Fractals

Modeling fluid flow and transport in fracture network is of considerable importance in a wide range of science and engineering. In this work, a probability model of the effective permeability for fracture network is developed based on fractal theory and Monte Carlo method. The proposed numerical model is expressed as a function of the fractal dimension, the minimum/maximum trace length, fracture orientation as well as random number, and the validity is validated by comparison with the analytical fractal model. It is found that the effective permeability increases with the increase of the fractal dimension and fracture density of fracture network, as well as decreases with increase of the dip angle of fractures. The proposed Fractal Monte Carlo method can significantly reduce the computational load compared with conventional numerical techniques and also permits to relate the other flow and transport properties in fracture networks.

The sandstone micro-structure is one of the most important parameters for quantitative evaluation of groundwater/oil/gas resources and prediction of flow rates of water/oil/gas. In this study, we applied seven low-permeability sandstone samples obtained from North China to study the structures of pore and solid phase based on digital core technology and the fractal theory. In this work, rock images were collected by scanning electron microscopy (SEM), and then image software (image pro plus) was applied to convert the SEM images into binary images and calculated the perimeter and area of pores and particles. Using the fractal medium criterion, we analyzed the fractal characteristics of pore and solid phases, and then we extended the criterion from the pore phase to both pore and solid phases. This paper explores the relationship among the pore and solid phase fractal characteristics, porosity, pore size, particle size and fractal dimensions for pores and particles. The results were compared with the experimental data and showed that the solid phase satisfies the fractal distribution, while the pore phase dissatisfies the fractal distribution. It was also found that the minimum and maximum pore and particle sizes, magnification times have a significant effect on the results. In addition, by introducing the diameter and mean hydraulic radius into Ergun–Wu equation, the permeabilities of the samples were calculated.

The point within a triangle that meets the minimum sum of the distances to the three vertices is called the Fermat point. The aim of this paper is to obtain the Fermat average distance on a self-similar fractal tree in terms of the self-similar measure and the technique of finite pattern.

The maximum matching problem is of great interest in many areas, for example, statistical physics and theoretical computer science. However, precise determining of maximum matchings in general graphs is a challenge and calculationally intractable. In this paper, we study analytically maximum matching problem in the pseudofractal scale-free web. We derive exact expressions for size and number of maximum matchings of the network based on self-similar structure and iterative construction of network, and obtain asymptotic incremental constant for the number of maximum matchings. The obtained results are helpful to understand the structural characteristics of scale-free networks with pseudofractality.

The Caputo–Fabrizio (C–F) definition, which solves the singularity problem in the Caputo definition, has been preliminarily applied in the field of circuit system modeling. However, the complex characteristics of the C–F definition-based circuit systems are still understudied. Therefore, this paper proposes a C–F definition-based fractional-order RLC (CF-FORLC) circuit model and analyzes its basic characteristics. First, the effects of different component orders on the performance parameters including the impedance, quality factor, and bandwidth are analyzed, which gives insights into the design of CF-FORLC. Then, the analytical solutions and the frequency-domain characteristics of CF-FORLC with different capacitance and inductance orders under arbitrary input are derived. Finally, the data of actual circuits are fitted to obtain the parameters of the CF-FORLC model, and the orders of the C–F definition-based capacitors and inductors are estimated. The results of the comparative experiments show that the proposed modeling scheme can improve the consistency of the dynamic performance of the model with that of the actual circuit. In addition, the proposed CF-FORLC model shows higher accuracy and flexibility with more adjustable parameters.

In this paper, we relax the definition of Rényi information dimension. The power law of the Entropy-Layer in the Galton board is discovered and we calculate its information fractal dimension. When the Galton board is extended to bias or three-dimensional space, we get the same fractal features. In addition, according to the connection between Pascal’s triangle and the Poisson distribution, we find constrained Poisson distribution groups with the same information dimension. This is the first time the information entropy is utilized to explore the fractal features of the Galton board and Pascal’s triangle.

To explore the influence of liquid nitrogen (LN${}_2)$ cooling on the structural damage and failure characteristics of coal, a series of structural experiments, uniaxial compression tests, and crack morphology characterization experiments were carried out in this study. Based on the fractal theory, the pore structure damage and failure characteristics of the coal before and after the LN₂ cooling treatment were compared. After the LN₂ treatment, the structural damage inside the coal increases. Compared to the untreated coal, the fractal dimensions of the total pores, seepage pores, and adsorption pores in the LN₂-treated coal decrease by 0.88%, 0.24%, and 5.16%, respectively. The results indicate that the LN₂ cooling treatment can increase the connectivity of pores, especially adsorption pores. The internal structure damage of the coal induced by the LN₂ treatment reduces its compressive strength and elastic modulus by 27.86% and 32.62%, respectively. Simultaneously, it can be found that the fragments fractal dimension of the coal decreases by 9.53%, and the fractal dimension of the induced fracture surface increases from 2.134 to 2.151 after the LN₂ treatment. By the LN₂ treatment, the distribution of fragments becomes more uniform, and the complexity of the generated fracture surface increases. Therefore, the LN₂ cooling treatment can increase the effective seepage path and the contact area between the crack and the coal matrix to promote the gas flow and enhance the gas desorption in the coal matrix. The experimental results also show that the damage of the single LN₂ treatment on the coal is limited. LN₂ cycling treatment is suggested to use to improve the efficiency of the coalbed methane extraction because that it can give full play to the multiple cracking effects of LN₂ on the coal.

In this work, new $(3+1)$-dimensional Korteweg–de Vries (KdV) equation and modified KdV (mKdV) equation as well as the corresponding fractional forms are presented. These two equations are derived for the first time relying on the extended $(3+1)$-dimensional zero curvature equation. In addition, symmetries and conservation laws are displayed. Meanwhile, one-soliton solution for the time fractional of $(3+1)$-dimensional KdV equation and mKdV equation is obtained.

The main goal of this work is to develop a robust framework for an exhaustive description of essential properties of a fractal object. For this purpose, the inherent features of fractal sets are scrutinized. The topological, metrological, morphological, and topographical attributes of fractal systems are delineated. The criteria of the fractal connectedness are established. The characteristics of the fractal connectivity and ramification are ascertained. The index of the fractal loopiness is introduced. The quantifications of the fractal heterogeneity, lacunarity, and anisotropy are briefly sketched out. A set of key attributes which enable a proper characterization of fractal system are suggested.

In this study, the fractal nonlinear dispersive Boussineq-like equation is investigated by variational perspective for the first time. The fractal variational principle of the fractal Boussineq-like equation is established via fractal semi-inverse method (FSM). Based on the fractal variational principle, variational transform method (VTM) is proposed to find the exact fractal solitary wave solution for the fractal nonlinear dispersive Boussineq-like equation. The numerical examples illustrate the novel scheme is very simple and efficient. Moreover, some physical properties of fractal solitary wave solutions are illustrated by some simulation graphics.

First, we define what we named the generalized $n$-polynomial convex mappings as a generalization of convex mappings, investigate their meaningful properties, and establish two Hermite–Hadamard’s-type integral inequalities via the newly proposed mappings in the frame of fractal space as well. Second, in accordance with the discovered identity with a parameter, we present certain improved integral inequalities with regard to the mappings whose first-order derivatives in absolute value belong to the generalized $n$-polynomial convexity. As applications, on the basis of local fractional calculus, we acquire three inequalities in view of special means, numerical integrations, as well as probability density mappings, respectively.

First, we construct a reformative version of the power-mean integral inequality in the sense of fractal space. Second, we define what we named as the generalized $(s,P)$-convex mappings, and investigate some related properties. Moreover, in accordance with the derived midpoint-type integral identities on fractal space, we establish certain improvements of the midpoint-type integral inequalities for mappings whose first-order derivatives in absolute value belong to the generalized $(s,P)$-convex mappings. As applications in association with local fractional calculus, we acquire three inequalities considering $\nu$-type arithmetic mean and $\beta$-logarithmic mean, numerical integration, as well as probability density mappings, respectively.

The statistical simulation of atmospheric carbon dioxide based on time series analysis plays an important role in the development of refined carbon emission assessment and traceability. A new statistical method based on the random Weierstrass–Mandelbrot function is proposed to simulate the high-Reynolds-number and high-Péclet-number scalar turbulence in the atmospheric flow. In this paper, we focus on the carbon dioxide, which is the most important scalar for the global climate changes. The new statistical method is used to simulate the time series of carbon dioxide concentrations in the high-frequency inertial subrange. Results show that the simulated power spectral densities are almost the same as the observed ones in the inertial subrange. Compared with the Gaussian distribution, the simulated probability density functions are more similar to the observed ones with the skewness characteristic. More importantly, the fractal characteristics of the time series that cannot be simulated by traditional methods in the meteorological engineering can also be well simulated by the new method.

Let $D=\{(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}),(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}),(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}),(\begin{array}{c}\hfill -1\hfill \\ \hfill -1\hfill \end{array})\}$ and $M\in M_2(\mathbb{Z})$ be an expansive matrix. In this paper, we have proved that there exists $\mathrm{\Lambda }$ such that the set ${{ℰ}}_{\mathrm{\Lambda }}:=\{e^{2\pi i〈\lambda ,x〉}:\lambda \in \mathrm{\Lambda }\}$ is an orthonormal basis of $L^2({\mu }_{M,D})$ for a matrix which has all even entries. Also, we have found a spectrum $\mathrm{\Lambda }$ of $L^2({\mu }_{M,D})$ for some specific $M$ and $D$.

The well-known Sierpiński square is a fractal generated by iterated function system (IFS). In this paper, we focus on a class of fractal networks created by IFS. We show a universal approach to solve the average geodesic distance of these fractal networks.

The physical/probabilistic meanings of fractional derivatives or integrals are very basic and important problems in fractional calculus. In this paper, we investigate the probabilistic interpretations of the general fractional derivatives with memory effect and their connections with cut-off distributions. Some examples like the cut-off Rayleigh distribution and the Atangana–Gómez fractional derivative are especially discussed. This paper shows that the kernels of the general fractional derivatives with memory effect may be not the same as the density functions of the physical processes described. For example, the kernel of Atangana–Gómez fractional derivative is a Gaussian function but its corresponding probabilistic distribution is a cut-off Rayleigh distribution.

Many electrical systems can be characterized more authentically by fractional-order dynamic systems. The Atangana–Baleanu and the Caputo–Fabrizio fractional derivatives have solved the singularity problem in Caputo derivative. This work uses Atangana–Baleanu and Caputo–Fabrizio fractional derivatives to model the fractional-order mutual inductance in the frequency domain. To use the fractional mutual inductance in circuit design, the T-model equivalent circuits are presented with different fractional derivatives. The fractional impedance matching networks based on proposed fractional mutual coupling circuits are simulated as an application. The impedance characteristics of networks with different fractional orders are analyzed. The results indicate that the proposed fractional mutual coupling circuits based on Atangana–Baleanu and Caputo–Fabrizio fractional derivatives can be applied to the complex electrical systems to increase the design degree of freedom, which provides more choices for describing the nonlinear characteristics of the system.

The fractal necklaces in ${ℝ}^d$ ($d\ge 2$) introduced in this paper are a class of connected fractal sets generated by the so-called necklace IFSs, for which a lot of basic topology questions are interesting. We give two subclasses of fractal necklaces and prove that every necklace in these two classes has no cut points. Also, we prove that every stable self-similar necklace in ${ℝ}^2$ has no cut points, whilst an analog for self-affine necklaces is false.

This paper is a continuation of our work [J. O. González Cervantes and J. Bory Reyes, A quaternionic fractional Borel–Pompeiu type formula, Fractal 30(1) (2022) 2250013], where we introduced a fractional operator calculus related to a fractional $\psi$-Fueter operator in the one-dimensional Riemann–Liouville derivative sense in each direction of the quaternionic structure, that depends on an additional vector of complex parameters with fractional real parts. This allowed us also to study a pair of lower order fractional operators and prove the associated analogues of both Stokes and Borel–Pompieu formulas for holomorphic functions in two complex variables.

The terms fractional differentiation and fractal differentiation are recently combined to form a new fractional differentiation operator. Several kernels, such as the power-law and the Mittag-Leffler function, are employed to investigate these novel operators. Both fractional-order and fractal dimensions are found in these developed operators. In the present investigation, we have applied these new operators to study certain chaotic attractors with Mittag-Leffler and power-law kernels. These chaotic models are solved using an effective numerical procedure called the Adams–Bashforth technique, based on two- and three-step Newton interpolation polynomials. In this analysis, the order of the fractional derivative is assumed to be the exponential, trigonometric, and hyperbolic forms of variables. The solutions obtained by the numerical procedure with two different kernels are portrayed in terms of chaotic plots.

This paper probes into change of box dimension for an arbitrary fractal continuous function after Hadamard fractional integration. For classic calculus, we know that a function’s differentiability increases one-order after integration, and decreases one-order after differentiation. But for fractional integro-differentiation, this similar fundamental problem is still open [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217–229; M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math. 76 (1996) 265–275; Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438 (in Chinese); Y. S. Liang, Progress on estimation of fractal dimensions of fractional calculus of continuous functions, Fractals 26 (2019), doi:10.1142/S0218348X19500841] (see conjecture 1.1), although it is believed that the roughness of a fractal function is increased or decreased accounts $\alpha$ after $\alpha$-order fractional differentiation or integration. This paper partly answers this problem. Particularly, the estimation of box dimension of Hadamard fractional integral is of methodology for considering similar fractional integration. In this paper, it is proved that the upper box dimension of the graph of $f(x)$ does not increase after $\alpha$-order Hadamard fractional integral $D_H^{-\alpha }f(x)$, i.e.

Let $\psi :\mathbb{N}\to {ℝ}^{+}$ be a function satisfying $\psi (n)\to \infty$ as $n\to \infty$. Write

This paper proposes a kind of tree-like convergent microchannel heat sink (TCMCHS) and numerically investigates effects of the outlet-to-inlet width ratio $\alpha$ of the convergent microchannel, the width ratio of the inlet of the daughter microchannel to the outlet of the parent microchannel $\gamma$ (D-P width ratio), the channel length ratio $\beta$, the initial branching number $N$, and the branching levels $m$ on the maximum temperature $T_{max}$, the total thermal resistance $R_T$, the pressure drop $\mathrm{\Delta }P$ and the coefficient of performance (COP) of the TCMCHS with rectangular channel cross-section under the size limitations of fixed total channel volume. The results display that both the maximum temperature and the total thermal resistance drop with the decrease of the outlet-to-inlet width ratio $\alpha$; however, the pressure drop $\mathrm{\Delta }P$ of TCMCHS first drops and then increases with the rise of the outlet-to-inlet width ratio $\alpha$. The different influences of the outlet-to-inlet width ratio $\alpha$ on the hydraulic and thermal performances lead to first increased and then decreased COP with the growth of the outlet-to-inlet width ratio $\alpha$. Moreover, the COP declines with the increasing length ratio $\beta$, initial branching number $N$ and the branching level $m$, but first rises when $\alpha$ is small and then declines when $\alpha$ is large with the rising D-P width ratio $\gamma$. The findings of this paper provide a feasible scheme for the improvement of the thermal and hydraulic performances of MCHS by using tree-like convergent microchannel.

This research explores the predictability of the Brazilian inflation monthly price indexes time series by information theory quantifiers. Given this, we apply the Bandt and Pompe method to estimate the information theory quantifiers, specifically the Permutation entropy ($H_s$) and the Fisher information measure ($F_s$). Based on the values of both complexity measures, we construct the 2D plane called the Shannon-Fisher causality plane, which allows us to explore the disorder and quantify the randomness inherent to the temporal evolution of the significant Brazilian inflation indexes. Also, we apply $H_s$ and $F_s$ to rank the Brazilian inflation indexes based on the complexity hierarchy. An overview shows that the Brazilian inflation indexes display lower entropy which implies in higher predictability. The sliding window approach reveals that Brazilian inflation indexes (IPA, IPCA, IGP, IPC, and INPC) present decrease related to efficiency. The only exception is the INCC, which shows an increase in efficiency.

In this paper, we study the iterated function systems (IFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. We define a weak* separation condition (W*SC), which is strictly weaker than the weak separation condition of Lau and Ngai. By assuming the bounded distortion property, we show that the W*SC is equivalent to the identity limit criterion of Zerner for such IFSs. In particular, in the one-dimensional case, we show that the W*SC is also equivalent to Ahlfors–David regularity, the Assouad dimension is strictly less than $1$ and positivity of the $s$-dimensional Hausdorff measure, where $s$ is the zero of the topological pressure.

Decoding of the coupling among the brain and heart activations is an important research area in network physiology. We studied the coupling of brain and heart activations for 48 subjects who performed the NASA Revised Multi-Attribute Task Battery II under three different activity level conditions. During the experiment, the physical activity of subjects was manipulated by changing the speed of a stationary bike (including no movement, 50rpm, and 70rpm) or a treadmill (including no movement, 3km/h, and 5km/h), while their physiological signals were recorded. We analyzed the complex structure of electroencephalogram (EEG) and R-R signals using fractal theory and sample entropy. The results demonstrated that the alterations of the complex structures of EEG and R-R signals are strongly correlated, which indicates the coupling between brain and heart activations. This method of analysis can be applied to evaluate the coupling between different organs.

We investigated the correlation among brain and leg muscle activations by analyzing Electroencephalogram (EEG) and Electromyogram (EMG) signals in different conditions. Twelve subjects performed four tasks, including (1) quarter turns, (2) U-turns, (3) bypass obstacles, and (4) repeating quarter turns and U-turns two times. Then, we quantified the alterations of the complexity of these signals by computing the fractal dimension and sample entropy. The results showed that EEG and EMG signals in the case of the first task are more complex than the second task, in which they are more complex than the third task. Furthermore, the brain and muscle signals show the least complexity in the case of the fourth task. Moreover, we found strong correlations in the variations of fractal dimension ($r=0.9835$) and sample entropy ($r=0.9168$) between EEG and EMG signals in various tasks. Therefore, brain and muscle activations are strongly correlated in different tasks. Similar analyses can be conducted in the case of other organs to decode their correlations.