Search Results (597)

The acidification modification of coal seams is a significant technical measure for transforming coalbed methane reservoirs and enhancing the permeability of coal seams, thereby improving the extractability of coalbed methane. However, the acids currently used in fracturing fluids are predominantly inorganic acids, which are highly corrosive and can contaminate groundwater reservoirs. In contrast, organic acids are not only significantly less corrosive than inorganic acids but also readily bind with the coal matrix. Some organic acids even exhibit complexing and flocculating effects, thus avoiding groundwater contamination. This study focuses on the 1/3 coking coal from the Guqiao Coal Mine of Huainan Mining Group Co., Ltd., in China. It systematically investigates the fractal characteristics and chemical structure of coal samples before and after pore modification using four organic acids (acetic acid, glycolic acid, oxalic acid, and citric acid) and compares their effects with those of hydrochloric acid solutions at the same concentration. Following treatment with organic acids, the coal samples exhibit an increase in surface fractal dimension, a reduction in spatial fractal dimension, a decline in micropore volume proportion, and a rise in the proportions of transitional and mesopore volumes, and the structure of the hydroxyl group and oxygen-containing functional group decreased. This indicates that treating coal samples with organic acids enhances their pore structure and chemical structure. A comparative analysis reveals that hydrochloric acid is more effective than acetic acid in modifying coal pores, while oxalic acid and citric acid outperform hydrochloric acid, and citric acid shows the best results. The findings provide essential theoretical support for organic acidification modification technology in coalbed methane reservoirs and hydraulic fracturing techniques for coalbed methane extraction. Full article

Ubiquitiform is a new theory of finite-order self-similar physical structure and it is more reasonable to describe real engineering surfaces by ubiquitiform rather than fractal. In this paper, by introducing the frequency truncation criterion, a new analytical expression of the two-dimensional W–M function based on the ubiquitiform theory is firstly derived and constructed and the two-dimensional ubiquitiformal curve characterization under different contact characteristic parameters is achieved. On this basis, the anisotropic three-dimensional surface W–M function with ubiquitiformal features is constructed, and the evolution law of the anisotropic three-dimensional surface morphology under the regulation of the ubiquitiformal complexity is investigated. Then, an improved adaptive box counting algorithm is proposed, and the lower limit of the metric scale in the self-similarity region of the asperities on the rough surface is determined and then the computation method of the ubiquitiformal complexity is established. At last, the validity and accuracy of the method are confirmed by the Koch curves. Key findings include: (1) higher ubiquitiformal complexity D corresponds to increased surface irregularity and complexity; (2) the characteristic scale factor G affects surface height only; (3) reducing the lower limit of metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ increases surface undulation frequency, revealing finer details. This research provides a rationale and quantitative guidance for the matching design of critical joint interfaces in modern precision machinery. Full article

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order ($\alpha$, where $0<\alpha \le 1$). This approach is based on the semiclassical approximation and the principles of the Maslov method. The fractal time derivative is introduced in the framework of $F^{\alpha }$ calculus. The Fisher–KPP equation is decomposed into a system of nonlinear equations that describe the dynamics of interacting quasiparticles within classes of trajectory-concentrated functions. A key element in constructing approximate quasiparticle solutions is the interplay between the dynamical system of quasiparticle moments and an auxiliary linear system of equations, which is coupled with the nonlinear system. General constructions are illustrated through examples that examine the effect of the fractal parameter ($\alpha$) on quasiparticle behavior. Full article

Viscoelastic materials are commonly used in civil engineering, biomedical sciences, and polymers, where understanding their creep and relaxation behaviors is essential for predicting long-term performance. This paper introduces an operator-based method for modeling viscoelastic materials, providing a unified framework to describe both creep and relaxation functions. The method utilizes stiffness and compliance operators, offering a systematic approach for analyzing viscoelastic problems. The operator-based method enhances the mathematical duality between the creep and relaxation functions, providing greater physical intuition and understanding of time-dependent material behavior. It directly reflects the intrinsic properties of materials, independent of input and output conditions. The method is extended to dynamic problems, with complex modulus and compliance derived through operator representations. The fractal tree model, with its constant loss factor across the frequency spectrum, demonstrates potential engineering applications. By incorporating a damage-based variable coefficient, the model now also accounts for the accelerated creep phase of rocks, capturing damage evolution under prolonged loading. While promising, the current method is limited to one-dimensional problems, and future research will aim to extend it to three-dimensional cases, integrate experimental validation, and explore broader applications. Full article

This article investigates and analyzes the diverse patterns of Julia sets generated by new classes of generalized exponential and sine rational functions. Using a generalized viscosity approximation-type iterative method, we derive escape criteria to visualize the Julia sets of these functions. This approach enhances existing algorithms, enabling the visualization of intricate fractal patterns as Julia sets. We graphically illustrate the variations in size and shape of the images as the iteration parameters change. The new fractals obtained are visually appealing and attractive. Moreover, we observe fascinating behavior in Julia sets when certain input parameters are fixed, while the values of n and m vary. We believe the conclusions of this study will inspire and motivate researchers and enthusiasts with a strong interest in fractal geometry. Full article

Full-body movement involving multi-segmental coordination has been essential to our evolution as a species, but its study has been focused mostly on the analysis of one-dimensional data. The field is poised for a change by the availability of high-density recording and data sharing. New ideas are needed to revive classical theoretical questions such as the organization of the highly redundant biomechanical degrees of freedom and the optimal distribution of variability for efficiency and adaptiveness. In movement science, there are popular methods that up-dimensionalize: they start with one or a few recorded dimensions and make inferences about the properties of a higher-dimensional system. The opposite problem, dimensionality reduction, arises when making inferences about the properties of a low-dimensional manifold embedded inside a large number of kinematic degrees of freedom. We present an approach to quantify the smoothness and degree to which the kinematic manifold of full-body movement is distributed among embedding dimensions. The principal components of embedding dimensions are rank-ordered by variance. The power law scaling exponent of this variance spectrum is a function of the smoothness and dimensionality of the embedded manifold. It defines a threshold value below which the manifold becomes non-differentiable. We verified this approach by showing that the Kuramoto model obeys the threshold when approaching global synchronization. Next, we tested whether the scaling exponent was sensitive to participants’ gait impairment in a full-body motion capture dataset containing short gait trials. Variance scaling was highest in healthy individuals, followed by osteoarthritis patients after hip replacement, and lastly, the same patients before surgery. Interestingly, in the same order of groups, the intrinsic dimensionality increased but the fractal dimension decreased, suggesting a more compact but complex manifold in the healthy group. Thinking about manifold dimensionality and smoothness could inform classic problems in movement science and the exploration of the biomechanics of full-body action. Full article

Functional polymeric materials play a critical role in optimizing flocculation and sedimentation processes in mining tailings, where complex interactions with mineral surfaces govern polymer performance. This study examines the structure–performance relationship, which describes how the internal structure of aggregates (e.g., compactness, porosity and fractal dimension) influences sedimentation behavior, specifically for anionic polyacrylamide (SNF 704) in kaolin-quartz-pyrite suspensions at a pH of 10.5. Using focused beam reflectance measurement (FBRM) and static sedimentation tests, we demonstrate that pyrite exhibits the highest flocculant adsorption capacity, inducing a train-like polymer conformation on its surface. This reduces the formation of effective polymeric bridges, resulting in less compact and more porous aggregates that negatively impact sedimentation rates. Increasing the flocculant dosage improves the capture of fine particles; however, at high pyrite concentrations, rapid saturation of adsorption sites limits flocculation efficiency. Additionally, the fractal dimension of the aggregates decreases with increasing pyrite content, revealing more open structures that hinder consolidation. These findings underscore the importance of optimizing polymer dosage and tailoring flocculant design to the mineralogical composition, thereby enhancing water recovery and sustainability in mining operations. This study highlights the role of structure–property relationships in polymeric flocculants and their potential for next-generation tailings management solutions. Full article

(1) Background: Condylar fracture healing pattern classification in children and adolescents is primarily based on the radiological assessment of condylar morphology; however, recent studies showed the presence of a poor correlation between post-treatment radiological findings and clinical temporomandibular joint (TMJ) dysfunction. The present case series aimed to correlate the condylar morphology, shape, and trabecular bone density with the skeletal asymmetry and the clinical recovery of two growing patients with unilateral condylar fractures undergoing orthopedic treatment with the Balters Bionator appliance. (2) Methods: Pre- and post-treatment (12 months) cone-beam computed tomography (CBCT) scans of two growing patients with unilateral condylar fracture were retrieved; both patients were treated with the Balters Bionator appliance for one year. Morphological evaluation of the condylar healing pattern was carried out on CBCT reconstructions of the mandible. Condylar remodeling and skeletal asymmetry were assessed through linear measurements performed on pre- and post-treatment CBCT scans; then, fractal analysis (FA) was employed to assess the condylar trabecular bone density on orthopantomographies (OPTs). Clinical and TMJ functional evaluation were retrieved from patients’ records from before and at the end of the treatment (12 months). (3) Results: Conservative treatment of condylar fractures in growing patients led to an increased bone density of the condylar heads, regardless of the post-treatment size and morphology of the injured condyles. Patient one presented an unchanged condylar morphology on the affected side, while patient two’s condyle was slightly spherical. The qualitative results were confirmed by quantitative measurements on CBCTs. The radiological healing patterns were associated with slightly different functional outcomes. Both patients also exhibited an improvement in skeletal asymmetry and TMJ function. (4) Conclusions: According to the findings in the present study, the condylar remodeling and bone apposition after conservative treatment of condylar fractures in growing patients can exhibit different radiological and functional outcomes. Indeed, an unchanged morphology of the condylar head is more likely to determine a physiological TMJ recovery. Full article

There are currently numerous types of water-flooding characteristic curves, most of which are derived from fundamental theories such as material balance, relative permeability, along with experimental results. A single exponential or power function expression cannot accurately characterize the complex flow characteristics of different types of reservoirs, and the equivalent relationships corresponding to production wells and entire oilfields remain unclear. Consequently, practical applications often encounter issues such as curve tailing, difficulty in determining linear segments, inability to identify anomalous points, and inaccuracies in dynamic fitting and prediction. This paper derives a novel water-flooding characteristic curve expression based on fractal theory, incorporating the fractal characteristics of two-phase oil–water flow in reservoirs, as well as the micro-level pore–throat flow features and macro-level dynamic laws of water flooding. The approach is analyzed and validated with real oilfield cases. This study indicates that fitting with the novel water-flooding characteristic curve yields high correlation coefficients and excellent fitting results, demonstrating strong applicability across various types of oilfields and water cut stages. It can more accurately describe the water-flooding characteristics under different reservoir conditions and rapidly predict recoverable reserves, offering significant application value in the dynamic analysis of oilfields and the formulation of development strategies. Full article

Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) that is used to describe the topological structural complexity or degree of disorder in networks. Topological indices, as graph invariants, provide quantitative descriptors for characterizing global structural properties. In this paper, we investigate two types of generalized Sierpiński graphs constructed on the basis of different seed graphs, and employ six topological indices—the first Zagreb index, the second Zagreb index, the forgotten index, the augmented Zagreb index, the Sombor index, and the elliptic Sombor index—to analyze the corresponding entropy. We utilize the method of edge partition based on vertex degrees and derive analytical formulations for the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy. This research approach, which integrates entropy with Sierpiński network characteristics, furnishes novel perspectives and instrumental tools for addressing challenges in chemical graph theory, computer networks, and other related fields. Full article

Background: The beat-by-beat fluctuation of heart rate (HR) in its temporal sequence (HR dynamics) provides information on HR regulation by the autonomic nervous system (ANS) and its dysregulation in pathological states. Commonly, linear analyses of HR and its variability (HRV) are used to draw conclusions about pathological states despite clear statistical and translational limitations. Objective: The main aim of this study was to compare linear and nonlinear HR measures, including detrended fluctuation analysis (DFA), based on ECG recordings by radiotelemetry in C57BL/6N mice to identify pathological HR dynamics. Methods: We investigated different behavioral and a wide range of pharmacological interventions which alter ANS regulation through various peripheral and/or central mechanisms including receptors implicated in psychiatric disorders. This spectrum of interventions served as a reference system for comparison of linear and nonlinear HR measures to identify pathological states. Results: Physiological HR dynamics constitute a self-similar, scale-invariant, fractal process with persistent intrinsic long-range correlations resulting in physiological DFA scaling coefficients of α~1. Strongly altered DFA scaling coefficients (α ≠ 1) indicate pathological states of HR dynamics as elicited by (1) parasympathetic blockade, (2) parasympathetic overactivation and (3) sympathetic overactivation but not inhibition. The DFA scaling coefficients are identical in mice and humans under physiological conditions with identical pathological states by defined pharmacological interventions. Conclusions: Here, we show the importance of tonic vagal function for physiological HR dynamics in mice, as reported in humans. Unlike linear measures, DFA provides an important translational measure that reliably identifies pathological HR dynamics based on altered ANS control by pharmacological interventions. Central ANS dysregulation represents a likely mechanism of increased cardiac mortality in psychiatric disorders. Full article

In this manuscript, we introduce the concept of fuzzy S-metric spaces and study some of their characteristics. We prove a fixed-point theorem for a self-mapping on a complete fuzzy S-metric space. To illustrate the versatility of our new ideas and related fixed-point theorems, we give examples to illustrate their use in a variety of domains, including fractal formation. These examples illustrate how the fuzzy S-contraction can be applied to iterated function systems, enabling the exploration of fractal forms under diverse contractive conditions. In addition, we solve the satellite web coupling problem by employing this coherent framework. Full article

Accurate identification of growth and development stages is critical for orthodontic diagnosis, treatment planning, and post-treatment retention. While hand–wrist radiographs are the traditional gold standard, the associated radiation exposure necessitates alternative imaging methods. Lateral cephalometric radiographs, particularly the maturation stages of the second, third, and fourth cervical vertebrae (C2, C3, and C4), have emerged as a promising alternative. However, the nonlinear dynamics of these images pose significant challenges for reliable detection. This study presents a novel approach that integrates chaotic functional connectivity (FC) matrices and fractal dimension analysis to address these challenges. The fractal dimensions of C2, C3, and C4 vertebrae were calculated from 945 lateral cephalometric radiographs using three methods: fast Fourier transform (FFT), box counting, and a pre-processed FFT variant. These results were used to construct chaotic FC matrices based on correlations between the calculated fractal dimensions. To effectively model the nonlinear dynamics, chaotic maps were generated, representing a significant advance over traditional methods. Feature selection was performed using a wrapper-based approach combining k-nearest neighbors (kNN) and the Puma optimization algorithm, which efficiently handles the chaotic and computationally complex nature of cervical vertebrae images. This selection minimized the number of features while maintaining high classification performance. The resulting AI-driven model was validated with 10-fold cross-validation and demonstrated high accuracy in identifying growth stages. Our results highlight the effectiveness of integrating chaotic FC matrices and AI in orthodontic practice. The proposed model, with its low computational complexity, successfully handles the nonlinear dynamics in C2, C3, and C4 vertebral images, enabling accurate detection of growth and developmental stages. This work represents a significant step in the detection of growth and development stages and provides a practical and effective solution for future orthodontic diagnosis. Full article

The Weierstrass function $W\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}^{n}cos\left(2\pi b^{n}x\right)$ is a function that is continuous everywhere and differentiable nowhere. There are many investigations on fractal dimensions of the Weierstrass function, and the investigation of its Hausdorff dimension is still ongoing. In this paper, we summarize past researchers’ investigations on fractal dimensions of the Weierstrass function graph. Full article

This paper introduces a novel chaotic finance system derived by incorporating a modeling uncertainty with an absolute function nonlinearity into existing financial systems. The new system, based on the works of Gao and Ma, and Vaidyanathan et al., demonstrates enhanced chaotic behavior with a maximal Lyapunov exponent (MLE) of 0.1355 and a fractal Lyapunov dimension of 2.3197. These values surpass those of the Gao-Ma system (MLE = 0.0904, Lyapunov dimension = 2.2296) and the Vaidyanathan system (MLE = 0.1266, Lyapunov dimension = 2.2997), signifying greater complexity and unpredictability. Through parameter analysis, the system transitions between periodic and chaotic regimes, as confirmed by bifurcation diagrams and Lyapunov exponent spectra. Furthermore, multistability is demonstrated with coexisting chaotic attractors for p = 0.442 and periodic attractors for p = 0.48. The effects of offset boosting control are explored, with attractor positions adjustable by varying a control parameter k, enabling transitions between bipolar and unipolar chaotic signals. These findings underline the system’s potential for advanced applications in secure communications and engineering, providing a deeper understanding of chaotic finance models. Full article