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A scaling law for the osmotic pressure of quasi-two-dimensional polymer melts as a function of concentration is obtained, which shows fractal characteristics. Structural properties such as the chains' contour length and their inner-monomer pair distribution function display fractal scaling properties as well. These predictions are confirmed with mesoscale numerical simulations. The chains are swollen and highly entangled, yet Flory's exponent is always ν = 1/2. The melt can be considered a fluid of "blobs" whose size becomes renormalized in terms of the contour's length while the fractal dimension df increases monotonically between 5/4 and 2, as the monomer concentration is increased. The semidilute scaling of the pressure is recovered when df = 1. Our results agree with recent experiments and with numerical reports on quasi-2d melts. This work provides a new paradigm to study and interpret thermodynamic and structural data in low-dimensional polymer melts, namely as fractal macromolecular objects. [ABSTRACT FROM AUTHOR]

Hardness of thin films is a noteworthy property in the electronic and mechanical industry and is generally observed to be dependent on the degree of roughening facilitated from surface heights' surface spatial heterogeneity at the mesoscopic observation scale. Nonetheless, owing to enhanced scale fluctuations and higher-order central moments, conventional parameters provide limitations and errors in capturing the spatial heterogeneity of surfaces. Herein, we have utilized scale-independent fractal parameters to analyze the spatial heterogeneity of surface heights in Ti1−xSixN ternary hard films deposited with varying Si doping concentrations using sputtering technique. The fractal dimension, lacunarity coefficient, Moran index, surface entropy, Otsu's separability, and fractal succolarity were computed to provide an overarching understanding of the surface heights' spatial heterogeneity. Principal component analysis was employed on the data sets to identify the parameter(s) accounting for the maximum variance and accordingly, the structure–property relation between spatial heterogeneity of surface and hardness is analyzed and discussed in the context of the fractal dimension of surface heights. The results indicate the possibility of mesoscopic surface engineering and, consequently, tuning of hardness and modulus of elasticity in Ti1−xSixN hard films by mere changing of surface spatial heterogeneity facilitated by the fractal dimension of surface heights. [ABSTRACT FROM AUTHOR]

Additive incorporation into polymers can enhance properties, such as the dielectric constant and thermal conductivity. The improved properties of polymer composites have largely been attributed to aggregate and network formation, but the influence of material texture requires clarification. Herein, the relationship between the texture of composite materials and thermal conductivity was investigated based on the fractal nature of composites consisting of polyvinylidene fluoride (PVDF) with β-Si3N4 (SN) as a high-thermal-conductivity filler. The SN particle groups were found to contribute to the formation of thermal conduction paths in the composite material and, thus, improve the thermal conductivity. The characteristics of the particle group clusters involved in SN particle percolation were elucidated by analyzing the interactions between multiple particles as a function of the amount of each fractal dimension. In addition, based on the statistical relationships obtained from the multifractal analysis, the strengths of the interactions between particles during thermal conduction path formation were estimated. Overall, the texture of the SN/PVDF material was characterized by local particle group formation (self-assembly) and subsequent global particle group network construction (self-organization), depending on the amount of SN added. These results suggest that multifractal dimensions can be used as material design indices for the texture of polymer composites. [ABSTRACT FROM AUTHOR]

In this work, we address the following question: Is it possible for a one‐dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by n$$ n $$, each one only able to bend on the set associated with the n$$ n $$‐th step in the construction of the Cantor set and compute the Γ$$ \Gamma $$‐limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension ln2/ln3$$ \ln 2/\ln 3 $$, which is the dimension of the Cantor set. [ABSTRACT FROM AUTHOR]

Phenol–formaldehyde resin can be used to improve oil recovery; its key lies in its aggregation behavior and blocking strength. However, the traditional phenol–formaldehyde resin used in the petroleum field is all prepared by phenol and formaldehyde. In order to get rid of the dependence on fossil resources and make full use of renewable biomass resources, we used the abandoned walnut shells of the unique agricultural and forestry crops in Gansu province, which contains lignin partially replaced phenol to synthesize the new sulfonated lignin-based phenol–formaldehyde resin (SLPFR). The results showed that the optimum conditions for microwave polymers were polystyrene substitution rate of 20 wt%, decomposition temperature of 160 ℃, decommissioning time of 20 min, and sodium hydroxide concentration of 0.3 mol/L. Infrared spectroscopes and scanning telescopes have shown that after disintegration, there was an increased concentration of phosphorus, which had more active spots and was more suitable for replacing phenylphenol-synthetic formaldehyde resins. LC–MS indicated the molecular mass of SLPFR and the possible structure of molecules, indicating the successful synthesis of SLPFR. In this work, the aggregation behavior and dispersion stability of SLPFR were investigated from the composition of formation water. The effects of metal cations (Na+, Mg2+, Ca2+) on the dispersion stability of SLPFR in formation water were determined by turbidimetry, and the effects of metal cations on the particle size and zeta potential of the SLPFR system were measured by dynamic light scattering method and electrophoretic light scattering method. The stability of the aggregate structure of the SLPFR system was calculated by combining the fractal dimension. In this paper, the surfactant + SLPFR system and partially hydrolyzed polyacrylamide (HPAM) + SLPFR system were designed for the specific conditions of oil reservoirs, and the effects of metal cations on the aggregation behavior and dispersion stability of these two systems were studied. Based on the above comprehensive analysis, aggregation models were constructed to describe the aggregation behavior and dispersion stability of the HPAM + surfactant + SLPFR system. This makes it possible to predict in real time the migration and plugging of SLPFR aggregates in formation water. [ABSTRACT FROM AUTHOR]

Rock burst prevention is a critical concern in deep rock engineering. Combining laboratory tests and numerical simulations, this work investigates the cutting-seam method and its mechanism for preventing strain rock bursts. In the laboratory test, the true triaxial loading system is used to simulate rock strain bursts. The test results show that the cutting-seam samples better preserve their integrity compared to the sample without the measure. The cutting seam effectively halts the development of penetrating cracks on the free face. After the rock burst, the number of fragments from the cutting-seam samples significantly reduces compared to the intact sample. The debris characteristic analysis shows a better fractal characteristic in the fragment size-number than the fragment size-mass, with the fractal dimension decreasing from the intact sample to the single-seam and further to the double-seam sample. Acoustic emission results demonstrate that the cutting seam reduces the stored strain energy and cumulative damage in the rock sample during the loading, and the double-seam effect is better than the single-seam case. Based on the laboratory tests, discrete element simulations are conducted to reveal that the position, depth, and spacing of the cutting seam influence its effectiveness in preventing rock bursts. The influence mechanism and the parameter selection are also discussed. Finally, simulations applied to an engineering project verify that the cutting-seam method alters the surrounding stress distribution of the tunnel, shifting stress concentration zones deeper and effectively reducing strain energy density near the seams. These findings are expected to enhance the understanding of the cutting-seam approach and provide theoretical support for its practical application. Highlights: A cutting-seam approach is proposed and tested to prevent the strain rock burst. The cutting seam physically halts the penetrating cracks and minimizes damage to the free surface. Different cutting-seam parameters and their influences are discussed through laboratory and numerical tests. The cutting-seam method shifts the stress concentration area deeper and reduces the strain energy density nearby. [ABSTRACT FROM AUTHOR]

Purpose: To quantify chorioretinal microvascular damage and recovery post-treatment in patients with acute syphilitic posterior placoid chorioretinitis (ASPPC) using fractal dimension (FD). Methods: Retrospective cohort study of patients with serologically confirmed syphilitic uveitis. We obtained optical coherence tomography angiography (OCTA) scans at baseline and follow-up after intravenous penicillin treatment and computed FD of the superficial capillary plexus (SCP), deep capillary plexus (DCP), and choriocapillaris (CC) using ImageJ. Results: We enrolled seven patients with ASPPC (11 eyes), and 17 control subjects (34 eyes). Pre-treatment averages of FD-SCP, FD-DCP, and FD-CC were: 1.672 (±0.115), 1.638 (±0.097), and 1.72 (±0.137); post-treatment: 1.760 (±0.071), 1.764 (±0.043), and 1.898 (±0.047). After treatment FD-CC increased in all 11 eyes with an average of 0.163 (p = 0.003); FD-DCP increased in 10 (91%) eyes with an average of 0.126 (p = 0.003); and FD-SCP increased in seven (64%) eyes with an average of 0.089 (p = 0.059). Compared to the post-treatment FD values in the syphilitic group, controls had similar FD-SCP (p = 0.266), FD-DCP (p = 0.078), and FD-CC (p = 0.449). Conclusions: CC and DCP are mostly affected in ASPPC with minimal changes in the SCP. All vascular layers FD recovered after completing antibiotic treatment. [ABSTRACT FROM AUTHOR]

To study the deformation failure and damage evolution law of weathered granite (WG) under different confining pressures, triaxial compression simultaneous acoustic emission (AE) tests were conducted on WG, and the mechanical and AE parameter characteristics of WG were obtained. It is shown that with the increase of confining pressure, the strength and modulus of elasticity of WG increase, and the failure pattern evolves from splitting failure to shear failure. Using the Mohr-Coulomb criterion, the cohesion c of WG is calculated to be 7.03 MPa, and the angle of internal friction φ is 61.95°. Analysing the AE peak frequency and energy at the characteristic points of the differential stress-axial strain curves, it is evident that the confining pressure limits crack evolution. The fractal characteristics of the WG AE ringing counts at each confining pressure are good, and the correlations between their ln r-ln C(r) curves and the fitted curves are all above 0.90. The WG at each confining pressure showed an increasing growth rate in the values of the damage variables and a rapid and substantial decrease in the fractal dimension before failure occurred. The study's results can provide a theoretical basis for the stability analysis of the WG surrounding rock. [ABSTRACT FROM AUTHOR]

A novel complexity index for axisymmetric forged components in forging processes is introduced based on their overall geometric shape and degree of deformation. This approach involves extracting the circumferential section area of axisymmetric forgings, characterizing the variation in this geometric feature using fractal dimension analysis, and calculating the area ratio between the cross-section and circumscribing rectangle of the forging. To verify this approach, H-shaped forgings were selected, and the proposed complexity index and three existing shape complexity factors were calculated. A hot forging process was performed virtually and practically to check the processing difficulty of H-shaped parts, and the analysis results from the simulation and experimental tests demonstrated that the complexity index was more reasonable for evaluating the difficulty of forging parts. To further validate the applicability of the complexity index, several published cases involving one-stage, two-stage, and multistage forging processes were investigated in detail. Finally, the potential application was discussed with published cases and an actual case from the industry. Regarding the actual five-stage forging process of the bushing part, the deviation of the complexity index calculated at each stage from the ideal value is within a small range between 1.3 and 6.9%, and the statistically shortest average die life of different stages reaches 190,000 pieces. The promising results could further demonstrate that the novel complexity index could work as a useful tool to support the process design of the multistage forging process. [ABSTRACT FROM AUTHOR]

Let k \ge 1 be an integer and f a holomorphic endomorphism of \mathbb {P}^k (\mathbb {C}) of algebraic degree d\geq 2. We introduce a dynamical volume dimension for ergodic f-invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than (k-1)\log d, a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k=1, but depends on the dynamics of f to incorporate the absence of an analogue of Koebe's theorem and the non-conformality of holomorphic endomorphisms for k\geq 2. If \nu is an ergodic f-invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of \nu. As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k\geq 1. Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents. [ABSTRACT FROM AUTHOR]

We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension N\ge 2, we show the C^{1, \alpha } regularity of the free boundary outside of a singular set of Hausdorff dimension at most N-3. In particular, we prove that the free boundaries are C^{1, \alpha } regular in dimension N=2, while in dimension N=3 the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension N=2, which are minimizing for one-phase functionals with weight functions in L^\infty that are arbitrarily close to a positive constant. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite dimensional subspaces of a Banach space. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation. [ABSTRACT FROM AUTHOR]

Thermal shock behaviors of porous alumina samples with different porosities (4.6–16.0 %) and mean grain sizes (2.72–5.02 μm) were investigated under different quenching temperatures. The slit island method was used to analyze the structures of the pores. The effect of the pore structure on the thermal shock resistance of alumina ceramics was described quantitatively. The results revealed that the thermal shock resistance decreased when the fractal dimension of the pores increased. Alumina ceramic strips with a porosity and mean grain size of 4.6 % and 5.02 μm, respectively, were fabricated using raw powders with a mean particle size of ∼0.3 μm. These samples possessed the highest residual strength ratio (∼46.16 % at 600 °C) and critical temperature difference (∼262 °C). In addition, the fractal characterization of cracks on the surfaces of the alumina ceramics after thermal shock was carried out using the box-counting dimension method. In addition, we investigated the correlation between the mechanical properties and fractal dimensions of the surface cracks. The box-counting dimension increased with crack propagation and negatively correlated with the residual strength. Furthermore, the propagation rate of the surface cracks decreased as the temperature difference increased, resulting in a decrease in the growth rate of the fractal dimension. [ABSTRACT FROM AUTHOR]

Here, we report room temperature deposition of nickel oxide (NiO) thin films on silicon and glass substrates by direct current (dc) magnetron sputtering using a metallic nickel target. The effect of sputtering power (20 W, 25 W, and 30 W) on the thin films, their surface scaling, fractal dimension, and optical properties are extensively investigated. Autocorrelation and height–height correlation functions were applied to AFM images to extract deep insights about the thin films' surfaces. Fractal dimension (D f) was extracted through the power spectral density (PSD) function. Various scaling exponents, including α, β, and 1/z, of the NiO film's surface were independently observed. The local roughness exponent, α, was approximately 0.81 for films deposited with 20 W sputtering power and decreased to 0.71 with higher sputtering power. The interface width (σ) scales with sputtering power (Sp) as ∼ Spβ, with a growth exponent (β) value of 1.17. The lateral correlation length (ξ) follows as ∼ Sp1/z with a 1/z value of 0.703. Additionally, optical parameters were recorded through UV–Vis. optical spectroscopy, and an attempt was made to correlate them with fractal parameters (D f & α). Optical absorption (reflection) increased (decreased) with increasing D f values. The minimum (maximum) reflection (absorption) was observed on the roughest surface (D f = 2.29). The calculated band gap decreased with increasing fractal dimension. This investigation suggests that sputtered surfaces with minimal reflectivity, band gap, and enhanced light-absorbing capacity could potentially be used as active solar layers for advanced optoelectronic devices. [ABSTRACT FROM AUTHOR]

The macroscopic physical and mechanical behavior, mesocosmic pore size distribution, and microscopic particle morphology of polyurethane concrete (PUC) during freeze–thaw conditions are thoroughly examined in this research. The results reveal that as the number of freeze–thaw cycles (F-T cycles) increases, the P-wave velocity of PUC drops constantly, the peak shear strength reduces gradually, but the porosity increases. A four-peak exponential function can be used to describe the pore dispersion features of PUC. With the increase in the number of F-T cycles, PUC showed an increasing trend in all characteristic pore radii, but only the volume fraction of macropores increased. The compressive and shear strength of PUC were positively correlated with the volume fraction of nanopores and micropores and negatively correlated with the volume fraction of macropores. It was proved that the macroporous content is the main reason for the decrease in the mechanical properties of PUC. The fractal dimension of PUC decreases with the increase in the number of F-T cycles, indicating that the complexity of the PUC pore structure decreases after F-T cycles. At the micro-meso-macro scale, the damage progression of PUC under freeze–thaw conditions demonstrates a gradual and mutually feeding connection. The damage evolution process can be broadly divided into three stages, beginning with the formation of micropores in the polyurethane agglomeration, progressing to the gradual germination and expansion of cracks in the polyurethane agglomeration, and finally to the connected sum expansion of cracks between the polyurethane agglomeration and the aggregates, resulting in significant deterioration of the macromechanical properties of PUC. [ABSTRACT FROM AUTHOR]

The evaluation of mechanical properties in materials derived from cements incorporating ultrafine fly ash was performed. An examination of the influences of curing time, fly ash concentration, and particle size on the compressive strength of materials derived from cements was executed. Upon the integration of ultrafine fly ash into materials derived from cements, the pore structure was scrutinized using the mercury intrusion technique, and the impact of fly ash concentration and particle size on diverse pore structure parameters was investigated. According to the findings, the quantity of macropores in materials derived from cements progressively diminished with the rise in fly ash concentration, and the porosity consistently decreased with the extension of curing time, culminating in an elevation of compressive strength. The proportion of gel pores in materials derived from cements contracted with the increment in fly ash concentration, resulting in a decline in compressive strength. Fly ash particles of reduced dimensions exhibit heightened compressive strength due to the diminution in particle size. Employing fractal theory, the fractal dimension of the pore surface in materials derived from cements was ascertained, and an outstanding correlation between the fractal dimension and the total pore surface area was discovered. Grounded in the connection concerning the compressive strength of materials derived from cements and assorted pore structure parameters, a suitable relationship model has been devised between the total pore volume, transitional pore volume, and compressive strength. At the same time, a fitting scope of the model had been determined. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to study the correlation between the thicknesses of the C–Ni films that have been prepared by RF-magnetron sputtering on quartz substrates and their three-dimensional (3D) micro morphology. In this work by AFM images, this paper studied stereo metric analysis of these films. Design/methodology/approach: The C–Ni films have been prepared by RF-magnetron sputtering on quartz substrates using a mosaic target consisting of pure graphite and strips of pure nickel approximately 2 cm2 attached to the graphite race track. The field emission scanning electronic microscopy (FESEM) images were used for the morphological characterization. Findings: The histogram peaks are zero for all samples and the histograms are almost symmetric around zero. Temperature did not have much effect on the degree of isolation, so all four diagrams have similar results. The qualitative observations through statistical parameters of the 3D surface texture revealed that the smoothest surface has been obtained for C-Ni films annealed at 500 °C (Sa, Sq, Sz and Sv have the lower values), while the most irregular topography has been found for C-Ni films annealed at 300 °C (the fractal dimension D = 2.01 ± 0.131). Originality/value: As shown in FESEM images, the size of the particles was increased for films deposited from 300 ºC to 800ºC; however, at 1000ºC, it decreased significantly. The histogram peaks are zero for all samples and the histograms were almost symmetric around zero. Also, the largest and lowest root mean heights (Sq) belong to films at 300 °C and 500 °C. Furthermore, the more irregular surface was found at 300 °C, and the more regular surface was found at 500 °C. As the temperature was increased to 800 °C, the values of the IAPSD function increased systematically, and then the values of the IAPSD function was decreased in the fourth sample. The surface skewness of samples annealed at 1000 °C was positive which confirms the lack of dominance of cavities on their surface with the highest amount of C-Ni films at 800 °C. [ABSTRACT FROM AUTHOR]

AbstractUtilizing the Matlab platform, two-dimensional diffusion-limited aggregation and three-dimensional diffusion-limited cluster aggregation models were established to simulate the hydrophobic flocculation of coal particles. The formation process of flocs aggregated by monomer particles and small groups was simulated from both plane and space perspectives. The gyration radius fractal dimension increased from 1.401 to 1.539 in the DLA model and from 1.623 to 1.729 in the DLCA model, indicating that the floc morphology gradually stabilized and the structure became increasingly regular during the hydrophobic flocculation process. The porosity of DLA increased from 0.762 to 0.851, indicating that the dendritic structure of flocs formed a spatial shielding effect on the planar structure, which prevented the particles from entering the flocs. In contrast, the porosity of DLCA first increased from 0.506 to 0.611 and then decreased, stabilizing at approximately 0.53, indicating that small groups filled the internal voids of flocs in the spatial structure, leading to a gradual increase in floc density. The surface fractal dimension and density of flocs were calculated using the surface fractal and hydrostatic sedimentation-terminal velocity theories, respectively, confirming the conclusions drawn from the simulation analysis. [ABSTRACT FROM AUTHOR]

This paper investigates a special Cantor set with different upper and lower box dimensions, and discusses its Hausdorff dimension. In addition, we extend the set and generate a class of sets with different upper and lower box dimensions, reaching a sufficiently large interval within the range of the box dimension. [ABSTRACT FROM AUTHOR]

This work showed the capability of handling large number of classes for classification with human cognition inspired methods. A cognition based techniques for both feature extraction, (self-similarity feature, Intensity Level Multi Fractal Dimension (ILMFD)) as well as classification purpose (decision tree clustering based multi-level Artificial Neural Network classifier-MLANN-DTC) were employed to implement facial recognition based object detection system. A DTC based approach reduces the search space time and also provides opportunity for very less amount of classes (a smaller part of the large number of classes) to be handled by the respective classifier for classification. It also mimics fast recognition capability of humans. In this work, two different databases were used for experiment, first one is our own collected facial images from rotation based video clips (117 persons and 40 facial images per person) named as NS database, and other is standard ORL database (40 persons and 10 facial images per person). In pre-processing step, the facial images were segmented to obtain facial part using context window based texture of pixels (CWTP) & back-propagation neural network (BPNN) based model and then a scale and rotation independent ILMFD feature was computed from each segmented image. Further, a combination of K-means and hierarchal clustering was used to build super classes. All classes' data were distributed among these 6 super classes (heuristically chosen) for own NS database and 3 for ORL database as per their similarity based on ILMFD features. Multi-level ANNs models were employed for all super classes and further their classification results were fed into decision clustering based model to obtain fine-tuned results, which showed significant improvement in terms of classification efficiency. This approach believes in center tendency of largest cluster to refer the actual class decision from multiple decisions obtain corresponding to multiple input data of the same class. In this work, the MLANN-DTC based proposed model has produced 89.542 ± 1.167% and 87.098 ± 2.066% classification efficiency (± standard deviation) for single input and for group based decision (decision clustering), 95.042 ± 0.719% and 89 ± 2.549% for NS and ORL database respectively. This improved classification results motivate its application for other object recognition and classification problems. The basic idea of this work also supports better handling of classification which deals with a large number of classes. [ABSTRACT FROM AUTHOR]

AbstractBased on fractal theory, the geometric size, surface morphology, and spatial structure of flocs in the hydrophobic flocculation process of coal particles were examined via particle size distribution detection, image analysis technology, and numerical simulation. The floc size distribution curve was classified and discussed via the dual-domain fractal rule, and the variation law of the composition and distribution of the coarse- and fine-grained coal particle flocs with the flocculation process was clarified. The various levels and characteristics of the floc growth geometry were explained by the analysis of the fractal dimension of the individual floc surface and the multiple spectra of the floc groups in the metallographic microscopy image. The hydrophobic flocculation process of coal particles was simulated via the diffusion-limited cluster aggregation model. The gyration fractal dimension was used to examine floc fragmentation and reconstruction and clarify the growth law of the floc spatial structure during the hydrophobic flocculation process. [ABSTRACT FROM AUTHOR]

Layered joints significantly affect the propagation and attenuation of engineering blasting stress waves and have a significant influence on rock breakage. In this study, the explosion crack propagation mechanism of a stratified rock mass with weak or no cementation is analyzed. Multilayer plexiglass plates are employed to simulate a layered rock mass, establishing an experimental model to investigate the vertical interaction between explosive charges and layered rock masses. The extent of ruptures and damage within the charging and sealing sections of the rock mass is analyzed. The explosion cracks at the bottom of the hole are relatively dense, whereas those at the hole opening are relatively sparse. The range of the crushing and fracture zones around the gun hole in the charging section is larger than that in the blocking section. Moreover, as the blocking section approaches the free plane, the crushing zone size decreases. A notable area of rapid damage escalation is identified between the blocking and charging sections, leading to a significant increase in damage intensity. The numerical analysis results indicate that the cumulative explosive energy within the charging section surpasses that of the blocking section, and the rate of energy accumulation is notably higher in the charging region. [ABSTRACT FROM AUTHOR]

Seismic activity in Aswan is influenced by the complex interactions of tectonic plates, the accumulation of stress, and the presence of geological fault systems. It revealed that epicenters are well distributed along four fault segments in a conjugate pattern, indicating a prominent E-W compressional stress. This research aims to explore the characteristics of seismicity and seismotectonics, with a focus on assessing their implications for risk reduction and disaster management in this densely populated region. A data review from the Egyptian National Seismological Network (ENSN) identified 464 earthquakes occurred between 2000 and 2021, with local magnitudes ranging from 0.3 to 4.4, and depths up to 25 km. The calculated Gutenberg-Richter b-value is approximately 0.87 ± 0.05, indicating a gradual stress accumulation. The current analysis shows a more consistent level of moderate seismic activity, unlike previous studies in Aswan region that reported a wide range of b-values from 0.554 to 1.07. This suggests that while earlier research captured a wider range of seismic behaviors, recent data indicates a stabilization in earthquake frequency and intensity. Additionally, the fractal dimension (Dc) calculated at 1.57 ± 0.04 shows an intermediate level of complexity and reflecting a clustering pattern of earthquakes. The variations in the b-value with different magnitudes and depths signify the involvement of active smaller faults, responsible for earthquakes up to magnitude 2.2, which then transition to fractured zones inducing earthquakes up to magnitude 2.5. This transition is followed by a decline in seismic activity, indicating regions that are potentially more likely to experience larger earthquakes. Moreover, stress disparities at various depths contribute to smaller earthquakes within the 5-10 km depth range. Return period analysis suggests that the earthquakes of magnitude 3.7 or higher are expected to occur approximately once every decade in Aswan. These findings are of utmost importance for earthquake risk reduction, hazard assessment, and the sustainable development of Aswan area. [ABSTRACT FROM AUTHOR]

This retrospective study aimed to assess different macular neovascular network characteristics in relation to changes in best corrected visual acuity (BCVA) over 3 and 12 months following treatment. Using optical coherence tomography angiography, we reviewed the medical records of 46 treatment-naïve patients with neovascular age-related macular degeneration (nAMD) who received intravitreal aflibercept injections. The change in BCVA from baseline to 3 months and 12 months after treatment was recorded. The mean vessels percentage area, junctions density, lacunarity, and fractal dimension were significantly correlated with the change of BCVA from baseline to month 3 (P = 0.003, 0.046, 0.007, and 0.005 respectively). Fractal dimension and vessels percentage area were correlated with the change of BCVA from baseline to month 12 (P = 0.023 and 0.023 respectively). The findings suggest that baseline characteristics of macular neovascular complexes may serve as predictors for BCVA changes following treatment with aflibercept in nAMD patients. [ABSTRACT FROM AUTHOR]

The Upper Cretaceous Qingshankou Formation's lacustrine shales in the Songliao Basin are among China's most promising shale oil reservoirs. To elucidate their pore and fractal characteristics, a comprehensive set of analyses encompassing total organic carbon (TOC), X-ray diffraction (XRD), and low-temperature N2 adsorption (LTNA), Rock-Eval pyrolysis experiments and two-dimensional nuclear magnetic resonance (2D-NMR) were conducted. Using the Frenkel-Halsey-Hill (FHH) method, fractal dimensions (D) were calculated, and their relationship with pore metrics and shale compositions were explored. Two distinct fractal dimensions, D1 (0 < P/P0 < 0.5) and D2 (0.5

Summary In this study, the alternating magnetic field (AMF) with frequency of 50, 100, 150, 200 and 250 Hz at magnetic field intensity of 5 mT was used to assist the freezing of tilapia fillets, with no magnetic field (NMF) as a control group. The evaluation of thawed tilapia fillets encompassed an assessment of freezing curve, texture, microstructure, moisture state and protein structure, followed by a subsequent analysis of their correlation. The findings showed that the application of a magnetic field during the freezing process yielded favourable outcomes in enhancing the overall quality of tilapia fillets. Among them, 200 Hz had the best effect. Under the condition, the frozen tilapia fillets assisted by magnetic field showed the shortest freezing time, the least decrease in hardness and elasticity and the best chewiness after thawing. The size of the ice crystals generated during freezing was small and uniform, and the fractal dimension of the frozen sections was the highest. Muscle tissue was the least damaged, which led to the improvement of its water‐holding capacity. Magnetic field‐assisted freezing was found to maintain the α‐helix and β‐sheet in the secondary structure of proteins. The correlation analysis showed that cohesion, elasticity, fractal dimension, FD, A2, A22 and β‐turn were closely related to the quality changes of tilapia fillets after magnetic field treatment. [ABSTRACT FROM AUTHOR]

The contact stiffness of the mechanical joint usually becomes the weakest part of the stiffness for the whole machinery equipment, which is one of the important parameters affecting the dynamic characteristics of the engineering machinery. Based on the three-dimensional Weierstrass–Mandelbrot (WM) function, the novel normal contact stiffness model of the joint with the bi-fractal surface is proposed, which comprehensively considers the effects of elastoplastic deformation of asperity and friction factor. The effect of various parameters (fractal dimension, scaling parameter, material parameter, friction factor) on the normal contact stiffness of the joint is analyzed by numerical simulation. The normal contact stiffness of the joint increases with an increase in the fractal dimension, normal load, and material properties and decreases with an increase in the scaling parameter. Meanwhile, the fractal parameters of the equivalent rough surface of the joint are calculated by the structural function method. The experimental results show that when the load is between 14 and 38 N∙m, the error of the model is within 20%. The normal contact stiffness model of the bi-fractal surface joint can provide a theoretical basis for the analysis of the dynamic characteristics of the whole machine at the design stage. [ABSTRACT FROM AUTHOR]

In this paper, we study the shrinking target problem regarding Q-Cantor series expansions of the formal Laurent series field. We provide the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the unit disk I. [ABSTRACT FROM AUTHOR]

Building upon Li & Britz (Li NZ, Britz T. 2024 On the scale-freeness of random colored substitution networks. Proc. Amer. Math. Soc. 152, 1377–1389 (doi: 10.1090/proc/16604)), this study aims to introduce fractal geometry into graph theory and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore fractal-like graphs, termed iterated graph systems (IGS) for the first time. While the concept of substitution is commonplace in fractal geometry and dynamical systems, its analysis in the context of graph theory remains a nascent field. By delving into the properties of these systems, including distance and diameter, we derive two primary outcomes. Firstly, within the deterministic IGS, we establish that the Minkowski dimension and Hausdorff dimension align through explicit formulae. Secondly, in the case of random IGS, we demonstrate that almost every Gromov–Hausdorff scaling limit exhibits identical Minkowski and Hausdorff dimensions analytically by their Lyapunov exponents. The exploration of IGS holds the potential to unveil novel directions. These findings not only, mathematically, contribute to our understanding of the interplay between fractals and graphs, but also, physically, suggest promising avenues for applications for complex networks. [ABSTRACT FROM AUTHOR]

This study analyzes the fractal aspects of the structures of S100 proteins to better understand their structural complexity. We take into account 33 solution structures and 18 crystal structures corresponding to human S100 proteins for the calculation of mass and surface fractal dimensions. The mass fractal dimension value is calculated as Dm = 1.54, confirming the extended conformation of the dimers of these proteins. The mean value of the surface fractal dimension is Ds = 2.35 ± 0.09 when computed using solution structures and Ds = 2.23 ± 0.05 when computed using crystal structures, revealing the surface irregularities of S100 proteins. Changes in surface fractal dimensions have been recorded for S100 proteins due to the changes in the pH of the environment, due to mutations in their sequences that alter how the protein folds, and/or due to their interactions with ions and/or ligands that reflect the structural rearrangements that occur upon binding. These changes can significantly influence the biological activity of the protein, making the fractal dimension of the surface a valuable parameter in studying protein functions, interactions, and potential therapeutic targeting. [ABSTRACT FROM AUTHOR]

A variety of methods for 3D object recognition and registration based on a deep learning pipeline have recently emerged. Nonetheless, these methods require large amounts of data that are not easy to obtain, sometimes rendering them virtually useless in real-life scenarios due to a lack of generalization capabilities. To counter this, we propose a novel local descriptor that takes advantage of the fractal dimension. For each 3D point, we create a descriptor by computing the fractal dimension of the neighbors at different radii. Our redmethod has many benefits, such as being agnostic to the sensor of choice and noise, up to a level, and having few parameters to tinker with. Furthermore, it requires no training and does not rely on semantic information. We test our descriptor using well-known datasets and it largely outperforms Fast Point Feature Histogram, which is the state-of-the-art descriptor for 3D data. We also apply our descriptor to a registration pipeline and achieve accurate three-dimensional representations of the scenes, which are captured with a commercial sensor. [ABSTRACT FROM AUTHOR]

Using Landsat TM/OLI remote sensing images and social statistical data from 1995, 2000, 2005, 2010, 2015, and 2020, construction land information in Beijing's main urban area was extracted with ArcGIS 10.4.1 and other software. Based on calculations of the expansion speed, expansion intensity, fractal dimension, and elasticity coefficient, the spatiotemporal expansion characteristics of the urban area of Beijing were analyzed to reveal the laws and driving forces of urban expansion in Beijing. The results showed that the urban construction land area in Beijing expanded by a factor of 0.53 from 1995 to 2020, and its expansion speed and intensity gradually slowed. The overall expansion trend is that the central urban area remains basically unchanged, while the peripheral areas are rapidly expanding, showing a trend of rapid growth first and then stable growth, and the urban layout is basically stable. The urban expansion of Beijing has led to increasingly complex, tortuous, and unstable boundaries. Overall, the center of gravity of Beijing is moving toward the northeast, and the elasticity coefficient of urban expansion is 1.67 times that of a reasonable coefficient. The intensity and direction of urban expansion in Beijing are most significantly related to population mobility. Research on the expansion of Beijing lies the foundation for the integration and coordinated planning of resources in the various districts of Beijing and provides a basis for its sustainable development. [ABSTRACT FROM AUTHOR]

In the modified formula for the cubic law, the roughness correction coefficient C serves as a parameter that indicates the characteristics of the roughness of a rock's surface. In this study, surface data for natural rock samples were acquired through high-precision 3D scanning and combined with publicly accessible CT scan data on rough rock fractures to generate spatial coordinates. The roughness correction coefficient C was calculated and analyzed using both formulaic and numerical methods. The analysis revealed significant effects of scale on the roughness correction coefficient for rock fractures within a size range of 5 to 10 cm (determined based on the actual sample size), with tensile fractures demonstrating greater variations compared to shear fractures. When calculating the roughness correction coefficient on the same fracture surface in different directions, significant directional effects were observed. Furthermore, elliptical fitting demonstrated favorable results. The conclusion drawn was that the roughness correction coefficient for fracture surfaces can be represented effectively using a tensor form, thereby simplifying the expression of directionality. Calculating and analyzing the fractal dimension of a rough surface further confirmed the existence of effects of size on roughness. [ABSTRACT FROM AUTHOR]

In this study, we analyze the changes over time in the complexity and structure of words used in article titles and the connections between articles in citation networks, focusing on the topic of artificial intelligence (AI) up to 2020. By measuring unpredictability in word usage and changes in the connections between articles, we gain insights into shifts in research focus and diversity of themes. Our investigation reveals correspondence between fluctuations in word complexity and changes in the structure of citation networks, highlighting links between thematic evolution and network dynamics. This approach not only enhances our understanding of scientific progress but also may help in anticipating emerging fields and fostering innovation, providing a quantitative lens for studying scientific domains beyond AI. [ABSTRACT FROM AUTHOR]

This study contributes to design studies by offering a novel approach to understanding the spatial layout of Jiangnan gardens through the lens of fractal geometry. Analyzing 106 gardens, we found that the ideal fractal dimension range for Jiangnan gardens is 1.148 ~ 1.276, with gardens in the 2500 ~ 7200 m2 range exhibiting the highest complexity (1.238 ~ 1.276). Additionally, for gardens ranging from 2500 ~ 20,000 m2, the maximum spacious space area stabilizes, no longer expanding indefinitely with overall area. This suggests a design principle of spatial proportion and balance. By quantifying spatial complexity and the contrast between spacious and profund spaces, the study provides a new method for evaluating garden design and can help students and designers better apply the principles of Jiangnan garden design. [ABSTRACT FROM AUTHOR]

The fractal function approximation is an important branch of fractal theory. Compared with the ordinary function approximation, its advantage is that it can not only improve the accuracy of the approximation, but also maintain the fractional-dimensional characteristics of the objective function. However, in the process of studying the fractal function approximation, it is inevitable to calculate the fractal dimension of the sum function after adding multiple functions. This paper systematically sorts out and computes the fractal dimension of sum functions for different cases, so as to make a good theoretical foundation for fractal function approximation. Finally, considering that the change of aircraft aerodynamic value has fractional dimension characteristics, the use of the fractional function to approximate the aerodynamic numerical function has an inherent advantage, and the use of fractional calculus for aerodynamic modeling can more accurately measure the maximum angle of attack. Therefore, the application of fractional calculus and fractal function approximation theory in aerodynamic numerical modeling has been discussed in detail at the end of this paper, and the corresponding fractional aerodynamic modeling process is also demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we estimate the oscillation of harmonic functions on the (N − 1)-dimensional Sierpiński gasket KN, where K3 is the classical Sierpiński gasket. By using this result, we prove that both the box dimension and the Hausdorff dimension of graphs of harmonic functions on KN are log2N. [ABSTRACT FROM AUTHOR]

Recently, many number of people various countries contracted the COVID-19 (C-19) disease. The identification of this harmful disease and other pneumonia diseases is an important process in the medical world. CT-Scan is mainly prescribed to predict the severity of lung diseases affected by C-19 and other pneumonia diseases. Generally, a CT-Scan image is represented as a grayscale image (GI). Biomedical-oriented GIs are more complex and they are too difficult to recognize the status of lung diseases directly from the experimental images. To address this type of medical problem, a multifractal approach can be applied to analyze and illustrate the GIs in detail. Therefore, the multifractal dimensional analysis is used to diagnose and explore the vehemence of the contamination levels in the lungs. In this study, the complexity of CT-Scan images of C-19 patients is analyzed in order to perform a comparison in terms of age and noise levels. It was also discovered that the intricate of CT-Scan images is considerably varied for patients aged 50 and above when compared to younger subjects. This comparative study indicates that the deadly virus affects elderly persons compared to youngsters. The proposed age-based classification is supported by image processing techniques, qualitative measures, and statistical tools. The obtained results are demonstrated graphically by Generalized Fractal Dimensions (GFD) spectrum, 3D visualization, ANOVA table, and box plots to expose the rate of discrimination of complexity in CT-Scan lung images. [ABSTRACT FROM AUTHOR]

The electrical activities of the reliable neuron models have different responses within intrinsic biophysical effects and can functionalize for asymmetric coexisting electrical activities under anti-monotonicity phenomenon. This manuscript presents mathematical analysis of neuron model based on Maxwell electromagnetic induction through newly proposed fractal-fractional differential and integral operators. The neuron model based on Maxwell electromagnetic induction changes with time along a fractal dimension that describes the cumulative chaotic phenomenon. The cumulative chaotic phenomenon of neuron model is mathematically modeled via exponential and Mittag–Leffler kernels with variable and fixed fractal and fractional orders. In order to exhibit fractal properties and memory effects, the neuron model is discretized by means of Adams–Bashforth-Moulton method that allows explicitly to compute the approximate solution of neuron model. The comparison of neuron model based on memory effect and fractal dimension have distinguished the evolution of neuron model at (i) variability of fractal order with fixed fractional order, (ii) variability of fractional order with fixed fractal order, and (iii) variability of fractal order as well fractional order. [ABSTRACT FROM AUTHOR]

Both the amount and spatial arrangement (configurational heterogeneity) of resources can affect population abundance and community diversity via influence on the growth, survival, reproduction, recruitment and movement of species. However, in most cases, it is difficult to separate the effects of resource amount from arrangement because these two attributes are often naturally correlated. In this study, we examined the configurational heterogeneity of resources (oviposition habitat—emergent rocks, ER) within rivers and decoupled the effects of resource amount from those due to the spatial arrangement on oviposition by eight species of aquatic insects (seven caddisflies and one mayfly). To capture the configurational heterogeneity of resources in 28 sites (riffles) across multiple streams in Australia and Scotland, we calculated fractal dimensions (DB) using the box‐counting technique. We then used simulated riffles to explore how numbers of ER, edginess (the proportion of ER along river margins) and patchiness (clustering of emergent rocks in the middle) separately and together affected the values of DB using asymptotic regression models. Finally, we used multiple regression to test whether the numbers of egg masses laid in natural riffles of each of the eight species were explained by the number of ER, fractal dimension or both. The distributions of ER in natural riffles were scale‐independent, self‐repeating patterns (i.e. they were fractal), and values of DB varied significantly among riffles. Variations in fractal dimensions among simulated riffles were significantly related to the number of ER, edginess and patchiness. However, in natural riffles, only the number of ER and patchiness affected DB. Egg mass abundances were related to the fractal dimensions of ER distributions in riffles in three species and to the number of ER in five species. The fractal dimensions of riffles are unlikely to be driven by large‐scale processes but instead may result from within‐riffle variability that influences rock movement, arrangement and emergence. Increased oviposition by aquatic insects in riffles with greater numbers of ER suggests that these species may be limited by the amount of oviposition resources. Of the species responding to fractal dimension, two species favoured tightly clustered ER whereas the third favoured ER in much looser clusters. It is feasible that aquatic insects can detect ER clusters from the air by responding to changes in the reflectivity of water (albedo) caused by turbulence around ER. That fractals can capture configurational heterogeneity of resources in streams suggests that this technique is useful to test for general ecological patterns across diverse stream systems. Whether configurational heterogeneity influences adults directly or indirectly, these results showed that the amount and arrangement of suitable oviposition habitat plays a role in determining egg mass densities, with potential consequences for larval densities. These patterns may have important community‐level and functional consequences and hence spatial arrangements should be considered when humans manipulate these resources in rivers. [ABSTRACT FROM AUTHOR]

Surface morphology plays a crucial role in friction between two contacting geomaterial surfaces, yet many questions remain unanswered regarding how detailed frictional responses deviate from analytical solutions for smooth surfaces due to the presence of roughness. In this study, we revisit the Cattaneo‐Mindlin problem for contacts between two fractally rough elastic or elasto‐plastic spheres generated based on ultra‐high degree (e.g., up to 2,000) spherical harmonics with the corresponding wavelength less than a thousandth of the mean grain diameter. Transverse contacts are simulated by finite element method, validated by the extended Cattaneo‐Mindlin solution to full slide regime for smooth sphere contacts. Extensive simulations are conducted to study contacts between two rough spheres with various surface geometries, micro friction coefficients, normal contact distances, relative roughness, fractal dimensions, and wavelength ranges. Out results indicate that: (a) the new analytical solution can approximately predict the macro contact response except for extremely high relative roughness and narrow wavelength range; (b) deviations induced by roughness from smooth sphere contacts can be neutralized by plasticity, high normal contact interference, and high micro friction coefficient; and (c) fractal dimension impacts frictional contacts less than relative roughness. The main cause of these phenomena can be credited to the underlying microscale contact information. Contact area and stress distributions and their evolutions provide concrete evidence of these observed behavior. This work provides a pathway for applying computational contact mechanics to many geophysical fields, such as the asperity model in earthquake science and the mechanics of granular materials. Plain Language Summary: Geophysical bodies, such as faults, rocks, and grains, exhibit surface roughness across length scales. Contacts between rough surfaces are usually simplified to a rough‐to‐flat contact scenario. To release these strong assumptions, for the first time, extensive finite element simulations are conducted for transverse contacts between two rough grains. Particularly, grain roughness is controlled by ultra‐high degree spherical harmonics, which enables depicting surfaces of diverse grains from silica sands up to asteroids. Our results highlight the importance of surface features in frictional contacts, and indicate that the effects of roughness can be alleviated by the plasticity, contact conditions, and high intrinsic friction coefficient. Furthermore, fractality, quantifying how much the surface fluctuates over length scales, impacts frictional contacts less than roughness amplitude. This study provides a pathway for applying computational contact mechanics to many geophysical applications involving frictional contacts, such as asperity models in earthquake science and mechanics of granular materials. Key Points: Transverse contacts are simulated between two fractal rough grains generated by ultra‐high degree spherical harmonics for the first timeCattaneo‐Mindlin solution is extended to the full slide regime, capable of predicting the critical partial‐to‐complete sliding transitionEffects of roughness can be generally neutralized by plasticity, high contact interference, and friction coefficient [ABSTRACT FROM AUTHOR]

Contact interface is essential for the dynamic response of the bolted structures. To accurately predict the dynamic characteristics of bolted joint structures, a fractal extension of the segmented scale model, i.e., the JK model, is proposed in this paper to comprehensively analyze the dynamic contact performance of engineering surfaces and revisit the multi-scale model based on the concept of asperities. The influence of asperity geometry, dimensionless material properties, and the elastic, elastoplastic, and full plastic mechanical models of a single asperity is established considering the asperity–substrate interaction. Then, a segmented scale contact model of rough surfaces is proposed based on the island distribution function in a strict sense. The mechanical contact process of determining rough surfaces is divided into small-scale, medium-scale, and large-scale stages. Moreover, cross-scale boundary conditions, i.e., al1′, al2′, and al3′, are provided through strict mathematical deduction. The results show that the real contact area and contact stiffness are positively correlated with fractal dimension and negatively correlated with fractal roughness. On a small scale, the contact damping decreases with an increase in load. In meso-scale and large-scale stages, the contact damping increases with the load. Finally, the reliability of the proposed model is verified by setting up three groups of modal vibration experiments. [ABSTRACT FROM AUTHOR]

The global incidence of cutaneous squamous cell carcinoma (cSCC), a prevalent and aggressive skin cancer, has risen significantly, posing a substantial public health challenge. This study investigates the tumor microenvironment (TME) of cSCC by focusing on the spatial distribution patterns of immune and vascular markers (CD31, CD20, CD4, and CD8) using fractal dimension (FD) analysis. Our analysis encompassed 141 cases, including 100 invasive cSCCs and 41 specimens with pre-invasive lesions exclusively, and the rest were peripheral pre-invasive lesions from the invasive cSCC class. The FD values for each marker were computed and compared between pre-invasive and invasive lesion classes. The results revealed significant differences in FD values between the two classes for CD20 and CD31 markers, suggesting distinct alterations in B cell distribution and angiogenic activity during cSCC progression. However, CD4 and CD8 markers did not exhibit significant changes individually. Still, the CD4/CD8 ratio showed a significant difference, suggesting a potential shift in the balance between T helper and cytotoxic T cell responses, impacting the immune landscape as lesions progressed from pre-invasive to invasive stages. These findings underscore the complexity and heterogeneity of the TME in cSCC and highlight the potential of FD analysis as a quantitative tool for characterizing tumor progression. Further research is needed to elucidate the implications of these differences in the clinical management of cSCC. [ABSTRACT FROM AUTHOR]

The document "Fractal Analysis and Fractal Dimension in Materials Chemistry" explores the application of fractal theory in understanding materials chemistry, including catalysis, chemical reactions, and material design. The research focuses on determining fractal dimensions of materials through micrograph analysis, adsorption isotherms, and small-angle X-Ray scattering. The articles in this special issue cover topics such as high-precision machine tools design, alumina ceramics synthesis, nanoparticle catalysts, porous materials, zeolite synthesis, fluid flow through porous media, and thin film technology. This collection provides valuable insights for researchers interested in fractal dimension determination, materials characterization, and design optimization. [Extracted from the 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 Rs is more obvious under a highly effective confining pressure, and the strain-softening degree of Rs can reach 0.669 when the average particle size d ¯ is 0.5 mm. The changes in the normalized modulus ratio Eu/Eu50 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 MFL 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 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. [ABSTRACT FROM AUTHOR]

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 (Df3) 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 DT3 displaying high accuracy and rationality. Compared with the traditional fractal dimension Df3, the proposed DT3 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 DT3 and oil or gas transfer, which can provide the theoretical foundation and technical support for oil and gas exploration. [ABSTRACT FROM AUTHOR]

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%). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 A B / F , Q C B , Q C V , Q E , Q F M I , Q G , Q M I , and Q N C I E . [ABSTRACT FROM AUTHOR]