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A multifunctional, multiphase-supported photocatalyst was fabricated via the combination of LCD three-dimensional (3D) printing, cold plasma discharge (CPD) technology, and surface chemistry to be used for the degradation of organic pollutants in water and for the generation of hydrogen gas. BiOBr was chosen as the main catalytic phase and was used in association with a phosphor, SrAl2O4:Eu2+,Dy3+ (SAED), and a metal–organic framework, Ni-BDC. Their immobilization was realized thanks to polydopamine (PDA), which is used as a coupling agent. While BiOBr was the most active photocatalytic phase, the phosphor acted as a light-storing component for photodegradation in the dark and overall, as a boost for photoactivity. PDA also boosted the photoactivity, in parallel with its crucial role in the immobilization of the SAED/Ni-MOF/BiOBr catalyst on fractal polymer support. An in situ complexation-assisted precipitation (ISCAP) process was used, allowing the formation of the catalytic phases with nano features. The resulting hybrid photocatalysts were tested through rhodamine B (RhB) dye photodegradation upon visible light irradiation or light/dark alternating conditions to simulate more realistic variable light illumination. We found that, although used in a very small amount, the Ni-MOF had a significant impact on the photoactivity, forming a Z-scheme heterojunction with BiOBr. Finally, H2 evolution testing further established the photocatalytic activity and versatility of the supported catalysts. [ABSTRACT FROM AUTHOR]

Introduction: Electrical stimulation of the brain has shown promising prospects in treating various brain diseases. Although biphasic pulse stimulation remains the predominant clinical approach, there has been increasing interest in exploring alternative stimulation waveforms, such as sinusoidal stimulation, to improve the effectiveness of brain stimulation and to expand its application to a wider range of brain disorders. Despite this growing attention, the effects of sinusoidal stimulation on neurons, especially on their nonlinear firing characteristics, remains unclear. Methods: To address the question, 50 Hz sinusoidal stimulation was applied on Schaffer collaterals of the rat hippocampal CA1 region in vivo. Single unit activity of both pyramidal cells and interneurons in the downstream CA1 region was recorded and analyzed. Two fractal indexes, namely the Fano factor and Hurst exponent, were used to evaluate changes in the long-range correlations, a manifestation of nonlinear dynamics, in spike sequences of neuronal firing. Results: The results demonstrate that sinusoidal electrical stimulation increased the firing rates of both pyramidal cells and interneurons, as well as altered their firing to stimulation-related patterns. Importantly, the sinusoidal stimulation increased, rather than decreased the scaling exponents of both Fano factor and Hurst exponent, indicating an increase in the long-range correlations of both pyramidal cells and interneurons. Discussion: The results firstly reported that periodic sinusoidal stimulation without long-range correlations can increase the long-range correlations of neurons in the downstream post-synaptic area. These results provide new nonlinear mechanisms of brain sinusoidal stimulation and facilitate the development of new stimulation modes. [ABSTRACT FROM AUTHOR]

The dynamics of thin-walled parts are highly affected by the clamping conditions. Clamping stiffness is a function of c lamping force and surface roughness profiles of the clamp and part. Since the surface profiles cannot be altered, estimating clamping stiffness as a function of the clamping force is essential to simulate vibrations of the machined thin-walled parts. This paper presents the modeling of clamping stiffness as a function of the applied clamping force and material properties. Surface profile parameters are estimated from the identified contact stiffnesses evaluated using the Finite Element (FE) model. The contact stiffnesses are either predicted directly from the proposed mechanics model of the part or estimated from the Fractal surface parameters. It is shown that an average clamping stiffness can be predicted from the Fractal surface parameters, or directly and more accurately from the static model of the clamped part. [ABSTRACT FROM AUTHOR]

The manuscript investigates the utilization of Koch curves and hexagonal ring-shaped geometry for an ultra-wideband fractal antenna, achieved through a combination of DGS, parasitic elements, and EC-SRR. The antenna's properties are investigated without and with DGS, Parasitic element, and EC-SRR in the hexagonal ring-shaped geometries for broader band characteristics. Simulation results show that the antenna designed without DGS resonates at five distinct frequencies, while with DGS, Parasitic element, and SRR, it resonates at six frequencies. An enhanced bandwidth of 10.42 GHz (99.43%) is revealed in the final antenna geometry, and the proposed antenna resonates at six frequencies, 3.4, 5.8, 8.5, 11, 14.2, and 15 GHz, with reflection coefficients of − 24.33, − 38.10, − 28.35, − 27.39, − 32.32, and − 15.93 dB, respectively. Combining a defected ground structure, parasitic element, and EC-SRR increases frequency bands and enhances the BW and reflection coefficient. With an overall dimension of 30 mm × 24 mm, the proposed antenna is suitable for wireless applications in the frequency ranges of 2.40–3.89 GHz and 5.33–15.75 GHz. The proposed antenna is fabricated, and the results are measured. It is found that simulated and measured results are in good agreement with each other. [ABSTRACT FROM AUTHOR]

In this paper, an analytical approach has been developed for electromagnetic field enhancement in fractal nanostructure under the quasi-static limit. A metallic fractal embedded in an unbounded, isotropic, and lossless chiral medium has been treated under this approach. In addition, the field enhancement is also noted by imposing surface plasmon resonance condition. To achieve the desired objective, Maxwell divergence equations are considered and written in an appropriate form by mean of fractional calculus. [ABSTRACT FROM AUTHOR]

Objectives: The present study aimed to investigate the incremental prognostic value of the right ventricular fractal dimension (FD), a novel marker of myocardial trabecular complexity by cardiac magnetic resonance (CMR) in patients with arrhythmogenic cardiomyopathy (ACM). Methods: Consecutive patients with ACM undergoing CMR were followed up for major cardiac events, including sudden cardiac death, aborted cardiac arrest, and appropriate implantable cardioverter defibrillator intervention. Prognosis prediction was compared by Cox regression analysis. We established a multivariable model supplemented with RV FD and evaluated its discrimination by Harrell's C-statistic. We compared the category-free, continuous net reclassification improvement (cNRI) and integrated discrimination index (IDI) before and after the addition of FD. Results: A total of 105 patients were prospectively included from three centers and followed up for a median of 60 (48, 66) months; experienced 36 major cardiac events were recorded. Trabecular FD displayed a strong unadjusted association with major cardiac events (p < 0.05). In the multivariable Cox regression analysis, RV maximal apical FD maintained an independent association with major cardiac events (hazard ratio, 1.31 (1.11–1.55), p < 0.002). The Hosmer–Lemeshow goodness of fit test displayed good fit (X2 = 0.68, p = 0.99). Diagnostic performance was significantly improved after the addition of RV maximal apical FD to the multivariable baseline model, and the continuous net reclassification improvement increased 21% (p = 0.001), and the integrated discrimination index improved 16% (p = 0.045). Conclusions: In patients with ACM, CMR-assessed myocardial trabecular complexity was independently correlated with adverse cardiovascular events and provided incremental prognostic value. Clinical relevance statement: The application of FD values for assessing RV myocardial trabeculae may become an accessible and promising parameter in monitoring and early diagnosis of risk factors for adverse cardiovascular events in patients with ACM. Key Points: • Ventricular trabecular morphology, a novel quantitative marker by CMR, has been explored for the first time to determine the severity of ACM. • Patients with higher maximal apical fractal dimension of RV displayed significantly higher cumulative incidence of major cardiac events. • RV maximal apical FD was independently associated with major cardiac events and provided incremental prognostic value in patients with ACM. [ABSTRACT FROM AUTHOR]

We propose and theoretically substantiate an algorithm for conducting a generalized Chaos game with an arbitrary jump on finite convex polygons of the extended hyperbolic plane whose components in the Cayley–Klein projective model are the Lobachevsky plane and its ideal domain. In particular, the defining identities for a point dividing an elliptic, hyperbolic, or parabolic segment in a given ratio are proved, and formulas for calculating the coordinates of such a point at a canonical frame of the first type are obtained. The results of a generalized Chaos game conducted using the advanced software package pyv are presented. [ABSTRACT FROM AUTHOR]

Earthquakes in the Himalayan region predominantly exhibit tectonic characteristics, stemming from fault slippage caused by the ongoing collision between the Indian plate and the Eurasian plate. The western Nepal and adjoining regions (28.5°N to 31.0°N and 78°E to 82.96°E) have hosted the damaging earthquake namely, the 1905 Kangra earthquake, the 1991 Uttarkashi earthquake, and the 1999 Chamoli earthquake in the past and need to understand the recent seismic activity of the region. In this study, we quantified seismic activity in the region from the fractal approach. The key idea of fractal method is based on the self-similarities of objects at different scales. Twelve windows of 100 seismic events with 25-event shifts were analyzed to observe variations in b-value and Dc value for the earthquakes (Mb ≥ 4.0) transpiring between 1964 and 2021. The three consecutive windows from 1991 to 2006 demonstrate a variation in b-value from 0.80 ± 0.07 to 0.90 ± 0.09 and a decrement in Dc value from 1.24 ± 0.03 to 1.20 ± 0.03, indicating a high clustering of earthquakes during the occurrence of major earthquakes, illustrated by the instances like the 1991 Uttarkashi earthquake, the 1999 Chamoli earthquake, and the magnitude 6.1 Mb earthquake in 2004. In addition, the study shows the negative correlation between b-value and Dc value for spatial and temporal distribution of earthquakes in the study region. This study demonstrates the use of b-value and Dc value for identifying seismicity patterns prior to major earthquakes. Continuous monitoring of these seismic patterns could be valuable for assessing the potential seismic hazard and risk throughout the entire Himalayan region. [ABSTRACT FROM AUTHOR]

3D digital models can facilitate the understanding of micro- and nanoscale pore structures and enhance the permeability prediction accuracy of geological materials. However, the reconstruction of digital models using the conventional Markov chain Monte Carlo (MCMC) algorithm cannot well reflect the microstructure of the heterogenous media and needs to be improved. In this study, referring to the box-counting method (BCM) of fractal theory, the conventional probability distribution function was extended to different scale probability functions, and the concept of value paths was introduced to control the reconstruction quality. Subsequently, a comparison to the conventional MCMC algorithm was conducted based on the statistical function and the pore fractal dimension to verify the effectiveness of the improved algorithm. Finally, the reconstruction model was applied in finite element simulation to predict the permeability of soft soil. The verification results showed that the improved MCMC algorithm can well realize the reconstruction of heterogeneous porous media since the squared correlation coefficients of statistical functions of the 3D reconstruction and the original image are above 0.98, and the pore fractal dimension of reconstructions using the improved MCMC was closer to the original image than that using the conventional MCMC. Furthermore, in permeability prediction, permeability simulations using images reconstructed with an improved MCMC algorithm are closer to the experimental values, with a maximum prediction error of 1.87 times (that of MCMC was 4.21 times) the corresponding experimental value. [ABSTRACT FROM AUTHOR]

Autism spectrum disorder (ASD) is a neurological condition characterized by impaired functional connectivity (FC) networks in the brain. There are several brain networks associated with ASD that have been studied for ASD diagnosis, but the results are inconsistent. A functional magnetic resonance imaging (fMRI) study was performed to address this gap by comparing brain networks among autistic individuals and individuals with typical development (TD) using data from the ABIDE-I and ABIDE-II databases. Blood oxygen level-dependent (BOLD) time series were extracted from 236 regions of interest (ROI) in fMRI data using three atlases: Gordon's, Harvard Oxford, and Diedrichsen. Consequently, 27,730 nonlinear features are extracted from FC matrices, including fractals, non-fractals, and Pearson correlation coefficients (PCC). A parametric and nonparametric classifier was used to analyze the top 0.1%, 0.3%, 0.5%, 0.7%, 1%, 2%, and 3% of features based on the XGBoost feature ranking algorithm. In the study, we found that non-fractal brain FC measures can accurately identify ASD and TD more effectively than fractal and PCC measures. Classifiers performed well, with FC features at the top 0.3%. The classification model at OHSU was more accurate than the model at other sites. There was 100% accuracy at a single site and 96.17% accuracy at all sites using a multilayer perceptron classifier with non-fractal features. The classifier model shows that Cingulo-Parietal Task Control (13.6%), Retro-Splenial Temporal (RST) (13.3%), and Salience (11.6%) are significant contributors. The optimal performance was observed in features derived from networks such as RST and default (4 connections), auditory and Fronto-Parietal Task Control (3 connections), Cingulo-Opercular Task Control (COTC), and ventral attention (3 connections) within COTC (3 connections), visual and COTC (3 connections), and cerebellum and COTC (3 connections). Based on the results, non-fractal-based FC has value in distinguishing ASD from TD using resting-state fMRI. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new class of generalized enriched cyclic contraction and establish a related fixed point theorem in the setting of a Banach space. As an application to the fractals, we develop a new iterated function system (IFS) consisting of enriched cyclic (bn, ...n(t), βn(t))-contractions. [ABSTRACT FROM AUTHOR]

Research has demonstrated the significant influence of surface roughness on electrical conductivity and streaming potential. This article presents a model for the flow potential in a porous medium with rough fractures, based on fractal theory and representative elementary volume theory. Expressions for the model's permeability, conductivity, and coupling coefficient between flow and potential are derived. The results indicate that both the relative roughness and structural parameters of the model have certain effects on the conductivity and coupling coefficient. By comparing experimental data from previous literature with the predicted values from the model, the correctness and effectiveness of the model are validated. [ABSTRACT FROM AUTHOR]

Predicting the invasion depth of superficial esophageal neoplasms (SENs) is crucial for an indication toward endoscopic treatment. We aimed to develop imaging biomarkers to predict the invasion depth of SENs using the intrapapillary capillary loop (IPCL) patterns. Texture and fractal analyses were performed on IPCL demarcations. Logistic regression models were built to distinguish narrow‐band images (NBIs) of T1b from T1a and high‐grade dysplasia (HGD). Classification performances of a base model composed of the age, sex, height, and body mass index of the patient, and a morphometric model comprising the texture and fractal metrics along with the base model parameters were compared. NBIs of 39 patients including T staging of 6 HGD, 22 T1a, and 11 T1b were analyzed. In the logistic regression by a multivariable generalized linear model classifying histological depths, only texture entropy was significantly and positively associated with the invasion depth. Lower texture entropy of IPCL demarcations was significantly associated with diagnoses of T1b (odds ratio: 0.144, 95% confidence interval: 0.019–0.687) in comparison to the stratum with higher texture entropy. The performance of the morphometric model represented by areas under the receiver operating curves (0.795) was better than that of the base model (0.562). We suggest morphometric analyses, particularly texture entropy, have the potential as image biomarkers to aid in the prediction of SEN invasion depth using IPCL patterns. [ABSTRACT FROM AUTHOR]

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z2 −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White's test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers. [ABSTRACT FROM AUTHOR]

The novelty of this work is that it is the first to introduce complex-valued suprametric spaces and apply it to Fractal Generation and mixed Volterra–Fredholm Integral Equations. In the realm of fuzzy logic, complex-valued suprametric spaces provide a robust framework for quantifying the similarity between fuzzy sets; for instance, utilizing a complex-valued suprametric approach, we compared the similarity between fuzzy sets represented by complex-valued feature vectors, yielding quantitative measures of their relationships. Thereafter, we establish related fixed point results and their applications in algorithmic and numerical contexts. The study then delves into the generation of fractals, exemplified by the Barnsley Fern fractal, utilizing sequences of affine transformations within complex-valued suprametric spaces. Moreover, this article presents two algorithms for soft computing and fractal generation. The first algorithm uses complex-valued suprametric similarity for fuzzy clustering, iteratively assigning fuzzy sets to clusters based on similarity and updating cluster centers until convergence. The distinctive pattern of the Barnsley Fern fractal is produced by the second algorithm's repetitive affine transformations, which are chosen at random. These techniques demonstrate how well complex numbers cluster and how simple procedures can create complicated fractals. Moving beyond fractal generation, the paper addresses the solution of mixed Volterra–Fredholm integral equations in the complex plane using our results, demonstrating numerical illustrations of complex-valued integral equations. [ABSTRACT FROM AUTHOR]

The traditional Langmuir equation displays drawback in accurately characterizing the methane adsorption behavior in coal, due to it assuming the uniform surface of coal pores. Additionally, the decay law of gas adsorption capacity with an increasing coal reservoir temperature remains unknown. In this study, the fractal adsorption model is proposed based on the fractal dimension (Df) of coal pores and the attenuation coefficient (n) of the adsorption capacity. The principles and methods of this fractal adsorption model are deduced and summarized in detail. The results show that the pore structures of the two coal samples exhibit obvious fractal characteristics, with the values of fractal dimensions (Df) being 2.6279 and 2.93. The values of adsorption capacity attenuation coefficients (n) are estimated as −0.006 and −0.004 by the adsorption experiments with different temperatures. The proposed fractal adsorption model presents a greater theoretical significance and higher accuracy than that of the Langmuir equation. The accuracy of the fractal adsorption model with temperature effect dependence is verified, establishing a prediction method for methane adsorption capacity in deep coal reservoirs. This study can serve as a theoretical foundation for coalbed methane exploration and development, as well as provide valuable insights for unconventional natural gas exploitation. [ABSTRACT FROM AUTHOR]

Previous research has demonstrated that interpersonal dynamics are fractal, and that conflict is a key control parameter that drives fractal complexity. The present study aimed to extend this line of research to examine the putative fractal structure of conflict dynamics over time, and the role that this self-organizing fractal structure may play in the resilience of romantic relationships. An experience sampling methodology was used to assess levels of conflict, satisfaction, and commitment in the dating relationships of undergraduate students, three times per day for 30 days. Hypothesis 1 was supported, with conflict ratings over time generally conforming to an inverse power-law distribution (IPL) distribution. Hypothesis 2 was supported as well, with better IPL fits measured as variance accounted for (R2), predicting higher levels of satisfaction and commitment over the 30 days. Hypothesis 3 showed mixed support, with moderate network linkages (i.e., soft assembly) between conflict and satisfaction and commitment predicting higher IPL fits (the linkage of satisfaction and commitment did not predict IPL fit as predicted). Hypothesis 4 predicted that IPL fit would interact with mean conflict, buffering the impacts of conflict on mean satisfaction and commitment across the 30 days. This hypothesis was not supported; however, several statistical factors may have obscured the buffering effects of higher IPL fit and so results may be inconclusive. These methodological factors, and others, are discussed along with the potential theoretical and practical implications of the current results. [ABSTRACT FROM AUTHOR]

In this investigation, we study a class of analytic functions of type Carathéodory style in the open unit disk connected with some fractal domains. This class of analytic functions is formulated based on a kind of Langevin differential equations (LDEs). We aim to study the analytic solvability of LDEs in the advantage of geometric function theory consuming the geometric properties of the Julia fractal (JF) and other fractal connected with the logarithmic function. The analytic solutions of the LDEs are obtainable by employing the subordination theory. [ABSTRACT FROM AUTHOR]

The MOF‐5@clinoptilolite (MOF@CP) composites are successfully synthesized using polyvinylpyrrolidone (PVP) for adsorption separation of CO2/CH4, CO2/N2, and CH4/N2. The effects of the PVP amounts on the dispersity of MOF‐5 on CP random lamellas of the MOF@CPs are evaluated via various characterizations. Meanwhile, their single‐component gas adsorption isotherms, breakthrough experiments, and cycling test are measured. The results elucidate that the used PVP amount has a significant influence on the particle size and the uniformity of MOF‐5 dispersed on CP random lamellas. The MOF‐5‐loaded amount in MOF@CP is estimated to be up to 42.52 wt.%. Especially, the surface fractal evolutions indicated the surfaces of MOF@CP became from rough to smooth with the increase of PVP. The CO2/CH4, CO2/N2, and CH4/N2 selectivity factors of MOF@CP are higher than that of CP, displaying a better separation performance. Their cycling test revealed that MOF@CPs could be used repetitively, highlighting efficiency for CO2 and CH4 separation. Meanwhile, the MOF‐5@CP stability under moisture is preliminarily investigated, showing higher moisture resistance stability than MOF‐5. Additionally, the grand canonical Monte Carlo simulations demonstrated the adsorption separation mechanism of the prepared MOF@CPs. [ABSTRACT FROM AUTHOR]

Fractal branching networks are structures that are ubiquitous in nature, possessing multiple transport characteristics, and have been the subject of long-term attention by many researchers. Currently, in the field of textiles, research on the ability of tree-like structures to enhance the water transport property of fabrics is mostly focused on the thickness direction of the fabric. However, there is little research on the liquid water transfer rate in the horizontal direction. In this article, the water transport performance of liquid water in tree-like fractal nonwoven fabrics with different shapes (tree-like and rectangular) was investigated, and the liquid water transport rate was measured using the horizontal wicking method. It was found that when the sum of the cube of the branch width was equal to the cube of the main stem width, the water transport efficiency of liquid water in the fabric is optimal. At the same time, a summary of the experimental data was made by comparing the water transport property of fabrics with corresponding equal areas but different widths or heights. It was found that for nonwoven fabrics with the same area size, tree-like structures can transport liquid water to a farther distance in the same amount of time. Furthermore, the liquid water transport ability of tree-like nonwoven fabrics in the horizontal direction was simulated using finite element analysis in COMSOL Multiphysics, and the results were in good agreement with the experimental values. [ABSTRACT FROM AUTHOR]

Abstract Interparticle pore space and vugs are two different scales of pore space in vuggy porous media. Vuggy porous media widely exists in carbonate reservoirs, and the permeability of this porous media plays an important role in many engineering fields. It has been shown that the change of effective stress has important effects on the permeability of vuggy porous media. In this work, a fractal permeability model for vuggy porous media is developed based on the fractal theory and elastic mechanics. Besides, a Monte Carlo simulation is also implemented to obtain feasible values of permeability. The proposed model can predict the elastic deformation of the fractal vuggy porous media under loading stress, which plays a crucial role in the variations of permeability. The predicted permeability data based on the present fractal model are compared with experimental data, which verifies the validity of the present fractal permeability model for vuggy porous media. The parameter sensitivity analysis indicates that the permeability of stress-sensitivity vuggy porous media is related to the capillary fractal dimension, capillary fractal tortuosity dimension, Young’s modulus, and Poisson’s ratio.

Abstract Underground coal seam mining significantly alters the stress and energy distribution within the overlying rock, leading to eventual structural degradation. Therefore, it is imperative to quantitatively identify the temporal and spatial characteristics of stress evolution of overlying rock caused by mining. This paper introduces a novel rock stress model integrating entropy and a spatial–temporal cube. Similar material model tests are used to identify the abrupt entropy changes within the mining rock, and the trend analysis is carried out to describe the spatial–temporal evolution law of stress during mining. Experimental findings indicate elevated stress levels in the unmined rock preceding and following the panel, as well as within specific rock strata above it. Definitively, dynamic stress arches within the surrounding rock of the stope predominantly bear and distribute the load and pressure from the overlying rock, and each stress mutation is accompanied by a sudden stress entropy change. Over time, z-score shows that the noticeable reduction in mining-induced overburden stress becomes increasingly pronounced, especially in the water-conducting fracture zone. The model's bifurcation set serves as the comprehensive criterion for the entropy-induced sudden changes in the rock system, signifying overall failure.

Abstract Background The aim of the study was to investigate the changes occurring in the mandibular condyle by using mentoplate together with rapid maxillary expansion (MP-RME) treatment in the correction of skeletal class III relationship, using fractal analysis (FA). Methods The sample consisted of 30 individuals (8–11 years) diagnosed with skeletal Class III malocclusion who underwent MP-RME treatment. Archival records provided cone-beam computed tomography (CBCT) images taken at two intervals: before MP-RME treatment (T0) and after treatment (T1). The CBCT images were obtained using standardized settings to ensure consistency in image quality and resolution. The trabecular structures in the bilateral condyles at both T0 and T1 were analyzed using FA. The FA was performed on these condylar images using the Image J software. The region of interest (ROI) was carefully selected in the condyle to avoid overlapping with cortical bone, and the box-counting method was employed to calculate the fractal dimension (FD). Statistical analysis was conducted to compare the FD values between T0 and T1 and to evaluate gender differences. The statistical significance was determined using paired t-tests for intra-group comparisons and independent t-tests for inter-group comparisons, with a significance level set at p 0.05). However, a significant gender difference was observed in FA values, with males exhibiting higher FA values in the left condyle compared to females at both T0 and T1 (p

Objective: Familial Mediterranean fever (FMF) is an inflammatory disease and chronic inflammation may affect bone turnover and metabolism. This study aimed to compare the morphological, fractal, and textural features of the mandibular bone in FMF patients with healthy controls on panoramic radiographs. Methods: Fifty patients with FMF and, age- and sex-matched 50 healthy controls were included in the study. Morphological evaluation of the mandibular cortex on digital panoramic images of a total of 100 individuals was performed using the mandibular cortical index (MCI). For fractal dimension (FD) and texture analysis of trabecular bone, regions of interest with a size of 50x50 pixels were selected from the trabecular bone region between the roots of the second premolar and first molar teeth. The boxcounting method was applied to calculate the FD. Since the pixel gray-scale levels of these regions showed different distributions, pre-processing was performed with histogram equalization for texture analysis. First-order and gray-level co-occurrence matrix-based second-order features of panoramic images were calculated and their textural characterizations were obtained. Results: The MCI values of the mandibular cortex did not significantly differ between the case and control groups (p>0.05). FD values for the trabecular bone were 1.43 in the case group and 1.44 in the control group, and there was no significant difference between them (p>0.05). First and second-order textural features of trabecular bone did not differ statistically significantly between the case and control groups (p>0.05). Conclusion: Morphological, fractal, and textural features of the mandibular bone did not differ on panoramic radiographs between FMF patients and healthy controls.

The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.

A hexagon-shaped fractal multiple-input multiple-output (MIMO) antenna for WLAN, WiMax, and C band applications is proposed. The antenna consists of two hexagon-shaped self-similar fractal structures placed at a 6 mm gap with a reduced ground plane along with a ground stub at the centre. The proposed design covers the 2.37–7.82 GHz band with band rejections at 2.54–2.87 GHz, 3.56–3.83 GHz and 6.26–7.13 GHz. The notches are introduced to avoid interference between different application bands. The parametrically optimised reduced height of the ground plane and the strategical placement of a stub at the centre of this ground plane provide notch bands and reduce the mutual coupling below −17 dB throughout the operating bands. The proposed antenna gain varies between 2–3 dBi in the operating bands. The diversity parameters like envelope correlation coefficient (ECC) and diversity gain, channel capacity loss (CCL) and Total Active Reflection Coefficient (TARC) are calculated and found to be well below the acceptable range. A prototype of the proposed design is fabricated and measured. The measurement results are in good agreement with their simulated counterparts. [ABSTRACT FROM AUTHOR]

Background: The aim of the study was to investigate the changes occurring in the mandibular condyle by using mentoplate together with rapid maxillary expansion (MP-RME) treatment in the correction of skeletal class III relationship, using fractal analysis (FA). Methods: The sample consisted of 30 individuals (8–11 years) diagnosed with skeletal Class III malocclusion who underwent MP-RME treatment. Archival records provided cone-beam computed tomography (CBCT) images taken at two intervals: before MP-RME treatment (T0) and after treatment (T1). The CBCT images were obtained using standardized settings to ensure consistency in image quality and resolution. The trabecular structures in the bilateral condyles at both T0 and T1 were analyzed using FA. The FA was performed on these condylar images using the Image J software. The region of interest (ROI) was carefully selected in the condyle to avoid overlapping with cortical bone, and the box-counting method was employed to calculate the fractal dimension (FD). Statistical analysis was conducted to compare the FD values between T0 and T1 and to evaluate gender differences. The statistical significance was determined using paired t-tests for intra-group comparisons and independent t-tests for inter-group comparisons, with a significance level set at p < 0.05. Results: The analysis revealed no statistically significant differences in the trabecular structures of the condyles between T0 and T1 (p > 0.05). However, a significant gender difference was observed in FA values, with males exhibiting higher FA values in the left condyle compared to females at both T0 and T1 (p < 0.05). Specifically, the FA values in the left condyle increased from a mean of 1.09 ± 0.09 at T0 to 1.13 ± 0.08 at T1 in males, whereas in females, the FA values remained relatively stable with a mean of 1 ± 0.09 at T0 and 1.03 ± 0.11 at T1. Conclusion: The findings indicate that MP-RME therapy does not induce significant alterations in the trabecular structure of the mandibular condyle. These results suggest the treatment's safety concerning the structural integrity of the condyle, although the observed gender differences in FA values warrant further investigation. [ABSTRACT FROM AUTHOR]

Mathematical modelling of surface roughness is of significant interest for a variety of modern applications, including, but not limited to, tribology and optics. The most popular approaches to modelling rough surfaces are reviewed and critically examined. By providing counterexamples, it is shown that approaches based solely on the use of the fractal geometry or power spectral density have many drawbacks. It is recommended to avoid these approaches. It is argued that the surfaces that cannot be distinguished from the original rough surfaces can be synthesised by employing the concept of the representative elementary pattern of roughness (REPR), i.e., the smallest interval (or area) of a rough surface that statistically represents the whole surface. The REPR may be extracted from surface measurement data by the use of the "moving window" technique in combination with the Kolmogorov–Smirnov statistic. [ABSTRACT FROM AUTHOR]

The structure of the proton exhibits Fractal behavior at low x, where x is the fraction of the proton's momentum carried by the interacting partons. This Fractal behavior is characterized by self-similar properties at different scales and can be quantified using the concept of Fractal dimension. An investigation into the Fractal properties of the proton structure at low x is critical for understanding the fundamental properties of the strong force and developing a more comprehensive understanding of the hadron structure. Fractals, characterized by self-similar patterns across scales, demonstrate a direct correlation between their Fractal dimension and entropy, where higher Fractal dimensions correspond to increased informational content. Furthermore, it is essential for designing high-energy physics experiments and developing more accurate models of subatomic particle interactions. This paper has a fresh look at the self-similar properties of the proton structure at low x. Our study involves the use of the xFitter framework to parameterize the proton structure functions with a Fractal formalism at low x. We also examine how the inclusion of new data affects the results of our analysis. [ABSTRACT FROM AUTHOR]

The nonlinear variability present during steady-state gait may provide a signature of health and showcase one's walking adaptability. Although treadmills can capture vast amounts of walking data required for estimating variability within a small space, gait patterns may be misrepresented compared to an overground setting. Smartphones may provide a low-cost and user-friendly estimate of gait patterns among a variety of walking settings. However, no study has investigated differences in gait patterns derived from a smartphone between treadmill walking (TW) and overground walking (OW). This study implemented a smartphone accelerometer to compare differences in temporal gait variability and gait dynamics between TW and OW. Sixteen healthy adults (8F; 24.7 ± 3.8 years) visited the laboratory on three separate days and completed three 8-minute OW and three TW trials, at their preferred speed, during each visit. The inter-stride interval was calculated as the time difference between right heel contact events located within the vertical accelerometery signals recorded from a smartphone while placed in participants front right pant pocket during walking trials. The inter-stride interval series was used to calculate stride time standard deviation (SD) and coefficient of variation (COV), statistical persistence (fractal scaling index), and statistical regularity (sample entropy). Two-way analysis of variance compared walking condition and laboratory visits for each measure. Compared to TW, OW displayed significantly (p < 0.01) greater stride time SD (0.014 s, 0.020 s), COV (1.26 %, 1.82 %), fractal scaling index (0.70, 0.79) and sample entropy (1.43, 1.63). No differences were found between visits for all measures. Smartphone-based assessment of gait provides the ability to distinguish between OW and TW conditions, similar to previously established methodologies. Furthermore, smartphones may be a low-cost and user-friendly tool to estimate gait patterns outside the laboratory to improve ecological validity, with implications for free-living monitoring of gait among various populations. • Smartphones can enable gait monitoring outside the lab and for clinical purposes. • Treadmills reduce gait variability and dynamics compared to overground setting. • Context of walking must be considered within the interpretation of the results. [ABSTRACT FROM AUTHOR]

The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically investigated that it has fourth order of convergence. The type of convergence is also discussed, which is global under an appropriate choice of the initial matrix. Numerical experiments are reported based on input matrices of different sizes as well as various stopping termination levels with comparisons to methods of the same nature and same number of matrix–matrix multiplications. The simulation results confirm the efficiency of the proposed scheme in contrast to its competitors of the same nature. [ABSTRACT FROM AUTHOR]

Tree-like branching networks are widespread in nature and have found wide applications in engineering, where Murray's law is generally adopted to optimally design tree-like systems, but it may become invalid in some cases. Here we give an energy approach to the analysis of the law and re-find Li–Yu's law for the optimal ratio of the square root of 2 with a suitable constraint. When the cross-section of each branch is considered as a fractal pattern, a modified Murray's law is obtained, which includes the original Murray's law for a Peano-like pore and Li–Yu's law for cylindrical branches, furthermore a useful relationship between the diameter and length of each hierarchy is obtained, which is contrary to the tree-like fractal patterns, and the new hierarchy is named as "fractal Murray tree", which also has many potential applications in science, engineering, social science and economics. This paper is intended to serve as a foundation for further research into the fractal Murray tree and its applications in various fields. [ABSTRACT FROM AUTHOR]

To explore the dynamic mechanical response of layered rocks under different impact pressures, dynamic impact experiments were performed on five sets of layered rock-like specimens at different dips with a splitting Hopkinson compression bar. The effects of impact pressure on the compressive properties of layered rocks are researched. The fracture patterns are also discussed regarding fragmentation morphology, fractal characteristics, and energy dissipation. The findings suggest that the dynamic compressive strength shows the "U" shape variation with the dip angle θ, firstly declining and then enlarging. The dynamic compressive strength grows with increased impact pressure at different dip angles. The dynamic compressive strength of the layered rock at different dip angles increased the fastest with the change of impact pressure from 0.2 MPa to 0.3 MPa. The fractal dimension D tends to enlarge with increasing impact load. There is a highly linear relationship between them. The D of the sample fragments increases slower as the θ increases. In general, the absorbed energy declines first and then enlarges. With the higher impact pressure, the percentage of the total absorbed energy increases for each θ, besides the energy gains. The broken effect of a high dip angle specimen is better when crushing rock at low impact pressure. And when that is high, it is easy to cut the surrounding rock vertically or parallel to the layer direction. [ABSTRACT FROM AUTHOR]