Fractals

The cross-correlation between carbon spot and futures markets reflects the risk conduction mechanism between the two markets. Deeply depicting and analyzing this risk conduction mechanism is of great significance for investors to carry out risk management strategies. Considering the nonlinear and asymmetric characteristics of cross-correlation between carbon spot and futures markets, this paper applies multifractal cross-correlation analysis method to investigate the cross-correlation between carbon spot and futures markets. Firstly, through Empirical Mode Decomposition (EMD)-Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) detection, it is found that there is an obvious cross-correlation between carbon spot and futures markets, and the cross-correlation has multifractal characteristic. Secondly, by using EMD-time delay-DCCA method, we find the conduction direction between carbon spot and futures markets is bidirectional, and the futures market has a greater impact on the spot market in the short term. Thirdly, through using EMD-MF-ADCCA method, we find the cross-correlation between the two markets is asymmetric, and the cross-correlation between the two markets is more significant when carbon market is in a downward trend than in an upward trend. Fourthly, through constructing EMD-time delay-ADCCA model, we find that there is a two-way asymmetric conduction effect between carbon spot and futures markets when the lag period is short, and when the carbon market is in a downward trend, the conduction effect between the two markets is stronger. However, with the extension of time lag, the conduction effect of the two markets no longer presents obvious asymmetric characteristics.

Joint multifractality has been studied in many areas of applied sciences, but few studies focused on the sources of joint multifractality, especially the cross-correlations between two records which play a crucial role in the presence of joint multifractality. To test the effect of cross-correlations on joint multifractality, we propose a de-cross-correlation method via the shifting technique based on the framework of multifractal detrended cross-correlation analysis (MF-DCCA), namely, multifractal detrended de-cross-correlation analysis (MF-DDA). It keeps the original detrending procedure of MF-DCCA. The proposed method is validated via some simulation and real data including multifractal random walk (MRW), daily water levels, and daily stock returns. Results of validation show that the MF-DDA can detect the effects of cross-correlations between simulation series and between real series, and are consistent with the simulation setting of MRW and the actual situation of real data. This proposed method can be extended to other similar joint multifractal analysis methods.

In this paper, we study shape preserving aspects of bivariate $\alpha$-fractal functions. Its specific aims are: (i) to solve the range restricted problem for bivariate fractal approximation (ii) to establish the fractal analogue of lionized Weierstrass theorem of bivariate functions (iii) to study the constrained approximation by ${{𝒞}}^r$-bivariate $\alpha$-fractal functions (v) to investigate the conditions on the parameters of the iterated function system in order that the bivariate $\alpha$-fractal function $f^{\alpha }$ preserves fundamental shapes, namely, positivity and convexity (concavity) in addition to the smoothness of $f$ over a rectangle (vi) to establish fractal versions of some elementary theorems in the shape preserving approximation of bivariate functions.

For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(\psi )$ of complex numbers which are well approximated with the given bound $\psi$ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound $\psi$ and its Hausdorff dimension is equal to that of the $\psi$-approximable set $W(\psi )$ of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case.

Many techniques are currently available to test time series data for fractal patterns, but concerns have been raised about the fact that they do not always yield consistent results. Furthermore, fractal and nonfractal patterns, such as seasonal variation, tend to be heavily correlated in time series data. It is therefore essential that fractal patterns are clearly distinguished from nonfractal ones in the estimation process. The study is concerned with this capability in the following techniques: detrended fluctuation analysis (DFA), re-scaled range analysis (R/S), Higuchi’s fractality dimension, smoothed spectral regression, Geweke & Porter Hudak’s (GPH) estimator, the spectral periodogram (Sperio) estimate, Whittle’s estimate and the fractional differencing parameter, as well as the conventional time series diagnostics (unit root test, autocorrelation function (ACF) plots). A brief overview is provided of each fractal estimation method and results are presented of their application to three existing time series data sets: annual recordings of the water discharge level of the River Nile (663 years), daily number of births to teens in the state of Texas (5 years) and daily attendance rates in one urban school (4 years). Results indicate that all techniques indicate fractality in two of the three data sets (River Nile flow, school attendance). The results for the births to teens data were equivocal with a strong suggestion of both seasonal and fractal patterns in the ACF plot, but only seasonality in the modeling results. On the other hand, the ACF plot of the school attendance data hints at seasonality that is not picked up in the estimation. For a better perspective on these results, a simulation study was undertaken testing the ability of these techniques to distinguish seasonal and fractal processes under variable input conditions. The results of these simulations show largely accurate fractality estimates in the absence of seasonality. However, fractality estimates tend to be inflated when seasonal variability is present in the data. An integrated analytical approach is therefore recommended for these types of data, triangulating the results from the time series diagnostics with the parameter estimates from formal modeling and making seasonal adjustments to the data if necessary.

Magnetic field effects have been instrumental to unveil several exotic phenomena in two-dimensional (2D) materials. Here, we show that graphene exhibits self-similar transport once the material is nanostructured with a magnetic field in complex fashion. In specific, when magnetic barriers are arranged according to the Cantor set rules. In this study, the charge carriers are described as quantum relativistic particles through an effective low-energy Hamiltonian. The transmission, transport and thermoelectric properties are computed with the transfer matrix method, the Landauer–Büttiker formalism and the Cutler–Mott formula, respectively. Self-similarity is reflected in the conductance spectra and Seebeck coefficient for different structural parameters such as generation number, the intensity of the magnetic field, the height of the barrier and the total length of the system. Moreover, well-defined scaling rules, which described fairly good the scalability between self-similar patterns, are obtained. We also compare the self-similar patterns of magnetic complex structures with the corresponding ones to magnetic-electric complex structures, finding better scalability for the former. It is worth mentioning that as far as we have corroborated the breaking symmetry associated to the magnetic field is paramount for the self-similar transport. So, magnetic complex structures constitute an excellent option to corroborate the peculiar phenomenon of self-similar transport.

This paper aims to study a two-scale population growth model in a closed system by He–Laplace method together with the fractional complex transform (FCT). The two-scale derivative is described with the help of He’s fractional derivative. The FCT approach is used to convert differential equation of the two-scale fractal order in its traditional partner, which is then readily solved by He–Laplace iterative scheme. The results are computed as a series of easily computed components. The validation of the proposed methodology is illustrated by a quantitative comparison of numerical results with those obtained using other techniques. The results show that the proposed method is fast, accurate, straightforward, and computationally reasonable.

The Efficient Market Hypothesis (EMH) can be considered the central pillar of support of the Modern Economic-Financial Theory. However, in the last few years, the EMH has been strongly contraried due to empirical evidence related to long-memory, fractal dimension and fat-tailed that were critical factors in formulating a theory opposed to EMH called Fractal Market Hypothesis (FMH). The purpose of this paper is to test the weak form of EMH for nine constituent assets of the Ibovespa index based on the Multifractal Detrended Fluctuations Analysis (MF-DFA). Our findings show overall scaling behavior approaching the uncorrelated regime, a lower degree of multifractality, the dominance of high fractal exponents, and a broad probability density function as the source of multifractality. Given this, Brazilian assets satisfied the EMH in the Fama’s sense because the multifractal properties inherent of time series returns for Brazilian assets tend to follow a random walk.

Based on the statistical geometry theory and the fractal theory, a theoretical model to predict the pressure drop of fluid flowing through porous media has been derived in this work. This model is expressed as a function of specific surface, porosity, tortuosity, the pore fractal dimension, maximum pore size, which have clear physical meaning. To verify this model, experiments have been carried out with air using glass beads of three samples. For each sample, the present model fits the experimental data well. Besides, a comparison between the Ergun equation and present model has been made, the result shows that the present model fits the experimental data much better than the Ergun equation and has good accuracy in the range of $200<\mathrm{\text{Re}}<1000$.

Blockchain is a related FinTech asset but it is not the same technology. Basically, Blockchain is a decentralized and distributed digital ledger used to record Bitcoin transactions. The goal of this work is to employ multi-scale analysis to examine self-similarity in EDC Blockchain digital asset. Specifically, market technical data are examined; namely, open, high, low, and close. The resulting generalized Hurst exponent (GHE) estimates revealed that all Blockchain technical indicators exhibit multi-scale dynamics. In addition, short and long dynamics are different. It is concluded that market technical indicators associated with Blockchain provide valuable information for traders.

This paper aims to study the dynamic response of the fractal Fangzhu’s forced device to elucidate how the harvester device’s surface fractal patterned influences water molecules’ motion due to the increase or reduction of the wave velocity. It is also investigated under which conditions the oscillation amplitude favors the complete water removal from Fangzhu fractal surface device. Based on the approximate fractal frequency–amplitude relation derived using an ancient Chinese algorithm, and by varying the value of the fractal dimension parameter linked to the material surface morphology, it was found that the Fangzhu water harvester device can efficiently capture microscopic water droplets if the surface morphology is such that the fractal dimension $\alpha$ is less than one. This condition can be used to develop superhydrophobic surfaces mimicking fractal patterned surface structures.

The permeability of coal seams directly determines the disaster prevention effect of water injection. It is of great significance to realize the accurate fractal expression of coal structure and construct a fractal permeability model for elucidating the dynamic evolution law of the permeability of coal during and after water injection. Therefore, a rough Menger sponge is used as the physical model to characterize the structure of the coal in this paper. Combined with the structure fractal dimension calculated by the increasing fractal construction method, a new fractal permeability model is established to describe the seepage process of water injection. The influence of the tortuosity and roughness of the pore surfaces on the seepage characteristics is discussed, and the theoretical permeability is compared with the results of NMR experiments and existing models. The volume fractal dimension obtained by calculation is between 2.74 and 2.77, which decreases with increasing confining pressure and increases with increasing pore water pressure. The research shows that the fractal dimensions of the pore structure and volume can quantitatively describe the specific surface area of porous medium and are directly proportional to the specific surface area. With the increase in the structure fractal dimension, the fluid flow resistance increases, and the theoretical permeability decreases. The results of the permeability comparison show that the variation law of the theoretical permeability is consistent with that of the liquid permeability and that the theoretical result is closer to the measured value because the tortuosity and roughness of the seepage channel surfaces is further considered in the model. The theoretical model provides reliable theoretical support for further perfecting seepage theory of porous media and improving the effect of coal seam water injection.

By using generalized fractional integrals, some Bullen-type inequalities are derived where the first derivative of considered functions is Lipschitzian, bounded or generalized $(s,m)$-preinvex. These results are also applied to the estimation of the moments of the random variable, Bullen-type quadrature formula and certain other special means. Meanwhile, three numerical examples provided in terms of the proposed method illustrate that the approach is effortless to implement and computationally attractive.

This paper mainly investigated the fractal characteristics of crack propagation in hydraulic fracturing and the effect of fractal dimension of induced fractures on the rock breakdown pressure. A series of fracturing experiments with water and nitrogen (N${}_2)$ as the fracturing fluids were conducted on coal, red sandstone and shale. The injection pressure curves, rock breakdown pressure and macro- and micro-morphologies of the induced fractures were analyzed. A new fractal crack initiation criterion was developed to evaluate the rock breakdown pressure in hydraulic fracturing. The validity of the new criterion was verified by comparing the theoretical and experimental rock breakdown pressures. Further, the difference between the water fracturing and N₂ fracturing was analyzed and the influence of crack fractal effect on the rock breakdown pressure was discussed by considering different initial permeabilities of rocks, injection rates and viscosities of fracturing fluids. The results show: (1) N₂ fracturing may reduce the rock breakdown pressure and induce more tortuous macro-fractures with larger fractal dimension; (2) ignoring the actual fractal dimension of induced fractures would cause the prediction values of the rock breakdown pressure to be larger; and (3) the influence degree of crack fractal dimension on the rock breakdown pressure is related to the initial permeability of rocks, injection rate and viscosity of fracturing fluids. Therefore, we suggest the crack fractal dimension is an important factor that should not be ignored in designing the fracturing operations.

In this paper, the fractal derivative is employed to define the fractal Blasius equation in a microgravity space arising in fluid dynamics, and its fractal variational principle is successfully established by the fractal semi-inverse transform method. The approximate analytical solution of the fractal Blasius equation is obtained by the variational iteration method. Our results have theoretical significance and practical application value. The example shows the approximate approach is reliable and accurate.

In this study, the local time and space structural derivative diffusion model is introduced to describe non-Fickian diffusion in heterogeneous media from ultraslow to superfast. Its equivalent heterogeneous diffusion process (HDP) model with time and space-dependent diffusion coefficient is also given. The connection and difference between the proposed model and the existing HDP models are compared. Different types of diffusion processes can be captured by selecting proper structural functions in the structural derivative model, which is determined based on the concentration of tracers, mean squared displacement (MSD) and diffusion coefficient of the heterogeneous media. The dynamics of the free cooling granular gases with constant restitution coefficient and velocity-dependent restitution coefficient is investigated to test the proposed model. The results show that the diffusion law of the granular particles is not uniform for the short and long-time scales, but one uniform diffusion law derived from the proposed model can well fit the data in different time scales. The patterns of the time-dependent diffusion coefficient are also simulated to deeply understand the complicated non-Fickian diffusion in complex system.

When a random environment with the Gaussian white noise function is considered, the Wick-type stochastic fractional quadratic-cubic nonlinear Schrödinger equation is used to govern the propagation of optical pulse in polarization-preserving fibers. Using a new strategy, namely combining the variable-coefficient fractional Riccati equation method with the fractional derivative, Mittag–Leffler function and Hermite transformation, some special fractional solutions with the Brownian motion function including fractional bright and dark solitons, and fractional combined soliton solutions are given. Under the influence of the stochastic effect from the stochastic Brownian motion function portrayed by using the Lorentz chaotic system, some wave packets randomly appear during the propagation, and thus make fractional bright soliton travel wriggled in the both periodic dispersion system and the exponential dispersion decreasing system. However, the stochastic Brownian motion function has a more significant impact on the propagation of fractional bright soliton in the periodic dispersion system than that in the exponential dispersion decreasing system.

In this paper, we provide a complete characterization on when the Hausdorff and topological Hausdorff dimension of a given Bedford–McMullen carpet coincide. These two dimensions have a common value if and only if the carpet is either the Cartesian product of $[0,1]$ and a self-similar set, or it is one of the two diagonals of the unit square.

This work investigates the effect of surface roughness on gas diffusion behavior in the composite, which is composed of porous media embedded with a fractal-like tree network. Based on rough elements of wall surfaces of the tree network of this composite following the fractal scaling law, gas diffusion coefficient is expressed as a function of micro-structural parameters of this composite (such as the relative roughness, the porosity, the diameter ratio, the length ratio, the branching level and the branching angle). It is found that our model is more consistent with available experimental data than empirical correlations. In addition, the influence of the relative roughness and the other micro-structural parameters on gas diffusion has been analyzed. It is believed that the current model could reveal more physical mechanisms than empirical models.

In this paper, a novel fractional quantum logistic map (FQLM) based on fractional difference is proposed and image encryption algorithm based on it are exploited. First, the FQLM is proposed and its dynamical behaviors are observed with bifurcation diagram, phase portraits and largest Lyapunov exponent plots. After that, the ECC is utilized to cover up the encryption keys and the FQLM is used for the permutation and diffusion process of image encryption. Finally, the cryptosystem is used in image encryption and the algorithm is analyzed in four respects, showing a distinct advantage of the proposed cryptosystem over some of existing ones.

A difference equation can well describe a lattice problem, and its dynamical property was always modeled approximately by a differential-difference equation. This paper suggests a fractal differential-difference model by taking into account the lattice’s geometry. The fractal differential-difference Burgers equation and the fractal Klein–Gordon equation are used as examples to study the solution properties by the Lie group method, and various Lie algebras of the corresponding Lie transformation group are also obtained.

In this paper, we analyze 26 Chinese sectoral indices and evaluate the effects of the crisis caused by COVID-19 on its efficiency. We calculated the degree of multifractality in the pre- and post-COVID-19 period and found that it increases, albeit unevenly, for the economic sectors. The results suggest that global crises can affect the efficiency of the stock markets in an unequal way, with important implications for portfolio management, risk management, financial regulation and the development of predictive models.

The morphology of a shallow-water wave is affected by the unsmooth boundary, while its peak is rarely changed. This phenomenon cannot be explained by a differential model. This paper adopts a fractal modification of the Boussinesq equation, and its traveling solitary solution is studied through its fractal variational principle, the results reveal the basic properties of solitary waves in fractal space.

Let $k,b\ge 2$ be two positive integers. For $D=k\{0,1,\dots ,b-1\}$, it is well known that the self-similar measure ${\mu }_{k,b}$ defined by ${\mu }_{k,b}(\cdot )=\frac{1}{b}{\sum }_{i=0}^{b-1}{\mu }_{k,b}(kb(\cdot )-ki)$ is a spectral measure with a spectrum

The purpose of this paper is to study a generalization of strongly $\eta$-convex functions using the local fractional calculus developed by Yang [Advanced Local Fractional Calculus and Its Applications (World Science Publisher, New York, 2012)], namely generalized strongly $\eta$-convex function. Among other results, we obtain some Hermite–Hadamard and Fejér type inequalities for this class of functions.

This paper investigates the multifractal structure due to the recurrence under the $n$th Bowen metric on the dynamical system. It is shown that the set with the given lower and upper recurrence rates can have the full topological pressure, if the system $(X,d,T)$ with specification property is non-minimal, and $T$ is positively expansive.

In this work, the fractal variational principle of the fractal Ginzburg–Landau equation is established by employing fractal semi-inverse method, and its approximate analytical solution is obtained by fractal homotopy perturbation transform method. An example illustrates that the proposed technique is an excellent and fascinating method to deal with complex fractal models.

In this paper, we study the uniform Diophantine approximation in the nonautonomous dynamic system generated by the Cantor series expansions, which is formulated by considering the following set:

Assessment of the pressure drop that occurs in cable-in-conduit conductors (CICCs) requires a reliable predictive model. In this paper, a fractal model to predict the pressure drop of helium flows in CICCs is proposed based on the porous media analogy and fractal theory. A fractal friction factor correlation is also derived using the proposed pressure drop model. The good agreement between the predictions and the experimental data for conductors with $n_1=3$ verifies the validity of both the fractal pressure drop model and the friction factor correlation for such configuration conductors. The fractal correlation indicates that the friction factor of the cable bundle in a CICC is related not only to the void fraction but also to the average hydraulic diameter, the tortuosity of the fluid flow paths, the maximum pore size and the fractal dimension of the cable cross-section. This theoretical correlation does not contain any fitting constants and each parameter has a clear physical meaning.

Fractal-based approaches have become popular for describing the hydraulic functions. The SWCC and relative hydraulic conductivity were calculated and compared based on fractal models by the fractal dimension ($D)$ determined by small angle X-ray scattering (SAXS) and X-ray computed tomography (CT). Pore structure parameters including pore size, $D$, tortuosity and tortuosity fractal dimension, porosity, characteristic length were investigated. The SWCC and relative hydraulic conductivity were predicted, compared and verified. The results indicated that the $D$ of GMZ01 calculated by SAXS and CT were 2.60 and 2.72, respectively, and the difference was small. The $D$ of GMZ01 calculated by CT was 2.62. $D$ calculated by SAXS and CT has a large difference. Nitrogen adsorption revealed the effectiveness of SAXS in predicting porosity in the nanoscale self-similar interval. A SWCC fractal model considering the tortuosity fractal dimension was improved. The tortuosity of GMZ01, CaGMZ, GMZ07 and SY are calculated as 1.53, 1.47, 1.60 and 1.29, respectively. When the tortuosity was considered, the hydraulic conductivity became lower. For GMZ01, the SWCC predicted by the fractal dimension measured by SAXS was closer to the experimental data than the SWCC predicted by the fractal dimension measured by CT. The fractal model was difficult to describe SWCC under low matric suction, while the predicted SWCC had a good correlation with the experimental value in high matric suction. The relative hydraulic conductivity calculated by Xu and Yu model was larger than that calculated by Xu model.

As a fundamental research object, the minimum edge dominating set (MEDS) problem is of both theoretical and practical interest. However, determining the size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and it thus makes sense to find special graphs for which the MEDS problem can be exactly solved. In this paper, we study analytically the MEDS problem in the pseudofractal scale-free web and the Sierpiński gasket with the same number of vertices and edges. For both graphs, we obtain exact expressions for the edge domination number, as well as recursive solutions to the number of distinct MEDSs. In the pseudofractal scale-free web, the edge domination number is one-ninth of the number of edges, which is three-fifths of the edge domination number of the Sierpiński gasket. Moreover, the number of all MEDSs in the pseudofractal scale-free web is also less than that corresponding to the Sierpiński gasket. We argue that the difference of the size and number of MEDSs between the two studied graphs lies in the scale-free topology.

Metagenomics provides a new window into the structure and function of the human gut microbiome. Results suggest that certain properties of the adult gut microbial community are conserved across host populations and show remarkable resilience. The non-random clustering of taxa in gut microbiomes is evidence for the existence of functional networks in communities that may be involved in underlying dynamical processes giving rise to these patterns. Models for understanding the underlying structure (including enterotypes) and derived properties are in demand by researchers. We propose a simple random function system to model adaption and self-organization of the microbiome taxonomic space when fostering the optimal functioning of the system. The construction of this model is based on key facts of microbiota functioning, reported in recent studies. We aim to demonstrate the existence of a probability distribution as a microbiome attractor resulting from an intermittent adaption process. Its mathematical structural properties explain the stability of gut microbiota and its resilience to perturbation after occasional stress. The model is consistent with microbiome clustering results and provides precise mathematical meaning to reported gradients among enterotypes. The model also explains how intermittent perturbations, such as long-term dietary patterns, may affect microbiome structure; these results are consistent with reported experimental results. The mathematical facts implied by the model reveal an underlying mechanism that may explain gut microbiome structure and related experimental findings. Within this framework, stability and resilience properties of human gut microbiota are explained as a consequence of the model.

In this paper, a convection and heat transfer problem of power-law fluid in a three-dimensional porous media with complex evaporating surface is studied. The Buoyancy-Marangoni convection for non-Newtonian power-law fluids in porous media is solved using a compact high-order finite volume method. For this model, the left wall is kept at high temperature and high concentration, the right wall is affected by lower temperature and lower concentration, the upper wall is a complex evaporating surface. The Weierstrass–Mandelbrot function is used to approximate the shape of the evaporation interface, the analytical solution of its fractal dimension can be obtained by the pore area fractal dimension, and the volume percentage of liquid. The fluid in the porous cavity is a power-law fluid containing copper oxide nanoparticles. The solid material of the porous medium is aluminum foam. Numerical simulations can be used to determine Marangoni number, Rayleigh number and the pore area fractal dimension on the flow, heat transfer, and mass transfer rate.

The Fermat problem of triangle is interesting in plane geometry. Considering the Fermat problem within a fractal, we study the average Fermat distance of three random points on a self-similar tree by using the method of finite pattern.

To examine the overall properties of the China’s stock market affected by the COVID-19 pandemic, we provide a comprehensive study of the multifractal properties across stock market sectors applying multifractal detrended fluctuation analysis (MF-DFA). We focus on intra-day 5min transaction data of 10 sectors in the Shanghai Stock Exchange, and the time-depended MF-DFA was also used to explore the dynamical evolution of the multifractal characterize for the overall period. We found that all sectors exhibit stronger multifractality and the weak market efficiency during the period of COVID-19, MHII performed best with the lower multifractal properties and higher market efficiency, three worse performers were ITII, TBII, and EII. While ITII, TBII, MCII, and UII were the four fastest recovering sectors from the gloom of the epidemic, with the plummeted multifractal intensity and the significantly increased market efficiency for the post-phase of the pandemic. The results based on the time-varying MF-DFA confirmed that the Chinese stock market is not only affected by the COVID-19 epidemic domestic but the impact of the US stock market crash in March was also transmitted to it, causing those industries sensitive to the market to fluctuate subsequently.

In real world, many networks have self-similar structures at different length scales. In this paper, we propose a family of growing symmetrical networks based on toy-model from a deterministic initial graph. We aim to analyze its topological properties, such as the average path length, clustering coefficient and degree distribution, which uses iterative calculations and recursive methods.

In this paper, the Weyl–Marchaud fractional derivative of various fractal interpolation functions (FIFs) like linear FIF, quadratic FIF, hidden variable FIF and $\alpha$-FIF is investigated. Further, the fractal dimension of the quadratic FIF is estimated and it is compared with the order of the Weyl–Marchaud fractional derivative. Besides, this paper shows that the Weyl–Marchaud fractional derivative of all FIFs is again FIFs if the order of the fractional derivative meets the necessary condition.

In order to estimate relative permeabilities of brine–gas, a semi-analytical model based on fractal theory is proposed. The model simplifies porous media as treelike nanotubes, the single nanotube’s flow has been branched out into whole pore network of core sample through treelike fractal theory. Two-phase flow description in sole nanotube, based on a setting of gas bubbles pushing continuous water phase ahead, solves pressure drop around brine and gas bubbles by Stokes’ viscous force, Laplace equation and Fairbrother–Stubbs bubble film thickness formula, Bretherton bubble pressure drop. By substituting the other three classic formulas into Bretherton expression, the iteration equation of pressure drop around a single gas bubble is acquired, the number of gas bubbles is multiplied and the gas phase pressure drop is calculated. Through merging small bubbles into a large bubble, the relative permeabilities of two phases can be solved by introducing Poiseuille’s law and Darcy’s law. After getting the fundamental data, pore size distribution data and viscosity of two phases, relative permeability curves can be drawn, revealing comprehensive elements including wettability, fluid viscosity, saturation, morphological characteristics. The calculated curves cross each other on the right half of the $x$-axis, meet the features of real production, flowback difficulties of hydraulic liquid and well production dropping quickly, also show identical trend with experimental data. Through a series of sensitivity analysis, to some extent, the influence on relative permeabilities is compared of different fractal models, different film thickness, bubble length, water sorption layer thickness, elastic deformation of pores.

Hydraulic permeability in hydrate-bearing sediments largely controlling the rate of gas production from hydrate deposits is widely estimated by various modified models based on the Kozeny–Carman equation. However, the effect of hydrate saturation on the shape factor in the Kozeny–Carman equation is not fully understood. In this study, a fractal model of the shape factor is theoretically derived to show physical relations between the shape factor and fractal parameters of fluid occupied pores. Then, verification of the model is performed, and effects of intrinsic porosity, intrinsic pore area and tortuosity fractal dimensions on the hydrate saturation dependent shape factor are analyzed. Finally, how weighted specific surface areas evolve with hydrate saturation is theoretically discussed, and results are further applied to modify the Kozeny–Carman equation. It is shown that the fractal model has a good performance on the shape factor estimation, and value of the shape factor is largely controlled by porosity, pore area and tortuosity fractal dimensions of fluid occupied pores in hydrate-bearing sediments. Multiplying product of the shape factor, the squared specific surface area, and the squared hydraulic tortuosity linearly increases with increasing hydrate saturation in general for the pore-filling hydrate pore habit but linearly decreases due to the presence of grain-coating gas hydrates. This linear relation permits a feasible modification of the Kozeny–Carman equation for hydrate-bearing sediments, and values of the model parameter are suggested for pore-filling and grain-coating hydrate pore habits.

An accurate and reliable mathematical model for water inrush can help ensure operational safety in coal mines. In this study, a numerical model for water inrush caused by collapsed columns that couples mechanical rock deformation with flow in porous media based on a fractal treelike fracture network was developed. In particular, the proposed model considers collapsed columns and mining damage zones as fractal dual-porosity media. The model incorporates Darcy’s law, the Brinkman equation, and the Navier–Stokes equation to govern the groundwater flow. The Brinkman equation was adopted to simulate the groundwater flow in collapsed columns, for which the non-Darcy coefficient can be derived based on damage criteria for the fracture zone. To simulate the dynamics of water inrush in collapsed columns, the finite element method was used to solve the governing equations for the groundwater flow, solid mechanics, and rock damage. The proposed model was validated against two sets of field data, and the results showed that dynamic changes in the non-Darcy coefficient could be used to identify the occurrence of water inrush, which can be employed to improve mining safety.

Understanding the fundamental mechanisms of vapor condensation on rough surfaces is crucial to a wide range of industrial applications. A hybrid thermal lattice Boltzmann model of the condensation heat transfer process on downward-facing rough surfaces characterized by the Cantor fractal is developed and numerically analyzed to investigate the condensation phase change behaviors on rough hydrophobic and hydrophilic surfaces. The dynamic behaviors of vapor condensation, including the evolutions of vapor–liquid interface, heat flux, condensate mass, and temperature distribution, on the hydrophilic and hydrophobic rough surfaces are presented and compared with corresponding smooth surfaces. The results indicate that the rough surface preferred a filmwise condensation under hydrophilic conditions but a hybrid dropwise–filmwise condensation under hydrophobic conditions. On the rough hydrophobic surface, the liquid film can rapidly adsorb droplets, maintaining a high-efficiency dropwise condensation. The absorption of droplets accelerates the liquid film growth and detachment process on the rough hydrophobic surface, which reduces the time-averaged thermal resistance of the filmwise region. These two behaviors together enhance condensation heat transfer on the downward-facing rough hydrophobic surface. Besides, stable dropwise condensation could also be formed on smooth hydrophilic surfaces and has better heat transfer performance than corresponding hydrophobic surfaces under the same heat transfer condition.

For any $x\in (0,1]$, let

This paper performs a systematic investigation into the temporal evolution of daily death cases of COVID-19 worldwide lethality considering 90 countries. We apply the information theory quantifiers, more specifically the Permutation entropy $(H_s)$ and Fisher information measure $(F_s)$ to construct the Shannon-Fisher causality plane (SFCP), which allows us to quantify the disorder and evaluate randomness present in the time series of daily death cases related to COVID-19 in each country. Moreover, we employ $H_s$ and $F_s$ to rank the COVID-19 lethality in these countries based on the complexity hierarchy. Our findings reveal that the countries that are located farther from the random ideal position ($H_s=1$, $F_s=0$) in the SFCP such as Taiwan, Vietnam, New Zealand, Singapore, Monaco, Iceland, Thailand, Bahamas, Cyprus, Australia, and Norway are characterized by a less entropy and low disorder, which leads to high predictability of the COVID-19 lethality. Otherwise, the countries that are located near to the random ideal position ($H_s=1$, $F_s=0$) in the SFCP such as Ecuador, Czechia, Iraq, Colombia, Belgium, Italy, Philippines, Iran, Peru, and Japan are characterized by high entropy and high disorder, which implies low predictability of the COVID-19 lethality. We also employ two cluster techniques to analyze the similarity considering the temporal evolution of COVID-19 worldwide lethality for the countries investigated. Based on the values of $H_s$, $F_s$ and our cluster analysis, we suggest that this health crisis will only be adequately combated through global adherence to scientific exchange and technology sharing to homogenize the actions to combat the COVID-19.

This paper concentrates on discussing the properties of Riemann–Liouvile fractional (RLF) calculus of two special continuous functions. The first type proves the non-differentiability of a special continuous function that does not satisfy Hölder condition, and the second type uses fractal iteration to construct a fractal function defined on $[0,1]$ with unbounded variation. Then we calculate RLF integral and RLF derivative of this special function, and give the corresponding numerical calculation results and the corresponding function image.

In this paper, we mainly investigate relationship between fractal dimension of continuous functions and orders of Weyl fractional integrals. If a continuous function defined on a closed interval is of bounded variation, its Weyl fractional integral must still be a continuous function with bounded variation. Thus, both its Weyl fractional integral and itself have Box dimension one. If a continuous function satisfies Hölder condition, we give estimation of fractal dimension of its Weyl fractional integral. If a Hölder continuous function is equal to 0 on $ℝ/[a,b]$, a better estimation of fractal dimension can be obtained. When a function is continuous on $ℝ$ and its Weyl fractional integral is well defined, a general estimation of upper Box dimension of Weyl fractional integral of the function has been given which is strictly less than two. In the end, it has been proved that upper Box dimension of Weyl fractional integrals of continuous functions is no more than upper Box dimension of original functions.

Porous fins with temperature-dependent internal heat generation are frequently used to improve performance in a wide range of heat transfer and porous media applications. Thermal analysis of the porous fin fractal model with temperature-dependent heat generation is generated using fractal derivatives and investigated analytically using a novel Maclaurin series method (MSM). Nonlinear temperature distribution in a porous longitudinal fin is produced by the MSM. The porous fin solution is demonstrated using the Sierpinski fractal, which is based on time-dependent heat generation. The effects of the convection parameter, porosity, internal heat production, and generation number parameter on the dimensionless temperature distribution are discussed. MSM results are graphically and tabularly compared to existing solution methods such as HPM, CM, CSCM, LWCM, and GWRM. A comparison study reveals that MSM is a very reliable, accurate, and effective addition in the field of differential equations.

The unsmooth boundary of a cavity for the stage of filling glass fiber-reinforced polymer melt affects the flow property and the shape of the free interface’s shape, as a result, it will affect greatly the process, the degree and the location of the front interface, and the weld lines’ property and product quality as well. Here, the two-scale fractal theory is adopted to figure out the basic property of the unsmooth boundary, and governing equations are established in a fractal space. The numerical results can provide insight into the filling process and reveal the main factors affecting the shape of the free interface. The fractal interface is illustrated by the least-square method in different fractal spaces based on numerical simulation results. The fractal cavity sheds a new light on controlling the interface.

For calculating the fractal dimension, the standard method uses a compass that with a specific radius, an arc is drawn to find the points that intercept the curve. An important consideration when making a measurement using the compass or ruler method is to consider the impact of the observer. Most of the rules are read by the user and are therefore very susceptible to misreading or visual errors. The main contribution of this work lies in the development of an algorithm for calculating the fractal dimension of fluctuating continuous functions, over a fractally divided space, from which it is possible to obtain also the integral of the function with an acceptable precision using the trapezoid rule compound.

The tree-like branching network widely exists in nature and human society. The heat and mass transfer of tree-like branching network has been studied for a long time. The research indicates that a tree-like branching network has excellent heat and mass transfer performance. In this paper, a model of spontaneous imbibition in the tree-like branching network with fractal roughened surfaces is proposed. Based on the series-parallel equivalent model, the analytical expression of the time-varying imbibition height and the expression of the equilibrium height are derived. It is found that the structural parameters of the branching network have an important influence on the imbibition process, including the diameter in the 0th branching level, the length ratio, the diameter ratio, the branching number, the total number of branching levels, the relative roughness and the branching angle. For instance, the equilibrium height increases with the increase in the total number of branching levels, but the time required to reach the equilibrium height also increases with the increase in the total number of branching levels. It is also seen that the equilibrium height increases with the increase in fractal dimension of roughened elements and length ratio, and decreases with the increase in diameter ratio. In addition, an interesting phenomenon is that the time required to reach the equilibrium height increases with the increase in branching angle, but the equilibrium height remains the same. Every parameter of the presented model has a clear physical meaning, revealing the imbibition mechanism of the tree-like branching network.

In this paper, we present new Hermite–Hadamard–Fejér type inequalities via Riemann–Liouville fractional integrals for the function $g(\tau )h(\tau )$, where $g(\tau )$ is convex function and $h(\tau )$ is symmetric with respect to $\frac{a+b}{2}$. Our results in some special cases yield the well-known classic Hermite–Hadamard–Fejér type inequalities. At the end of the paper, the applications of our results are given.

We propose a new hyperchaotic complex system. The hyperchaotic complex system has a distinctive chaotic attractor, which is called butterfly-like attractor. We find that the state of the system has a complex evolution process with the change of parameters, and the system will produce at least two kinds of attractors, which we call parametric attractors. We systematically study the mechanics, symmetry, dissipation, Lyapunov exponents and Poincaré maps of the hyperchaotic complex system. We extend the hyperchaotic complex system to fractional order, and discuss the influence of order and parameters’ change on fractional order new hyperchaotic system, and compare it with integer order. What is interesting is that there exist parametric attractors in both integer order and fractional order of the new system. Based on the integer order and fractional order, we further analyze the new hyperchaotic complex system of hybrid order, propose the definition of complex chaotic family and carry out mechanical analysis.

In this work, space-time fractal model about nonlinear KdV-Burgers–Kuramoto (NKBK) equation which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters has been put forward through coupling fractional complex transform (FCT) via local fractional derivative (LFD) for the first time. These measures are considered in the sense of local derivative operators. Analytical approximate solutions of the model are obtained by local fractional reduced differential transform method (LFRDTM). The obtained results related to physical phenomenon in Cantorian time-space reveal that the suggested project is easy to use and the calculation is more precise. The graphical representation of special solution of LFNKBK yields interesting and better physical consequences of NKBK with LFD.

The determination of the generalized fractal dimensions (FDs) from MRI tractograms is affected by image resolution and the number of fiber tracts. In this work, we demonstrate that only certain combinations of image resolution, number of fiber tracts and exponent of the generalized dimensions are able to capture structural changes in human cerebral white matter. MRI tractography was carried out for a different number of fiber tracts and was discretized on different grid sizes. Generalized FDs were evaluated for two groups of healthy subjects with different age distribution. For the box-counting dimension ($q=0$) the highest difference between two age groups was found for a matrix size of 2048³ pixel with a strong dependence on the number of fiber tracts. The correlation dimension found the highest differences for a resolution of 1024³ pixel, largely independent of the number of fiber tracts. The correlation dimension evaluated for tractograms discretized on a 1024³ grid is a robust measurement to capture structural changes in human cerebral white matter by means of FD.

Tortuous pores are ubiquitous in natural porous media; however, the linear effects of tortuosity on the liquid flow behaviors are usually considered through capillary bundle models, which lead to inaccurate permeability prediction. Additionally, in the nanopores, due to the existence of heterogeneous viscosity and boundary slip, the effects of tortuosity on nanoconfined liquid flow behaviors are more complicated. In this paper, we comprehensively use the theoretical models and a nanoscale multi-relaxation-time lattice Boltzmann method (MRT-LBM) to characterize water flow behaviors in the tortuous nanopores, where heterogeneous viscosity and boundary slip are considered. The results show that the increase of tortuosity causes more negative effects, which results in a decreasing apparent permeability. The conventional model of the influence of tortuosity on the apparent permeability is linear$k_{\mathrm{\text{app}}}=\frac{k_{\mathrm{\text{ini}}}}{\tau }$, which greatly underestimates the effect of tortuosity on apparent permeability. Then, a new fractal apparent permeability model $k_{\mathrm{\text{app}}}=\frac{k_{\mathrm{\text{ini}}}}{A+Bln(\tau -C)}$ based on modified tortuous effects is obtained by fitting the results from the MRT-LBM simulations. Based on the new fractal apparent permeability models, the results show that with an increasing fractal tortuosity dimension, the apparent permeability decreases rapidly first and then this trend becomes moderate. Our works can provide a more accurate basic model for studying liquid flow in porous media based on the capillary bundle model or pore network model.

In this paper, we mainly construct two continuous functions defined on $I=[0,1]$ which have only one unbounded variation point, and show that both Box and Hausdorff dimensions of these functions are 1. We prove that their Riemann–Liouville fractional integrals are bounded variation function. We also show that these functions repair inaccuracies of the example in [N. Liu and K. Yao, Fractal dimension of a special continuous function, Fractals 26(4) (2018) 1850048; Y. S. Liang, Definition and classification of one-dimensional continuous functions with unbounded variation, Fractals 25(5) (2017) 1750048]. In the proofs, we mainly use Leibniz’s alternating series test and the properties of logarithmic function.

This study employs the Caputo–Fabrizio and Atangana–Baleanu fractional derivatives to determine the impedance and admittance model of fractional capacitor and inductor. The analog implementation circuits are proposed aiming at fractional-order electric elements based on these two derivatives, which can be widely used in a variety of electrical systems using new fractional operators. Constant phase capacitor and inductor are approximated by the Oustaloup algorithm and the recursive net-grid-type analog circuit, respectively. Based on that, approximation circuits of fractional electric components under Caputo–Fabrizio and Atangana–Baleanu definitions are given. For the purpose of judging whether the implementation topology of fractional-order capacitor and inductor is accurate, taking fractional RC and RL circuit defined by Caputo, Caputo–Fabrizio and Atangana–Baleanu derivatives as examples, the comparison of numerical and circuit simulations is carried out. The correctness of the analog implementation circuits using the Caputo–Fabrizio and Atangana–Baleanu definitions is verified. Fractional-order RC charging circuit experiments based on Caputo, Caputo–Fabrizio and Atangana–Baleanu derivatives are taken as examples. Several experiments with different fractional-order and circuit parameters are carried out. The validity of the implementation methods is ulteriorly proved with experiment data.

An important research area in physiological and sport sciences is the analysis of the variations of the muscle reaction due to changes in walking speed. In this paper, we investigated the effect of walking speed variations on leg muscle reaction by the analysis of Electromyogram (EMG) signals at different walking inclines. For this purpose, we benefited from fractal theory and sample entropy to analyze how the complexity of EMG signals changes at different walking speeds. According to the results, although fractal theory could not show a clear trend between the variations of the complexity of EMG signals and the variations of the walking speed, however, based on the results, increasing the speed of walking in the case of different inclines is mapped on to the decrement of the sample entropy of EMG signals. Therefore, sample entropy could decode the effect of walking speed on the reaction of leg muscle. This analysis method could be applied to analyze the variations of other physiological signals of humans durin walking.

Analysis of leg muscle activation and gait variability during locomotion is an important area of research in physiological and sport sciences. In this paper, we analyzed the coupling between the alterations of leg muscle activation and gait variability in single-task and dual-task walking. Since leg muscle activation in the form of electromyogram (EMG) signals and gait variability in the form of stride interval time series have complex structures, fractal theory and approximate entropy were used to evaluate their correlation at various walking conditions. Sixty subjects walked at their preferred speed for 10 min under the single-task condition and for 90s under the cognitive dual-task condition, and we evaluated the variations of the fractal dimension and approximate entropy of EMG signals and stride interval time series. According to the results, dual-task walking caused reductions in the complexity of EMG signals and stride interval time series than single-task walking. This technique can be used to evaluate the correlation between other organs during different locomotion.

Due to the exponential growth in the use of systems with applications of blockchain technology, this paper develops a funding system, with donations and offers of shares, through the Ethereum platform with blockchain technology. Given the benefits that blockchain provides data protection and has high security, this paper offers a decentralized donation expert system using smart contracts that makes fully reliable donation systems to attract more funds to this urgent global health issue. Smart contracts provide faithful donations and meet the characteristics of being versatile, accessible, and sustainable to combat the COVID-19 pandemic. This expert system found the Merkle grid as an optimum method to work efficiently on the blockchain. The expert system proved to be steady and efficient by using an essential test dataset. A reliable donation system expects more donors and investors since a sustainable and reliable approach is always a milestone. The primary purpose of developing this system is to attract donors to bring down the COVID-19 pandemic by providing a faithful donation system.