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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]

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]

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]

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]

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.

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 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]

Designers have been fascinated by exploring new geometries made by high-performance structures. In more specific terms, biological systems have always been proven to be characterised by sophisticated structures with adapting properties to nature challenges. Insightful analyses have shown how these natural structures are dominated by characteristics such as high energy absorption and elevated strength-weight proportion. Fractal geometries are examples of bio-inspired mathematical objects whose complex 3D structures can be obtained only by advanced manufacturing systems, such as additive manufacturing (AM). This study investigates the feasibility and energy absorption properties of a novel fractal structure based on a 3D Greek cross (3D-CFS). The structure was designed with different volume fractions and produced by powder bed fusion (PBF) AM processes in polyamide (PA12) and thermoplastic polyurethane (TPU). The 3D-CFS properties are investigated under quasi-static and dynamic compression tests. The analysis revealed that for certain geometrical parameters, the manufacturing of the structures is constrained by the sintered powder entrapped in the structure. However, in the case of powder-free structures, the results showed a high impact resistance and cushioning capability. Overall, in terms of specific energy absorption (SEA), the TPU structures showed values between 2.5 and 3.5 kJ/kg, while PA12 ones are between 7.5 and 17.4 kJ/kg, making the 3D-CFS structure compatible with personal protective equipment (PPE) applications. Compared to the literature data on cellular structures made by AM, 3D-CFS performs considerably better. Also, PA12 3D-CFS is better, with a SEA value up to 170% higher than that of a typical material employed for head PPE (e.g. EPS-60 SEA equal to 2.76 kJ/kg). In contrast, TPU 3D-CFS looks more promising in the case of multiple impact conditions. [ABSTRACT FROM AUTHOR]

In stealth applications, there is a growing emphasis on the development of radar-absorbing structures that are efficient, flexible, and optically transparent. This study proposes a screen-printed metamaterial absorber (MMA) on polyethylene terephthalate (PET) substrates using indium tin oxide (ITO) as the grounding layer, which achieves both optical transparency and flexibility. These materials and methods enhance the overall flexibility and transparency of MMA. To address the limited transparency caused by the silver nanoparticle ink for the top pattern, a metal mesh was incorporated to reduce the area ratio of the printed patterns, thereby enhancing transparency. By incrementing the fractal order of the structure, we optimized the operating frequency to target the X-band, which is most commonly used in radar detection. The proposed MMA demonstrates remarkable performance, with a measured absorption of 91.99% at 8.85 GHz and an average optical transmittance of 46.70% across the visible light spectrum (450 to 700 nm), indicating its potential for applications in transparent windows or drone stealth. [ABSTRACT FROM AUTHOR]

This paper considers the possibility of using monofractal and multifractal analysis of acoustic signals to detect water leaks through gate valves. Detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MF-DFA) were used. Experimental studies were conducted on a ½-inch nominal diameter wedge valve, which was fitted to a ¾-inch nominal diameter steel pipeline. The water leak was simulated by opening the valve. The resulting leakage rates for different valve opening conditions were 5.3, 10.5, 14, 16.8, and 20 L per minute (L/min). The Hurst exponent for acoustic signals in a hermetically sealed valve is at the same level as a deterministic signal, while the width of the multifractal spectrum closely matches that of a monofractal process. When a leak occurs, turbulent flow pulsations appear, and with small leak sizes, the acoustic signals become anticorrelated with a high degree of multifractality. As the leakage increases, the Hurst exponent also increases and the width of the multifractal spectrum decreases. The main contributor to the multifractal structure of leak signals is small, noise-like fluctuations. The analysis of acoustic signals using the DFA and MF-DFA methods enables determining the extent of water leakage through a non-sealed gate valve. The results of the experimental studies are in agreement with the numerical simulations. Using the Ansys Fluent software (v. 19.2), the frequencies of flow vortices at different positions of gate valve were calculated. The k-ω SST turbulence model was employed for calculations. The calculations were conducted in a transient formulation of the problem. It was found that as the leakage decreases, the areas with a higher turbulence eddy frequency increase. An increase in the frequency of turbulent fluctuations leads to enhanced energy dissipation. Some of the energy from ordered processes is converted into the energy of disordered processes. [ABSTRACT FROM AUTHOR]

The physiological loss of muscle mass and strength with aging is referred to as "sarcopenia", whose combined effect with osteoporosis is a serious threat to the elderly, accounting for decreased mobility and increased risk of falls with consequent fractures. In previous studies, we observed a high degree of inter-individual variability in paraspinal muscle fatty infiltration, one of the most relevant indices of muscle wasting. This aspect led us to develop a computerized method to quantitatively characterize muscle fatty infiltration in aging and diseases. Magnetic resonance images of paraspinal muscles from 58 women of different ages (age range of 23–85 years) and physio-pathological status (healthy young, pre-menopause, menopause, and osteoporosis) were used to set up a method based on fractal-derived texture analysis of lean muscle area (contractile muscle) to estimate muscle fatty infiltration. In particular, lacunarity was computed by parameter β from the GBA (gliding box algorithm) curvilinear plot fitted by our hyperbola model function. Succolarity was estimated by parameter µ, for the four main directions through an algorithm implemented with this purpose. The results show that lacunarity, by quantifying muscle fatty infiltration, can discriminate between osteoporosis and healthy aging, while succolarity can separate the other three groups showing similar lacunarity. Therefore, fractal-derived features of contractile muscle, by measuring fatty infiltration, can represent good indices of sarcopenia in aging and disease. [ABSTRACT FROM AUTHOR]

The Upper Permian Longtan Formation in the Laoshan Uplift of the South Yellow Sea Basin hosts thick and distinctive tight sandstones. However, a comprehensive understanding of its pore structure and reservoir classification remains lacking. This study investigates the fully cored well, CSDP-2, utilizing thin section analysis, scanning electron microscopy, energy spectrum analysis, X-ray diffraction, high-pressure mercury intrusion, and nuclear magnetic resonance to characterize its petrophysical properties, pore space, and movable fluid characteristics. Additionally, fractal principles are further employed to examine reservoir heterogeneity and conduct a quantitative assessment, considering the complexity of tight sandstone pore structures. The findings reveal that the sandstones predominantly comprise feldspathic litharenites, with an average porosity of 1.567% and permeability of 0.099 mD, primarily containing intragranular pores. Two distinct sets of pores with significantly different sizes (r < 2 μm; r > 6 μm) were identified, displaying relatively high fractal dimensions and discrete distribution. Movable fluids primarily occupy pores with radii > 0.019 μm, reflecting pronounced overall heterogeneity. The reservoir was classified into three categories utilizing permeability, median radius, and movable fluid saturation as key evaluation parameters, with Class I representing a relatively high-quality reservoir. These findings advance our understanding of the pore development mechanism of tight sandstone reservoirs and provide geological evidence for further hydrocarbon exploration in this study area. [ABSTRACT FROM AUTHOR]

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 matrixbased 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. [ABSTRACT FROM AUTHOR]

This paper investigates a type of iterated implicit function systems composed of equations F n (x , y) = c , where F n (x , y) is a continuous function, and c is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation F n (x , y) = c containing the implicit function or being α n -contractive about y. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated. [ABSTRACT FROM AUTHOR]

AI-driven mineral prospectivity mapping (MPM) is a valid and increasingly accepted tool for delineating the targets of mineral exploration, but it suffers from noisy and unrepresentative input features. In this study, a set of fractal and multifractal methods, including box-counting calculation, concentration–area fractal modeling, and multifractal analyses, were employed to excavate the underlying nonlinear mineralization-related information from geological features. Based on these methods, multiple feature selection criteria, namely prediction–area plot, K-means clustering, information gain, chi-square, and the Pearson correlation coefficient, were jointly applied to rank the relative importance of ore-related features and their fractal representations, so as to choose the optimal input feature dataset readily used for training predictive AI models. The results indicate that fault density, the multifractal spectrum width (∆α) of the Yanshanian intrusions, information dimension (D1) of magnetic anomalies, correlation dimension (D2) of iron-oxide alteration, and the D2 of argillic alteration serve as the most effective predictor features representative of the corresponding ore-controlling elements. The comparative results of the model assessment suggest that all the AI models trained by the fractal datasets outperform their counterparts trained by raw datasets, demonstrating a significant improvement in the predictive capability of fractal-trained AI models in terms of both classification accuracy and predictive efficiency. A Shapley additive explanation was employed to trace the contributions of these features and to explain the modeling results, which imply that fractal representations provide more discriminative and definitive feature values that enhance the cognitive capability of AI models trained by these data, thereby improving their predictive performance, especially for those indirect predictor features that show subtle correlations with mineralization in the raw dataset. In addition, fractal-trained models can benefit practical mineral exploration by outputting low-risk exploration targets that achieve higher capturing efficiency and by providing new mineralization clues extracted from remote sensing data. This study demonstrates that the fractal representations of geological features filtered by multi-criteria feature selection can provide a feasible and promising means of improving the predictive capability of AI-driven MPM. [ABSTRACT FROM AUTHOR]

In this paper, the possibility of using monofractal and multifractal analysis of acoustic signals of pipelines to detect leaks is considered. An experimental stand has been created to study the fractal characteristics of acoustic signals of pipelines with "slit" type defects. During the experiments, defects of the "slit" type pipeline with dimensions of 2 mm, 8 mm, and 20 mm were modeled. Detrended fluctuation analysis (DFA) and the multifractal detrended fluctuation analysis (MF-DFA) were used. As a result of the experimental studies, it was found that the occurrence of leakage leads to the occurrence of anticorrelated vibrations in a pipeline with multifractal properties. The analyses of acoustic signals by DFA and MF-DFA methods make it possible to reliably determine the leakage. The Hurst exponent and the width of the multifractal spectrum can serve as indicators of the occurrence of leaks in pipelines. [ABSTRACT FROM AUTHOR]

Most geophysical and geochemical data are commonly acknowledged to exhibit fractal and multifractal properties, but the fractal characteristics of polarizability have received limited attention from the literature. The present study demonstrates that the polarizability data of the graphite deposits have fractal characteristics and introduces the fractal method for its quantitative analysis to indicate and predict the properties of graphite deposits. The results show that the concentration-area (C-A) method is superior to classical interpolation in anomaly extraction but inferior to the spectrum-area (S-A) method in the coverage region. Because the type of graphite ore is sedimentary-metamorphic in this area, the graphite ore-bodies can be regarded as a special stratum, which is different from most metal deposits, and the anomaly of graphite ore are shown in the background mode of the S-A method. The high values of the background mode effectively indicate the potential areas where the graphite-bearing strata occur, while observing a decrease in the power-law exponent (β) of the background mode as the width of ore-bodies increases. The validity of this conclusion was confirmed based on the vertical profiles of the predicted area, and the uncharted ore vein was thereby identified. Furthermore, it was found that the anomaly mode can serve as a grade indicator of graphite ore rather than delineating the fault. By integrating the background and anomaly modes of the S-A method, we can quantitatively predict and effectively identify high-grade targets from sedimentary deposits containing minerals in future exploration. [ABSTRACT FROM AUTHOR]

We study the group TA of rearrangements of the airplane limit space introduced by Belk and Forrest (2019). We prove that TA is generated by a copy of Thompson's group F and a copy of Thompson's group T, hence it is finitely generated. Then we study the commutator subgroup [TA; TA], proving that the abelianization of TA is isomorphic to Z and that [TA; TA] is simple, finitely generated and acts 2-transitively on the so-called components of the airplane limit space. Moreover, we show that TA is contained in T and contains a natural copy of the basilica rearrangement group TB studied by Belk and Forrest (2015). [ABSTRACT FROM AUTHOR]

Groundwater potential in Morocco’s Souss-Massa mountainous basin (SMMB) is being identified using geospatial tools and geological data. We deployed four mathematical models, namely Data-Driven Multi-index Overlay (DMIO), Geometric Average (GA), Support Vector Machine (SVM), and Logistic Regression (LR), to establish data-driven patterns among the nine influencing factors, primarily drainage density, permeability, slope, distance to rivers, elevation, lineament density, distance to lineaments, intersection node density, and rainfall. Based on the Concentration-Area (C-A) fractal approach, the findings of the four models were developed and classified into five levels of potentiality ranging from very low to very high. The regions designated as having high and very high potentialities for the DMIO, GA, SVM, and LR models, respectively, account for 22.44 %, 9.80 %, 19.36 %, and 26.77 % of the overall basin. We validated the models by calculating each model's area under the ROC curve (AUC). The estimated AUC values are more than 70 %, suggesting the model performs well. The four models' performance was compared, revealing that the SVM model outperforms the others. Gravimetric data shows that possible groundwater zones closely coincide with gravimetric lineaments. The findings of this study can provide valuable insights to decision-makers, allowing them to improve decision-making processes and develop holistic groundwater resource management in the Souss-Massa mountainous basin (SMMB).

Recycled aggregate concrete (RAC) has multiple interface transition zones (ITZs) due to its complex composition, providing convenient ion transfer channels. Therefore, it is significant to explore the changes of the internal pore structure in RAC under ion erosion conditions for quantifying and predicting ion transfer mechanisms. This paper innovatively utilizes the seepage effect for investigating the evolution of pore structures in RAC under sulfate attack. We conduct a systematic experimental study on the transport of sulfate ions in RAC with different fly ash (FA) substitution rates under dry-wet cycles. Combining backscattered electron imaging (BSE), permeation and fractal characteristics, this paper deeply explore the relationship between pore structure feature, sulfate percolation mechanism and the macroscopic properties of RAC incorporated with different amount of FA.

The study is devoted to the consideration of the terms, which are the most frequently used and significant for the formation of meaning in linguistics. The relevance of the study is, on the one hand, determined by the variety of approaches to their definition and interpretation, and, on the other hand, by the necessity to specify them to facilitate academic communication and linguistic analysis. The purpose of this work is to consider a number of such concepts and substantiate our approach to them and their interpretation. The study uses both general theoretical and linguistic research methods. Particular attention is paid to phenomenological analysis while considering the terms themselves, as well as semantic, categorical-constitutive, positional and pragmatic analysis while studying empirical material. There are considered the concepts of text, discourse, dicteme as a text unit, language personality of an author, or a narrator and a reader/listener. The article analyzes differences between the text and discourse concepts. The work substantiates heuristic potential of a dicteme as a minimum text unit. The dicteme is studied both - from elementary and holistic approaches, as a fractal that forms the space of the whole text according to synergistic laws. The study emphasizes the ideological role of a linguistic personality both at the stages of creation and consumption of a text. There is carried out a linguistic analysis of an English literary text in order to identify denotative and conceptual information in its ideological and aesthetic manifestation in it. The study highlights the role of strong positions of text in the transfer of conceptual information and considers the function of influence in an artistic and media works, which, in the first case, is mainly emotive in nature, and, in the second, manipulative. It underlines the interdependence of the conceptual priorities of the creator and consumer of a media text.

The paper introduces a fractal geometry utilizing an Octagonal Split Ring Resonator (SRR) hybrid DRA as its basis. The primary antenna consists of a conventional Octagonal patch, which is then scaled down by 0.5 in the first iteration and further by 0.25 in the second and then the DRA is utilized to enhance its performance. Remarkably compact at 6 × 6.2 cm, the antenna demonstrates effectiveness across frequencies of 5.1, 6.8, and 7.2 GHz notably; it maintains a gain above 3.3 dBi and directivity above 4.1 dBi in all operational bands. Analysis employing surface current and return loss substantiates the impact of the fractal design with DR. These simulations and measured results affirm the proposed antenna's suitability for robust C band wireless applications, offering promising performance in antenna parameters.

We develop the local convergence of the six order Contraharmonic-mean Newton’s method (CHN) to solve Banach space valued equations. Our analysis approach is two fold: The first way uses Taylor’s series and derivatives of higher orders. The second one uses only the first derivatives. We examine the theoretical results by solving a boundary value problem also using the examples relating the proposed method with other’s methods such as Newton’s, Kou’s and Jarratt’s to show that the proposed method performs better. The conjugate maps for second-degree polynomial are verified. We also calculate the fixed points (extraneous). The article is completed with the study of basins of attraction, which support and further validate the theoretical and numerical results.

Abstract The integration of biology, computer science, and statistics has given rise to the interdisciplinary field of bioinformatics, which aims to decode biological intricacies. It produces extensive and diverse features, presenting an enormous challenge in classifying bioinformatic problems. Therefore, an intelligent bioinformatics classification system must select the most relevant features to enhance machine learning performance. This paper proposes a feature selection model based on the fractal concept to improve the performance of intelligent systems in classifying high-dimensional biological problems. The proposed fractal feature selection (FFS) model divides features into blocks, measures the similarity between blocks using root mean square error (RMSE), and determines the importance of features based on low RMSE. The proposed FFS is tested and evaluated over ten high-dimensional bioinformatics datasets. The experiment results showed that the model significantly improved machine learning accuracy. The average accuracy rate was 79% with full features in machine learning algorithms, while FFS delivered promising results with an accuracy rate of 94%.

ABSTRACTIn the field of synthetic biology, a promissory technoscience, researchers use risk-based and speculative scenarios to anticipate synthetic biology futures. Yet, there is a sense in the field of history repeating as synthetic biologists repeatedly deal with similar challenges. Perhaps a focus on contemplating the unexpected and attempts to deal with complex futures comes with a failure, paradoxically, to prepare sufficiently for the expected. This study develops a conceptual framework derived from fractals and analyses data from ethnographic involvement in the field synthetic biology to reimagine how patterns in synthetic biology practices repeat at different orders of scale. This study begins with the ‘pressure testing’ of a biofoundry and shows how synthetic biology repeats at different scales of organisation, interaction and time. The case is made for a possible fractal anticipation which would have the capacity to identify patterns and support innovators, social scientists and researchers to be better prepared for encountering similar developments at different orders of magnitude.

The simulation and characterisation of randomly rough surfaces is an important topic in surface science, tribology, geo- and planetary sciences, image analysis and optics. Extensions to general random processes with two continuous variables are straightforward. Several surface generation algorithms are available, and preference for one or another method often depends on the specific scientific field. The same holds for the methods to estimate the fractal dimension D. This work analyses six algorithms for the determination of D as a function of the size of the domain, variance, and the input value for D, using surfaces generated by Fourier filtering techniques and the random midpoint displacement algorithm. Several of the methods to determine fractal dimension are needlessly complex and severely biased, whereas simple and computationally efficient methods produce better results. A fine-tuned analysis of the power spectral density is very precise and shows how the different surface generation algorithms deviate from ideal fractal behaviour. For large datasets defined on equidistant two-dimensional grids, it is clearly the most sensitive and precise method to determine fractal dimension. [ABSTRACT FROM AUTHOR]

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω -continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems. [ABSTRACT FROM AUTHOR]

The restorative and mental state enhancing effects of brief mindfulness-based interventions (MBIs) and restorative environments such as nature has been supported in the research literature. However, regular adoption of these practices is limited by practical constraints and motivational barriers. The current study addressed these challenges by introducing two novel approaches which utilise the immersive and interactive qualities of virtual reality (VR). This included an interactive MBI and an abstract restorative environment using fractal-like imagery. These approaches were explored using a comparative evaluation of two short (6 min) VR interventions: Passive VR (applying principles from restorative interventions) and Interactive VR (implementing a focused attention form of mindfulness meditation). A mixed methods approach revealed increased state mindfulness, reduced mental fatigue, and enhanced aspects of mood (calm/relaxation, anxiety) consistently between conditions. Between group differences revealed additional benefits for cognition (focus), mood (happiness and sadness), and motivational value with the interactive intervention. The abstract environment, used in both interventions, maintained comparable levels of perceived restoration with a nature VR control condition. The results provide preliminary evidence supporting the use of interactive approaches for mindfulness interventions and abstract versions of restorative environments. [ABSTRACT FROM AUTHOR]

We develop the local convergence of the six order Contraharmonic-mean Newton's method (CHN) to solve Banach space valued equations. Our analysis approach is two fold: The first way uses Taylor's series and derivatives of higher orders. The second one uses only the first derivatives. We examine the theoretical results by solving a boundary value problem also using the examples relating the proposed method with other's methods such as Newton's, Kou's and Jarratt's to show that the proposed method performs better. The conjugate maps for second-degree polynomial are verified. We also calculate the fixed points (extraneous). The article is completed with the study of basins of attraction, which support and further validate the theoretical and numerical results. [ABSTRACT FROM AUTHOR]

A sriyantra/srichakra-shaped wideband fractal-based monopole antenna is designed and substrate material analysis is performed to determine its ability to meet the needs of S-, C-, X-, and Ku-band communication applications. The proposed model design is inspired by sriyantra, which is believed to be a source of energy to the universe according to Indian mythology. This work is focused on the integration of fractal geometry with sriyantra/srichakra for providing a linkage between philosophical belief and a modern communication system with experimental validation. The designed antenna performance characteristics are analysed on various substrates of rigid, flexible, and textile materials to identify the suitable model as per the requirements of specified applications. The proposed model provides 17.7 GHz bandwidth, a peak realized gain of 8.66 dB, and peak efficiency of 95% with RT/duroid 5880 substrate material. The design with polyimide for the proposed structure provides bandwidth of 14.8 GHz, peak gain of 6.18 dB, and peak efficiency of 90%. The same design was implemented on the fabric substrate material of bed sheet cotton, achieving bandwidth of 12.5 GHz, peak gain of 6.39 GHz, and peak efficiency of 92%. The proposed srichakra structure of the fractal monopole antenna on three substrate materials provides wide bandwidth with considerable gain and efficiency in S-, C-, X-, and Ku-band communication applications. The experimental validation provides a good correlation with the simulation results obtained from the CST (computer simulation technology) tool for applicability in communication systems. [ABSTRACT FROM AUTHOR]

In the open literature, there are numerous studies on the normal and extreme (flood and drought) behavior of wet and dry periods based on the understanding of the standard precipitation index (SPI), which provides a series of categorizations by considering the standard normal (Gaussian) probability distribution function (PDF). The numerical meaning of each categorization assessment is quite lacking in terms of future predictions of wet and dry period duration based on historical records. This paper presents a new approach for calculating possible formations of future wet and dry period durations based on historical records through an effective fractal geometric forecasting approach. The essence of the proposed methodology is based on the number of dry periods (steps) of non-overlapping monthly duration along consecutive broken line paths in the SPI classification for wet and dry period durations. It has been observed that the plot of periods on double logarithmic paper falls along a straight line against the number of such periods, implying a power function, which is the essence of fractal geometry. Extending the empirically derived straight line provides the number of periods that may occur in the future over a range of SPI levels. This methodology is referred to as SPI fractal (SPIF), and the classic SPI classification is converted into SPIF wet and dry polygons, which provide additional information about the drought period number within a valid polygonal area, compared to the classic SPI results. The wet and dry period features of any hydro-meteorology time series are constrained in SPIF polygons. The application of the methodology was carried out on monthly rainfall records on the European side of the Istanbul Florya meteorological station in Turkey. [ABSTRACT FROM AUTHOR]

When describing the tribological behaviour of technical surfaces, the need for full-length scale microtopographic characterization often arises. The self-affine of surfaces and the characterisation of self-affine using a fractal dimension and its implantation into tribological models are commonly used. The goal of our present work was to determine the frequency range of fractal behaviour of surfaces by analysing the microtopographic measurements of an anodised aluminium brake plunger. We also wanted to know if bifractal and multifractal behaviour can be detected in real machine parts. As a result, we developed a new methodology for determining the fractal range boundaries to separate the nano- and micro-roughness. To reach our goals, we used an atomic force microscope (AFM) and a stylus instrument to obtain measurements in a wide frequency range (19 nm–3 mm). Power spectral density (PSD)-based fractal evaluation found that the examined surface could not be characterised by a single fractal dimension. A new method capable of separating nano- and micro-roughness has been developed for investigating multifractal behaviour. The presented procedure separates nano- and micro-roughness based on the geometric characteristics of surfaces. In this way, it becomes possible to specifically examine the relationship between the micro-geometry that can be measured in each wavelength range and the effects of cutting technology and the material structure that creates them. [ABSTRACT FROM AUTHOR]

Specifying the role of genetic mutations in cancer development is crucial for effective screening or targeted treatments for people with hereditary cancer predispositions. Our goal here is to find the relationship between a number of cancerogenic mutations and the probability of cancer induction over the lifetime of cancer patients. We believe that the Avrami–Dobrzyński biophysical model can be used to describe this mechanism. Therefore, clinical data from breast and ovarian cancer patients were used to validate this model of cancer induction, which is based on a purely physical concept of the phase-transition process with an analogy to the neoplastic transformation. The obtained values of model parameters established using clinical data confirm the hypothesis that the carcinogenic process strongly follows fractal dynamics. We found that the model's theoretical prediction and population clinical data slightly differed for patients with the age below 30 years old, and that might point to the existence of an ancillary protection mechanism against cancer development. Additionally, we reveal that the existing clinical data predict breast or ovarian cancers onset two years earlier for patients with BRCA1/2 mutations. [ABSTRACT FROM AUTHOR]

The integration of biology, computer science, and statistics has given rise to the interdisciplinary field of bioinformatics, which aims to decode biological intricacies. It produces extensive and diverse features, presenting an enormous challenge in classifying bioinformatic problems. Therefore, an intelligent bioinformatics classification system must select the most relevant features to enhance machine learning performance. This paper proposes a feature selection model based on the fractal concept to improve the performance of intelligent systems in classifying high-dimensional biological problems. The proposed fractal feature selection (FFS) model divides features into blocks, measures the similarity between blocks using root mean square error (RMSE), and determines the importance of features based on low RMSE. The proposed FFS is tested and evaluated over ten high-dimensional bioinformatics datasets. The experiment results showed that the model significantly improved machine learning accuracy. The average accuracy rate was 79% with full features in machine learning algorithms, while FFS delivered promising results with an accuracy rate of 94%. [ABSTRACT FROM AUTHOR]

The main objective of this work is the characterization of the seismic activity linked to the Pinar Fault in the region of Western Cuba, through the use of mathematical methods that allow characterizing and/or quantifying the space-time relationship in said seismic series. Along with this analysis, an interpretation and correlation of the results obtained with those derived from previous research is carried out. To solve the problem of this work, four fundamental parameters are determined through the application of mathematical-statistical methods: the Poisson entropy, the estimation of the generalized dimensions Dq, the b value of the Gutenberg-Richter relationship and the Hurst exponent. The integration of the criteria offered by these parameters contributes to the interpretation of the neotectonics of the region from a seismic point of view and in turn allows the creation of more complete seismic records than the current ones. Among the main results obtained is the demonstration that the seismic series of the region complies with Hurst's hypothesis, where most natural phenomena behave as statistical trends with stochastic noise and not as random walks. This facilitates the relationship between the research results and the recent phenomena that govern neotectonics in the area. [ABSTRACT FROM AUTHOR]

A review of the most important concepts of the last 100 years on the topic of fascia and its fundamental importance to tissue and organ structure and function is provided as a basis for the author's commentary on the self-organizing nature of fascia. Implications for clinical applications in medicine, in both pathophysiology and the treatment of disease, follow from these anatomic insights. Organizing principles of nature put forth by D'Arcy Thompson, Buckminster Fuller, Benoit Mandelbrot, and Adrian Bejan set the stage for understanding tissue and fascial form. The author presents videos from the operating room of living anatomy illustrating the concept of a basic fascia segment, which differentiates according to need in the various locations of the body. [ABSTRACT FROM AUTHOR]

The quantification of the irregular morphology and distribution pattern of mineral grains is an essential but challenging task in ore-related mineralogical research, allowing for tracing the footprints of pattern-forming geological processes that are crucial to understanding mineralization and/or diagenetic systems. In this study, a large model, namely, the Segmenting Anything Model (SAM), was employed to automatically segment and annotate quartz, lepidolite and albite grains derived from Yichun rare-metal granite (YCRMG), based on which a series of fractal and multifractal methods, including box-counting calculation, perimeter–area analysis and multifractal spectra, were implemented. The results indicate that the mineral grains from YCRMG show great scaling invariance within the range of 1.04~52,300 μm. The automatic annotation of mineral grains from photomicrographs yields accurate fractal dimensions with an error of only 0.6% and thus can be utilized for efficient fractal-based grain quantification. The resultant fractal dimensions display a distinct distribution pattern in the diagram of box-counting fractal dimension (Db) versus perimeter–area fractal dimension (DPA), in which lepidolites are sandwiched between greater-valued quartz and lower-valued albites. Snowball-textured albites, i.e., concentrically arranged albite laths in quartz and K-feldspar, exhibit characteristic Db values ranging from 1.6 to 1.7, which coincide with the fractal indices derived from the fractal growth model. The zonal albites exhibit a strictly increasing trend regarding the values of fractal and multifractal exponents from core to rim, forming a featured "fractal-index banding" in the radar diagram. This pattern suggests that the snowball texture gradually evolved from rim to core, thus leading to greater fractal indices of outer zones, which represent higher complexity and maturity of the evolving system, which supports a metasomatic origin of the snowball texture. Our study demonstrates that fractal analyses with the aid of a large model are effective and efficient in characterizing and understanding complex patterns of mineral grains. [ABSTRACT FROM AUTHOR]

This study presents multistability mechanisms that can enhance the energy harvesting performance of a piezoelectric energy harvester (PEH) with geometrical nonlinearities. To configure triple potential wells, static bifurcation diagrams in the structural parameter plane are depicted. On this basis, the key structural parameter is considered, of which three reasonable values are then chosen for comparing and evaluating the performances of the triple-well PEH under them. Then, intra-well responses and the corresponding voltages of the system are investigated qualitatively. A preliminary analysis of the suitable energy-harvesting conditions is carried out, which is then validated by numerical simulations of the evolution of coexisting attractors and their basins of attraction with variations in the excitation level and frequency. It follows that, under a low-level ambient excitation, the intra-well responses around the trivial equilibrium dominate the energy-harvesting performance. When the level of the environmental excitation is very low, which one of the three values of the key structural parameter is the best for improving the performance of the PEH system depends on the range of the excitation frequency; when the excitation level increases sufficiently to induce inter-well responses, the maximum one is the best for improving the performance of the PEH. The findings provide valuable insights for researchers working in the structure optimization and practical applications of geometrically nonlinear PEHs. [ABSTRACT FROM AUTHOR]

The temporal evolution of the global mean sea level (GMSL) is investigated in the present analysis using the monthly mean values obtained from two sources: a reconstructed dataset and a satellite altimeter dataset. To this end, we use two well-known techniques, detrended fluctuation analysis (DFA) and multifractal DFA (MF-DFA), to study the scaling properties of the time series considered. The main result is that power-law long-range correlations and multifractality apply to both data sets of the global mean sea level. In addition, the analysis revealed nearly identical scaling features for both the 134-year and the last 28-year GMSL-time series, possibly suggesting that the long-range correlations stem more from natural causes. This demonstrates that the relationship between climate change and sea-level anomalies needs more extensive research in the future due to the importance of their indirect processes for ecology and conservation. [ABSTRACT FROM AUTHOR]

Rainfall, or more generally the precipitation process (flux), is a clear example of chaotic variables resulting from a highly nonlinear dynamical system, the atmosphere, which is represented by a set of physical equations such as the Navier–Stokes equations, energy balances, and the hydrological cycle, among others. As a generalization of the Euclidean (ordinary) measurements, chaotic solutions of these equations are characterized by fractal indices, that is, non-integer values that represent the complexity of variables like the rainfall. However, observed precipitation is measured as an aggregate variable over time; thus, a physical analysis of observed fluxes is very limited. Consequently, this review aims to go through the different approaches used to identify and analyze the complexity of observed precipitation, taking advantage of its geometry footprint. To address the review, it ranges from classical perspectives of fractal-based techniques to new perspectives at temporal and spatial scales as well as for the classification of climatic features, including the monofractal dimension, multifractal approaches, Hurst exponent, Shannon entropy, and time-scaling in intensity–duration–frequency curves. [ABSTRACT FROM AUTHOR]

Superconducting nanowire single-photon detectors (SNSPDs) have become a mainstream photon-counting technology that has been widely applied in various scenarios. So far, most multi-channel SNSPD systems, either reported in literature or commercially available, are polarization sensitive, that is, the system detection efficiency (SDE) of each channel is dependent on the state of polarization of the to-be-detected photons. Here, we reported an eight-channel system with fractal SNSPDs working in the wavelength range of 930 to 940 nm, which are all featured with low polarization sensitivity. In a close-cycled Gifford-McMahon cryocooler system with the base temperature of 2.2 K, we installed and compared the performance of two types of devices: (1) SNSPD, composed of a single, continuous nanowire and (2) superconducting nanowire avalanche photodetector (SNAP), composed of 16 cascaded units of two nanowires electrically connected in parallel. The highest SDE among the eight channels reaches 96−5+4%, with the polarization sensitivity of 1.02 and a dark-count rate of 13 counts per second. The average SDE for eight channels for all states of polarization is estimated to be 90 ± 5%. It is concluded that both the SNSPDs and the SNAPs can reach saturated, high SDE at the wavelength of interest, and the SNSPDs show lower dark-count (false-count) rates, whereas the SNAPs show better properties in the time domain. With the adoption of this system, we showcased the measurements of the second-order photon-correlation functions of light emission from a single-photon source based on a semiconductor quantum dot and from a pulsed laser. It is believed that this work will provide new choices of systems with single-photon detectors combining the merits of high SDE, low polarization sensitivity, and low noise that can be tailored for different applications.