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.     

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.     

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.     

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.     

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.     

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.     

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.     

A semi-Markov process with an inverse Gaussian distribution as sojourn time

Let X_n denote the state of a device after n repairs. We assume that the time between two repairs is the time τ taken by a Wiener process {W(t), t ? 0}, starting from w₀ and with drift μ < 0, to reach c ∈ [0, w₀). After the nth repair, the process takes on either the value X_n?1 + 1 or X_n?1 + 2. The probability that X_n = X_n?1 + j, for j = 1, 2, depends on whether τ ? t₀ (a fixed constant) or τ > t₀. The device is considered to be worn out when X_n ? k, where k ∈ {1, 2, …}. This model is based on the ones proposed by Rishel (1991) [1] and Tseng and Peng (2007) [2]. We obtain an explicit expression for the mean lifetime of the device. Numerical methods are used to illustrate the analytical findings.     

The concept of fractal experiments: New possibilities in quantitative description of quasi-reproducible measurements

In this paper the authors suggest a new conception of the so-called fractal (self-similar) experiment. Under the fractal experiment (FE) one can imply a cycle of measurements that are subjected by the scaling transformations F(z) → F(zξ^m) in contrast with conventional scheme F(z) → F(z + mT) (m = 0,1,…, M–1), where z defines the controllable (input) variable and can be associated with time, complex frequency, wavelength and etc., T – mean period of time between successive measurements and m defines a number of successive measurements. One can connect a fractal experiment with specific memory effect that arises between successive measurements. The general theory of experiment for quasi-periodic measurements proposed in [1] after some transformations can be applied for the set of the FE, as well. But attentive analysis shown in this paper allows generalizing the previous results for the case when the influence of uncontrollable factors becomes significant. The theory developed for this case allows to consider more real cases when the influence of dynamic (unstable) processes taking place during the cycle of measurements corresponding to some FE is becoming essential. These experiments we define as quasi-reproducible (QR) fractal experiments.The proposed concept opens new possibilities in theory of measurements and numerous applications, especially in different nanotechnologies, when the influence of the scaling factor plays the essential role. This concept allows also to introduce the so-called intermediate model (IM) which can serve as an unified platform for reconciliation of the proposed microscopic theory with reliable experiments “refined” from the influence of the random noise and apparatus function. We forced to consider a modified model experiment in order to demonstrate some common peculiarities that can be appeared in real cases. We know only couple of similar examples of experiments that are close to the proposed concept. Mechanical relaxation and dielectric spectroscopy (based on measurements of the complex susceptibility ε(jω)) represent the branches of physics related to consideration of mechanical and electric relaxation phenomena in different heterogeneous materials. The dielectric spectroscopy can be considered as an instructive example for better understanding of the proposed concept.In cases, when the microscopic model is absent the results of measurements can be expressed in terms of the fitting parameters associated with the generalized Prony spectrum (GPS) belonging to the IM. The authors do hope that this new approach will find an interesting continuation in various applications of different nanotechnologies.     

Calculation of a static potential created by plane fractal cluster

In this paper we demonstrate new approach that can help in calculation of electrostatic potential of a fractal (self-similar) cluster that is created by a system of charged particles. For this purpose we used the simplified model of a plane dendrite cluster [1] that is generated by a system of the concentric charged rings located in some horizontal plane (see Fig. 2). The radiuses and charges of the system of concentric rings satisfy correspondingly to relationships: r_n = r₀ξⁿ and e_n = e₀bⁿ, where n determines the number of a current ring. The self-similar structure of the system considered allows to reduce the problem to consideration of the functional equation that similar to the conventional scaling equation. Its solution represents itself the sum of power-low terms of integer order and non-integer power-law term multiplied to a log-periodic function [5], [6]. The appearance of this term was confirmed numerically for internal region of the self-similar cluster (r₀ ≪ r ≪ r_N−1), where r₀, r_N−1 determine the smallest and the largest radiuses of the limiting rings correspondingly. The results were obtained for homogeneously (b > 0) and heterogeneously (b < 0) charged rings. We expect that this approach allows to consider more complex self-similar structures with different geometries of charge distributions.     

Effects of CO2 concentration and moisture content of sugar-free media on the tissue-cultured plantlets in a large growth chamber

The dynamic fluctuations of CO₂ concentration in the tissue culture growth chamber after transplantation of petunia, chrysanthemum and tomato plantlets were recorded with a real-time control system to determine the critical CO₂ concentration levels of 35 μl l^?1 at which CO₂ enrichment is needed. The experimental data showed that the tissue-cultured plantlets of petunia, chrysanthemum and tomato had the same CO₂ concentration dynamics. The results indicated that CO₂ enrichment was proper on the second day after transplantation. Petunia plantlets were used to conduct experiments under PPFD of 80 μmol m^?2 s^?1, and CO₂ concentrations of 350 ± 50 μl l^?1, 650 ± 50 μl l^?1 and 950 ± 50 μl l^?1 as well as medium moisture contents of 60%, 70% and 80%, with the result that plantlets grew better under CO₂ concentration of 650 ± 50 μl l^?1 than under the other two concentrations with all the different media water contents. Three media water contents under the same CO₂ concentration produced plantlets with the same quality. The impacts of CO₂ concentrations on plantlets are more important than those of the media water contents. Sugar-free tissue culture, as compared with the conventional culture, showed that CO₂ enrichment to 350 ± 50 μl l^?1 can promote the growth of the cultured plantlets. Sugar-free tissue culture produced healthy plantlets with thick roots, almost equivalent to the common plantlets.     

Net-convergence and weak separation axioms in (L, M)-fuzzy topological molecular lattices

In this paper, we used the concept of (L, M)-fuzzy remote neighborhood system to study and establish the convergence theory of molecular nets. Next, we introduce the T_i-axioms (i = ?1, 0, 1, 2) in (L, M)-fuzzy topological molecular lattices, and discuss some of their characterizations. Finally, we show that the T_i-axioms (i = ?1, 0, 1, 2) are preserved under homeomorphisms.     

GNGA for general regions: Semilinear elliptic PDE and crossing eigenvalues

We consider the semilinear elliptic PDE Δu + f(λ, u) = 0 with the zero-Dirichlet boundary condition on a family of regions, namely stadions. Linear problems on such regions have been widely studied in the past. We seek to observe the corresponding phenomena in our nonlinear setting. Using the Gradient Newton Galerkin Algorithm (GNGA) of Neuberger and Swift, we document bifurcation, nodal structure, and symmetry of solutions. This paper provides the first published instance where the GNGA is applied to general regions. Our investigation involves both the dimension of the stadions and the value λ as parameters. We find that the so-called crossings and avoided crossings of eigenvalues as the dimension of the stadions vary influences the symmetry and variational structure of nonlinear solutions in a natural way.     

Fractal discrimination of random fractal aggregates and its application in biomarker analysis for blood coagulation

A recent rheological study has established that the fractal dimension, d_f, of an incipient clot, formed at the Gel Point (sol–gel transition) of coagulating blood is a significant new biomarker of haemostasis. In whole healthy blood, incipient clots show a clearly defined value of d_f = 1.7 within a narrow range, which represents a new ‘healthy index’ for normal clotting. The addition of unfractionated heparin significantly prolongs the onset of clot formation with a corresponding reduction of d_f as a function of heparin dose. However, as clots mature they exhibit (i) an expected increase in d_f and (ii) a significant increase to spread of these values, i.e. d_f’s in the range 2.0–2.5, limiting the use of d_f as a discriminant of clot microstructure.The present study, details how and why the spectral dimension, d_s, can be used to accommodate this shortcoming and allow discrimination of mature forms of clot microstructure in indistinguishable in terms of their fractal dimension. To elucidate why d_s permits discrimination a numerical experiment was conducted on computationally generated random fractal aggregates (RFAs) with a priori set value of d_f. Starting from RFAs with a d_f of 1.7, mature RFAs are evolved from these incipient templates by two differing growth processes achieving a final d_f of 2.1. Fractal and statistical analysis of the mature RFAs reveals, for the first time, that their differing internal structure is manifest in the magnitude of d_s. The potential clinical significance of these findings is discussed in terms of the possibility of exploiting the incipient clot’s ability to template the internal arrangement of the mature clot to better predict long term clot susceptibility to lysis.     

Simulation and circuit implementation of 12-scroll chaotic system

Based on the typical Chua’s circuit and the latest research results of multi-scroll system, a model of the new system is constructed to produce multi-scroll chaotic attractors, replacing the typical Chua’s diode with the combination of sign function. The major method is to make equilibrium point located in the center of two adjacent breakpoints, and keep scrolls and bond orbits alternated with each other. The chaos generation mechanism is studied by analyzing the symmetry and invariance, the existence of the dissipation and attractor, the system equilibrium and stability. The fractal dimension, the K–S entropy, the time domain waveform and the initial value sensitivity are applied to verifying the chaotic behaviors. The numerical simulations show that the system generates n-double (n = 1, 2, 3, 4, 5, 6) scroll chaotic attractors. Finally, the design of the hardware circuit produces at a maximum of 12-scroll hardware experimental results. Theoretical analysis, numerical simulation and hardware experimental results are full matched, which further proves the existence of the system and the physical realization.     

Time-delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott-rotor system

Time-delay is an unavoidable phenomenon in active control systems. Measuring of the system states, processing of the measured signals, executing the control laws, conditioning and enforcing the control actions are the main reasons of time-delayed systems. This paper studies the vibration control of a horizontally suspended Jeffcott-rotor system having cubic and quadratic nonlinearities via time-delayed position-velocity controller. The intervals of the time-delays (τ₁ and τ₂) at which the system response is stable has been studied. The τ₁ − τ₂ plane is constructed to illustrate the area at which the system solutions are stable. The influences of the controller gains on the stable-solutions area in τ₁ − τ₂ plane are explored. The analysis revealed that the time-delay increases the vibration amplitudes and can destabilize the system solution in the case of negative position feedback control, while at positive position feedback control it improves the vibration suppression performance. The time-delays mechanism in stabilizing and destabilizing the dynamical systems is explained. Then, we proposed a simple and concrete method to determine the optimal value for time-delays that can improve the vibrations suppression efficiency. The acquired analytical results are confirmed numerically and the optimal working conditions of the system are concluded. Finally, a comparison with the papers that published previously is included.     

Vacuum polarization in light two-electron atoms and ions

A general theory of the vacuum polarization in light atomic and muon-atomic systems is considered. We derive the closed analytical expression for the Uehling potential and evaluate corrections on vacuum polarization for the 1¹S-state of the two-electron ³He and ⁴He atoms and for some two-electron ions, including the Li⁺, Be²⁺, B³⁺ and C⁴⁺ ions. The correction for vacuum polarization in two-electron He atoms has been evaluated as ΔE_ueh ≈ 7.253 ± 0.0025 × 10⁻⁷ a.u. The analogous corrections in the two-electron He-like ions rapidly increase with the nuclear charge Q (ΔE_ueh ≈ 2.7061 × 10⁻⁶ a.u. for the Li⁺ ion and ΔE_ueh ≈ 2.3495 × 10⁻⁵ a.u. for the C⁴⁺ ion). The corresponding corrections have also been evaluated for the electron–nucleus and electron–electron interactions.     

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c₁(τ) and using a limiting argument for c = c₁(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?₂f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?₂f(0, 0) = 0.     

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.