Symmetry, Volume 11, Issue 6 (June 2019) – 110 articles

One of the pillars of experimental science is sampling. Based on the analysis of samples, estimations for populations are made. There is an entire science based on sampling. Distribution of the population, of the sample, and the connection among those two (including sampling distribution) provides rich information for any estimation to be made. Distributions are split into two main groups: continuous and discrete. The present study applies to continuous distributions. One of the challenges of sampling is its accuracy, or, in other words, how representative the sample is of the population from which it was drawn. To answer this question, a series of statistics have been developed to measure the agreement between the theoretical (the population) and observed (the sample) distributions. Another challenge, connected to this, is the presence of outliers - regarded here as observations wrongly collected, that is, not belonging to the population subjected to study. To detect outliers, a series of tests have been proposed, but mainly for normal (Gauss) distributions—the most frequently encountered distribution. The present study proposes a statistic (and a test) intended to be used for any continuous distribution to detect outliers by constructing the confidence interval for the extreme value in the sample, at a certain (preselected) risk of being in error, and depending on the sample size. The proposed statistic is operational for known distributions (with a known probability density function) and is also dependent on the statistical parameters of the population—here it is discussed in connection with estimating those parameters by the maximum likelihood estimation method operating on a uniform U(0,1) continuous symmetrical distribution. Full article

Symmetry pervades nature, but asymmetry is also rather common. Deviations from genetically programmed symmetry are usually associated with internal or external developmental disturbances and may therefore be related to imperfections in physiological processes. In this study, we test the hypotheses that the photosynthetic efficiency of individual leaves of a plant is negatively related to their asymmetry. We measured chlorophyll fluorescence in leaves of three woody species (Betula pubescens, Populus tremula and Salix caprea) in early and late summer in two localities situated ca. 1000 km apart, and we quantified the asymmetry of these leaves by a multivariate measure based on the relative positions of several landmarks. Contrary to our expectation, we found that the photochemical efficiency of photosystem II was positively correlated with leaf fluctuating asymmetry; this effect was weak but consistent across the studied plant species, localities and seasons. Our finding adds to limited evidence that within-plant variation in leaf asymmetry is associated with variation in leaf physiology. Irrespective of the underlying mechanisms, which remain unknown, the results suggest that trees may benefit even more from their asymmetric leaves, at least in terms of photosynthesis, than they do from their more symmetric leaves. Full article

Magnetohydrodynamics (MHD) deals with the analysis of electrically conducting fluids. The study of nanofluids by considering the influence of MHD phenomena is a topic of great interest from an industrial and technological point of view. Thus, the modified MHD mixed convective, nonlinear, radiative and dissipative problem was modelled over an arc-shaped geometry for Al₂O₃ + H₂O nanofluid at 310 K and the freezing temperature of 273.15 K. Firstly, the model was reduced into a coupled set of ordinary differential equations using similarity transformations. The impact of the freezing temperature and the molecular diameter were incorporated in the energy equation. Then, the Runge–Kutta scheme, along with the shooting technique, was adopted for the mathematical computations and code was written in Mathematica 10.0. Further, a comprehensive discussion of the flow characteristics is provided. The results for the dynamic viscosity, heat capacity and effective density of the nanoparticles were examined for various nanoparticle diameters and volume fractions. Full article

An online neutrosophic similarity-based objectness tracking with a weighted multiple instance learning algorithm (NeutWMIL) is proposed. Each training sample is extracted surrounding the object location, and the distribution of these samples is symmetric. To provide a more robust weight for each sample in the positive bag, the asymmetry of the importance of the samples is considered. The neutrosophic similarity-based objectness estimation with object properties (super straddling) is applied. The neutrosophic theory is a new branch of philosophy for dealing with incomplete, indeterminate, and inconsistent information. By considering the surrounding information of the object, a single valued neutrosophic set (SVNS)-based segmentation parameter selection method is proposed, to produce a well-built set of superpixels which can better explain the object area at each frame. Then, the intersection and shape-distance criteria are proposed for weighting each superpixel in the SVNS domain, mainly via three membership functions, T (truth), I (indeterminacy), and F (falsity), for each criterion. After filtering out the superpixels with low response, the newly defined neutrosophic weights are utilized for weighting each sample. Furthermore, the objectness estimation information is also applied for estimating and alleviating the problem of tracking drift. Experimental results on challenging benchmark video sequences reveal the superior performance of our algorithm when confronting appearance changes and background clutters. Full article

Given a graph $G=(V,E)$ , a function $f:V\to \{0,1,2,\cdots \}$ is said to be a total dominating function if ${\sum }_{u\in N\left(v\right)}f\left(u\right)>0$ for every $v\in V$ , where $N\left(v\right)$ denotes the open neighbourhood of v. Let $V_i=\{x\in V:f\left(x\right)=i\}$ . We say that a function $f:V\to \{0,1,2\}$ is a total weak Roman dominating function if f is a total dominating function and for every vertex $v\in V_0$ there exists $u\in N\left(v\right)\cap (V_1\cup V_2)$ such that the function $f^{\prime }$ , defined by $f^{\prime }\left(v\right)=1$ , $f^{\prime }\left(u\right)=f\left(u\right)-1$ and $f^{\prime }\left(x\right)=f\left(x\right)$ whenever $x\in V\backslash \{u,v\}$ , is a total dominating function as well. The weight of a function f is defined to be $w\left(f\right)={\sum }_{v\in V}f\left(v\right).$ In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by ${\gamma }_{tr}\left(G\right)$ , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on ${\gamma }_{tr}\left(G\right)$ and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard. Full article

In this paper we define the degenerate Carlitz-type $(p,q)$ -Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type q-Euler numbers and polynomials. We also give some theorems and exact formulas, which have a connection to degenerate Carlitz-type $(p,q)$ -Euler numbers and polynomials. Full article

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): ^c $D_{0^{+}}^{q}u\left(t\right)=f(t,u\left(t\right)),t\in [0,T]$ , ${\displaystyle u^{\left(k\right)}\left(0\right)={\xi }_k,u\left(T\right)=\sum _{i=1}^m{{\beta }_i}_{RL}{I}_{0^{+}}^{p_i}u\left({\eta }_i\right),}$ where $n-1<q<n,n\ge 2,m,n\in \mathbb{N}$ , ${\xi }_k,{\beta }_i\in \mathbb{R}$ , $k=0,1,\dots ,n-2$ , $i=1,2,\dots ,m$ , and ^c $D_{0^{+}}^q$ is the Caputo fractional derivatives, $f:[0,T]\times C\left(\right[0,T],E)\to E$ , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be $\mathbb{R}$ (with the absolute value) or $C\left(\right[0,T],\mathbb{R})$ with the supremum-norm. _RL $I_{0^{+}}^{p_i}$ is the Riemann–Liouville fractional integral of order $p_i>0$ , ${\eta }_i\in (0,T)$ , and ${\displaystyle \sum _{i=1}^m{\beta }_i{\eta }_i^{p_i+n-1}\frac{\Gamma \left(n\right)}{\Gamma (n+p_i)}\ne T^{n-1}}$ . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results. Full article

The default COSMO-RS (Conductor like Screening Model for Real Solvents) approach is incapable of accurate computation of C60 solubility in net solvents. Additionally, there is no adequate selection of single or multiple reference solvent, which can be applied to the whole population of 180 solvents for improving prediction of mole fraction at saturated conditions. This failure cannot be addressed to inaccurate data of the Buckminster fusion, although they pose a challenge for experimental measurement due to intense sublimation of C60 at elevated temperatures and the possibility of solvates precipitation. However, taking advantage of the richness of experimental data of fullerene solubility, it is possible to identify the source of errors expressed in terms of fluidization affinity. Classification of solvents according to the value of this fluidization term allowed for formulation of a consonance solvents approach, which enables accurate prediction of C60 solubility using the single reference solvent method. Full article

The aim of this work is to develop a “learning model” which outranks countries according to their confrontation of historical macroeconomic indicators for a given period of time with the spreads at the end of that time and to formulate a forward-looking investment strategy regarding government bonds for the following time period. The mechanism of identifying investment opportunities among government bonds is based on the multiple criteria decision making technique, and we look to the Promethee II method as a symmetry approach to country ordering. The spread is defined as the difference between the yield to maturity of the 10-year government bond of a country and the Germany government bond with the same maturity. In this paper, an optimization approach based on three models is developed to find the weights of importance for macroeconomic characteristics, together with a sensitivity analysis on changes in these characteristics. The method was applied to 17 European countries characterized by 16 macroeconomic characteristics. The originality of this paper lies in the two-stage approach to the investment strategy construction based on criteria weights optimization with stability intervals for their values. Full article

In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann–Hilbert investigated. We construct a $2\times 2$ Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann–Hilbert (R–H) problem. Then, we mainly use the symmetry relationship of potential matrix Q to analyze the zeros of $det{P}^{+}$ and $det{P}^{-}$ ; the N-soliton solutions of the NLS-like equation are expressed explicitly by a particular R–H problem with an unit jump matrix. In addition, the single-soliton solution and collisions of two solitons are analyzed, and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically. Furthermore, on the basis of the R–H problem, the evolution equation of the R–H data with the perturbation is derived. Full article

The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional Hölder spectra, either directly or via some Legendre transforms on either directional scaling function or anisotropic scaling functions. We obtained general upper bounds for directional Hölder spectra. We also showed optimal results for two large classes of examples of deterministic and random anisotropic self-similar tools for possible modeling turbulence (or cascades) and textures in images: Sierpinski cascade functions and fractional Brownian sheets. Full article

Coefficient of variation (CV) is a simple but useful statistical tool to make comparisons about the independent populations in many research areas. In this study, firstly, we proposed the asymptotic distribution for the ratio of the CVs of two separate symmetric or asymmetric populations. Then, we derived the asymptotic confidence interval and test statistic for hypothesis testing about the ratio of the CVs of these populations. Finally, the performance of the introduced approach was studied through simulation study. Full article

The impact of shallow tunnel construction on the surrounding environment is often considered as a symmetric half-plane problem with circular holes. In this research, the analytical solutions of the ground stresses and displacements of a shallow circular tunnel in an elastic half-plane under arbitrary distributed loads on ground surface were derived, based on the complex variable method. Then, an application was implemented to analyze the potential plastic zone induced by shallow tunneling adjacent to the ground surface structures. The verification of the results obtained from the proposed analytical prediction model was carried out using the numerical simulations. Additionally, the influences of different boundary condition (different magnitudes and ranges of arbitrary distributed loads and different symmetric boundary conditions of the tunnel perimeter) on the distribution characteristics of the potential plastic zones were analyzed. In general, the results showed that the larger the pile loads and the closer the relative position between the tunnel and distributed loads, the more distinct the coalesced trends of the potential plastic zones around the tunnel and the potential plastic zones around the distributed loads. Full article

In this paper, we surveyed the influence of learn effectiveness in a computer course under the factors of learning attitude and learning problems for students in senior-high school. We followed the formula for a regression line as R = A + BX +ε and simulated on SPSS platform with symmetry to obtained the results as follows: (1) In learning attitude, both the cognitive-level and behavior-level, are positively correlated with satisfaction. This means the students have cognitive-level and behavior-level more positively correlated with satisfaction in computer subjects and have a high degree of self-learning effectiveness. (2) In learning problems, the female students had higher learning effectiveness than male students, and the students who practiced on the computer on their own initiative long-term each week had higher learning effectiveness. Full article

Exploiting wind energy, which is a complex process in urban areas, requires turbines suitable for unfavorable weather conditions, in order to trap the wind from different directions; Savonius turbines are suitable for these conditions. In this paper, the effect of overlap ratios and the position of blades on a vertical axis wind turbine is comprehensively investigated and analyzed. For this purpose, two positive and negative overlap situations are first defined along the X-axis and examined at the different tip speed ratios of the blade, while maintaining the size of the external diameter of the rotor, to find the optimum point; then, the same procedure is done along the Y-axis. The finite volume method is used to solve the computational fluid dynamics. Two-dimensional numerical simulations are performed using URANS equations and the sliding mesh method. The turbulence model employed is a realizable K-ε model. According to the values of the dynamic torque and power coefficient, while investigating horizontal and vertical overlaps along the X- and Y-axis, the blades with overlap ratios of HOLR = +0.15 and VOLR = +0.1 show better performances when compared to other corresponding overlaps. Accordingly, the average C_m and C_p improvements are 16% and 7.5%, respectively, compared to the base with a zero overlap ratio. Full article

In many real world problems, science fields such as biology, computer science, data mining, electrical and mechanical engineering, and signal processing, researchers aim to compare and classify several regression models. In this paper, a computational approach, based on the non-parametric methods, is used to investigate the similarities, and to classify several linear and non-linear regression models with symmetric errors. The ability of each given approach is then evaluated using simulated and real world practical datasets. Full article

In this paper, we study asymptotic semicircular laws induced both by arbitrarily fixed $C^{*}$ -probability spaces, and p-adic number fields ${\left\{{\mathbb{Q}}_p\right\}}_{p\in \mathcal{P}}$ , as p→ ∞ in the set $\mathcal{P}$ of all primes. Full article

In this study we compared the biomonitoring potential of various types of flower asymmetry indices in Iris pumila (Dwarf Bearded Iris). We chose 197 naturally growing clones from the arid steppe habitat in the largest sandy area in Europe (Deliblato Sands Nature Reserve), and we transplanted two replicates of each clone to a polluted highway site with a heavy traffic flow. After a period of acclimatization, lower levels of photosynthetic pigment concentrations and higher stomatal density and specific leaf area in transplants verified that the chosen highway site was indeed more stressful and therefore suitable for estimation of the flower asymmetry biomonitoring potential. We analyzed radially and bilaterally symmetrical flower structures (radial fluctuating asymmetry (RA) and bilateral fluctuating asymmetry (FA)) on three perianth parts—falls, standards, and styles—and calculated various asymmetry indices based on linear and geometric morphometrics. Despite utilizing a heavily polluted environment and fairly large sample sizes, only one asymmetry index was significantly higher on the polluted site with demonstrated stressful effects on utilized plants, indicating that flower asymmetry was not an efficient method for biomonitoring in the case of I. pumila RA and FA indices. Full article

Neutrosophic triangular norms (t-norms) and their residuated lattices are not only the main research object of neutrosophic set theory, but also the core content of neutrosophic logic. Neutrosophic implications are important operators of neutrosophic logic. Neutrosophic residual implications based on neutrosophic t-norms can be applied to the fields of neutrosophic inference and neutrosophic control. In this paper, neutrosophic t-norms, neutrosophic residual implications, and the residuated lattices derived from neutrosophic t-norms are investigated deeply. First of all, the lattice and its corresponding system are proved to be a complete lattice and a De Morgan algebra, respectively. Second, the notions of neutrosophic t-norms are introduced on the complete lattice discussed earlier. The basic concepts and typical examples of representable and non-representable neutrosophic t-norms are obtained. Naturally, De Morgan neutrosophic triples are defined for the duality of neutrosophic t-norms and neutrosophic t-conorms with respect to neutrosophic negators. Third, neutrosophic residual implications generated from neutrosophic t-norms and their basic properties are investigated. Furthermore, residual neutrosophic t-norms are proved to be infinitely ∨-distributive, and then some important properties possessed by neutrosophic residual implications are given. Finally, a method for producing neutrosophic t-norms from neutrosophic implications is presented, and the residuated lattices are constructed on the basis of neutrosophic t-norms and neutrosophic residual implications. Full article

The quasi-cancellativity of Abel Grassmann‘s groupoids (AG-groupoids) are discussed and two conjectures are partially solved. First, the following conjecture is proved to be true: every AG-3-band is quasi-cancellative. Moreover, a new notion of AG-(4,1)-band is proposed, and it is also proved that every AG-(4,1)-band is quasi-cancellative. Second, the notions of left (right) quasi-cancellative AG-groupoids and power-cancellative AG-groupoids are proposed, and the following results are obtained: for an AG*-groupoid or AG**-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative; for a power-cancellative and locally power-associative AG-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative. Finally, a general result is proved, that for any AG-groupoid, if it is left quasi-cancellative then it is right quasi-cancellative. Full article

Traffic data are the basis of traffic control, planning, management, and other implementations. Incomplete traffic data that are not conducive to all aspects of transport research and related activities can have adverse effects such as traffic status identification error and poor control performance. For intelligent transportation systems, the data recovery strategy has become increasingly important since the application of the traffic system relies on the traffic data quality. In this study, a bidirectional k-nearest neighbor searching strategy was constructed for effectively detecting and recovering abnormal data considering the symmetric time network and the correlation of the traffic data in time dimension. Moreover, the state vector of the proposed bidirectional searching strategy was designed based the bidirectional retrieval for enhancing the accuracy. In addition, the proposed bidirectional searching strategy shows significantly more accuracy compared to those of the previous methods. Full article

Spontaneous mirror symmetry breaking (SMSB), a phenomenon leading to non-equilibrium stationary states (NESS) that exhibits biases away from the racemic composition is discussed here in the framework of dissipative reaction networks. Such networks may lead to a metastable racemic non-equilibrium stationary state that transforms into one of two degenerate but stable enantiomeric NESSs. In such a bifurcation scenario, the type of the reaction network, as well the boundary conditions, are similar to those characterizing the currently accepted stages of emergence of replicators and autocatalytic systems. Simple asymmetric inductions by physical chiral forces during previous stages of chemical evolution, for example in astrophysical scenarios, must involve unavoidable racemization processes during the time scales associated with the different stages of chemical evolution. However, residual enantiomeric excesses of such asymmetric inductions suffice to drive the SMSB stochastic distribution of chiral signs into a deterministic distribution. According to these features, we propose that a basic model of the chiral machinery of proto-life would emerge during the formation of proto-cell systems by the convergence of the former enantioselective scenarios. Full article

This study presents a filtering and sampling structure based on symmetrical second order systems working on half-period. It is shown that undamped second order oscillating systems working on half-period could provide: (i) a large attenuation coefficient for an alternating signal (due to the filtering second order system), and (ii) a robust sampling procedure (the slope of the generated output being zero at the sampling time moment). Unlike previous studies on the same topics, these results are achieved without the use of an additional integrator. Full article

In the present paper, we solve the problem of determining the fuzzy distance between two subsets of a fuzzy metric space. We address the problem by reducing it to the problem of finding an optimal approximate solution of a fixed point equation. This approach is well studied for the corresponding problem in metric spaces and is known as proximity point problem. We employ fuzzy weak contractions for that purpose. Fuzzy weak contraction is a recently introduced concept intermediate to a fuzzy contraction and a fuzzy non-expansive mapping. Fuzzy versions of some geometric properties essentially belonging to Hilbert spaces are considered in the main theorem. We include an illustrative example and two corollaries, one of which comes from a well-known fixed point theorem. The illustrative example shows that the main theorem properly includes its corollaries. The work is in the domain of fuzzy global optimization by use of fixed point methods. Full article

The control of heated fluid is of interest in many fields of engineering, such as boiler and heat exchanger design. The broken symmetry of a thermo-physical system within a multi-sized media could be used to control its physical characteristics. In the current study, the effects of magnetohydrodynamic (MHD) forces and nanoparticles on boiling in a subcooled region inside an upright annular pipe have been investigated. The effect of magneto hydrodynamics on the base fluid (liquid water) was measured, and different nanoparticle concentrations were employed as the working fluids. The magnetic field perpendicular to fluid flow is used to control the liquid water and vapor water phase motion. The governing equation of motion and conservation of energy in both phases is solved with the aid of correlation for vaporization and condensation of nucleate boiling on the wall. The results of the mathematical simulation are in suitable agreement with the results of previous experiments. As associated with pure water, the results with dilute Nanofluids presented that the application of nanoparticles homogenized the temperature difference through the fluid and vapor phase. The results show that the MHD controller is a powerful method to decrease the amplitude of the vaporization and resulted in oscillations. Full article

To accurately and efficiently detect tool wear values during production and processing activities, a new online detection model is proposed called the Residual Dense Network (RDN). The model is created with two main steps: Firstly, the time-domain signals for a cutting tool are obtained (e.g., using acceleration sensors); these signals are processed to denoise and segmented to provide a larger number of uniform samples. This processing helps to improve the robustness of the model. Secondly, a new deep convolutional neural network is proposed to extract features adaptively, by combining the idea of a recursive residual network and a dense network. Notably, this method is specifically tailored to the tool wear value detection problem. In this way, the limitations of traditional manual feature extraction steps can be avoided. The experimental results demonstrate that the proposed method is promising in terms of detection accuracy and speed; it provides a new way to detect tool wear values in practical industrial scenarios. Full article

Decision making under uncertainty describes situations that consider a profound lack of knowledge, where the functional form is completely unknown, and often, the relevant input and output variables are unknown as well. Data, being the vital input of decision making, contain a dissimilar level of imprecision that necessitates different approaches for making a proper and legitimate decision. In this article, we propose the concept of the intuitionistic type-2 fuzzy set (IT2FS). Several arithmetic operations on IT2FS such as union, intersection, complement, containment, etc., are defined, and the related algebraic properties of IT2FS are also studied. Subsequently, we define two new operators, namely the necessity operator and the possibility operator, to convert an IT2FS into an ordinary T2FS, and then discuss some of their basic properties. Moreover, in this study, two distance measures, the Hamming distance and Euclidian distance of IT2FS, are proposed, and their applications are illustrated with an example. Full article

It is well known that there are some unfavorable shortcomings in the ordinary operational rules (OORs) of intuitionistic fuzzy number (IFN), and there exists a close and forceful connection between the intuitionistic fuzzy set (IFS) and Dempster-Shafer Theory (DST). We can utilize this relationship to present a transparent and fruitful semantic framework for IFS in terms of DST. In the framework of DST, an IFN can be converted into a basic probability assignment (BPA) and operations on IFNs can be represented as operations on a belief interval (BI), which can break away from the revealed shortcomings of the OORs of the IFN. Although there are many operators to aggregate the IFN, the operator to aggregate the BPA is rare. The Maclaurin symmetric mean (MSM) operator has the advantage of considering interrelationships among any number of attributes. The power average (PA) operator can reduce the influences of extreme evaluation values. In addition, for measuring the difference between IFNs, we replace the Hamming distance and Euclidean distance with the Jousselme distance (JD). In this paper, we develop an intuitionistic fuzzy power MSM (IFPMSM_DST) operator and an intuitionistic fuzzy weighted power MSM (IFPWMSM_DST) operator in the framework of the DST and provide their favorable properties. Then, we propose a novel method based on the proposed operators to solve multi-attribute decision-making (MADM) problems without intermediate defuzzification when their attributes and weights are both IFNs. Finally, some examples are utilized to demonstrate that the proposed methods outperform the previous ones. Full article

Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle scattering (SAS) curves from deterministic fractal models with a single scaling factor have allowed the obtaining of valuable fractal properties but they are insufficient to describe non-uniform structures with rich scaling properties such as fractals with multiple scaling factors. To extract additional information about this class of fractal structures we performed an analysis of multifractal spectra and SAS intensity of a representative fractal model with two scaling factors—termed Vicsek-like fractal. We observed that the box-counting fractal dimension in multifractal spectra coincide with the scattering exponent of SAS curves in mass-fractal regions. Our analyses further revealed transitions from heterogeneous to homogeneous structures accompanied by changes from short to long-range mass-fractal regions. These transitions are explained in terms of the relative values of the scaling factors. Full article