Fractals

Detection and maintenance of wind turbine blades are essential, as they are constantly exposed to a hostile environment and are easily damaged. Defective repairs, lightning damage, and damaged dust guards are the most common faults found in our database. These faults decrease the performance of the wind generator. Although visual site inspections are common, they are inefficient due to long downtime periods. This document proposes a systematically designed expert system for the classification of visual faults from a database of typical faults in a wind farm in the Isthmus of Tehuantepec region, México. Convolutional neural networks are used for this purpose.

This study presents numerical simulations on fluid flow under high stress and intense hydraulic activity and focuses on complex reservoirs with porous and fractured microstructures based on fractal geometry theory. A Triple-Fracture Network model, which includes pores, micro-fractures and macro-fractures, is categorized into fractal fracture networks and fractal pore systems based on fractal geometry theory. Using the Monte Carlo method, we construct macroscopic randomly distributed fractures to examine the impact of macro-fracture topological structures on flow dynamics. Fractal permeability models for micro-fracture and micro-pore systems are developed, incorporating variables such as fracture lengths and pore sizes. The proposed approach partitions porous media into macroscopically discrete fractures and microscopically characterized pores, utilizing fractal geometry theory to analyze hydro-mechanics coupling. The validity of the proposed model is confirmed through comparisons with analytical solutions and available models as well as the published field data. This results indicate that the spatial distribution of the effective stress and macroscopic fractures significantly affects the fractal dimensions of microscopic fractures and pores. It is found that the density of macro-fractures influences the fracture pressure, while the initial porosity and proportionality constant $\beta$ of micro-fractures impact the pressure distributions in the tested domains. It is found that the maximum length of micro-fractures correlates positively with permeability. This model introduces a novel perspective on random macro-fractures and fluid coupling in microstructures, enhancing the understanding of macro- and micro-hydrodynamics interactions.

The extremely low porosity and permeability in tight reservoirs make diffusion effects significant and non-negligible. Accurately estimating diffusion coefficients is critical for predicting CO₂-enhanced oil recovery and CO₂ storage (CCUS–EOR) in such reservoirs. However, predicting effective vapor–liquid phase diffusion in porous media and incorporating it into large-scale compositional simulations remains a major challenge. This study combines fractal theory with Fick and Maxwell–Stefan diffusion frameworks, proposing a simplified fractal diffusion model that incorporates fractal parameters and porosity to better characterize CO₂ and in situ oil diffusion in heterogeneous tight reservoirs. Numerical models were developed using MATLAB to simulate both immiscible and miscible CO₂ flooding within multiphase, multicomponent flow equations while considering diffusion effects. The results reveal that diffusion significantly increases the area influenced by CO₂ and improves the uniformity of CO₂ concentration along the flow path. Confined phase behavior in micro- and nanopores enhances diffusion, increasing CO₂ mole fractions in displacement zones. In immiscible flooding, CO₂ extraction of lighter oil components results in higher residual oil density compared to models that neglect diffusion. This fractal-based model for heterogeneous effective diffusion, combined with insights into the role of diffusion in CO₂ flooding in tight reservoirs, offers important implications for developing more effective CCUS–EOR strategies.

We innovatively introduce a new group of integral operators called multiplicative $k$-Atangana–Baleanu fractional integrals. Following the path of this research, we discuss their ${}^{\ast }$integrability, continuity, boundedness together with linearity, and successfully construct two Hermite–Hadamard–Mercer inequalities employing this multiplicative fractional integrals, focusing on endpoint- and midpoint-type, respectively. Subsequently, we further propose three identities, involving trapezoid-Mercer-, midpoint-Mercer-, and parameterized-Mercer-type, and combining the function ${\mathrm{\Lambda }}^{\ast }$ is multiplicatively convex or, for some fixed $q>1$, the function ${(\mathrm{\text{ln}}{\mathrm{\Lambda }}^{\ast })}^q$ is convex, and we formulate a series of corresponding fractional Mercer-type inequalities. Intended to enhance the readers’ deeper understanding of the results, we offer two instances accompanied by graphical representations, facilitating the validity of the inequalities gained in the current paper. Finally, some applications in multiplicative differential equation, the quadrature formula and special means are presented as well.

Higher-dimensional nonlinear models can describe more complex evolutionary mechanisms. In this paper, we considered the time fractional $(3+1)$-dimensional modified extended Zakharov–Kuznetsov equation with the sense of the Riemann–Liouville fractional derivative in plasma physics. In the first place, the existence of symmetry of this studied equation through the symmetry scheme was proved. Then, the optimal system to the time fractional $(3+1)$-dimensional modified extended Zakharov–Kuznetsov equation was also constructed. Subsequently, the time fractional higher-dimensional equation was reduced into the lower-dimensional fractional differential equation with the help of the Erdélyi–Kober fractional operators. Last, some conservation laws by using a new conservation theorem were also given. These novel results provide a window for us to discover this high-dimensional nonlinear equation.

A Caputo fractional-order derivative is used in a Quadrotor Unmanned Aerial Vehicle (QUAV) chaotic system to transform it into fractional ordered QUAV (FoQUAV) chaotic system for the first time. Therefore, this study explores three control (adaptive, active, and passive) strategies applied on the FoQUAV system. Further, utilizing Lyapunov theory, we established global stability around the equilibrium point and conducted numerical simulations for validation and comparison. In this work, adaptive control technique is used by considering $a_f$, $a_g$, and $a_h$ as unknown parameters and are derived using updated law. Similarly, active control technique is derived using two sub-controllers including ${\widehat{\mu }}_i=A_i+N_i$ with $A_i$; linear and $N_i$; nonlinear controllers, respectively. Finally, a feedback linearization-based, passive controller is used, which has predefined control input and converges to a stable equilibria. Moreover, this work also revealed, during performing numerical simulations, that the open loop system stabilizes at times $t\approx 4$ s, $t\approx 60$ s, and $t\approx 90$ s, respectively, for adaptive, active, and passive controllers. Finally, the passive controller face several challenges during stabilizing the yaw of a considered system and overall is proved that adaptive controller perform best with respect to time consumption and quick performance in comparison.

In 2023, Xi et al. introduced the Hausdorff dimension of a family of networks which inspired by the potential theoretic methods in fractal geometry. In this paper, we will construct a class of colored substitution networks and obtain its Hausdorff dimension using the self-similarity.

Network classification plays a crucial role in various domains like social network analysis and bioinformatics. While Graph Neural Networks (GNNs) have achieved significant success, they struggle with the problem of over-smoothing and capturing global information. Additionally, GNNs require a large amount of data, hindering performance on small datasets. To address these limitations, we propose a novel approach utilizing Deng’s entropy, capturing network topology and node/edge information. This entropy is calculated at multiple scales, resulting in an entropy sequence that incorporates both local and global features. We embed the networks by combining the entropy sequences for edges and nodes into a matrix, which then are fed into a bidirectional long short-term memory network to perform network classification. Our method outperforms GNNs in the bioinformatics, social, and molecule domains, achieving superior classification power on nine out of eleven benchmark datasets. Further experiments with both real-world and synthetic datasets highlight its exceptional performance, achieving an accuracy of 97.24% on real-world complex networks and 100% on synthetic complex networks. Additionally, our approach proves effective on datasets with a small number of networks and unbalanced classes and excels at distinguishing between synthetic and real-world networks.

The study of correlations between different organs under various conditions is a prominent field in biomedical science and engineering. This paper explores the relationship between brain and muscle activities during rest and various limb movements including plantar flexion and knee flexion. We employed complexity measures, calculating the fractal dimension (FD) and sample entropy (SampEn) of electroencephalogram (EEG) and electromyogram (EMG) signals, which serve as indicators of brain and muscle activities, respectively. Our analysis focused on how the complexity variations in these signals correlate across different tasks. The results revealed opposite trends in the complexity of EEG and EMG signals. Specifically, the complexity of EEG signals increased from initial rest to final rest, plantar flexion, and knee flexion, suggesting heightened neural activity likely due to motor planning and execution. Conversely, the complexity of EMG signals decreased, indicating more synchronized and consistent muscle contractions during these movements, reflecting coordinated motor control and reduced variability in muscular activity. This analytical approach can be applied to study the correlations between different organs’ reactions and brain activity across various tasks.

Analyzing the skin’s response to different external conditions is a significant area of research in biomedical engineering. This paper examines the variations in skin activity across different emotional states. We employed the concepts of complexity, calculating the approximate entropy (ApEn) and sample entropy (SampEn) of galvanic skin response (GSR) signals from 12 subjects who were exposed to 21 stimulus videos (three sessions), in total, categorized under six basic emotions: surprise, anger, fear, happiness, sadness, disgust, and an additional neutral state. The results demonstrated significant variations in the complexity of GSR signals across these emotional states. The methodology used in this study can also be extended to analyze other physiological signals, such as EEG, EMG, and ECG, to potentially understand the responses of various organs like the brain, muscles, and heart to different emotional stimuli.

By blending the Internet of Things (IoT) into artificial intelligence and blockchain with social issues (decentralized IoT for smart society SSIoTX), education can become more intelligent, accessible, and personalized. Such technologies customize learning for the needs of each learner, allowing progress to be monitored in real-time and modifying the content per cognitive profile determined by cognitive skills. Furthermore, AI, together with SSIoTX, can be of great help to students who have some disability by giving them some sensor devices that make them even more accessible in their education and integrated into classrooms. Additionally, they drive student collaboration and allow teachers to assess how their lesson creates teaching outcomes. SSIoTX can tackle education disparities and make school resources more efficient, contributing to learning environments that are even better and healthier. In summary, it is far from the perspective that blockchain, IoT, and AI technologies have a high social impact since they are bringing education to more and making it inclusive. The paper’s core is developing a deep convolution learning model (YOLO V7) to recognize the cognitive profile patterns based on cognitive skills quizzes on an SSIoTX ecosystem. Cognitive profiles are processing information, problem-solving, and decision-making. Cognitive skills are memory, attention, reasoning, and social cognition. Since the SSIoTX expert system is on the blockchain, a decentralized application (DAPP) has been built in IoTX.

Understanding the correlation between the brain’s activity and the physiological responses of other organs under varying conditions is a crucial area of research that holds significant potential for advancing our knowledge of human physiology. In this study, we focused on investigating the interaction between the heart and brain by employing advanced complexity analysis techniques, specifically examining the fractal dimension and approximate entropy of electroencephalogram (EEG) and R-R interval time series. The analysis was conducted on data collected from 12 subjects who were observed under three distinct conditions: baseline (normal resting state) and two collaborative activities performed both with and without the presence of noise. Our findings revealed that the complexity patterns of EEG and R-R signals showed similar trends in alterations across all conditions, suggesting a strong coupling between the brain’s and heart’s responses. This observed coupling highlights the potential for a coordinated physiological interaction between these two critical systems. Furthermore, our approach, which successfully decoded the heart–brain correlation, offers a promising framework for extending this analysis to explore correlations between the brain and other organs, thereby contributing to a deeper understanding of the complex networks that underlie human health and adaptive physiological responses.

The realm of alternative investments, comprising commodities, real estate, private equity, and other non-traditional asset classes, presents a rich landscape for investors seeking diversification and enhanced returns. Amidst the complexities of these markets, fractal analysis emerges as a powerful tool for unraveling the intricate dynamics and exploring diversification benefits. This paper delves into the application of fractal theory in alternative investments, assessing its efficacy in analyzing asset price movements, correlations, and portfolio diversification. Through a comprehensive review of the literature and empirical evidence, we elucidate how fractal analysis techniques can deepen our understanding of alternative investment markets and inform strategic decision-making. Key findings highlight the ability of fractal analysis to capture nonlinear dependencies, self-similarity, and multi-scale dynamics across various asset classes. The study demonstrates how these insights enhance portfolio resilience, optimize asset allocation, and improve risk-adjusted returns. Additionally, this research contributes novel perspectives on integrating fractal methods with portfolio management and risk diversification strategies.

The interplay between the brain and facial muscles involves a continuous exchange of information, where neural signals originating from the brain direct facial muscle reactions in response to various stimuli. In this research, we employed an information-based approach to study the brain–facial muscle relation in response to various conditions. To do so, we exposed 13 participants (3 F, 10 M, 19–22 years old) to three distinct types of music, each with varying levels of complexity. During these sessions, we recorded both Electroencephalogram (EEG) and Electromyogram (EMG) signals to capture neural and muscular responses, respectively. Subsequently, we utilized Shannon entropy to study the contained information in these signals. Our results demonstrated that music with higher complexity levels significantly altered the contained information in the signals. Furthermore, we identified a strong correlation in the alterations in the contained information among these signals. This suggests a significant connection among facial muscle and brain activities. This approach holds promise for investigating similar relationships among other physiological organs and brain activities, offering insights into broader patterns of physiological interactions.

This study explored the relationship between brain and skin activities by investigating the contained information in electroencephalogram (EEG) and galvanic skin response (GSR) signals. Eleven subjects (4M, 7F, 18–22 years) were monitored while listening to various genres of music (relaxing, pop, and rock), while their EEG and GSR were recorded, and then we computed the Shannon entropy of recorded signals. The results revealed that rock music had the most pronounced impact on altering the information content of the signals compared to other musical genres. Additionally, a strong correlation (0.8926) was detected among the changes in contained information in both signals. This indicates a significant correlation between skin and brain activities. Importantly, this analytical approach shows potential for studying the correlation between brain activity and other bodily organs in diverse scenarios, providing insights into physiological responses under various conditions.

Investigating the disparities in brain activity between healthy individuals and those with intellectual and developmental disorders (IDD) is pivotal in biomedical research. In this study, we delved into these differences by employing a complexity-based analysis of electroencephalogram (EEG) signals. Specifically, we computed the fractal dimension (FD), approximate entropy, and sample entropy (SampEn) of EEG signals for both control (healthy) subjects and IDD individuals during rest and stimulation with music. Our findings revealed that IDD subjects exhibited higher complexity in both rest and stimulation scenarios compared to control subjects, indicative of heightened brain activity complexity among IDD individuals. This analysis not only sheds light on the variations in brain activity associated with IDD, but also underscores its potential applicability in assessing brain activity alterations across various disorders compared to healthy counterparts.

The major goal of this work is to seek the exact wave solutions, and give the bifurcation and chaotic analysis of the time-fractional Benjamin Ono equation in the conformable sense for shallow-water waves. First, the traveling wave transformation is used for the considered model and the variational principle (VP) is extracted by the semi-inverse method (SIM). Based on the VP, the Hamiltonian is derived. Applying Galilean transformation, the corresponding planar dynamical system is obtained, and the bifurcation and chaotic analysis are presented in detail. In the end, the variational method, which is based on the VP and Ritz method, and the Hamiltonian-based method are employed to develop the abundant wave solutions, including the bright solitary, dark solitary and periodic wave solutions. The graphic depictions of the obtained diverse wave solutions are presented with the aid of Maple. In the meantime, the impact of the fractional order on the structure of the extracted waveforms are elaborated. As far as we all know, the findings of this research have not been reported and can enable us to gain a deeper understanding of the nonlinear dynamics of the considered equation.

Financial markets are characterized by complex and often unpredictable dynamics, presenting significant challenges for investors, analysts, and policymakers. In recent years, fractal theory has emerged as a powerful tool for understanding the intricate patterns and behaviors exhibited by financial time series data. This paper provides a comprehensive review of the application of fractal theory in financial time series analysis, examining its theoretical foundations, empirical applications, and practical implications. Through a synthesis of relevant literature, we explore the utility of fractal techniques such as fractal dimension estimation, detrended fluctuation analysis (DFA), and multifractal analysis in quantifying the long-range dependence, self-similarity, and scaling properties of financial time series. Additionally, we discuss the implications of fractal dynamics for risk management, portfolio optimization, and market microstructure analysis, highlighting opportunities for future research and innovation in this evolving field.

This paper investigates tangent measures in the sense of Preiss for homogeneous Cantor sets on $ℝ$ satisfying the convex open set condition, a similar problem has been discussed in our previous work but with the strong separation condition. Through the dynamics of “zooming in” on any typical point, we derive an explicit and uniform formula for the tangent measures associated with this category of homogeneous Cantor sets on $ℝ$.

Aspects of both hybrid and fractional calculus are combined in the (Proportional Caputo-Hybrid) $P_{\mathrm{\text{cap}}}$ operators, which are helpful in solving differential equations with non-integer orders and modeling a variety of complicated phenomena in science and engineering. In this paper, we establish the $P_{\mathrm{\text{cap}}}$ operators via multiplicative calculus which are termed as multiplicative $P_{\mathrm{\text{cap}}}$ operators. we initially formulate two H.H (Hermite–Hadamard)-type inequalities applicable to multiplicative (geometric) convex function via multiplicative $P_{\mathrm{\text{cap}}}$ operators. Subsequently, by leveraging certain characteristics of multiplicative convex functions, we present novel inequalities related to multiplicative convex function via multiplicative $P_{\mathrm{\text{cap}}}$ operators also demonstrating two novel identities applicable to multiplicatively differentiable functions. By leveraging these identities, we then establish inequalities of trapezoid and midpoint types specifically designed for multiplicatively convex functions. Additionally, we explore applications of these findings to special functions and special means.

The main aim of this paper is to present a class of Ostrowski-type inequalities for generalized $s$-$\phi$-convex functions on fractal sets, which have important applications in mathematical modeling and analysis in fields where fractal structures are prevalent, such as in signal processing, image analysis, and complex systems. For this purpose, we first define the $s$-$\phi$-convex function on fractal sets and discuss some of its interesting properties. Then, we derive a generalized integral identity for $n$th-order locally differentiable functions, and using it, we derive some new Ostrowski-type inequalities for generalized $s$-$\phi$-convex functions in a local fractal operator environment. In addition, we further illustrate the accuracy of our results with some numerical examples.

Gaining a comprehensive insight into how shale’s pore structure changes due to the extraction of oil from smaller pores is pivotal for understanding the impact of movable oil in shale plays, particularly when estimating reserves. The fractal dimension, a parameter reflecting pore complexity, can be quantitatively employed to explore this aspect. In this research endeavor, we focused on assessing variations in pore structure resulting from oil removal in four samples from the Fengcheng shale Formation in Mahu Sag, China. To achieve this, we utilized two diverse probing techniques: Gas adsorption and small-angle X-ray scattering (SAXS). These methodologies allowed us to analyze pore structure alterations both before and after oil extraction. By comparing data gathered through these approaches, we calculated and compared fractal dimensions across pre- and post-extraction conditions. The outcomes revealed notable differences in fractal dimensions obtained from the two methods. Before extraction, the disparity in fractal dimensions calculated from these techniques ranged from 9.6% to 18.3% across the samples. Post-extraction, this variation decreased to a range of 1.6% to 25.1%. These differences originate from the inherent physics governing each method’s operational mode. Furthermore, both techniques indicated an increase in fractal dimension post-extraction, implying heightened pore complexity. In terms of the gas adsorption method, the increase in fractal dimensions across studied samples ranged from 6.385 to 13.435. Comparatively, for the SAXS method, this increase varied between 2.22% and 18.61%. These findings suggest that the movable oil, whether fully or partially occupying pore spaces or being adsorbed onto mineral/organic matter surfaces, could potentially reduce pore complexity. This study pioneers the role of adsorbed or freely moving oil within both connected and isolated pores of shale plays which is crucial for accurate reserve estimation and economic evaluation of shale plays. By elucidating the intricate interactions between oil and pore structures, our research lays the groundwork for more precise assessments in this critical domain.

The average path length of a network serves to elucidate the network’s fluency and coherence by providing insights into how efficiently and effectively information or interactions can traverse the network and it is extensively studied in the context of network science. The Fermat distance among three nodes $i$, $j$, and $k$, denoted as $f(i,j,k)$, was defined as the shortest total path length between a node $p$ and nodes $i$, $j$, and $k$. The corresponding average Fermat distance also plays an important role in describing the connectedness of a network.

In this paper, we study a class of polygon networks with pseudo-fractal structure and analyze the average path length. Moreover, we derive the average Fermat distance in two ways. Interestingly, we find the ratio of asymptotic average Fermat distance to asymptotic average path length is exactly 3/2 and these metrics grow linearly with the order of the polygon networks.

In this work, a predator–prey model is considered to study its dynamics in a stochastic environment with the addition of predator migration. First, conditions on the existence of positive equilibria, along with their stability, and the emergence of limit cycles in the corresponding deterministic system are derived. Then, two stochastic properties are established: The ultimate boundedness of solutions and the existence of an ergodic stationary distribution for the system under consideration. The theoretical results are supported by numerical simulations, which reveal the significant influence of migration on the dynamical behaviors of both deterministic and stochastic systems.

In this paper, for a typical function $\mathrm{\Phi }\in \mathcal{𝒞}(Y)$ defined in an uncountable compact metric space $Y$, we give the lower Hewitt–Stromberg dimension of graphs $G_{\mathrm{\Phi }}(Y)=\{(y,\mathrm{\Phi }(y))|y\in Y\}$ of the function $\mathrm{\Phi }$. Moreover, we investigate the decomposition of functions within $\mathcal{𝒞}([0,1])$ based on the lower box dimension and the lower Hewitt–Stromberg dimension, revealing significant disparities compared to the context of the packing dimension. Second, we present some results on the lower Hewitt–Stromberg dimension of graphs of sums and products of continuous functions. The main proof is that for a given real number $1\le \beta \le 2$, some real-valued continuous functions in $\mathcal{𝒞}([0,1])$ can be decomposed into the sum and product of two continuous real-valued functions, and the lower Hewitt–Stromberg dimension of the graph for each function is $\beta$.

This paper focuses on considering a combined method for solving a fractional oscillation equation arising from a micro/nanobeam-based micro-electromechanical system under the magnetic force. It is an efficient method based upon the fractional complex transformation and the spreading residue harmonic balance method. The fractional oscillator is transformed as the conventional oscillation system by using the fractional complex transformation. The approximated periodic solutions and frequencies are given by applying the spreading residue harmonic balance method. Sensitive analysis of the approximated frequencies concerning different amplitudes and parameters is provided, which may be helpful for realizing the system stability. Numerical comparisons with some existing methods such as Runge–Kutta method and He’s frequency formulation approach are shown to confirm the efficiency of the proposed method.