, and Hennenberg's surface, and Riemann's family of singly periodic genus-one minimal surfaces.     

The average path length of scale free networks

In this paper, the exact solution of average path length in Barabási–Albert model is given. The average path length is an important property of networks and attracts much attention in many areas. The Barabási–Albert model, also called scale free model, is a popular model used in modeling real systems. Hence it is valuable for us to examine the average path length of scale free model. There are two answers, regarding the exact solution for the average path length of scale free networks, already provided by Newman and Bollobas respectively. As Newman proposed, the average path length grows as log(n) with the network size n. However, Bollobas suggested that while it was true when m = 1, the answer changed to log(n)/log(log(n)) when m > 1. In this paper, as we propose, the exact solution of average path length of BA model should approach log(n)/log(log(n)) regardless the value of m. Finally, the simulation is presented to show the validity of our result.     

The non-traveling wave solutions and novel fractal soliton for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients

A method is proposed by extending the linear traveling wave transformation into the nonlinear transformation with the (G′/G)-expansion method. The non-traveling wave solutions with variable separation can be constructed for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients via the method. A novel class of fractal soliton, namely, the cross-like fractal soliton is observed by selecting appropriately the arbitrary functions in the solutions.     

On the natural solution of an impulsive fractional differential equation of order q ∈ (1, 2)

This paper is motivated from some recent papers treating the impulsive Cauchy problems for some differential equations with fractional order q ∈ (1, 2). A better definition of solution for impulsive fractional differential equation is given. We build up an effective way to find natural solution for such problems. Then sufficient conditions for existence of the solutions are established by applying fixed point methods. Four examples are given to illustrate the results.     

Some properties for the intersection of Moran sets with their translates

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for the classification purpose. In this paper, we pursue this study on a class of Moran sets with their rational translates. We also get the fractal structure of intersection I(x, y) of a class of Moran sets with their rational translates, and the formula of the box-counting dimension. We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from shifting vector with the expansion in fraction of (x, y). Concretely, when (x, y) has a finite expansion in fraction, a very brief calculation formula of the measure is given.     

Dynamics of chaotic maps for modelling the multifractal spectrum of human brain Diffusion Tensor Images

The multifractal spectra of 3d Diffusion Tensor Images (DTI) obtained by magnetic resonance imaging of the human brain are studied. They are shown to deviate substantially from artificial brain images with the same white matter intensity. All spectra, obtained from 12 healthy subjects, show common characteristics indicating non-trivial moments of the intensity. To model the spectra the dynamics of the chaotic Ikeda map are used. The DTI multifractal spectra for positive q are best approximated by 3d coupled Ikeda maps in the fully developed chaotic regime. The coupling constants are as small as α = 0.01. These results reflect not only the white tissue non-trivial architectural complexity in the human brain, but also demonstrate the presence and importance of coupling between neuron axons. The architectural complexity is also mirrored by the deviations in the negative q-spectra, where the rare events dominate. To obtain a good agreement in the DTI negative q-spectrum of the brain with the Ikeda dynamics, it is enough to slightly modify the most rare events of the coupled Ikeda distributions. The representation of Diffusion Tensor Images with coupled Ikeda maps is not unique: similar conclusions are drawn when other chaotic maps (Tent, Logistic or Henon maps) are employed in the modelling of the neuron axons network.