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.     

Multifractal scaling of crack images from pyroligneous acid dried sludge

Tannery effluent (sludge, wastewater) is treated by natural way. The waste sludge has been taken for two treatment process. The alkali chemicals are neutralized by pyroligneous acid which is obtained by pyrolysis process of wood. This sludge is sent out for drying. The dried sludge contains some crack pattern formation. Photographs were used to record two sludge cracking surfaces. Experiment has been performed to study the fracture pattern formation in thin film sludge. We studied changes of crack surface of a sludge by image analysis and also assessed applicability of fractal scaling to crack surfaces. The calculated crack surface dimension shows that the fracture surface exhibit fractal structure. Image size was 256 × 256 pixels. MFA (multifractal analysis) was carried out to the method of moments, i.e., the probability distribution was estimated for moments ranging from ?150 < q < 150 and the generalized dimension were calculated from the log/log slope of the probability distribution for the respective moments over box sizes. Generalized dimension D(q) were attained for this box size range, which are capable of characterizing heterogeneous spatial crack structure. Multifractal spectra analyzed two fracture surface of the image and results were indicated that the width of spectra increases due to pyroligneous acid. Multifractal method is sensitive enough to measure the fracture distribution and can be seen as a different approach for changing research of crack images of manure sludge drying.     

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.     

Geometrical multifractality of the perimeter of DLA clusters

The geometrical multifractality of diffusion-limited aggregation (DLA) clusters is investigated by evaluating the D_q spectrum for q⩾0 using the standard box-counting technique. Using the cluster points themselves as input to the algorithm, deviations were found from the expected multifractal scaling. However on examining the geometric scaling properties of the cluster perimeter, such deviations were found to be significantly reduced, thus allowing a reliable D_q spectrum to be calculated.     

Describing some characters of serine proteinase using fractal analysis

In this paper we calculated the fractal dimensions of four proteins, chymotrypsin, elastase, trypsin and subtilisin, which are made up of about 220–275 amino acids and belong to the family of serine proteinase by using three definitions of fractal dimension i.e. the chain fractal dimension (D_L), the mass fractal dimension (D_m) and the correlation fractal dimension (D_c). We also analyzed the relationship between fractal dimension and space structure or secondary structure contents of proteins. The results showed that the values of fractal dimensions are almost same for the global mammalian enzymes (chymotrypsin, elastase and trypsin), but different for the global subtilisin. This demonstrated that the more similar structures, the more equal fractal dimensions, and if the fractal dimensions of proteins are different from each other, the three dimensional structures should not be similar. On the other hand, the detailed structures and fractal dimensions of the active sites of four enzymes are extraordinarily similar. Therefore, the fractal method can be applied to the elucidation of the proteins evolution.     

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.     

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.     

A random multifractal tilling

We develop a multifractal random tilling that fills the square. The multifractal is formed by an arrangement of rectangular blocks of different sizes, areas and number of neighbors. The overall feature of the tilling is an heterogeneous and anisotropic random self-affine object. The multifractal is constructed by an algorithm that makes successive sections of the square. At each n-step there is a random choice of a parameter ρ_i related to the section ratio. For the case of random choice between ρ₁ and ρ₂ we find analytically the full spectrum of fractal dimensions.     

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.     

Multifractal rainfall extremes: Theoretical analysis and practical estimation

We study the extremes generated by a multifractal model of temporal rainfall and propose a practical method to estimate the Intensity–Duration–Frequency (IDF) curves. The model assumes that rainfall is a sequence of independent and identically distributed multiplicative cascades of the beta-lognormal type, with common duration D. When properly fitted to data, this simple model was found to produce accurate IDF results [Langousis A, Veneziano D. Intensity–duration–frequency curves from scaling representations of rainfall. Water Resour Res 2007;43. doi:10.1029/2006WR005245]. Previous studies also showed that the IDF values from multifractal representations of rainfall scale with duration d and return period T under either d → 0 or T → ∞, with different scaling exponents in the two cases. We determine the regions of the (d, T)-plane in which each asymptotic scaling behavior applies in good approximation, find expressions for the IDF values in the scaling and non-scaling regimes, and quantify the bias when estimating the asymptotic power-law tail of rainfall intensity from finite-duration records, as was often done in the past. Numerically calculated exact IDF curves are compared to several analytic approximations. The approximations are found to be accurate and are used to propose a practical IDF estimation procedure.     

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.     

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.     

Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations

This paper first introduces the so-called quasi-continuous random dynamical system (RDS) on a separable Banach space. The quasi-continuity is weaker than all the usual continuities and thus is easier to check in practice. We then establish a necessary and sufficient condition for the existence of random attractors for the quasi-continuous RDS. We also give a general method to obtain the random attractors for the RDS on the Banach space Lq(D) for q?2. As an application, it is shown that the RDS generated by the stochastic reaction-diffusion equation possesses a finite-dimensional random attractor in Lq(D) for any q?2, a comparison result of fractal dimensions under the different Lq-norms is also obtained.     

Tsallis statistics and turbulence

In order to reveal the underlying statistics describing properly the fully developed turbulence, the probability density function of the local dissipation is derived by taking extremal of a generalized entropy (Tsallis entropy) under the two constraints, i.e., one is the normalization of probability and the other is to fix the intermittency exponent being constant. The generalized entropy includes the Boltzmann–Gibbs entropy as a special case where the Tsallis index q is equal to 1. The multifractal spectrum f(α) corresponding to the probability density function is determined self-consistently in the sense that all quantities can be determined by the observed value of the intermittency exponent. It is shown that the scaling exponents ζ_m of velocity structure function derived by making use of f(α) explains experimental data very well. It is also revealed that the asymptotic expression of ζ_m for m≫1 has a log term. The Tsallis index q turns out to be 0.380 which manifests itself that the system of fully developed turbulence has a nonextensive character.     

On “P” property and the column-W property

P-matrices play an important role in the well-posedness of a linear complementarity problem (LCP). Similarly, the well-posedness of a horizontal linear complementarity problem (HLCP) is closely related to the column-W property of a matrix k-tuple.In this paper we first consider the problem of generating P-matrices from a given pair of matrices. Given a matrix pair (D, F) where D is a square matrix of order m and matrix F has m rows, “what are the conditions under which there exists a matrix G such that (D + FG) is a P-matrix?”. We obtain necessary and sufficient conditions for the special case when the column rank of F is m ? 1. A decision algorithm of complexity O(m²) to check whether the given pair of matrices (D, F) is P-matrisable is obtained. We also obtain a necessary and an independent sufficient condition for the general case when rank(F) is less than m ? 1.We then generalise the P-matrix generating problem to the generation of matrix k-tuples satisfying the column-W property from a given matrix (k + 1)-tuple. That is, given a matrix (k + 1)-tuple (D₁, … ,D_k, F), where D_js are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D₁ + FG₁, … ,D_k + FG_k) satisfies the column-W property. As in the case of P-matrices we obtain necessary and sufficient conditions for the case when rank(F) = m ? 1. Using these conditions a decision algorithm of complexity O(km²) to check whether the given matrix (k + 1)-tuple is column-W matrisable is obtained. Then for the case when rank(F) is less than m ? 1, we obtain a necessary and an independent sufficient condition.For a special sub-class of P-matrices we give a polynomial time decision algorithm for P-matrisability. Finally, we obtain a geometric characterisation of column-W property by generalising the well known separation theorem for P-matrices.     

Chaotic vibrations of flexible infinite length cylindrical panels using the Kirchhoff–Love model

Treated as continuous deformable systems with an infinite number of degrees of freedom, flexible infinite length cylindrical panels subject to harmonic load are studied. Using the finite difference method with respect to spatial coordinates, the continuous system is reduced to lumped one governed by ordinary differential equations. These equations are transformed to a normal form and then solved numerically using the fourth order Runge–Kutta method. In order to trace and explain vibrational behaviour, dependencies w_max(q₀) and Lyapunov exponents are calculated for panels with parameter value k_x = 48. The corresponding charts of the control parameters {q₀, ω_q} are also reported. Novel scenarios yielding chaotic dynamics exhibited by cylindrical panels are illustrated and discussed.     

Generalized fractal dimensions: equivalences and basic properties

Given a positive probability Borel measure μ on , we establish some basic properties of the associated functions τ_μ^±(q) and of the generalized fractal dimensions D_μ^±(q) for . We first give the equivalence of the Hentschel–Procaccia dimensions with the Rényi dimensions and the mean-q dimensions, for q>0. We then use these relations to prove some regularity properties for τ_μ^±(q) and D_μ^±(q); we also provide some estimates for these functions, in particular estimates on their behaviour at ±∞, as well as for the dimensions corresponding to convolution of two measures. We finally present some calculations for specific examples illustrating the different cases met in the article.     

A note on the 3D4(q) generalized hexagons

The ³D₄(q) generalized hexagons are characterized by means of the regulus condition.     

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).