Fractals

In this paper, we study geodesics in the Sierpinski carpet and Menger sponge, as well as in a family of fractals that naturally generalize the carpet and sponge to higher dimensions. In all dimensions, between any two points we construct a geodesic taxicab path, namely a path comprised of segments parallel to the coordinate axes and possibly limiting to its endpoints by necessity. These paths are related to the skeletal graph approximations of the Sierpinski carpet that have been studied by many authors. We then provide a sharp bound on the ratio of the taxicab metric to the Euclidean metric, extending Cristea’s result for the Sierpinski carpet. As an application, we determine the diameter of the Sierpinski carpet taken over all rectifiable curves. For other members of the family, we provide a lower bound on the diameter taken over all piecewise smooth curves.

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

Talking is the most common type of human interaction that people have in their daily life. Besides all conducted studies on the analysis of human behavior in different conditions, no study has been reported yet that analyzed how the brain activity of two persons is related during their conversation. In this research, for the first time, we investigate the relationship between brain activities of people while communicating, considering human voice as the mean of this connection. For this purpose, we employ fractal analysis in order to investigate how the complexity of electroencephalography (EEG) signals for two persons are related. The results showed that the variations of complexity of EEG signals for two persons are correlated while communicating. Statistical analysis also supported the result of analysis. Therefore, it can be stated that the brain activities of two persons are correlated during communication. Fractal analysis can be employed to analyze the correlation between other physiological signals of people while communicating.

The multiphase flow behavior in shale porous media is known to be affected by multiscale pore size, dual surface wettability, and nanoscale transport mechanisms. However, it has not been fully understood so far. In this study, fractal model of gas–water relative permeabilities (RP) in dual-wettability shale porous media for both injected water spontaneous imbibition and the flow back process are proposed using fractal geometry. The shale pore structure is described as tortuous with different pore sizes and morphologies including slit pore, equilateral triangle, circular pore and square pore. The proportion of each pore morphology can be obtained from SEM/FIB-SEM pore structure characterization results. Injected water spontaneous imbibition after hydraulic fracturing is modeled as the capillary force dominated process and injected water flow back is modeled as a non-wetting gas phase drainage process in inorganic matter. The organic pores are deemed to be not accessible by injected water. The boundary slip of water and free gas flow in the inorganic matrix are considered while both free gas flow and adsorbed gas flow are modeled in organic matter. The proposed gas–water RP fractal model is verified via comparisons with the available experimental data and is discussed in detail. Study results reveal that gas phase RP increases with increasing pore fractal dimensions and tortuosity fractal dimensions, whereas it decreases with increasing Total Organic Carbon (TOC) volumes. Water phase RP decreases with increasing of pore fractal dimensions and tortuosity fractal dimensions, whereas it increases with increasing TOC volumes.

Surface finish of machined workpiece is one of the factors to evaluate the performance of machining operations. There are different factors such as machining parameters that affect the surface finish of machined workpiece. Tool wear is an unwanted machining issue that highly affects the surface finish of machined workpiece. In a similar way, different parameters (e.g. cutting speed, feed rate and depth of cut) also affect tool wear. In this research, we investigated how the complex structure of machined workpiece is related to the complex structure of tool wear. For this purpose, we benefited from the fractal analysis. The experiments were conducted based on the variations of machining parameters (depth of cut, feed rate and spindle speed), and accordingly the fractal dimension of machined surface was analyzed versus the fractal dimension of tool wear. Based on the obtained results, the complexity of machined surface is related to the complexity of tool wear. Fractal analysis could be applied to other machining operations to analyze the complex structures of machined surface and tool and potentially make a relationship between them.

This paper extends the $(2+1)$-dimensional Broer–Kaup equations in continuum mechanics to its fractional partner, which can model a lot of nonlinear waves in fractal porous media. Its derivation is demonstrated in detail by applying He’s fractional derivative. Using the semi-inverse method, two variational principles are established for the nonlinear coupled equations, which up to now are not discovered. The variational formulations can help to study the symmetries and find conserved quantities in the fractal space. The obtained variational principles are proved correct by minimizing the functionals with the calculus of variations, and might find potential applications in numerical simulation. The procedure reveals that the semi-inverse method is highly efficient and powerful, and can be generalized to other nonlinear evolution equations with fractal derivatives.

Superfast diffusion exists in various complex anisotropic systems. Its mean square displacement is an exponential function of time proved by several theoretical and experimental investigations. Previous studies have studied the superfast diffusion based on the time-space scaling local structural derivatives without considering the memory of dynamic behavior. This paper proposes a nonlocal time structural derivative model based on the Caputo fractional derivative to describe superfast diffusion in which the structural function is a power law function of time. The obtained concentration of the diffusive particles, i.e. the solution of the structural derivative model is a double-sided exponential distribution. The derived mean square displacement is a Mittag–Leffler function of time, which generalizes the exponential case. To verify the feasibility of the model, the charge and energy transfer at nanoscale interfaces in solar cells and the dynamics of the dripplons between two graphene sheets are employed. Compared with the existing models, the fitting results indicate that the proposed model is more accurate with higher credibility. The properties of the nonlocal structural derivative model with different structural functions are also discussed.

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann–Liouville fractional integral of continuous functions satisfying the Hölder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann–Liouville fractional integral and the Hölder condition equals to one, the Riemann–Liouville fractional integral of the function will be Lipschitz continuous. If the corresponding sum is strictly larger than one, the Riemann–Liouville fractional integral of the function is differentiable. Estimation of fractal dimension of the derivative function has also been discussed. Finally, the Riemann–Liouville fractional derivative of continuous functions satisfying the Hölder condition exists when order of the Riemann–Liouville fractional derivative is smaller than order of the Hölder condition. Upper Box dimension of the function has been proved to be increasing at most linearly.

In this paper, a comparative study of natural transform decomposition method and new iterative method is presented. The proposed methods are tested upon nonlinear fractional order foam drainage problem and fractional order modified regularized long-wave equation. The solutions obtained by the proposed methods have been compared with the classical solutions and the solution obtained by Adomian decomposition method. Furthermore, the efficiency and reliability of the proposed methods are shown with the help of numerical and graphical results. The fractional order derivatives are defined in Caputo’s sense whose order belongs to the closed interval [0,1]. The results reveal that the methods are quickly convergent and yield encouraging results.

Fractal model of gas diffusion in porous nanofibers with rough surfaces is derived, in which the porous structure is assumed to be composed of a bundle of tortuous capillaries whose pore size distribution and surface roughness follow the fractal scaling laws. The analytical expression for gas relative diffusion coefficient is a function of the relative roughness and the other microstructural parameters (porosity, the fractal dimension for pore size distribution and tortuosity, the maximum and minimum pore diameter and the characteristic length). The proposed fractal model is validated by comparison with available experimental data and correlations. At the same time, the effect of microstructural parameters of porous fibrous materials on gas diffusion has been studied in detail. It is believed that the current model may be extended to porous materials other than fibrous materials.

During the last years, civil infrastructure has experienced an increasing development to satisfy the society’s demands such as communication, transportation, work and living spaces, among others. In this sense, the development and application of methods to guarantee the structure optimal operation, known as Structural Health Monitoring schemes, are necessary in order to avoid economic and human losses. Modern schemes employ the structure vibration response as any damage will modify the structure physical properties, which will be reflected in the vibration response. Thus, by measuring the waveform changes of the response, the structure condition can be determined. Considering this fact, this paper investigates the effectiveness of Katz fractal dimension, Higuchi fractal dimension, Box fractal dimension, Petrosian fractal dimension, and Sevcik fractal dimension which are nonlinear measurements to extract features of vibration signals in order to determine the health condition of a 3D 9-bay truss-type bridge. The obtained results show that the algorithms corresponding to Higuchi and Petrosian fractal dimension algorithms exceed the other nonlinear measurements in efficiency to discriminate between a healthy structure and a damage produced by corrosion.

In this paper, we mainly research on Hadamard fractional integral of Besicovitch function. A series of propositions of Hadamard fractional integral of $\mathrm{\text{sin}}x$ have been proved first. Then, we give some fractal dimensions of Hadamard fractional integral of Besicovitch function including Box dimension, $\mathit{\text{K}}$-dimension and Packing dimension. Finally, relationship between the order of Hadamard fractional integral and fractal dimensions of Besicovitch function has also been given.

Coronavirus disease (COVID-19) is a pandemic disease that has affected almost all around the world. The most crucial step in the treatment of patients with COVID-19 is to investigate about the coronavirus itself. In this research, for the first time, we analyze the complex structure of the coronavirus genome and compare it with the other two dangerous viruses, namely, dengue and HIV. For this purpose, we employ fractal theory, sample entropy, and approximate entropy to analyze the genome walk of coronavirus, dengue virus, and HIV. Based on the obtained results, the genome walk of coronavirus has greater complexity than the other two deadly viruses. The result of statistical analysis also showed the significant difference between the complexity of genome walks in case of all complexity measures. The result of this analysis opens new doors to scientists to consider the complexity of a virus genome as an index to investigate its danger for human life.

Let $I$ be the unit matrix and $D=\left\{\right.\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill \hfill \end{array}\right),\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill \hfill \end{array}\right),\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill \hfill \end{array}\right),\left(\begin{array}{c}\hfill -1\hfill \\ \hfill -1\hfill \\ \hfill \hfill \end{array}\right)\left\}\right.$. In this paper, we consider the self-similar measure ${\mu }_{\rho I,D}$ on ${ℝ}^2$ generated by the iterated function system ${\{{\varphi }_d(x)=\rho I(x+d)\}}_{d\in D}$ where $0<\left|\rho \right|<1$. We prove that there exists $\mathrm{\Lambda }$ such that $E_{\mathrm{\Lambda }}=\left\{e^{2\pi i<\lambda ,x>}:\lambda \in \mathrm{\Lambda }\right\}$ is an orthonormal basis for $L^2({\mu }_{\rho I,D})$ if and only if $\left|\rho \right|=1/(2q)$ for some integer $q>0$.

This paper deals with the problem of Lyapunov inequalities for local fractional differential equations with boundary conditions. By using analytical method, a novel Lyapunov-type inequalities for the local fractional differential equations is provided. A systematic design algorithm is developed for the construction of Lyapunov inequalities.

The homogeneous perfect sets introduced by Wen and Wu [Hausdorff dimension of homogeneous perfect sets, Acta. Math. Hungar. 107 (2005) 35–44] is an important class of Moran sets. In this paper, we obtain the Assouad dimension and Assouad spectrum formulas for homogeneous perfect set under suitable condition. In the proof an Assouad spectrum formula for a large class of fractal sets is established.

In study of gas migration process in coal seam, the permeability evolution rule with time has been one of hot topics in the area of coal seam for decades. At present, in view of the time-varying rule of coal seam permeability, the influence of microstructure parameters and adsorption effect are seldom considered simultaneously. In this paper, the fractal seepage model coupled with coal deformation, and the adsorption expansion effect of coal is proposed. A multi-field coupling model is established by considering the influence of matrix and fracture structure. The influence of structural parameters and adsorption constant of main fracture on the time-varying curve of macro permeability are analyzed, including: (1) fractal dimension for fracture lengths; (2) maximum fracture length; (3) Langmuir adsorption strain constant; (4) Langmuir adsorption pressure constant. The results show that, under the influence of in situ stress and adsorption effect, the permeability of coal seam decreases with time. The present results show that the permeability is proportional to the fractal dimension for fracture lengths and the maximum fracture length. The uniformity of permeability distribution is found to be directly proportional to the pressure constant of Langmuir and inversely proportional to the volume constant of Langmuir.

Coronavirus disease (COVID-19) is a pandemic disease that has had a deadly effect on all countries around the world. Since an essential step in developing a vaccine is to consider genomic variations of a virus, in this research, we analyzed the variations of the coronavirus genome between different countries. For this purpose, we benefit from complexity and information theories. We analyzed the variations of the fractal dimension and Shannon entropy of genome walks for two-hundred samples of coronavirus genomes from 10 countries, including the Czech Republic, France, Thailand, USA, Japan, Taiwan, China, Australia, Greece, and India. The result of the analysis showed the significant variations (P-value=0.0001) in the complexity and information content of genome walks between different countries, and therefore, we conclude that the structure of the coronavirus genome is significantly different among different countries. This is a novel and very significant investigation that should be considered for developing a vaccine for this deadly virus.

Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears as a particular case of the linear self-affine functions for specific values of the scale parameters. We study the ${{ℒ}}^p$ convergence of this type of interpolants for $1\le p<\infty$ extending in this way the results available in the literature. In the second part, the affine approximants are defined in higher dimensions via product of interpolation spaces, considering rectangular grids in the product intervals. The associate operator of projection is considered. Some properties of the new functions are established and the aforementioned operator on the space of continuous functions defined on a multidimensional compact rectangle is studied.

The spread of infectious disease, virus epidemic, fashion, religion and rumors is strongly affected by the nearest neighbor hence underlying morphologies of the colonies are crucial. Likewise, the morphology of naturally grown patterns ranges from fractal to compact with lacunarity. We analyze the contact process on the fractal clusters simulated by generalized Diffusion-limited Aggregation (g-DLA) model. In g-DLA model, randomly walking particle is added to the cluster with sticking probability $p$ depending on the local density of occupied sites in the neighborhood of radius $r$ from the center of active site. It takes values $1$, $\alpha$ and ${\alpha }^2$ ($0<\alpha \le 1$) for highly dense, moderately dense and sparsely occupied regions, respectively. The corresponding morphology varies from fractal to compact as $\alpha$ varies from $1$ to $0$. Interestingly, the contact process on the g-DLA clusters shows clear transition from active phase to absorbing phase and the exponent values fall between 1-d and 2-d in directed percolation (DP) universality class. The local persistence exponents at transition are studied and are found to be much smaller than that for 1-d and 2-d DP cases. We conjecture that infection in the fractal cluster does not easily reach far-flung or remote areas at the periphery of the cluster.

Complicated gas–water transport behaviors in nanoporous shale media are known to be influenced by multiple transport mechanisms and pore structure characteristics. More accurate characterization of the fluid transport in shale reservoirs is essential to macroscale modeling for production prediction. This paper develops the analytical relative permeability models for gas–water two-phase in both organic and inorganic matter (OM and IM) of nanoporous shale using the fractal theory. Heterogeneous pore size distribution (PSD) of the shale media is considered instead of the tortuous capillaries with uniform diameters. The gas–water transport models for OM and IM are established, incorporating gas slippage described by second-order slip condition, water film thickness in IM, surface diffusion in OM, and the total organic carbon. Then, the presented model is validated by experimental results. After that, sensitivity analysis of gas–water transport behaviors based on pore structure properties of the shale sample is conducted, and the influence factors of fluid transport behaviors are discussed. The results show that the gas relative permeability is larger than 1 at the low pore pressure and water saturation. The larger pore pressure causes slight effect of gas slippage and surface diffusion on the gas relative permeability. The larger PSD fractal dimension of IM results in larger gas relative permeability and smaller water relative permeability. Besides, the large tortuosity fractal dimension will decrease the gas flux at the same water saturation, and the surface diffusion decreases with the increase of tortuosity fractal dimension of OM and pore pressure. The proposed models can provide an approach for macroscale modeling of the development of shale gas reservoirs.

In this paper, we mainly discuss continuous functions with certain fractal dimensions on $[0,1]$. We find space of continuous functions with certain Box dimension is not closed. Furthermore, Box dimension of linear combination of two continuous functions with the same Box dimension maybe does not exist. Definitions of fractal functions and local fractal functions have been given. Linear combination of a fractal function and a local fractal function with the same Box dimension must still be the original Box dimension with nontrivial coefficients.

For any $x\in (0,1]$, let the infinite series ${\mathrm{\Sigma }}_{n=1}^{\infty }\frac{1}{d_1(x)d_2(x)\cdots d_n(x)}$ be the Engel expansion of $x$. Suppose $\psi :\mathbb{N}\to {ℝ}^{+}$ is a strictly increasing function with ${lim}_{n\to \infty }\psi (n)=\infty$ and let $E(\psi )$, $E_{sup}(\psi )$ and $E_{inf}(\psi )$ be defined as the sets of numbers $x\in (0,1]$ for which the limit, upper limit and lower limit of $\frac{log{d}_n(x)}{\psi (n)}$ is equal to $1$. In this paper, we qualify the size of the set $E(\psi )$, $E_{sup}(\psi )$ and $E_{inf}(\psi )$ in the sense of Hausdorff dimension and show that these three dimensions can be different.

This paper presents a thorough study of a time-dependent nonlinear Schrödinger (NLS) differential equation with a time-fractional derivative. The fractional time complex transform is used to convert the problem into its differential partner, and its nonlinear part is then discretized using He’s polynomials so that the homotopy perturbation method (HPM) can be applied powerfully. The two-scale concept is used to explain the substantial meaning of the fractional time complex transform and the solution.

Under the realistic background that the capital market nowadays is a fractal market, this paper embeds the detrended cross-correlation analysis (DCCA) into the return-risk criterion to construct a Mean-DCCA portfolio model, and gives an analytical solution. Based on this, the validity of Mean-DCCA portfolio model is verified by empirical analysis. Compared to the mean-variance portfolio model, the Mean-DCCA portfolio model is more conducive for investors to build a sophisticated investment portfolio under multi-time-scale, improve the performance of portfolios, and overcome the defect that the mean-variance portfolio model has not considered the existence of fractal correlation characteristics between assets.

The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the ‘thickest’ and ‘thinnest’ parts of the set. Less extreme versions of these dimensions for sets have been introduced, including the upper and lower quasi-Assouad dimensions, $𝜃$-Assouad spectrum, and $\mathrm{\Phi }$-dimensions. In this paper, we study the analogue of the upper and lower $\mathrm{\Phi }$-dimensions for measures. We give general properties of such dimensions, as well as more specific results for self-similar measures satisfying various separation properties and discrete measures.

Currently, sparse code multiple access (SCMA) is a commonly used multiple-access technique, and it is a strong candidate for implementation as part of the fifth generation (5G) of wireless mobile communications. Although several design methods are available for SCMA codebooks, we propose a new method that optimizes point-to-point distances within the same codeword and from codebook-to-codebook for the same carrier based on singular value decomposition (SVD). A neural network-based receiver is proposed for detecting and decoding SVD–SCMA codewords. The simulation results show an improvement in the bit error rate (BER) compared to that for methods such as low-density signatures (LDS), SCMA, and multidimensional SCMA (MD-SCMA).

SARS-CoV-2 is a deadly virus that has affected human life since late 2019. Between all the countries that have reported the cases of patients with SARS-CoV-2 disease (COVID-19), the United States of America has the highest number of infected people and mortality rate. Since different states in the USA reported different numbers of patients and also death cases, analyzing the difference of SARS-CoV-2 between these states has great importance. Since the generated RNA walk from the SARS-CoV-2 genome includes complex random fluctuations that also contain information, in this study, we employ the complexity and information theories to investigate the variations of SARS-CoV-2 genome between different states in the USA for the first time. The results of our analysis showed that the fractal dimension and Shannon entropy of genome walk significantly change between different states. Based on these results, we can conclude that the SARS-CoV-2 genomic structure significantly changes between different states, which is resulted from the virus evolution. Therefore, developing a vaccine for SARS-CoV-2 is very challenging since it should be able to fight various structures of the virus in different states.

Severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) is the most dangerous type of coronavirus and has infected over 25.3 million people around the world (including causing 848,000 deaths). In this study, we investigated the similarity between the genome walks of coronaviruses in various animals and those of human SARS-CoV-2. Based on the results, although bats show a similar pattern of coronavirus genome walks to that of SARS-CoV-2 in humans, decoding the complex structure of coronavirus genome walks using sample entropy and fractal theory showed that the complexity of the pangolin coronavirus genome walk has a 94% match with the complexity of the SARS-CoV-2 genome walk in humans. This is the first reported study that found a similarity between the hidden characteristics of pangolin coronavirus and human SARS-CoV-2 using complexity-based analysis. The results of this study have great importance for the analysis of the origin and transfer of the virus.