Fractals

In this paper, we investigated the impact of the COVID-19 pandemic on the cryptocurrency market. The direction of information transfer between the 67 digital cryptocurrency markets was evaluated, in particular Bitcoin, Ethereum, Litecoin, and Ripple, and we determined which of them were the most influential in the markets. The comparison of the first half of 2019 (outside the pandemic of COVID-19) against the first semester of 2020 (during the COVID-19 pandemic) was used to analyze the pandemic influence. We found two distinct behaviors: (i) in 2019, Bitcoin, as the primary capitalization bond, presented a more substantial transfer of information than the other cryptocurrencies toward Bitcoin $\to$ Ripple (0.0541), followed by Litecoin $\to$ Ripple (0.0522); (ii) in 2020, the most substantial transfers of information occurred from Ethereum to other cryptocurrencies (Litecoin, Bitcoin, and Ripple, in that order). In this period, the weakest transfers happened from Litecoin $\to$ Ripple and in the opposite direction, with equal value (0.0104). Our results indicate that there was a change in the direction of the information flow between the investigated cryptocurrencies, where ETH became the dominant cryptocurrency during the period of turbulence caused by the COVID-19 pandemic.

This study evaluates the Brazilian agricultural commodities market and the dollar–real exchange price variation using the multifractal detrended fluctuations analysis methodology. We investigated the period from January 1, 2019 to September 25, 2019, outside the COVID-19 pandemic, and from January 1, 2020 to September 25, 2020, during the COVID-19 pandemic. We verified the fluctuations of commodities and dollar–real exchange prices during the pandemic caused by COVID-19 showed a record price. The results of Hurst exponent and multifractal parameters ${\alpha }_0$, $w$, and $r$ indicate that during the COVID-19 pandemic, sugar was the most efficient commodity, while pork the less one. Compared to the identical months in 2019, the dollar–real exchange was the most efficient market, while ethanol was the least efficient.

In this work, the Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model is described by the local fractional derivative (LFD) on Cantor sets. A novel algorithm is presented to seek the exact traveling wave solution of the nondifferentiable type for the local fractional Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model based on the variational theory, which is called variational wave transform method (VWTM). This new algorithm provides a new idea for seeking the exact traveling wave solutions in fractal space with simplicity and efficiency. The physical properties of traveling wave solutions are described by some 3D simulation figures.

In this work, we derive a new fractional $(3+1)$-dimensional Jimbo–Miwa equation via the local fractional derivative. With the help of the traveling wave transform of the non-differentiable type, the local fractional $(3+1)$-dimensional Jimbo–Miwa equation is converted into a nonlinear local fractional ordinary differential equation. Then, a new method named as Mittag-Leffler function-based method is used to construct the exact traveling wave solutions for the first time. Four families (eight sets) of the traveling wave solutions are obtained and the behaviors of the solutions on Cantor sets are presented via the 3D plot. The results show that the proposed method is a powerful tool to study the traveling wave theory of the local fractional equations.

A fractal interpolation function (FIF) is a special type of continuous function defined on a compact interval $I$ of $ℝ$ which interpolates a certain data set and whose graph is of fractal nature. But an $\alpha$-fractal (interpolation) function is a special type of FIF which is a fractal analogue corresponding to any continuous function $f$ defined on the interval $I$. In this paper, the definite integral of the $\alpha$-fractal function $f^{\alpha }$ corresponding to any continuous function $f$ on the interval $I$ is estimated although there is no explicit form of $\alpha$-fractal function till now. Some results related to the definite integral of $f^{\alpha }$ are established. Also, the flipped $\alpha$-fractal function $f_F^{{\alpha }_F}$ corresponding to the continuous function $f$ is constructed and a result is proved that relates the definite integrals of the fractal functions $f_F^{{\alpha }_F}$ and $f^{\alpha }$.

Tree bifurcation network has always been a hot issue concerning scientists all over the world. However, the research on the heat conduction properties of damaged tree-like bifurcation networks is insufficient, and many issues remain unsolved. Therefore, this paper systematically studies the heat conduction phenomenon of a new damaged tree-like bifurcation network with fractal roughened surfaces (FRS), and derives the equivalent thermal conductivity of damaged network. According to the study, the dimensionless thermal conductivity (DTC) shows a downward trend with an increase in roughness. In addition, it can be seen that an increase in the number of damaged channels leads to a downward trend in the optimal thermal conductivity, but a slow upward trend in the critical diameter. Another interesting phenomenon is that the factors leading to the change in the optimal thermal conductivity do not include the total number of bifurcation levels, bifurcation number or the fractal dimension of length distribution. The model established in this paper does not contain any empirical constants to ensure that each parameter has its physical significance, revealing the heat transfer mechanism of tree-like bifurcation network more profoundly.

The substitution network is a model of evolving self-similar networks. Using the technique of finite pattern, we obtain the asymptotic formula of the average Fermat distances on two-colored substitution networks.

This paper investigates certain new weighted Young- and Pólya–Szegö-type inequalities for unified fractional integral operators via an extended generalized Mittag-Leffler function. A large quantity of usable classical inequalities in the literature are included in the main results of this paper. Meanwhile, two types of new generalized weighted fractional integral operators are introduced to establish some new weighted Young- and Pólya–Szegö-type inequalities. As applications, several estimates of Chebyshev-type weighted unified fractional integral inequalities with two unknown functions are obtained by employing the Heaviside unit step function. Finally, some relations between main results and known inequalities for different kinds of fractional integral operators are provided.

In this paper, the bidirectional wave model is described by using the local fractional derivative (LFD) on Cantor sets for the first time. A novel algorithm is established to obtain the exact traveling-wave solution of the non-differential type for the local fractional bidirectional wave model (LFBWM), which is called the local fractional wave method (LFWM). The advantages of the LFWM are simple, efficient and accurate. The LFWM sheds a new light on solving the local fractional wave equations (LFWE) in physics and engineering. Finally, the physical properties of the obtained exact traveling-wave solution of the non-differential type are elaborated by some simulation figures.

We construct a family of evolving networks modeled by some fractal carpet which is not symmetric. We investigate the scale-free effect and small-word effect of our fractal networks, including the power-law of the cumulative degree distribution, the average clustering coefficient and the average geodesic distance.

In this paper, we present the weighted iterated friendship graphs and study the trapping problem on the weighted iterated friendship graphs. It can be found that for $0<r<1$ and $r=1$, the relationship between the average trapping time (ATT) and network size is sublinear and linear, respectively. By controlling the parameters of the weighted iterated friendship graphs, the models are changed to the self-similar weighted networks. The average shortest weighted path (ASWP) in the self-similar weighted friendship graphs is studied. The results show that when $0<r<1$, the ASWP is bounded, and when $r=1$, the ASWP is linearly related to the order of the networks.

Fractals play an important role in nonlinear science. The most important parameter when modeling a fractal is the fractal dimension. Existing information dimension can calculate the dimension of probability distribution. However, calculating the fractal dimension given a mass function, which is the generalization of probability, is still an open problem of immense interest. The main contribution of this work is to propose an information fractal dimension of mass function. Numerical examples are given to show the effectiveness of our proposed dimension. We discover an important property in that the dimension of mass function with the maximum Deng entropy is $\frac{ln3}{ln2}\approx 1.585$, which is the well-known fractal dimension of Sierpiski triangle. The application in complexity analysis of time series illustrates the effectiveness of our method.

In this paper, we prove the existence of a common fixed point for generalized weakly Zamfirescu-type mappings. Finally, we give an application to our main result in the fractal space which proves the existence of a common attractor for a new kind of deterministic fractal.

This paper derives a new fractional Fokas system with the aid of the local fractional derivative. A novel technology named the extended rational fractal sine–cosine method is presented for the first time ever to develop the abundant traveling wave solutions. Two families (six sets) of the traveling wave solutions on Cantor set are successfully constructed. The dynamic behaviors of the solutions on Cantor sets are provided via numerical simulations. The obtained results in this work strongly prove that the proposed approach is simple but effective, which is expected to shed a new light on the study of the traveling wave theory of the local fractional equations.

In this paper, we mainly investigate continuous functions with unbounded variation on closed intervals. Given the increasing number of proposals and definitions of different kinds of functions with fractal structure in the scope of fractal functions, it is important to introduce a systematic classification and discuss elementary definitions of fractal functions. Different examples of fractal functions with certain fractal dimensions have been given. Then, we have introduced different kind of fractal function such as singular fractal function, irregular fractal function, regular fractal function and complete regular fractal function. We also introduce fractal functions of integer dimension with one and two, respectively. Other continuous and discontinuous functions with fractal structure have been given. Applications of fractal functions in interpolation, approximation and relationship between fractional calculus have been investigated. Finally, we studied fractal dimension spaces with different fractal dimensions.

In this paper, first, the structure of social networks was investigated from fractal perspective and the fractal network was built from online social networks. Then inspired by the flock effect, the characteristics of “separation”, “cohesion” and “alignment” in information propagation were studied from the inherent granularity structure of fractal communities. Accordingly, the agreement degree and controversy degree between community members at different granularities were defined. And the “Seeking Common Ground While Shelving Differences” propagation rule in fractal network was presented. On this basis, a new S2I propagation model was proposed. From this model, communicators were divided into two states, I and OI. People in status I have a tendency to express support for a message. While those in state OI have a tendency to oppose it. The information propagation process and characteristics were analyzed further by using S2I model. Finally, the effectiveness of S2I model was verified by comparative experiments of simulation and real data analysis.

A novel analytical model is derived based on fractal geometry theory to characterize the permeability of power-law fluids through fractured porous media with fractal asperities on surfaces. The proposed model is expressed as a function of structure parameters of fractured porous media, fractal dimensions for porous matrix and for fracture as well as the index $n$ of power-law fluid behavior. The influences of these parameters on the permeability are analyzed in detail. It is found that the relationship between permeability and roughness exhibits nonlinear characteristics. A higher power-law fluid behavior index significantly affects the reduction rate of permeability with relative roughness. Moreover, permeability is sensitive to the perturbation due to the rate of aperture to length. The proposed fractal model can clearly reveal more physical mechanisms for power-law fluid in fractured porous media than the existing models. Compared with the experimental data available in the literature, the validity of model predictions is verified.

In this paper, we propose the multifractal detrending fluctuation analysis based on $L_p$-norm constrain (MF-DFN). We assess the performance of the proposed algorithm by constructing a $p$-model based multiplicative cascades time series. We calculate the generalized $Hurst$ exponent $H(q)$, $Rényi$ exponent $\tau (q)$, singularity exponent $\alpha$ and multifractal spectrum $f(\alpha )$ for multifractal detrended fluctuation analysis (MF-DFA) and MF-DFN, respectively. We notice that under different norm constraints, the distribution of multifractal characteristics is quite different. Appropriate norm constraints can make the multifractal characteristics of time series described more accurately. Based on the analytical solution curve, the distribution of different multifractal numerical curves determines the correct selection of norm constraint value. In this study, using a combination of norms $0.01$ and $4$, the depicted numerical curves almost overlap the theoretical curve, which shows that the proposed MF-DFN is superior to MF-DFA. To eliminate the influence of specific time series, we reset parameters $n_{max}$ and $P_1$ in $p$-model. Various experimental results also show the effectiveness of the proposed MF-DFN. In addition, we also verify the efficiency of the proposed MF-DFN by using the practical application. The experimental results show that our proposed method plays a significant role in ECG classification.

In this research, we study some algebraic and topological properties of generalized $m$-convex functions on fractal sets and establish a new Jensen type inequality. Also, we obtain some new Hermite–Hadamard type inequalities for local fractional integrals of this class of functions. As applications, we present some inequalities with generalized special means and generalized probability density functions.

In comparison with numerous available results with regard to bifurcations on account of fractional-order neural networks (FONNs) with the retarded delays, this paper is concerned with the stability and bifurcation of FONN with neutral delays. Delay-induced bifurcation results are firstly, procured and the bifurcation diagrams with respect to different delays are nicely plotted. Then it reveals that both time delay and fractional order have important effects in affecting the stability and bifurcation of the developed FONN. It further detects that FONN involving neutral delays can postpone the onset of bifurcations of the developed FONN compared with FONN with the retarded delays. It hints that neutral delays have a vital impact on the dynamics of FONNs in comparison with FONN with the retarded delays, which has outstanding performance in maintaining stability of FONNs. Eventually, numerical examples are addressed to check the availability of the developed results. The obtained criterion can be evaluated as an important contribution to the stability issue of the considered neural networks (NNs).

The Sharma–Tasso–Olver equation can well describe the wave motion in physics, however, it becomes ineffective when the boundary is non-smooth, so a modification of the equation is urgently needed. In this study, we derive a new fractal Sharma–Tasso–Olver equation that can model the wave motion with the non-smooth boundary by applying He’s fractal derivative. By means of the semi-inverse method, we successfully establish its fractal generalized variational principle, which provides the conservation laws in an energy form in the fractal space and reveals the possible solution structures of the equation. The obtained generalized variational principle can be used for the numerical and analytical studies of the solitary wave properties in the fractal PDEs.

The cooling curves of some natural and artificial systems show a series of anomalies (such as sudden drops in temperature or non-homogeneous behavior) that can hardly be modeled with simple laws covering their entire duration. Via three algorithms, we developed a procedure which consists of a stepwise fitting of a fractional Newtonian law of cooling model to empirical curves that show such anomalies. We used the Caputo fractional derivative to construct models for phenomena which show such anomalies — Mpemba effect and the cooling performance of household utensils — which we present in detail. In this manner, we were able to construct continuous functions that represent quite well the experimental data, as opposed to the functions commonly obtained by an integer-order model of Newton’s cooling law. Although these piecewise models are purely empirical descriptions, they can be predictive tools of great interest to product designers in the food industry and might provide elements to gain a deeper understanding of an unintelligible phenomenon such as the Mpemba effect.

We study Bohr’s formula for fractal Schrödinger operators on the half line. These fractal Schrödinger operators are defined by fractal measures with overlaps and a locally bounded potential that tends to infinity. We first derive an analog of Bohr’s formula for these Schrödinger operators under some suitable conditions. Then we demonstrate how this result can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, m-fold convolution of Cantor-type measures, and a family of graph-directed self-similar measures that are essentially of finite type.

This paper proposes a method for the fractal characterization on fracture volume in coal based on CT scanning experiment, and derives the relation among the fractal dimension $D$, fracture volume and length scale. The principle and methodology were deduced in detail, and the rationality was examined by the classical Mandelbrot fractal equation on fractal structure in porous media. The results demonstrate fracture volume is the comprehensive embodiment of the height, length, and aperture of fracture. Consequently, the estimated fractal dimension $D$ can more comprehensively symbolize the fractal characteristic on fracture volume of coal in three-dimensional space than the fractal dimension for fracture number. Therefore, it is worth to be further studied on establishing the analytical equation for fluids transport characteristic parameters in porous media based on $D$.

It is the purpose of this paper that uniform continuity of fractional order integral of fractal interpolation function (FIF) is researched. First, FIF generated by affine transformation is constructed. Second, the combination between Riemann–Liouville fractional order integral and FIF is defined. Finally, the continuity and the uniform continuity of fractional order integral of FIF are proved by compactness theorem of numbers sequence or finite covering theorem. The result shows that fractional order integral of FIF is uniformly continuous on a closed interval.

A new method to extract image features and to evaluate fractality is proposed based on two-dimensional (2D) continuous wavelet transform (CWT). Multi-directional 2D CWT coefficients are used to determine the direction and magnitude of image intensity gradients directly unlike other methods using gradient components in the horizontal and vertical directions. Image feature points are detected by comparing candidate directional 2D CWT coefficients at candidate points and their neighbors instead of gradient magnitudes or 2D CWT moduli used by traditional methods. It enables us to extract multiscale image features including line singularities such as corners which are recognized to be hardly extracted by traditional methods. This offers an advantage condition to study fractal objects consisting of lots of line singularities. It is suggested that the detected multiscale image features can reflect multiscale fractal measures to be used to evaluate fractality, that is, self-similarity across scales. From this suggestion, the method to evaluate fractality and calculate fractal dimensions using multiscale image features is proposed. The proposed method is applied to theoretical fractal models to show that the method is convenient and effective in extracting image features of them consisting of many line singularities and calculating their fractal dimensions. Finally, the method is validated using a case study dealing with fractality evaluation of geoscientific objects such as coastlines and stream networks from the digital elevation model (DEM) data.

For the family of rational maps $F_{\lambda }(z)=z^n+\lambda /z^n$ where $n\ge 2,$ it is known that there are infinitely many Mandelpiński necklaces ${{𝒮}}^k$ with $k\in \mathbb{N}$ around the McMullen domain surrounding the origin in the parameter $\lambda$-plane. In this paper, we prove the existence of infinitely many these rings with a number of ${{𝒮}}^k$ for fixed $k$ outside the Mandelpiński necklace ${{𝒮}}^1$. The ring ${{𝒮}}^2$ is a simple closed curve meeting ${{𝒮}}^1$ at $n-1$ points, such that it passes through exactly $n^2-1$ centers of Sierpinski holes and $n(n-1)$ superstable parameter values. For each $k\ge 3$, ${{𝒮}}^k$ passes through precisely alternating $2n$ superstable parameter values and the same number of centers of Sierpiński holes. There exist $(n-1)$ disjoint rings ${{𝒮}}^{k+1}$ not meeting ${{𝒮}}^{k-1}$ and surrounding the centers of Sierpiński holes lying on ${{𝒮}}^k$, in the exterior and interior of a curve ${{𝒮}}^{k-1}$, respectively. The number of such rings ${{𝒮}}^k$ for fixed $k$ is $2^{k-3}{(n-1)}^{k-1}$.

Initial Crypto-Token Offering (ICO) strategy has worked so far to fund any project. In this scheme, the ICO launches a token playing the role of a stockholder’s share; the more the people buy tokens, the more the funding for the projects. The stakeholders also earn money if the project gains more reliability and more people buy this token. In other words, an ICO strategy works based on either selling or buying of tokens in the areas of security, utility, equity, etc. However, this paper’s focus is only on using oracles and Hyperledger, which means that there is no need to launch a token, instead it uses the network blockchain benefits, particularly oracles and the Ethereum virtual machine, i.e. not the Ethereum blockchain platform but its virtual machine. Since this is a proposal for governmental or corporate usage, the Hyperledger and oracles strategy fits better for this application. Funds for this use-case are gathered in the healthcare field, specifically to reduce the COVID-19 pandemic. The system’s reliability is the core for attracting investors and donors to fund the system if it guarantees that resources will get to the right destiny. DAPP is based on a smart contract aligned with the smart societies concept to ensure system sustainability.

Our facial muscles react due to exposure to various stimuli. For example, when we sniff an odor, our facial muscles react. Since the brain controls the activities of muscles, therefore, a coupling should exist between their activities. This paper studies the coupling between the activations of the brain and facial muscles in auditory stimulation by the complexity-based analysis of Electroencephalogram (EEG) and Electromyogram (EMG) signals. Three pieces of music were chosen according to the differences in their complexities. We calculated the fractal dimension and sample entropy of EEG and EMG signals for 13 subjects in rest and response to different music. The results showed strong couplings among the alterations of the complexities of these signals and music, which indicate the correlation between brain and facial muscle activities. This complexity-based analysis could be further extended to the case of other physiological signals to decode the correlation between the responses of different organs and the brain.