Search Results (2,831)

This article starts from the premise that one of the global control strategies of the Permanent Magnet Synchronous Motor (PMSM), namely the Direct Torque Control (DTC) control strategy, is characterized by the fact that the internal flux and torque control loop usually uses ON–OFF controllers with hysteresis, which offer easy implementation and very short response times, but the oscillations introduced by them must be cancelled by the external speed loop controller. Typically, this is a PI speed controller, whose performance is good around global operating points and for relatively small variations in external parameters and disturbances, caused in particular by load torque variation. Exploiting the advantages of the DTC strategy, this article presents a way to improve the performance of the sensorless control system (SCS) of the PMSM using the Proportional Integrator (PI), PI Equilibrium Optimizer Algorithm (EOA), Fractional Order (FO) PI, Tilt Integral Derivative (TID) and FO Lead–Lag under constant flux conditions. Sliding Mode Control (SMC) and FOSMC are proposed under conditions where the flux is variable. The performance indicators of the control system are the usual ones: response time, settling time, overshoot, steady-state error and speed ripple, plus another one given by the fractal dimension (FD) of the PMSM rotor speed signal, and the hypothesis that the FD of the controlled signal is higher when the control system performs better is verified. The article also presents the basic equations of the PMSM, based on which the synthesis of integer and fractional controllers, the synthesis of an observer for estimating the PMSM rotor speed, electromagnetic torque and stator flux are presented. The comparison of the performance for the proposed control systems and the demonstration of the parametric robustness are performed by numerical simulations in Matlab/Simulink using Simscape Electrical and Fractional-Order Modelling and Control (FOMCON). Real-time control based on an embedded system using a TMS320F28379D controller demonstrates the good performance of the PMSM-SCS based on the DTC strategy in a complete Hardware-In-the-Loop (HIL) implementation. Full article

In the last few years, the conjunctivitis adenovirus disease has been investigated by using the concept of mathematical models. Hence, researchers have presented some mathematical models of the mentioned disease by using classical and fractional order derivatives. A complementary method involves analyzing the system of fractal fractional order equations by considering the set of symmetries of its solutions. By characterizing structures that relate to the fundamental dynamics of biological systems, symmetries offer a potent notion for the creation of mechanistic models. This study investigates a novel mathematical model for conjunctivitis adenovirus disease. Conjunctivitis is an infection in the eye that is caused by adenovirus, also known as pink eye disease. Adenovirus is a common virus that affects the eye’s mucosa. Infectious conjunctivitis is most common eye disease on the planet, impacting individuals across all age groups and demographics. We have formulated a model to investigate the transmission of the aforesaid disease and the impact of vaccination on its dynamics. Also, using mathematical analysis, the percentage of a population which needs vaccination to prevent the spreading of the mentioned disease can be investigated. Fractal fractional derivatives have been widely used in the last few years to study different infectious disease models. Hence, being inspired by the importance of fractal fractional theory to investigate the mentioned human eye-related disease, we derived some adequate results for the above model, including equilibrium points, reproductive number, and sensitivity analysis. Furthermore, by utilizing fixed point theory and numerical techniques, adequate requirements were established for the existence theory, Ulam–Hyers stability, and approximate solutions. We used nonlinear functional analysis and fixed point theory for the qualitative theory. We have graphically simulated the outcomes for several fractal fractional order levels using the numerical method. Full article

Spherical glass beads weaken the influences of particle morphology, surface properties, and microscopic fabric on shear strength, which is significant for revealing the relationship between macroscopic particle friction mechanisms and the particle size distribution of sand. This paper explores the shear mechanical properties of glass beads with different particle size ratios under different confining pressures. It obtains the particle size ratio and fractal dimension D through an optimal mechanical response. Simultaneously, we explore the range of the fractal dimension D under well-graded conditions. The test results show that the strain-softening degree of R_s is more obvious under a highly effective confining pressure, and the strain-softening degree of R_s can reach 0.669 when the average particle size $\overline{d}$ is 0.5 mm. The changes in the normalized modulus ratio E_u/E_u50 indicate that the particle ratio and arrangement are the fundamental reasons for the different macroscopic shear behaviors of particles. The range of the peak effective internal friction angle φ is 23 °~35 °, and it first increases and then decreases with the increase in the effective confining pressure. As the average particle size increases, the peak stress ratio M_FL and the peak effective internal friction angle φ first increase and then decrease, and both can be expressed using the Gaussian function. The range of the fractal dimension D for well-graded particles is 1.873 to 2.612, and the corresponding average particle size $\overline{d}$ ranges from 0.433 to 0.598. Under the optimal mechanical properties of glass beads, the particle size ratio of 0.25 mm to 0.75 mm is 23:27, and the fractal dimension D is 2.368. The study results provide a reference for exploring friction mechanics mechanisms and the optimal particle size distributions of isotropic sand. Full article

This research work employs a powerful analytical method known as the Riccati Modified Extended Simple Equation Method (RMESEM) to investigate and analyse chaotic soliton solutions of the (1 + 1)-dimensional Complex Quintic Swift–Hohenberg Equation (CQSHE). This model serves to describe complex dissipative systems that produce patterns. We have found that there exist numerous chaotic soliton solutions with periodic and axial perturbations to the intended CQSHE, provided that the coefficients are constrained by certain conditions. Furthermore, by applying a sophisticated transformation, the provided transformative approach RMESEM transforms CQSHE into a set of Nonlinear Ordinary Differential Equations (NODEs). The resulting set of NODEs is then transformed into an algebraic system of equations by incorporating the extended Riccati NODE to assume a series form solution. The soliton solutions to this system of equations can be found as periodic, hyperbolic, exponential, rational-hyperbolic, and rational families of functions. A variety of 3D and contour visuals are also provided to graphically illustrate the axially and periodically perturbed dynamics of these chaotic soliton solutions and the formation of fractals. Our findings are noteworthy because they shed light on the chaotic nature of the framework we are examining, enabling us to better understand the dynamics that underlie it. Full article

Faults, as a kind of fracture tectonics, play a role in reservoir closure or provide oil and gas transportation channels. The accurate understanding of the distribution characteristics of faults is significant for oil and gas exploration. The traditional fractal dimension for fault number (D_f3) cannot comprehensively characterize the complexity and heterogeneity of fault network distribution. In this paper, a fractal characterization method on three-dimensional (3D) tortuosity of fault tectonics is proposed based on 3D seismic exploration. The methodology is described in detail to establish the model on the fractal dimension for the 3D tortuosity of fault tectonics. The results show the proposed method of estimation of the D_T3 displaying high accuracy and rationality. Compared with the traditional fractal dimension D_f3, the proposed D_T3 can comprehensively characterize the fractal characteristics of faults network systems in the 3D space. This study achieves a breakthrough in the fractal characterization of the 3D tortuosity of fault tectonics. It is worth further study for establishing an analytical fractal equation based on the D_T3 and oil or gas transfer, which can provide the theoretical foundation and technical support for oil and gas exploration. Full article

The accurate characterization of soil structure is fundamental to groundwater science, environmental ecology, and Earth systems science. To address the challenge of quantifying the high spatial variability of large-scale soil structures, this study used a laser particle size analyzer to measure the distribution of soil particle size in 207 samples from ten profiles across the Daqing and Ziya River basins in the North China Plain. The quantified soil structure, expressed as soil fractal dimension D, was derived using monofractal theory. Various spatial analysis techniques, including Moran’s I index, correlation analysis heat maps, the Kolmogorov–Smirnov one-sample test, and geostatistical semivariogram function, were jointly applied to investigate the spatial variability of soil structural fractals across different depths in the piedmont plain–coastal areas of the two river basins. The results indicate the following: (1) Quantitative analysis confirms that under the influence of piedmont alluvial and fluvial dynamics, soil D values homogenize from the piedmont to the coastal areas, with decreasing particle size differences closer to the coast. However, the spatial variability of the soil structural fractals in the Ziya River Basin was greater than that in the Daqing River Basin. (2) The combined effects of climate change, regional differences, and human activity led to greater spatial variability in the soil structural fractals in the Ziya River Basin than in the Daqing River Basin. The correlation between D values and burial depth was strongest in the Xianxian profile (−0.78), whereas the spatial correlation was strongest in the Hengshui and Dacheng profiles (−0.47). (3) The greatest spatial variability in soil D values occurred at depths of 1–2 m, with a coefficient of variation of 23.595%, which was significantly higher than those at depths of 0–1 (14.569%) and 2–3 m (16.284%). Full article

In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. Based on the invariant properties of fractal operators, we discovered that the symmetry evolution laws of functional fractal trees in the physical fractal space can reveal the intrinsic correlations between special functions. This article explores the fractional-order correlation between special functions inspired by bone fractal operators. Specifically, the following contents are included: (1) showing the intrinsic expression in the convolutional kernel function of bone fractal operators and its correlation with special functions; (2) proving the following proposition: the convolutional kernel function of bone fractal operators is still related to the special functions under different input signals (external load, external stimulus); (3) using the bone fractal operators as the background and error function as the core, deriving the fractional-order correlation between different special functions. Full article

Mathematical models of heat and moisture transfer for anisotropic materials, based on the use of the fractional calculus of integro-differentiation, are considered because such two-factor fractal models have not been proposed in the literature so far. The numerical implementation of mathematical models for determining changes in heat exchange and moisture exchange is based on the adaptation of the fractal neural network method, grounded in the physics of processes. A fractal physics-informed neural network architecture with a decoupled structure is proposed, based on loss functions informed by the physical process under study. Fractional differential formulas are applied to the expressions of non-integer operators, and finite difference schemes are developed for all components of the loss functions. A step-by-step method for network training is proposed. An algorithm for the implementation of the fractal physics-informed neural network is developed. The efficiency of the new method is substantiated by comparing the obtained numerical results with numerical approximation by finite differences and experimental data for particular cases. Full article

In this research, the doping of SrTiO₃ with Mn⁴⁺ was performed in order to evaluate the potential application as a photocatalyst for the degradation of organic dye pollutants. Since photocatalytic activity depends on grain microstructure, fractal analysis was used to estimate the Hausdorff dimension to provide a more thorough investigation of Mn@SrTiO₃ morphology. Structural analysis by infrared spectroscopy indicated the incorporation of Mn⁴⁺ into the SrTiO₃ lattice, while by using x-ray diffraction, the crystallite size of 44 nm was determined. The photocatalytic activity test performed on complex ethyl violet organic dye revealed potential for Mn@SrTiO₃ application in water treatment. Based on fractal regression analysis, a good estimate was obtained for the reconstruction of grain shape, with a Hasudorff dimension of 1.13679, which was used to find the best kinetics model for the photodegradation reaction. The experimental data showed a nearly linear fit with fractal-like pseudo-zero order. These findings and applications of fractal dimensions could contribute to future characterizations of photocatalysts, providing a deeper understanding of surface properties and their influence on photocatalytic activity. Full article

Objectives: To assess the impact of the presence or position (buccal/palatal) of impacted canines on trabecular bone density using fractal analysis (FA) on cone-beam computed tomography (CBCT) images, and to compare the results with a control group without impacted canines. Methods: This retrospective study included 41 patients with unilateral impacted canines (30 palatal, 11 buccal) and a control group of 39 patients who underwent surgically assisted rapid maxillary expansion. All patients had CBCT images recorded for diagnostic and treatment purposes. Cross-sectional CBCT images were obtained between the first and second premolars on both sides of the patients’ maxilla. From these images, fractal dimension (FD) was measured in a 20 × 20 pixel region of interest in the trabecular bone using the ImageJ software. Results: The FD values were significantly higher on the impacted side in the impacted canine group (p = 0.02). Within the impacted canine group, a significant increase in FD was observed on the impacted side in the buccal-impacted subgroup (p = 0.02), while no significant difference was observed in the palatal-impacted subgroup (p > 0.05). Conclusions: According to the results of our study, there is an association between the position of the impacted canine and trabecular bone density. An increased trabecular bone density may play a role in the etiology of buccally impacted canines. Clinicians should consider anchorage planning, and appropriate force level, during the forced eruption of buccally impacted canines with high surrounding bone density, to minimize undesirable movements and achieve optimal treatment outcomes. Full article

Considering that the degradation of ball screw grease involves fractal characteristics, which exhibit long-term dependency and autocorrelation, a multivariate accelerated degradation modeling and reliability assessment method based on the fractional Brownian motion process model is proposed in this paper. Firstly, a nonlinear accelerated degradation model of grease is established using fractional Brownian motion, considering the heterogeneity of samples as well as the memory effect and long-term dependence in the deterioration process, and realizing parameter estimation. Secondly, a multivariate reliability evaluation model is established by considering multivariate performance indicators in combination with the Frank copula function. Finally, the effectiveness and potential engineering application value of this method are verified through actual degradation data of the grease. Full article

Shale gas is a prospective cleaner energy resource and the exploration and development of shale gas has made breakthroughs in many countries. Structure deformation is one of the main controlling factors of shale gas accumulation and enrichment in complex tectonic areas in southern China. In order to estimate the shale gas capacity of structurally deformed shale reservoirs, it is necessary to understand the systematic evolution of organic pores in the process of structural deformation. In particular, as the main storage space of high-over-mature marine shale reservoirs, the organic matter pore system directly affects the occurrence and migration of shale gas; however, there is a lack of systematic research on the fractal characteristics and deformation mechanism of organic pores under the background of different tectonic stresses. Therefore, to clarify the above issues, modular automated processing system (MAPS) scanning, low-pressure gas adsorption, quantitative evaluation of minerals by scanning (QEMSCAN), and focused ion beam scanning electron microscopy (FIB-SEM) were performed and interpreted with fractal and morphology analyses to investigate the deformation mechanisms and structure of organic pores from different tectonic units in Silurian Longmaxi shale. Results showed that in stress concentration areas such as around veins or high-angle fractures, the organic pore length-width ratio and the fractal dimension are higher, indicating that the pore is more obviously modified by stress. Under different tectonic backgrounds, the shale reservoir in Weiyuan suffered severe denudation and stronger tectonic compression during burial, which means that the organic pores are dominated by long strip pores and slit-shaped pores with high fractal dimension, while the pressure coefficient in Luzhou is high and the structural compression is weak, resulting in suborbicular pores and ink bottle pores with low fractal dimension. The porosity and permeability of different forms of organic pores are also obviously different; the connectivity of honeycomb pores with the smallest fractal dimension is the worst, that of suborbicular organic pores is medium, and that of long strip organic pores with the highest fractal dimension is the best. This study provides more mechanism discussion and case analysis for the microscopic heterogeneity of organic pores in shale reservoirs and also provides a new analysis perspective for the mechanism of shale gas productivity differences in different stress–strain environments. Full article

In this paper, we introduce an innovative approach to multi-focus image fusion by leveraging the concepts of fractal dimension and coupled neural P (CNP) systems in nonsubsampled contourlet transform (NSCT) domain. This method is designed to overcome the challenges posed by the limitations of camera lenses and depth-of-field effects, which often prevent all parts of a scene from being simultaneously in focus. Our proposed fusion technique employs CNP systems with a local topology-based fusion model to merge the low-frequency components effectively. Meanwhile, for the high-frequency components, we utilize the spatial frequency and fractal dimension-based focus measure (FDFM) to achieve superior fusion performance. The effectiveness of the method is validated through extensive experiments conducted on three benchmark datasets: Lytro, MFI-WHU, and MFFW. The results demonstrate the superiority of our proposed multi-focus image fusion method, showcasing its potential to significantly enhance image clarity across the entire scene. Our algorithm has achieved advantageous values on metrics $Q_{AB/F}$, $Q_{CB}$, $Q_{CV}$, $Q_E$, $Q_{FMI}$, $Q_G$, $Q_{MI}$, and $Q_{NCIE}$. Full article

Calcareous sand is a crucial construction material for island and reef development and reinforcing it using Microbially Induced Calcite Precipitation (MICP) technology is a promising new method. This study employed 3D scanning technology to assess changes in the particle size and morphology of MICP-treated, coated calcareous sand particles. Single-particle crushing tests were conducted to analyze their crushing strength, crushing energy, crushing modes, and fragment fractal dimensions. The results indicated that MICP treatment significantly increased particle size, surface area, and volume, while reducing flatness. At a cementation solution concentration of 1 mol/L, both crushing strength and crushing energy were optimized. The coated particles exhibited three crushing modes: explosive crushing, mixed crushing, and splitting crushing. Thicker coatings led to a tendency for particles to break into larger fragments through the mixed and splitting crushing modes. Fractal analysis revealed that coating thickness directly affects the local crushing characteristics of the particles. Full article

This paper presents a mathematical analysis of semantic convergence in transformer-based language models, drawing inspiration from the concept of fractal self-similarity. We introduce and prove a novel theorem characterizing the gradient of embedding similarity across layers. Specifically, we establish that there exists a monotonically increasing function that provides a lower bound on the rate at which the average cosine similarity between token embeddings at consecutive layers and the final layer increases. This establishes a fundamental property: semantic alignment of token representations consistently increases through the network, exhibiting a pattern of progressive refinement, analogous to fractal self-similarity. The key challenge addressed is the quantification and generalization of semantic convergence across diverse model architectures and input contexts. To validate our findings, we conduct experiments on BERT and DistilBERT models, analyzing embedding similarities for diverse input types. While our experiments are limited to these models, we empirically demonstrate consistent semantic convergence within these architectures. Quantitatively, we find that the average rates of semantic convergence are approximately 0.0826 for BERT and 0.1855 for DistilBERT. We observe that the rate of convergence varies based on token frequency and model depth, with rare words showing slightly higher similarities (differences of approximately 0.0167 for BERT and 0.0120 for DistilBERT). This work advances our understanding of transformer models’ internal mechanisms and provides a mathematical framework for comparing and optimizing model architectures. Full article