Search Results (1,616)

The Yang local fractional calculus was developed to analyze discontinuous mappings. The results obtained within this framework are equally useful in the classical sense. The study of integrals and their approximation rules is an interesting field of research. Among them, the Gaussian quadrature rules are more useful due to their accuracy. In this manuscript, we will explore the error inequalities of the four-point Lobatto–Legendre Quadrature rule incorporating fractal calculus. Our approach is based on the generation of inequalities through a generalized fractal identity. First, we develop an auxiliary result. Then, the applications of various classes of mappings are defined over ${\mathbb{R}}^{\mathsf{t}}$ and auxiliary results, and we develop several new estimates. Additionally, an Artificial Neural Network ($\mathit{A}\mathcal{N}\mathcal{N}$) framework is used to analyse the profile and computational stability of the derived inequalities. Lastly, we have focused on the applicable analysis of the proposed results. This is the first study carried out on Lobatto-type inequalities within fractal space. Full article

Additively manufactured AlSi10Mg aluminum alloy was investigated at two-layer build orientations: 0° and 90°. Impact-induced fractures were generated per the ASTM standard Charpy test. The resulting fracture surfaces were analyzed using the multi-image-based fractal analysis method. We used a digital microscope to analyze the fracture surface and examined compression, neutral, and tension zones on the surface. We found that the crack propagated symmetrically across the surface. The results showed that the compression zone has the lowest fractal dimension compared to the tension and neutral zones. The 0° orientation samples have a higher fractal dimension than the 90° orientation samples. Full article

Existing analytical solutions for Brownian coagulation of fractal aggregates typically assume a constant geometric standard deviation during derivation to achieve closed-form expressions. This simplification introduces notable systematic deviations and limits their applicable ranges. The present study develops a power-law relaxation approximation to address this issue and derives corresponding analytical solutions for both the continuum and free-molecular regimes using the log-normal method of moments. The proposed analytical solutions reduce to the existing analytical expressions as the relaxation coefficient approaches zero and converge to the asymptotic solutions as the relaxation coefficient approaches infinity. This demonstrates that the two conventional models are unified within a single theoretical framework. The relaxation solutions are validated against numerical reference results across a wide range of mass fractal dimensions and initial geometric standard deviations. The average relative errors remain below 2.1% for all test cases, confirming that the proposed formulations achieve substantially higher accuracy than existing analytical solutions. The mass fractal dimension exerts only a slight effect on Brownian coagulation in the continuum regime, whereas it acts as a dominant factor in the free-molecular regime. For the latter regime, smaller fractal dimensions lead to larger collision cross-sections and substantially accelerate the size growth of aggregates. Full article

The variation features of metallic element grades can reflect the enrichment level of mineral deposits. Thus, quantitative characterization of the spatial patterns of metallogenic elements is fundamental to studying ore-forming processes and implementing mineral exploration. This study applies fractal theory and self-developed MATLAB computational scripts to process Cu grade datasets from 28 drillholes within the Pulang porphyry copper deposit, Yunnan Province. Both rescaled range (R/S) analysis and correlation integral methods were applied to clarify the spatial patterns of Cu grades in drill-cores. The calculated Hurst exponents ranged from 0.510 to 0.636, which demonstrated the persistent variation of Cu grades along the vertical direction of drillholes. This work further explored the correlation between Cu mineralization and fluctuations in correlation dimension (D_C), with D_C values spanning 0.011–2.873. Results indicate steep fractal gradient zones host high-grade copper ore bodies, and fractal dimension is a robust indicator to trace the migration of hydrothermal fluids. The Hurst exponents of Cu grade sequences correlate strongly with mineralization intensity, and ore-bearing veins extend continuously throughout all sampled drillholes. Accordingly, fractal gradients can be utilized to depict prospective zones for favorable mineralization in uncharted regions. This methodology may be applicable to other structurally controlled mineral deposits where similar fracture-controlled mineralization occurs, though further testing on different deposit types is needed. Full article

This study investigates the visual complexity of two opposing historic street façades on Sadrettin Konevi Street in Erzurum using an integrated framework that combines fractal analysis, user perception, expert evaluation, and TOPSIS-based decision-making. Fractal analysis was first performed using the box-counting method to calculate the façade fractal dimension (FD) values. In the second stage, perceived visual complexity was evaluated through a survey involving 100 architects and architecture-related professionals. In the third stage, visual complexity criteria identified from the literature were refined using the Delphi technique, reducing them to six key street-scale criteria. These criteria were then weighted using the Analytic Hierarchy Process (AHP) based on expert pairwise comparisons, and expert-based evaluation scores of the street façades were calculated accordingly. Finally, TOPSIS was used to integrate findings from fractal analysis, user perception, and expert evaluation into a unified comparative framework. The results demonstrated strong agreement among the three assessment approaches. The façade with the higher fractal dimension (FD) value (1.7574) was also perceived as more visually complex by most participants (67%) and achieved the highest expert-based weighted score. Rather than providing a universally validated model, the proposed framework is intended as a proof-of-concept that illustrates how computational, perceptual, and expert-based approaches can be integrated to assess visual complexity. Because the framework was demonstrated using only two opposing historic street façades, the findings should be interpreted as preliminary and case-specific rather than universally generalizable. Full article

Cyclic high-pressure gas impact is a promising waterless stimulation method for enhancing permeability in deep low-permeability coal seams. However, the nonlinear fracture evolution, cumulative fatigue damage, and coupled fracturing mechanisms during repeated gas impacts remain insufficiently understood. In this study, cyclic high-pressure gas impact tests were conducted on unconfined synthetic rock-like specimens under gas pressures of 5 MPa and 7.5 MPa. The macroscopic crack networks induced by repeated impacts were quantitatively characterized using digital image processing and box-counting fractal analysis. Ultrasonic P-wave velocity measurements were used to reconstruct the spatial evolution of internal damage after each impact, and an empirical Weibull statistical damage model was established to describe the nonlinear fatigue degradation process. In addition, two-dimensional LS-DYNA numerical simulations were performed to reveal the transient stress wave propagation and stress-field evolution during cyclic impacts. The results show that fracture propagation under cyclic gas impacts exhibits a distinct nonlinear pattern, characterized by slow early-stage damage incubation followed by rapid late-stage crack coalescence. The fractal dimension of the surface crack network increased markedly after repeated impacts, reaching a maximum of 1.51 under the 7.5 MPa condition. Ultrasonic damage analysis further indicates that, based on path-averaged evaluations, apparent damage is more pronounced near boundaries at the lower pressure, whereas higher pressure induces severe structural degradation along the central measurement paths, with a maximum damage value of 0.47. The combined experimental and numerical results suggest that the initial impacts generate transient stress waves and cumulative microcracking, thereby progressively weakening the rock matrix. This progressive degradation subsequently enables quasi-static gas wedging to drive macroscopic crack propagation and coalescence. These findings provide a preliminary phenomenological baseline for understanding cyclic gas-induced cracking, providing a preliminary basis for understanding waterless reservoir stimulation by cyclic gas impacts. Full article

This paper considers the problem of early detection of late blight (Phytophthora infestans) in tomatoes based on computer vision and machine learning methods. The main purpose of the study was to develop a representative dataset of images of tomato leaves and an approach to extracting informative features for classifying the stages of disease development. A new dataset was generated using tomato plants grown under greenhouse conditions, with leaf images subsequently captured under controlled laboratory conditions, including five stages of late blight progression with variability in imaging devices, lighting conditions, and temporal disease dynamics. To improve the quality of image analysis, a preprocessing stage was applied, including conversion to grayscale, median filtering, and binarization using the Otsu method. In addition to the traditional textural features, fractal analysis was used to quantify the structural complexity of the affected leaf areas. To verify the information content of the selected features, classification experiments were conducted using Random Forest, XGBoost, and Support Vector Machine models, and the quality was evaluated using accuracy, precision, recall, and F1-score metrics. The results showed that the combination of textural and fractal features contributes to a more accurate distinction between the stages of disease. The developed dataset and the proposed approach can be used in further research on plant disease diagnosis, agricultural monitoring, and precision farming systems although it should be acknowledged that the dataset is limited to greenhouse settings, and field-scale generalizability requires further validation. Full article

This study presents a unified topological and multifractal framework for the characterization of leaf surface complexity in Theobroma grandiflorum. By combining laser scanning confocal microscopy (LSCM)-derived three-dimensional profilometry with Minkowski functionals and multifractal analysis, we quantitatively distinguished the structural organization of adaxial and abaxial surfaces beyond conventional morphological descriptions. The analysis of the Minkowski functionals revealed distinct connectivity regimes and threshold-dependent transitions, indicating differences in surface topology and percolation behavior. These findings were further supported by multifractal spectra, which exhibited a broader distribution of singularities for the abaxial surface, reflecting increased heterogeneity and structural complexity. The introduction of the normalized differential parameter ΔP proved to be an effective strategy for directly quantifying morphological asymmetry, while radar plots enabled an integrated visualization of multivariate descriptors. From a physical perspective, the observed differences are consistent with the functional specialization of leaf surfaces, where the abaxial side exhibits greater structural complexity associated with gas exchange and environmental interaction, while the adaxial surface remains more compact and protective. Overall, the proposed approach advances the application of fractal and topological methods to biological systems, offering a scalable and transferable framework for the analysis of complex natural surfaces. This methodology opens new perspectives for studies in plant morphology, taxonomy, and environmental adaptation, aligning with the broader scope of fractal and fractional analysis in complex systems. Full article

A technological approach to the formation of a 3D nanocomposition material based on hierarchical porous nickel oxide nanoparticles incorporated into porous silicon with a dendritic porous structure is proposed. Porous silicon was used as a 3D porous template, in the presence of which porous hierarchical nickel oxide nanoparticles were synthesized using a “green” synthesis method followed by annealing in an oxygen-containing atmosphere. The resulting materials were characterized using scanning electron microscopy, transmission electron microscopy, X-ray spectral microanalysis, X-ray diffraction, and the BET method. The potential of a developed composition based on porous hierarchical nickel and silicon oxide nanoparticles to enhance the sensitivity of adsorption gas sensors was assessed using impedance spectroscopy in the presence of a probe gas (isopropanol). Gas sensitivity measurements were conducted at room and elevated temperatures in the frequency range from 100 Hz to 500 kHz. Differences in the dependences of the real part of impedance on the imaginary part were revealed for the porNiO-porSi composition in Nyquist coordinates. The results are discussed in terms of percolation theory and fractal organization. Full article

Façade design has long oscillated between repetitive order and visual disorder, and computational tools now make controlled irregularity easy to generate but hard to build; comparatively few studies follow a stochastic envelope from algorithm to a completed building. This paper examines controlled randomness as a deliberate, constructible strategy for façade composition. A parametric Grasshopper workflow erodes a square grid of wall modules through a seeded random-reduction rule—a deterministic, rule-based procedure that uses no machine learning or artificial intelligence—whose intensity increases up the building, producing a depth-modulated, quarry-like relief realized with one ordinary product—40 × 20 cm concrete blocks in four thicknesses—set to a color-coded projection key. The method is documented through a completed mid-rise residential building in Aramoun, Mount Lebanon, and evaluated for aesthetics, environmental function, structure, and constructability. Across 200 seeds, box-counting fractal dimension and lacunarity show that the generated pattern occupies a narrow, reproducible band of complexity—stable to within about 0.5%—statistically distinct from both periodic order and free disorder. A module-by-module comparison of the as-built record against the model shows the designed pattern reproduced across all 2978 units without projection-class error, the only departures being sub-centimeter positional shifts in the base plane absorbed from a structural tolerance. Bounded by explicit rules and parameterized in a standard masonry block, randomness behaves as a controllable, buildable design parameter rather than noise. Full article

Despite the fact that oxygen plasma treatments are widely used to modify the surface properties of polytetrafluoroethylene (PTFE), the long-term stability of these surface modifications has not been fully investigated. Specifically, the roles of morphological restructuring and chemical modifications at the surface remain to be understood. In this work, we treat commercial PTFE samples using O₂ plasmas at different discharge pressures to investigate their surface modifications and subsequent ageing at atmospheric pressure. We provide direct evidence that ageing behavior is governed by nanoscale and microscale restructuring of the plasma-modified interface, revealing a novel link between morphology dynamics and wettability properties. To capture this surface evolution, the modified interface was characterized using water contact angle (WCA) measurements, scanning electron microscopy (SEM), Fourier-transform infrared (FTIR) spectroscopy, and mass spectrometry (MS). Initial plasma treatment enhances PTFE hydrophobicity, shifting the WCA from ${\theta }_c\simeq 105°$ to a highly hydrophobic state of ${\theta }_c\simeq 135°$. By monitoring the samples in contact with air over a 67-day period a gradual transition toward hydrophilicity was revealed, with WCAs stabilizing at ${\theta }_c\simeq 70°$ after approximately 20 days. SEM observations identified time-dependent morphological degradation of plasma-induced nanostructures, while qualitative and quantitative FTIR analysis—utilizing the Specified Area Under Band (SAUB) method—confirmed corresponding shifts in carbonyl and hydrocarbon indices. These results demonstrate that ageing kinetics are a direct function of plasma pressure. The transition is further supported by a phenomenological fractal model, which confirms a morphological shift from an initial fractal surface ($d_s\simeq 2.25$) toward a standard flat geometry ($d_s=2$). Furthermore, calculations indicate a sign reversal in solid-gas interface tension parameters, reflecting the changed chemical nature of the surface. We conclude that the loss of hydrophobicity is driven by a synergistic interplay between morphological relaxation and chemical restructuring. Full article

Synthetic fractional Brownian motion (FBM) images serve as standard data for assessing fractal dimension (FD) estimation techniques. The synthesis pipeline transforms continuous FBM matrices into 8-bit grayscale images using linear mapping and quantization, a process often regarded as benign and rarely documented. If this procedure distorts FD estimations, algorithm comparisons based on such benchmarks merge performance with preprocessing errors. We demonstrate that grayscale conversion induces systematic distortion in FD estimation. Identical matrices were initially processed using three linear mapping strategies with varying emphases (direct, 3σ statistical, external-coefficient) and subsequently assessed with four FD algorithms (two DBC variants, Higuchi, PSD). The results demonstrate that linear mapping significantly alters FD estimates. In particular, the FD regression slope of the PSD approach notably decreased from 0.9955 (direct mapping) to 0.6400 (external-coefficient mapping), whereas Higuchi displayed negligible sensitivity. The near-perfect log-log linearity ruled out scaling breakdown. The mapping strategies produce distinct grayscale statistical properties that amplify quantization residuals. We present the relative residual to measure this amplification. The relative residual correlates strongly with FD deviations for DBC and PSD methods (r up to 0.97), while showing limited association with the Higuchi estimator. These results violate the assumption of benchmark neutrality in FBM-based FD assessment. FD benchmarking studies should, therefore, report preprocessing strategies and minimize relative residuals to ensure algorithmic comparability. Full article

To characterize the cross-scale coupling between macroscopic deformation and microscopic rough-surface contact in metal O-rings, this study proposes a macro–micro contact analysis and leakage flow regime identification method for metal O-ring sealing interfaces. Finite element analysis was employed at the macroscopic scale to obtain the sealing contact width and pressure distribution, while microscopic rough-surface morphology was characterized using fractal theory. Based on asperity contact analysis, the equivalent leakage channel height was determined, and the Knudsen number (Kn) was introduced to identify the fluid flow regime within micro-gaps. The results show that the O-ring cross-section flattens into an elliptical shape under compression, while the contact pressure exhibits a saddle-shaped distribution, and micro-gap leakage channels remain present. As the compression ratio increased from 5% to 25%, the sealing contact width and contact pressure increased, whereas the leakage gap height decreased significantly, resulting in an increase in the Kn values from 0.00176 to 0.045, 0.07, 0.155, and 2.78. Consequently, the flow regime evolved from continuum flow through the transition regime to molecular flow. The findings reveal the cross-scale coupling mechanism between macro–micro contact behavior and leakage flow regime evolution, providing practical guidance for determining the preload level and selecting appropriate leakage models in the engineering design of metal O-ring seals. Full article

Rolling bearing fault features are often weak and easily affected by noise and interference. To improve fault feature extraction performance, this paper proposes an AHBKA-VME-SMHD method. First, the black-winged kite algorithm is improved by opposition-based learning, a Gompertz-based adaptive step size strategy, and an NGO-inspired random displacement strategy. Then, the improved algorithm is used to optimize the penalty factor and desired mode center frequency of VME, guided by a composite fitness function combining Higuchi fractal dimension and energy concentration index. Finally, SMHD is applied to enhance periodic impulsive components, and envelope spectrum analysis is used to identify fault characteristic frequencies. The proposed method is validated using simulated signals and two real-world bearing datasets, namely the CWRU and XJTU-SY datasets. The results show that the proposed method extracts clearer fault-related harmonics than the comparison methods. In addition, it obtains higher kurtosis and Gini index values and lower envelope spectrum entropy values, demonstrating its effectiveness for rolling bearing fault feature extraction. Full article

Conventional recurrent and convolutional-recurrent neural architectures are often effective in capturing local and sequential dependencies. However, they may be insufficient for representing hierarchical scale-dependent structures that occur in time series with fractal or multifractal properties. This study addresses this limitation by modifying the FractalNet-LSTM architecture through the modification of its fractal block. Rather than treating the fractal block as a fixed component, the proposed approach introduces parametric branching, which allows the number of parallel computational paths to be adapted to the structural complexity of the analyzed time series. The study evaluates the proposed architecture on three time series datasets from different domains. Prior to model training, the datasets were analyzed using methods to assess self-similarity, long-range dependence, and multifractal properties. The forecasting performance of the modified FractalNet-LSTM was compared with LSTM, BiLSTM, CNN-LSTM, and the classical FractalNet-LSTM model across several forecasting horizons. The experimental results indicate that the proposed architecture can improve forecasting accuracy for time series with pronounced fractal and multifractal characteristics. The advantage is most evident for datasets with heterogeneous scale-dependent behavior, while the improvement is less pronounced for weakly multifractal or near-monofractal series. For example, on the Industrial Boiler dataset, the proposed model increased R² from 0.8046 to 0.9195 for the 8-step horizon compared with the classical FractalNet-LSTM. On the Green Energy Demand dataset, R² increased from 0.9316 to 0.9808 for the 1-step horizon and from 0.7357 to 0.8445 for the 8-step horizon. The results indicate that shorter forecasting horizons may require fewer branches, whereas longer horizons and more complex multifractal structures may benefit from deeper or more expressive fractal blocks. Full article