Fractals

This paper investigates fractal dimension of product of continuous functions with Box dimension on $[0,1]$. For two continuous functions with different Box dimensions, the Box dimension of their product has been proved to be the larger one. Furthermore, the Box dimension of product of two continuous functions with the same Box dimension may not exist. Definitions of regular fractal and local fractal functions have been given. Product of a regular fractal function and a local fractal function with the same Box dimension must still be the original Box dimension.

Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a graph product, which is an extension of the pseudo-fractal scale-free web. We determine explicitly the independence number, the domination number, and the chromatic number. Moreover, we derive closed-form expressions for the number of acyclic orientations, the number of root-connected acyclic orientations, the number of spanning trees, as well as the number of perfect matchings for some particular cases.

The $(2+1)$-dimensional Boussinesq equation plays a key role in modeling the shallow water. In this work, we derive a new fractional $(2+1)$-dimensional Boussinesq equation based on the conformable fractional derivative for the first time. By means of the Hirota bilinear method, we obtain the $N$-soliton, breather and lump solutions. In addition, the abundant traveling wave solutions like bright solitary, dark solitary wave solutions are investigated by applying the variational method. The solutions are presented through the 3D plots and 2D contours by assigning the proper parameters. The corresponding physical interpretations are also elaborated. The findings in this work are expected to open some new horizons on the study of fractional PDEs in physics.

First, we establish the parametrized integral identity and its improved version via Atangana–Baleanu (AB) fractional integrals. For the focus of this paper, we utilize the resulting identities to derive a series of Simpson-like integral inequalities for mappings whose second-order derivatives belong to the $(s,P)$-convexity and $(s,P)$-concavity in absolute value. And a couple of outcomes, concerning the Simpson-like quadrature formulas, the $q$-digamma functions and the modified Bessel functions, are introduced as applications separately in the end.

This work suggests a fractional modification of the KdV–Kadomtsev–Petviashvili model with the beta-derivative to consider unsmooth boundary. Some new interesting solitary waves are found for the first time ever by the fractional sine–cosine method and the fractional ansatz method. These dynamical characteristics of new solitary waves are discussed by some three-dimensional (3D) figures, and the effect of the fractal parameters on the solitary waves traveling is also discussed and explained.

In this work, we used hourly data of high frequency of solar radiation from the entire Northeast region of Brazil. We used the Multifractal Detrended Fluctuation Analysis (MFDFA) method to analyze the characteristics of the solar radiation series in 137 meteorological stations from 2010 to 2022. For all analyzed series, the parameter ${\alpha }_0>0.5$ characterizes persistent series. The values of $r>1$ reveal asymmetry to the right, indicating that large fluctuations contributed to the multifractality process. The states of Maranhão and Bahia presented the highest values of spectrum width $W$, indicating greater complexity. We found that long-range correlations are the leading cause of multifractality observed in the dynamics of the series of solar radiation anomalies.

We take the novel Twitter-based economic uncertainty (TEU) to examine if it has cross-correlation characteristics with four major cryptocurrencies i.e. Bitcoin, Ethereum, Litecoin, and Ripple. To conduct a more thorough analysis, we apply multifractal detrended cross-correlation analysis (MFDCCA) on seasonal-trend decomposition using Loess (STL) decomposed series as well as without decomposed series on the daily data, ranging from 1 June 2011 to 30 June 2021. The findings of this study indicate that: (i) all pairs of TEU with cryptocurrencies are multifractal and have power-law behavior; (ii) the pairs of Ethereum and Bitcoin with TEU are found to be the most multifractal while Litecoin with TEU has the lowest multifractal characteristics; (iii) all STL decomposed series of cryptocurrency have persistent cross-correlation with TEU with the exception of Ethereum which has anti-persistent cross-correlation with TEU; (iv) all without decomposed series of cryptocurrencies show significant persistent cross-correlation characteristics with TEU; (v) the highest linkage is found for the pair of Bitcoin with TEU. Moreover, to reveal the dynamic characteristics in the cross-correlation of TEU with cryptocurrencies, the rolling window is employed for MFDCCA. These findings have important managerial and academic implications for policymakers, investors, and market participants.

In this study, a new fractional Zakharov–Kuznetsov equation (ZKE) within the local fractional derivative (LFD) is derived. Yang’s non-differentiable (ND) traveling wave transform is introduced, then two novel techniques namely the Mittag-Leffler function-based method (MLFBM) and Yang’s special function method (Y-SFM) are adopted to seek for the ND exact solutions for the first time. With the aid of the Mathematica software, the dynamic behaviors of the different solutions on the Cantor sets are illustrated via the 3D plots by assigning the appropriate parameters. The attained results confirm that the mentioned methods are effective and straightforward, which can be used to study the ND exact solutions of the local fractional partial differential equations (PDEs).

The nonlinear fractional Zoomeron equation with $M$-truncated fractional derivative, which is widely used in physics and engineering, is the major subject of this study. Using the fractional functional variable approach and fractional variational method, we effectively derived several new soliton solutions to the nonlinear fractional Zoomeron equation. The two concepts that are suggested can be used to quickly and effectively find the solutions to the same kinds of fractional evolution equations. Finally, various 3D and 2D simulation figures are plotted to show the physical characteristics of these acquired soliton solutions.

Behavior analysis of the neutron point kinetic model with Caputo and conformable derivatives (Khalil and Almeida operators) was performed. Hence, boundary thresholds that delimit the stability/instability zones within the anomalous diffusion exponent-reactivity parameter space were found. Stability criteria are established to limit the region of the values of the anomalous diffusion coefficient and reactivity parameters with which the oscillatory behavior of the neutron density does not exceed a value greater than 30% with respect to the value of the classical model. The parameter space map corresponding to the model with Caputo derivative shows a larger stability behavior zone than that obtained with the Khalil derivative defined in terms of a linear kernel. In a more general sense, the Almeida operator allows one to freely define the kernel function. A kernel of exponential type produces instabilities of different nature (significant increase in neutron density followed by a series of decreasing oscillations few moments after the start-up, or a rapid growth in neutron density resembling a Gaussian pulse appearing seconds after the start-up), as well as stability/instability zones of different shapes and sizes as the parameters in the kernel vary. Interestingly, it was possible to reduce the instability behavior zone with the exponential kernel and approximate its size with that of the zone predicted with the Caputo derivative.

In this research, we apply complexity-based techniques to study the activations of the brain while the subjects perform different types of locomotion, including walking, jogging, and running. Therefore, we can study the effect of locomotion speed (or toughness level) on brain’s reactions. For this purpose, we analyzed the fractal dimension and approximate entropy of electroencephalogram (EEG) signals recorded from subjects while they walked, jogged, and ran for 20 s in the case of each activity. The analysis of 21 recorded samples showed that the complexity of EEG signals increases by increasing the locomotion speed. This result indicates a higher level of processing in the brain while the subjects perform a harder task. This analysis can be extended to the case of other physiological signals to study the effect of the level of exercise on different organs’ activations.

In this research, we investigated the effect of changes in walking speed on variations of the complexity of electromyogram (EMG) signals recorded from the right and left legs of subjects. We specifically employed fractal theory and approximate entropy to analyze the changes in the complexity of EMG signals recorded from 13 subjects walked at 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 km/h on a flat surface. The results showed that by increasing of walking speed, the complexity of EMG signals decreases. The statistical analysis also indicated the significant effect of variations in walking speed on the variations of the complexity of EMG signals. This method analysis can be applied to other physiological signals of humans (e.g. electroencephalogram (EEG) signals) to investigate the effect of walking speed on other organs’ activations (e.g. brain).

Analysis of the brain activity to external stimulation is an important area of research in biomedical engineering. In this paper, for the first time, we analyzed the brain reaction to visual stimuli with different frequencies using three complexity methods. For this purpose, we utilized fractal theory, sample entropy, and approximate entropy to study the variations of the complexity EEG signals while subjects received visual stimuli at 7, 9, 11, and 13 Hz. The results showed that, in general, by moving from 9 Hz to 13 Hz stimuli, the complexity of EEG signals increases, except in the case of 11 Hz stimulus. The statistical analysis also supported the results of the analysis. The conducted analysis in this research can be performed in the case of other types of external stimuli to study how the brain reacts in different conditions.