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Inspired by natural bifurcated systems, a bionic microchannel system with tree-like topological structure is proposed, and applied for thermal management of proton exchange membrane fuel cell (PEMFC). The tree-like microchannel is optimized based on the minimization of flow resistance under steady laminar and pulsating flow conditions, respectively. Finite element simulations and microfluidic experiments are carried out for pulsating flow in the tree-like microchannel to determine its heat transfer performance. The results show that the optimized successive diameter ratio for the pulsating flow in a symmetric and dichotomous tree-like network is 0.707 (≈2−1/2), which is different from that of steady laminar flow (0.794≈2−1/3) known as Murray's law. Compared with steady flow, the application of sinusoidal-wave and rectangular-wave pulsating flow in a tree-like microchannel can further improve both the heat transfer efficiency and temperature uniformity. The tree-like microchannel with a length ratio of 2−2/3≈0.630 and diameter ratio of 2−1/3≈0.794 shows best performance under steady and pulsating flow modes. The present work has important theoretical significance for understanding the mechanism of natural bifurcated systems, and may provide theoretical basis and technical support for the industrial applications of bionic tree-like microchannel in thermal management of fuel cell and microelectronic device cooling etc. [Display omitted] • Bionic tree-like microchannel is designed for thermal management of PEMFC. • Optimized successive ratio of tree-like microchannel with pulsating flow is derived. • Thermal-hydraulic performance of bionic microchannel is investigated by FEM. • Mathematical model of bionic microchannel is validated by microfluidic experiment. [ABSTRACT FROM AUTHOR]

The coronal heating problem is considered one of the most puzzling problems in solar physics. It is based on the question of why the corona temperature reaches a temperature of over 2 × 10 6 K whereas the temperature of the surface of the sun has about 10 3 - 10 4 K. Since the corona is far away from the source of heat, it is realistic that the outer surface should be cooler. Solar physicists and astrophysicists have tried for several decades to solve this puzzle by supplying heat to the corona, despite several advances in MHD theory, computational and numerical simulations. Although these candidates' sources of energy hold a number of drawbacks, the problem is still open. A motivating approach has been proposed by Walsh which consists of supplying a time-dependent sinusoidal heating function to the heat to preserve the loop at coronal temperature. Yet, the loop temperature at ∼ 2 × 10 6 K is preserved and cools only if a critical frequency is exceeded. In this study, it was found that the MHD formulation of the Walsh approach in fractal dimensions may offer a potential solution to the problem. In particular, oscillations of the temperature in the space-fractal dimension close to unity relax quicker than those with smaller values. For time-fractal dimensions much smaller than unity, the temperature enters a state of steady state oscillations after a long time. Our analyses also reveal irregularities in the propagation of temperature over a long period of time. For low frequency and space-fractal dimensions close to unity and low time-fractal dimensions, the temperature enters a cycle of an oscillating phase by rising and falling gradually. It is probable that the loop will stay hot after a series of cycles, and extended periods of no heating are not possible for small frequencies. Additional features related to the propagation of the magnetic field in low-Reynolds numbers are discussed. [ABSTRACT FROM AUTHOR]

Fractals are important to explore the objects of our natural environment, such as mountains, trees, ferns, plants, lightning, rainbows, and snowflakes. Therefore, taking motivation from recent research about a new iteration scheme that converges more quickly than other popular hybrid iterative methods, we apply it to study the escape criterion. Using this criterion, we establish three algorithms. These algorithms are used to generate the important classes of fractals. Moreover, we present the analysis of how the fractal image changes with variations in different input parameters using proposed algorithms. More specifically, the complex function ℷ (z) = z m + b where m ≥ 2 , b ∈ ℂ is considered for the proof and application of the results. Also, we calculate the image generation time by considering different values of the involved parameters. Fascinating J-sets (i.e., Julia sets), M-sets (i.e., Mandelbrot sets), and multicorns (i.e., Mandelbrot sets of conjugate complex polynomial functions) at different values of m can be visualized. The successful application of a novel iteration scheme highlights the significance of this research over the existing one. [ABSTRACT FROM AUTHOR]

In this paper, we apply the variational approach for solving the time-space fractal Bogoyavlenskii equation. By the two-scale fractal complex transformation, the fractal modification of Bogoyarlenskii equation is equivalently rewritten as the original Bogoyavlenskii equation. By variational principle, different type solutions including fractal bright soliton solution, fractal bright-like soliton solution, fractal kinky-bright soliton solution and fractal periodic wave solution are provided, which are not touched in the existing literature. Numerical results are presented to show the propagation behaviours. [ABSTRACT FROM AUTHOR]

The interface leaking of the gasket-BPP sealing structure affects the efficiency, safety, and reliability of PEMFCs. Aiming to reveal the basic cause of interface leaking and predict the leakage rate, the contact behavior at the micro-scale had been researched by FEM with the fractal multi-scale method, which could eliminate the negative influence of a single observation scale on the gasket-BPP rough interface modeling. Furthermore, in order to relate the contacting behavior at the micro-scale to the leakage rate at the macro-scale of PEMFCs, the mapping model had been established to analyze the sealing performance of the entire interface through the self-affine and self-similar characteristics of the fractal multi-scale contacting model of the gasket-BPP. Not only macroscopic properties of gasket rubber materials are considered in this model, but also microscopic morphologies are taken in to investigate the interface leakage rate, which had not been studied in traditional macroscopic research. By comparing the compression curves and interface leakage rates of the flange sealing device in testing and calculating approaches, the accuracy and reliability of the mapping model had been verified, leading to the analysis and the prediction of the PEMFC interface leakage rate rapidly. • Gasket surfaces is modeled by using the real scanned position data of AFM. • Multi-scale contact behaviors have been studied by FEM. • Interface leakage rates have been tested by using flange sealing devices. • Numerical results show good agreements with experimental results. • PEMFC sealing performance is related to microscopic morphologies on gasket surfaces. [ABSTRACT FROM AUTHOR]

The Adjacent-Vertex-Distinguishing Total Coloring (AVDTC) conjecture asserts that every simple graph can be colored with an AVDTC using at most Δ + 3 colors. We show that this conjecture is satisfied for all the Sierpiński graphs S p n , their regularizations + S p n and + + S p n and the fractals obtained as limit space of these graphs. Moreover, we construct explicit total colorings (with at most Δ + 2 colors) for Sierpiński triangles graphs S T p n and Sierpiński triangle fractal with the limit space S T p ∞ , for p ∈ { 3 , 4 , 5 , 6 } , and prove that the AVDTC conjecture is also valid for these cases. [ABSTRACT FROM AUTHOR]

The science of complexity aims to answer the question of what rules nature chooses when assembling the basic constituents of matter and energy into structures and dynamical patterns that cascade through the entire hierarchy of scales in the Universe. A related phenomenon – natural language – can successfully mirror such structures as reflected by its ability to encode and transmit information about them and among them. It is thus legitimate to expect that natural language carries the essence of complexity. And indeed, in the human's speaking and writing it is particularly true that more is different. Natural language thus deserves a central place in the related quantitative study within the science of complexity. With this in mind the present review summarizes the main methodological concepts used in this domain and documents their applicability and utility in identifying universal as well as system-specific features of natural language in its written representation in several major Western languages. In particular, three main complexity-related current research trends in quantitative linguistics are exhaustively covered. The first part addresses the issue of word frequencies in texts and, in particular, demonstrates that taking punctuation into consideration largely restores scaling whose violation in the Zipf's law for the most frequent words is commonly modelled by the so-called Mandelbrot's correction. The second part introduces methods inspired by time series analysis, used in studying various kinds of long-range correlations in written texts. The related time series are generated on the basis of text partition into sentences or into phrases between consecutive punctuation marks. It turns out that these series develop features often found in signals generated by complex systems: the presence of long-range correlations along with fractal or even multifractal structures. Moreover, it appears that the distances between consecutive punctuation marks quite universally across languages comply with the discrete variant of the Weibull distribution, often appearing in survival analysis. In the third part, the application of the network formalism to natural language is reviewed, particularly in the context of word-adjacency networks whose structure reflects the word co-occurrence in texts. Various parameters characterizing topology of such networks can be used for classification of texts, for example, from a stylometric perspective. Network approach can also be applied in semantic analysis to represent a hierarchy of words and associations between them based on their meaning. Structure of such networks turns out to be significantly different from that observed in random networks, revealing genuine properties of language. Finally, punctuation appears to have a significant impact not only on the language's information-carrying ability but also on its key statistical properties, hence it seems recommended to consider punctuation marks on a par with words. [ABSTRACT FROM AUTHOR]

Growing evidence suggests the presence of white matter (WM) alterations in bipolar disorder (BD). In this study we aimed to investigate the state of WM structures, in terms of tissue integrity and morphological complexity, in BD patients compared to healthy controls (HC), in an attempt to better elucidate the microstructural changes associated with BD. We collected a dataset of 399 Diffusion Tensor Magnetic Resonance Imaging (167 BD and 232 healthy controls) images, acquired at five different sites, which was processed with Tract-Based Spatial Statistics (TBSS) and fractal analysis. The TBSS analysis demonstrated significantly lower FA values in the BD group. Diffusion abnormalities were primarily located in the temporo-parietal network. The Fractal Dimension (FD) analysis did not reveal consistent significant differences in the morphological complexity of WM structures between the groups. When the FD values of patients were considered individually, it is possible to notice some localized significant deviations from the healthy population. DTI sequences have not been harmonized before acquisition, samples' sizes are heterogeneous. This study, by applying both TBSS and FD analyses, allows to evaluate diffusion and structural alterations of WM at the same time. The evaluation of WM integrity from these two different perspectives could be useful to better understand the pathophysiological and morphological changes underpinning bipolar disorder. • We analyzed tissue integrity and complexity with diffusion imaging and fractals. • We found significant disruption in white matter of bipolar patients. • The fractal analysis found localized deviations in patients in respect to healthy. [ABSTRACT FROM AUTHOR]

Multi-task learning proved to be an effective strategy to increase the performance of a dense prediction network on a segmentation task, by defining auxiliary tasks to reflect different aspects of the problem and concurrently learning them with the main task of segmentation. Up to now, previous studies defined their auxiliary tasks in the Euclidean space. However, for some segmentation tasks, the complexity and high variation in the texture of a region of interest may not follow the smoothness constraint in the Euclidean geometry. This paper addresses this issue by introducing a new multi-task network, MTFD-Net , which utilizes the fractal geometry to quantify texture complexity through self-similar patterns in an image. To this end, we propose to transform an image into a map of fractal dimensions and define its learning as an auxiliary task, which will provide auxiliary supervision to the main segmentation task, towards betterment of left atrium (LA) segmentation in computed tomography (CT) images. To the best of our knowledge, this is the first proposal of a dense prediction network that employs the fractal geometry to define an auxiliary task and learns it in parallel to the segmentation task in a multi-task learning framework. Our experiments revealed that the proposed MTFD-Net model led to more accurate LA segmentations compared to its counterparts. • A segmentation network quantifying texture complexity through self-similar patterns. • The model learns fractal dimension map estimation in a multi-task learning framework. • The model led to improved performance for left atrium segmentation in CT images. [ABSTRACT FROM AUTHOR]

As one of the most fundamental clean energy sources, geothermal energy extraction and application has undergone extensive research and analysis. Understanding fluid seepage behavior in the pore and fracture microstructure of complex reservoir is fundamental to the extraction. In this study, we provide a novel method for measuring the interplay between pore microstructure and thermal–hydrological–mechanical coupling. The fractal geothermal model developed in this study was validated for correctness by comparison with field extraction data, published model, and analytical solution. For studying thermal conduction, seepage, and fracture-matrix interactions in geothermal reservoirs, the fractal seepage model proposed in this research is superior to the traditional cubic seepage model, according to the findings. Permeability increases by about 11.7% as fractal dimension increases by 0.4%. In addition, both the produced fluid temperature and the mean reservoir temperature are inversely proportional to the pore fractal dimension and initial porosity. We can infer that different structural parameters will have an array of effects on seepage resulting from geothermal extraction, which cannot be predicted using cubic permeability models. This provides new ideas for geothermal extraction practitioners. [ABSTRACT FROM AUTHOR]

We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in Mazurenko and Zarichnyi (2018) [24]. The main feature developed here is a metrization of the space of idempotent measures using the embedding of the space of idempotent measures to the space of fuzzy sets. The metric obtained induces a topology stronger than the canonical pointwise convergence topology. A key result is the existence of a bijection between idempotent measures and fuzzy sets and a conjugation between the Markov operator of an IFS on idempotent measures and the fuzzy fractal operator of the associated Fuzzy IFS. This allows to prove that the Markov operator for idempotent measures is a contraction w.r.t. the induced metric and, from this, to obtain a convergence theorem and algorithms that draw pictures of invariant measures as grayscale images. [ABSTRACT FROM AUTHOR]

At present, some image compression and encryption algorithms have been presented to protect digital images by combining chaos and compressed sensing (CS). However, some of them still suffer from security and efficiency shortcomings. To solve these issues, a multi-image compression and encryption based on 3D fractal cube, CS and chaos is introduced in this paper. Firstly, the multiple plain images are sparsified by the Haar Wavelet Transform (HWT), and measured by CS to obtain the measurement value matrices. Herein, a four-wing hyperchaotic system is adopted to generate the measurement matrix for CS. Subsequently, a 3D fractal cube (3D-FC) is defined and generated iteratively by constructing the initial 3D-FC. Based on it, the obtained measurement value matrices are shuffled to reduce the high correlation between the adjacent pixels. Next, the cipher images are obtained after the cipher-associated adaptive diffusion based on GF (28) is performed on the confused image. Moreover, we have conducted extensive experiments to verify the encryption efficiency, compression performance, and security. Both simulation results and performance analysis confirm that the present algorithm has satisfactory performance in these aspects. [ABSTRACT FROM AUTHOR]

The concept of fractal propagation, through the deployment of computer numerical controlled devices, has been discussed in this paper. For experimentation purposes, four of the most well known fractal patterns namely, Sierpinski gasket, Barnsley fern, Fractal tree and Koch snowflake curve have been chosen. This work shows the practical reproduction of before mentioned fractals on a physical media by deploying a 3-axis CNC plotter, as a part of the hardware, after proper python implementation as a part of the software. The mathematics and algorithm used in the production of the fractals have been discussed in detail and the method of deployment involves vectorisation of the fractals into a set of coordinates which are then passed on to the 3 axis CNC plotter plotter for production of the final output on a larger drawing surface. This paper aims at combining the individual concepts of fractal propagation and robotics in order to produce tangible fractal illustrations. [ABSTRACT FROM AUTHOR]

In this study, we introduce fractal bluff bodies to galloping energy harvesters (GEH) to improve the performance of wind energy harvesting. Three basic sectional shapes are proposed for GEH, from which four fractal bluff bodies are developed. An aero-electro-mechanical model with distributed parameters is established to elucidate the underlying mechanism. Then numerical simulations and wind tunnel tests are conducted based on a fabricated prototype. Compared with the conventional galloping energy harvester with a cuboid bluff body (GEH-CB), the GEH with a fractal bluff body (GEH-FB) owns a lower cut-in wind speed and can generate higher voltages. The galloping critical wind speed for GEH-FB is reduced by 0.4 m/s. Furthermore, the output voltage can increase by 51.97 % at the wind speed of 5 m/s. Especially, compared to the GEH-CB, the GEH-FB demonstrates a great advantage in terms of power density, being approximately 164.1 % higher. Three-dimensional computational fluid dynamics (3D-CFD) shows that the wake vortex area and the space between the adjacent vortices both increase, which explains the aerodynamic mechanism of a large aerodynamic force and a performance enhancement in wind galloping energy harvesting. Finally, some practical application tests exhibit the effectiveness of the proposed fractal galloping harvesters. • Fractal geometry is introduced to enhance the wind energy harvesting performance. • A distributed model is developed to predict the dynamical response. • The effects of similarity ratio and elements in fractal bluff bodies are investigated. • The aerodynamic-enhancement mechanism is revealed by macro- and micro-field analysis. [ABSTRACT FROM AUTHOR]

Several mathematical models have been established for describing the saturated seepage of fracture networks in low-permeability reservoirs, but there is still a lack of systematic research on unsaturated seepage models. Therefore, by combining these single connected fractures into the physical geometry of a parallel plate, a comprehensive spontaneous imbibition model at the fracture and core scales considering a rough wall geometry, streamline tortuosity, and heterogeneity of the fracture size distribution is derived by applying the classical cubic law describing flat flow and fractal geometry theory. Compared with existing models, our proposed model is verified by using experimental data on the spontaneous imbibition of a single fracture and fractured core available from the literature. Based on the established model, the mechanism through which various geometric elements control spontaneous imbibition is analysed at the fracture and core scales. The maximum fracture aperture plays an important role in controlling the wettability evolution behaviour at the core scale, and the fractal dimensions of fracture tortuosity and roughness are more sensitive to spontaneous imbibition in fractured cores than is the fractal dimension of the fracture size distribution. These research results are useful for evaluating the balance between reservoir damage and fracturing effects. [ABSTRACT FROM AUTHOR]

Global optimization of complex and high-dimensional functions remains a central challenge with broad applications in science and engineering. This study introduces a new optimization approach called quasi-random metaheuristic based on fractal search (QRFS), which harnesses the power of fractal geometry, low discrepancy sequences, and intelligent search space partitioning techniques. The QRFS uses fractals' inherent self-similarity and intricate structure to guide the solution space exploration. For the proposal, a deterministic but quasi-random element is used in the search process using low discrepancy sequences, such as Sobol, Halton, Hammersley, and Latin Hypercube. This integration allows the algorithm to systematically cover the search space while maintaining the level of diversity necessary for efficient exploration. The QRFS employs a dynamic strategy of partitioning the search space and reducing the population of solutions to optimize the use of function accesses, which causes it to adapt well to the characteristics of the problem. The algorithm intelligently identifies and prioritizes promising regions within the fractal-based representation, allocating computational resources where they are most likely to yield optimal solutions. Experimental evaluations on several benchmark problems demonstrate that QRFS consistently outperforms modern, canonical metaheuristics and variants of algorithms such as differential evolution (DE), particle swarm optimization (PSO), covariance matrix adaptive evolution strategy (CMA-ES), regarding solution quality. Besides, the algorithm shows remarkable scalability, which makes it suitable for high-dimensional optimization tasks. Overall, QRFS offers a robust and efficient approach to solving complex optimization problems in various domains, paving the way for improved decision-making in real-world applications. • QRFS optimizes using promising fractal regions. • QRFS dynamically partitions space, optimally using function accesses. • QRFS uses fractal self-similarity for exploration and exploitation. • Combines quasi-random determinism with metaheuristic stochasticity. • Statistical tests validate results, parametric and nonparametric. [ABSTRACT FROM AUTHOR]

The size-frequency distribution of many geological and geophysical variables, in relation to fractures, faulting and seismicity, is well described by a statistical distribution of the power law type which is characterized by its exponent. For earthquake magnitudes, the exponent is the well-known b -parameter of the Gutenberg-Richter scaling law. In this paper we: • provide a strict statistical derivation of the distribution law of earthquake magnitudes, • show that the maximum likelihood estimator of the b -parameter is unbiased, • demonstrate that the maximum likelihood estimator is invariant to the value chosen as the minimum magnitude threshold in so far as it is larger than the magnitude of completeness of the earthquake catalogue, and • provide a new estimator based on the minimization of the Kolmogorov-Smirnov statistic and provide a strategy for detecting and mapping the spatio-temporal variation of the b -parameter in seismic swarms. The findings are illustrated with simulated data and a case study with real data. • Strict statistical derivation of the probability density function of the magnitudes of earthquakes. • The paper shows that the Maximum Likelihood estimator of b is unbiased. • A new Kolmogorov-Smirnov statistic estimator of b. • Demonstration of the invariance of the Maximum Likelihood estimator with respect to the magnitude threshold. • Provides estimation results of b for small sample sizes and spatial smooothed estimates of b. [ABSTRACT FROM AUTHOR]

Gliomas are the brain tumors in glial cells, which are categorized into four numerical grades, I-II-III-IV, to quantize the aggressiveness and severity of the tumors; while divided into two major groups, high-grade (HG) and low-grade (LG), in general. Among many differences between these groups, one of the most distinct and characteristic features could be seen in the shape of the tumor boundaries by magnetic resonance imaging (MRI). Due to aggressive nature of the HG tumors in proliferation phase, the boundaries of HG tumors become more shape-wise complex compared to the LG tumors, which could be differentiated by analyzing the fractal complexity of the cell membranes. However, the complexity cannot be either manually calculated or estimated by eye inspection without a reference point with one single image or sometimes even with an image set. Therefore, we present an automated glioma grading framework to provide an insight on the grades with a novel contouring and fractal dimension analysis system. The primary component of the proposed system is an automated Nakagami imaging module with a specialized fuzzy c-means algorithm to contour the boundaries of the whole tumors. The contoured images, afterwards, are analyzed by the Minkowski–Bouligand and Hausdorff methods for two panning options to generate the fractal dimensions and to estimate the fractal complexities for classifying the gliomas The results are greatly encouraging that the overall classification accuracy is computed as 88.31 % using the basic support vector machines (SVM) classifier; while as 91.96 % with the arbitrary thresholding appended. The outcomes of this paper with implementable mathematical infrastructure would be very useful and beneficial as an expert system in intelligent and automatic glioma grading, for researchers and medical experts. • We proposed a novel automated glioma-grading framework. • 2D FLAIR MRI images consisting of HG and LG gliomas are analyzed. • Raw images are highlighted by Nakagami module and contoured by fuzzy 2-means. • Fractal dimensions of the contoured images are analyzed by two methods. • Overall accuracy is computed as ACC=91.96 % with a SVM classifier. [ABSTRACT FROM AUTHOR]

The first successful fabrication of metal halide Perovskite MAPbI 3 Film for NH 3 gas sensing using fractal electrode structure is reported. High performance NH 3 gas sensor was fabricated on ITO glass substrate by laser-patterning method. Gas sensors were fabricated using Hilbert, Peano, and Gosper curve structure electrodes, and their performance is compared to conventional interdigitated electrode. Gas sensing performance on Hilbert electrode design showed the highest response of 2.82 at 10 ppm NH 3 gas, which is 145 % times higher compared to conventional interdigitated structures. Simulation study demonstrates that the electric field intensity and number of nodes of the fractal electrode are higher than those of the traditional interdigitated electrode. This study demonstrates that fractal electrode is a feasible strategy for optimizing the potential performance of MAPbI 3 -based Perovskite gas sensor with low-cost fabrication with no extra processing. [Display omitted] • Fractal electrode improves MAPbI3-based NH 3 gas sensing response. • Complexity of the fractal curve results better edging effect. • High current density facilitates the electron transfer process. [ABSTRACT FROM AUTHOR]

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1] , for any L 2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α ≥ 5 / 4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces L t γ W x s , p , in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3 / p + 1 − 2 α , ∞ , p) and (2 α / γ + 1 − 2 α , γ , ∞). Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff H η ⁎ measure, where η ⁎ > 0 is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

To assess the complexity of fibrous dysplasia and ossifying fibroma lesions by fractal dimension (FD) in 2D and 3D cone-beam computed tomography (CBCT) images. Six cases of fibrous dysplasia were compared with six cases of ossifying fibroma in different images, thicknesses and regions of interest (ROI) of 2D and 3D images. FD was determined separately in hypondense and hyperdense areas using the box-counting method. The Student t-test was applied to determine the difference in FD between lesion groups. FD of hypodense areas in 2D images ranged from 1.2566 to 1.4532 in fibrous dysplasia and from 1.1540 to 1.3601 in ossifying fibroma, while FD of hyperdense lesion areas ranged from 1.1755 to 1.4132 for fibrous dysplasia and from 1.0480 to 1.2769 for ossifying fibroma images. Panoramic reconstruction and axial images of 0.25 mm thickness revealed significant differences between lesion groups in all ROI shapes (p < 0.05). FD of 3D images was higher in fibrous dysplasia (2.4999–2.7417) than in ossifying fibroma (1.9283–2.4144) in both hyperdense and hypodense areas, with statistical difference in almost all ROIs (p < 0.05). FD data suggest higher complexity of structures of fibrous dysplasia lesions than of structures of ossifying fibroma. [ABSTRACT FROM AUTHOR]

Combined computed tomography–derived myocardial blood flow (CTP-MBF) and computed tomography angiography (CTA) has shown good diagnostic performance for detection of coronary artery disease (CAD). However, fractal analysis might provide additional insight into ischemia pathophysiology by characterizing multiscale perfusion patterns and, therefore, may be useful in diagnosing hemodynamically significant CAD. The purpose of this study was to investigate, in a multicenter setting, whether fractal analysis of perfusion improves detection of hemodynamically relevant CAD over myocardial blood flow quantification (CTP-MBF) using dynamic, 4-dimensional, dynamic stress myocardial computed tomography perfusion (CTP) imaging. In total, 7 centers participating in the prospective AMPLIFiED (Assessment of Myocardial Perfusion Linked to Infarction and Fibrosis Explored with Dual-source CT) study acquired CTP and CTA data in patients with suspected or known CAD. Hemodynamically relevant CAD was defined as ≥90% stenosis on invasive coronary angiography or fractional flow reserve <0.80. Both fractal analysis and CTP-MBF quantification were performed on CTP images and were combined with CTA results. This study population included 127 participants, among them 61 patients, or 79 vessels, with CAD as per invasive reference standard. Compared with the combination of CTP-MBF and CTA, combined fractal analysis and CTA improved sensitivity on the per-patient level from 84% (95% CI: 72%-92%) to 95% (95% CI: 86%-99%; P = 0.01) and specificity from 70% (95% CI: 57%-82%) to 89% (95% CI: 78%-96%; P = 0.02). The area under the receiver-operating characteristic curve improved from 0.83 (95% CI: 0.75-0.90) to 0.92 (95% CI: 0.86-0.98; P = 0.01). Fractal analysis constitutes a quantitative and pathophysiologically meaningful approach to myocardial perfusion analysis using dynamic stress CTP, which improved diagnostic performance over CTP-MBF when combined with anatomical information from CTA. [Display omitted] [ABSTRACT FROM AUTHOR]

Different from the conventional calculation method, the discrete-time noise-suppression zeroing neural dynamic (DTNSZND) model is developed and investigated for online solving the time-varying/time-invariant scalar cube root problem with different disturbances (bounded/unbounded disturbances) from a control perspective in this paper. The solving precision and robustness of different DTNSZND models are generated via various difference rules are discussed and analyzed, and numerical simulations verify that the convergence property of the developed models are closely related to the accuracy of difference rules. The zero-stable, consistent and convergent of different DTNSZND models are demonstrated via theoretical analyses. Furthermore, numerical simulation and comparative results prove that the convergence, robustness and superiority of the presented different DTNSZND models for online solving time-varying/time-invariant cube root problems with different interferences in real domain or complex domain. Finally, different fractals are generated by utilizing the developed DTNSZND model to solve the cube root problem with different noises in complex domain, which are quite different from Newton fractals produced by Newton–Raphson iteration. Therefore, the proposed different DTNSZND models can be considered as a novel iterative algorithm to generate new fractal. [ABSTRACT FROM AUTHOR]

This study aimed to investigate the effects of arthrocentesis with intra-articular hyaluronic acid (HA) injection on mandibular condyles using fractal dimension (FD) analysis. Patients with temporomandibular joint (TMJ) internal derangement (ID) were divided into three groups according to how many times the arthrocentesis with HA injection was performed. FD analysis is a quantitative concept that provides information about trabecular bone microstructures. Regions of interest were selected from bone areas close to the articular surfaces of the right and left condyles on panoramic radiographs, which were taken before the procedure (T0) and at the sixth month (T1) after the procedure. Then, the FD values were calculated. All images were reviewed by the same researcher. A two-way repeated-measures analysis of variance was used to determine whether there was a significant interaction between groups, treatment sides, and time on the FD values. A total of 140 patients, including 118 patients who received bilateral arthrocentesis with HA injection and 22 control patients, were included in the study. The images of 20 randomly selected patients were re-evaluated after 2 weeks. As a result of the correlation analysis, there was no significant difference between the two measurements (P > 0.05). The main effect of time on the FD value was significant [F (1, 136) = 157.879, P < 0.001]. This effect was qualified by a significant time × group interaction effect [F (3, 136) = 18.533, P < 0.001]. The increases in the mean FD values on the right and left sides in all treatment groups between T0 and T1 times were significant (P < 0.001 for all), whereas changes in FD values on the right and left sides between T0 and T1 times were not significant in the control group (P = 0.164 and P = 0.557, respectively). After arthrocentesis with HA injections, the mean FD values increased in the mandibular condyles, depending on time in all treatment groups. Patients with TMJ ID are likely to have radiologically detectable degenerative changes and abnormalities in the condyles. The FD analysis method, which provides quantitative data, is recommended as an adjunct and guide for oral and maxillofacial surgeons in radiological examinations that should be performed together with clinical examination for the follow-up of microstructural changes in the condyle after arthrocentesis with HA injection. [ABSTRACT FROM AUTHOR]

Background: Incorporating variability within gait rehabilitation offers a promising approach to restore functional capacity. However, it's success requires adequate synchronization, a parameter that lacks report in most of the literature regarding cued gait training.Research Question: How changes to synchronization performance during fractal-like and isochronous cueing impacts gait variability measures?Methods: We asked twelve young male participants to walk in synchronization to two different temporally structure cueing (isochronous [ISO] and fractal [FRC]). We have also manipulated the cueing's tempo by increasing and decreasing it by 5% to manipulate synchronization, resulting in six conditions (stimuli [ISO,FRC] x tempo [SLOW, NORMAL, FAST]). The normal condition was set from an uncued trial through the participant's self-paced stride time. Synchronization performance (ASYNC) and gait variability (fractal scaling and coefficient of variation) were calculated from stride time data ( -ISIs,CV-ISIs). Repeated measures analysis of variance or Aligned Rank Transform were conducted to determine significant differences between metronome tempo and stimuli for the dependent variables RESULTS: Our results showed a FAST tempo decreases synchronization performance (ASYNC) and leads to lower -ISIs, for both ISO and FRC stimuli. This indicates that when an individual exhibits poor synchronization during cued gait training, his/her gait variability patterns will not follow the temporal structure of the presented metronome. Specifically, if the individual poorly synchronizes to the cues, the gait patterns become more random, a condition typically observed in older adults and neurological patients, which runs contrary to the hypothesis when using fractal-like metronomes.Significance: This study provides supporting evidence that measuring synchronization performance in cued training is fundamental for a proper clinical interpretation of its effects. This is particularly relevant for the recent and ongoing clinical research using fractal-like metronomes since the expected gait patterns are dependent on the synchronization performance. Randomized control trials must incorporate synchronization performance related measures. [ABSTRACT FROM AUTHOR]

Fractal fractional operators in Caputo and Caputo-Fabrizio sense are being used in this manuscript to explore the interaction between the immune system and cancer cells. The tumour-immune model has been investigated numerically and theoretically by the singular and non-singular fractal fractional operators. Via fixed point theorems, the existence and uniqueness of the model under the Caputo fractal fractional operator have been demonstrated. Using the fixed point theory, the existence of a unique solution has been derived under the Caputo-Fabrizio case. Through nonlinear analysis, the Ulam-Hyres stability of the model has been derived. For the singular and nonsingular fractal fractional operators, numerical results have been developed by Lagrangian-piece wise interpolation. We simulate the numerical results for the various sets of fractional and fractal orders to describe the relationship between immune and cancer cells under the novel operators with two different kernels. We compared the dynamics of the tumor-immune model using a power law and an exponential-decay kernel to explore that the nonsingular fractal fractional operator provides better dynamics for the considered model. [ABSTRACT FROM AUTHOR]

The Abelian sandpile model was the first example of a self-organized critical system studied by Bak, Tang and Wiesenfeld. The dynamics of the sandpiles occur when the grains topple over a graph. In this study, we allow the graph to evolve over time and change its topology at each stage. This turns out in the occurrence of phenomena impossible in the classical sandpile models. For instance, unstable configurations over evolutive graphs with a sink that never stabilize. We also experiment with the stabilization of configurations with a large number of grains at the center over evolutive graphs, this allows us to obtain interesting fractals. Finally, we obtain power laws associated with some evolutive sandpiles. • The concept of evolutive sandpiles is introduced allowing graphs to evolve over time. • This allows the occurrence of phenomena impossible in the classical sandpile models. • Stabilizing many grains on the center on evolutive grids unveils intriguing fractals. • Some power laws associated with evolutive graphs were obtained. [ABSTRACT FROM AUTHOR]

Fractal antenna is best known for its wide bandwidth (BW), compactness, and lower cost. Reconfigurable antennas are capable of modifying its frequency and radiation properties dynamically, in a controlled and reversible manner. Reconfigurable fractal antennas are combined of fractal antennas and RF switches. Adding reconfigurable elements to the fractal geometry further enhances the overall performance of the antenna, making it suitable for multiple applications. [Display omitted] In this communication, a band reconfigurable fractal antenna with modified partial ground is proposed for wireless applications. Four switches using PIN diodes supported by the antenna biasing circuit provide frequency reconfigurability, while star-shaped fractal geometry is used to achieve both miniaturisation and wideband functioning in the proposed antenna. The surface current distribution of the radiating patch is altered by the ON and OFF states of the PIN diode, which leads to the multiband resonance and reconfiguration features of the designed structure. The design has an overall electrical size of 70 × 45 × 1.6 mm3 and is made on a commonly available FR4 substrate with a thickness of 1.6 mm and a dielectric constant of 4.4. The three operational bands are as follows: case I, which covers 3.7–5 GHz (30 %); case II, which covers 1.7–3.5 GHz (71 %); and case III, which covers 1.4–4 GHz (96 %). Two distinct bands are obtained in cases I and II; and case III nearly covers the frequency band of cases I and II. Therefore, the impedance bandwidth (IBW) offers continuous wideband frequency coverage from 1.4 to 5 GHz (110 %). To sense the full band and then modify its bandwidth to choose the appropriate sub-band and prefilter out the others, the proposed antenna may switch between a wide operational band of 1.4–5 GHz with three distinct subbands. The antenna could potentially be useful for wireless communication systems in the future due to its frequency-selective feature and stable radiation patterns. [ABSTRACT FROM AUTHOR]

• Valuable elements in waste photovoltaic modules were effectively utilized. • More than 99 % of both aluminium and silver can be leached out. • Revised fractal kinetic model more accurately characterizes the kinetics of aluminium and silver leaching. Efficient recycling and reuse of resources are essential for global resource transformation. Researchers worldwide have focused on the proper disposal of waste photovoltaic (PV) modules produced by the rapidly developing PV industry. The utilisation of waste PV modules has high economic value and helps prevent the pollution of harmful elements in the environment to achieve sustainable development in the PV industry. This work proposes short 'rapid H 2 O 2 -H 2 SO 4 leaching' for the efficient recovery of aluminium and silver from waste PV modules. The extraction for aluminium and silver were 99.20 % and 99.01 %, respectively. Fractal geometry method was used to modify and optimise the leaching kinetics. The fitting calculation indicated that the leaching of aluminium and silver conformed to internal diffusion-controlled process. [ABSTRACT FROM AUTHOR]

• Analyzed the fractal characteristics and energy evolution of tuff fractures with different joint angles. • The damage modes and crack extension patterns of specimens with different joint angles were studied. • The crack evolution process of the specimen is reproduced by the finite discrete element method. Dynamic disturbances significantly affect the degradation of properties in rocks with natural joints. This study examines the impact of various joint angles on the fractal characteristics of fractures and energy evolution in tuff by conducting dynamic impact mechanical tests. Specimens were tested at joint angles ranging from 0° to 90° under constant strain rates. Analyses focused on the fractal dimensions of rock impact fractures, crack fractal dimensions, energy evolution, and numerical simulations to ascertain the influence of joint angles on the rock's mechanical properties. Findings indicate that the failure mode of tuff specimens transitions from tensile to mixed tensile-shear failure and then back to tensile failure as joint angles change. Concurrently, the fractal dimension of breakage in tuff varied between 1.751 and 2.263, with corresponding crack fractal dimensions ranging from 0.999 to 1.478. Notably, both the dynamic compressive strength and energy dissipation of jointed tuff first decrease and then increase as the joint angle broadens, forming a "V" shaped distribution. Utilizing finite discrete element numerical calculations to replicate the crack expansion and evolution process, the study corroborates the significant influence of joint angle on the rock's failure characteristics, which carries critical implications for rock engineering design. [ABSTRACT FROM AUTHOR]

• The initial damages have a greater influence on the acoustic emission response law. • The information dimension reflect the degree of confusion at different loading stages. • The larger the threshold value, the smaller the fractal probability. • The phenomenon of a sharp decrease in the information dimension is a precursor signal of coal rupture. Uniaxial compression experiments were conducted on coal samples with different initial damage degrees, and the acoustic emission (AE) signals of the whole deformation and destruction process were captured synchronously. The distribution patterns of the acoustic emission signals at different loading stages were discussed. The chaotic system was introduced to process the acoustic emission series. The results indicate that the acoustic emission responses of coal samples with different damage levels had obvious stage characteristics during the whole destruction process. According to the information dimension fitting curves, the coal samples had good fractal characteristics in the whole loading process, and the value of information dimension decreased with the increased of threshold value.The information dimension of the acoustic emission energy signals showed a dynamic evolution law of "increase-decrease-increase" with time, and the rapid decreased of the information dimension could be used as a stable and effective precursor for predicting the macro-damage of coal. [ABSTRACT FROM AUTHOR]

• Introduced a Roughness Parameter Method for direct estimation of fractal dimension. • RPM performs best for surfaces with FD values ranging from 2.65 to 2.9. • SDCCM accurately estimates FD for surfaces with FD values between 2.1 and 2.65. • RPM correlates well with surface topography parameters suggested by ISO 25178. • Effective FD determination for real-time surfaces using RPM and CCMs. The fracture process of any material is a natural response towards external loading. These fracture surfaces exhibit a fractal pattern that can be analyzed quantitatively by measuring its fractal dimension (FD). Subsequently, this FD value can be correlated with different properties of the material. Various versions of the cubic covering method (CCM) can accurately describe the fractal patterns present on irregular surfaces with minimum approximation. However, these methods are not direct methods for determining the FD, as they rely on the covering process of cubes rather than considering the irregular surface area. Considering this, a new approach known as the Roughness Parameter Method (RPM) was studied. This method involves the surface area of an irregular surface to measure surface roughness parameters for the direct estimation of the FD. Comparative analysis between direct and indirect methods to measure FD was carried out using Takagi surfaces and real-world irregular surfaces. [ABSTRACT FROM AUTHOR]

• The dynamic fracture development of shale are described, and the geometric parameters, topological and fractal characteristics of the fracture network are evaluated. • The dissipation law of rock failure energy is investigated, which reflects the essential characteristics of rock failure under impact load. • Combined with section scanning, stress simulation and energy consumption calculation, the dynamic load rock breaking mechanism is systematically analyzed. Shale fracture and energy dissipation are the basis of reservoir fracturing effect evaluation. The damage and fracture mechanism of shale under dynamic impact load is the key scientific concern of reservoir fracturing. To study the law of fracture, fracture propagation and energy dissipation of shale under dynamic impact, dynamic tests of water-based impact load under different impact velocities were carried out using impact rock breaking test platform based on air cannon. Topological and fractal theory, Weibull distribution model, space vector geometry method and Griffith fracture mechanics theory were used to systematically study the effects of impact velocity on fracture development and expansion, fragment mass and size distribution, damage degree of abutment residual and energy dissipation transformation. Scanning electron microscopy (SEM) was used to observe the micromorphology of the impact fracture surface. The results demonstrated that an increase in impact velocity led to the formation of a network fracture featuring main fractures, cavity fissure, bedding fractures, and crushing zones, thereby enhancing the fracture's capability to penetrate bedding. The apparent fracture network connectivity, topology structure and fractal dimension were positively correlated with impact velocity. The residual height of the shale decreased, resulting in an escalation of the damage degree from mild (0.43) to severe (0.75), while the average penetration angle decreases from 81.00° to 63.52°. The area of new surface experiences rapid initial growth followed by a more gradual increase, with dissipative energy showing a more pronounced increase compared to crushing energy. The micro-fracture mode underwent an evolution from brittleness to toughness and a transition from low energy consumption to high energy consumption. The propagation, reflection, and superposition of stress waves within the rock mass gave rise to a complex stress zone, ultimately causing the rock mass to break. These findings lay the groundwork for developing a stress-damage-energy-failure combined model for shale subjected to dynamic impact loads, thereby enhancing the development of intricate networks in shale reservoir and improving the effective extraction of shale gas. [ABSTRACT FROM AUTHOR]

[Display omitted] • Multifractality observed in geophysical logs due to long-range correlations. • We characterized stratigraphic variation using fractal analysis of geophysical logs. • The presence of hydrocarbon weakens log multifractal behaviour. We propose a novel fractal analysis-assisted approach for stratigraphic characterization to understand the in-situ subsurface geology of hydrocarbon reservoirs. By employing correlation dimension and multifractal detrended fluctuation analysis, we quantify the self-similarity and long-range dependencies in the geophysical logs, reflecting the unique sedimentation processes inherent in the stratigraphic formations of the Bhogpara oil field in the Upper Assam basin. The fractal properties of the logs associated with various stratigraphic units are used to establish a fractal-based stratigraphic signature. The hierarchy of the studied formations, based on the correlation dimension ranging from 1.98 to 1.13, is follows as: Girujan > Tipam > Barail > Kopili > Prang > Narpuh > Lakadong-Therria. Furthermore, the spectrum width values, was found to range from 0.70 to 2.04, reflecting the spatial correlation, geological complexity, and heterogeneity inherent in each formation as: Kopili > Prang > Narpuh > Barail > Girujan > Tipam > Lakadong-Therria. This method identifies common fractal patterns between wellbores and aids in determining the spatial continuity of stratigraphic units. Our approach effectively detects lateral variations in subsurface formations, linking well log fractality with depositional patterns through various transitional environments, such as lacustrine to shallow marine to fluvial, in presence of different fluids. The petroliferous formation exhibits lowest fractal dimension and weak multifractality. The research utilizes fractal analysis to improve stratigraphic correlation, enhancing subsurface characterization and reservoir delineation, particularly in complex geological environments, and enhancing reliability. [ABSTRACT FROM AUTHOR]

Consider a separable metric space (X , d) , and let (K (X) , d ˜) denote the space of non-empty compact subsets of X equipped with the Hausdorff metric. This paper aims to introduce and investigate the concepts of two general fractal dimensions and general dimensions within the framework of (K (X) , d ˜). In particular, we explore a relationship between the general fractal dimensions of a set Z of a self-similar sequence space and their counterparts in the space of compact subsets K (Z). [ABSTRACT FROM AUTHOR]

To date, the mixing of multiple species within microchannels relies primarily on infamously slow diffusive processes. Hence, the present study aims to numerically investigate the fundamental transportation and interaction of a dual-concentration fluid by employing fluid flow perturbations induced by a single instance of a 3D fractal geometry. In particular, a D3Q19 lattice-Boltzmann approach is employed to unravel the chaotic advection and the underlying flow dynamics of the multi-length-scale-induced mixing within the transitional flow regime at Reynolds number Re D = 300. Four obstacle configurations are explored in depth, namely, a control regular cube (RC), two fractal cubes (FC 2 β ) with varying blockage ratios (β), and a 45° tilted regular cube (RD). Results show that the FC 2 0.1312 with a reduced β still enhances the overall mixing performance by 10%, while the larger FC 2 0.2383 provides an augmentation of 59% compared to the RC. Furthermore, the effects of pertinent parameters: decay exponent, integral length scale, and energy spectra are studied. One key observation is that a higher decay exponent is correlated to a higher mixing index (MI) due to the precipitous collapse of turbulence intensity which dissipates energy to all scales of motion. The figure of merit (FoM) is employed to appraise the mixing performance of the obstacle relative to the penalty of blockage ratio, for which the FC 2 0.1312 yields the highest FoM. Through the rapid generation and subsequent decay of turbulence intensity, the multi-length scale FC 2 β s reshuffle the advective energy and express a preferred mixing mechanism transversely along the fluid interface, which attains a remarkable localized MI of 87% within ten obstacle diameters downstream from the obstacle position. Overall, this study may shed light on the underlying mechanism-of-action for the fundamental amplification of mixing within a microchannel, which allows for a potential application strategy that extracts highly homogenous mixtures from compact microchannel domains. Thermal mixing is also augmented using FC 2 0.2383 for a range of Reynolds number and shows that the method of assigning passive scalar quantities to the fluid provides a good two-way approximation for concentration/thermal mixing. • Numerical simulations of mixing in a microchannel using single 3D obstacles. • Flows simulated at Re = 300 using double distribution lattice Boltzmann method. • Maximum mixing enhancement of 59% achieved using fractal cube. • Fractal cube attained localized MI of up to 87% at highly homogenized regions. • Fractal cubes reshuffle advective energy spanwise and improve localized mixing. [ABSTRACT FROM AUTHOR]

In this paper, we explore the intriguing field of fractal calculus as it pertains to fractal curves and fractal sets. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for solving α -order differential equations. Notably, we extend our analysis to solve Fractal Bernoulli differential equations. The applications of our findings are then showcased through the solutions of problems such as fractal compound interest and the escape velocity of the earth in fractal space and time. Visual representations of our results are also provided to enhance understanding. • The fractal separable method and the integrating factor technique are suggested. • Fractal Bernoulli differential equations are given and solved. • The escape velocity of the earth in fractal space and time is studied. [ABSTRACT FROM AUTHOR]