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Sierpinski triangle graphs S n have often been mistaken for Sierpinski graphs S 3 n . Whereas the latter’s metric properties are by now well understood, the former graphs were mostly just considered as a pictorial representation of approximations to the Sierpinski triangle fractal. Therefore, we present here a new labeling for them which shows the relation, but also the differences to the more famous Sierpinski graphs proper. On the base of this labeling we describe an algorithm to obtain individual distances between vertices. This type of algorithm can then be extended to base- p Sierpinski triangle graphs S p n which are related to the class of classical Sierpinski graphs S p n , p ≥ 2 . Some of the metric properties of S p n can now be investigated for S p n as well; e.g., we characterize center and periphery of S p n .

The assurance of secret data transmitted over the Internet and confining its access to explicit grouped information has become a significant security and protection issue. To hide the presence of such information, a robust and blind color image steganography algorithm, using fractal cover images, Singular Value Decomposition (SVD), Integer Wavelet Transform (IWT), and Discrete Wavelet Transform (DWT) is proposed in this paper. The proposed algorithm conceals the color secret image intelligently inside a fractal cover image thus creating a unique stego-image that is virtually a replica of the original cover image. The novelty of this proposed blind image hiding method lies in the fact that a small file size of a unique fractal image can be used to conceal a large amount of information payload. The fractal cover images can be implemented without debasing its quality to make the shrouded information sufficiently strong against image processing attacks. The algorithm possesses features like robustness, imperceptibility, security, and greater hiding capacity due to the application of fractal compression, DWT-SVD, and IWT-SVD hybrid transform schemes. The secret image is inserted in U and V components of the fractal cover with a scaling factor based on the HVS model and a stego-image with sufficiently large keyspace is created. Furthermore, the proposed work compares the performance analysis and security offered by IWT- SVD and DWT- SVD hybrid combinations for different fractal covers using certain metrics like PSNR (Peak Signal to Noise Ratio), MSE (Mean Square Error), SSIM (Structural Similarity Index), BRISQUE (Blind/ Reference less Image Spatial Quality Evaluator) score and Bits per pixel (BPP). The results prove that image degradation is very less (less than 0.6%) in the proposed algorithm even when the secret image file size is more than double the file size of the fractal cover image which also saves the bandwidth and storage requirements.

In this paper, we present generalized Jensen-Mercer inequality for a generalized h-convex function on fractal sets. We proved Hermite-Hadamard-Mercer local fractional integral inequalities via integral operators pertaining Mittag-Leffler kernel. Also, we drive two new local fractional integral identities for differentiable functions. By employing these integral identities, we derive some new Hermite-Mercer type inequalities for generalized h-convex function in local fractional calculus settings. Finally, we give some examples to emphasize the applications of derived results. These results will be a significant addition to Jensen-type inequalities in the literature.

Chinese ice-ray lattices are perhaps one of the earliest and controlled designs of asymmetric and complex patterns applied as a traditional motif in windows. Such intricate and complex designs developed centuries back have created an evident curiosity to explore its underlying geometric rules. Some scholars used the Shape Grammar as a tool to explain and recreate similar patterns. The previous studies conceive the ice-ray lattice design as the iterative subdivisions of a polygon. However, they missed explaining this geometric quality through the discussion of fractal geometry, which can explain the shapes consuming self-similar or self-affine repetitions of itself at different scales. As a novel approach, this paper analytically focuses on the fractal characters of ice-ray lattice designs and uses fractal geometry as a unique tool for generating different types of ice-ray lattices. The significance of this study is the demonstration of the efficacy of fractal geometry and the simple geometric rule of IFS for analyzing and algorithmically modeling complex lattices and cracked-like patterns.

The pore structure of uranium-bearing sandstone is one of the critical factors that affect the uranium leaching performance. In this article, uranium-bearing sandstone from the Yili Basin, Xinjiang, China, was taken as the research object. The fractal characteristics of the pore structure of the uranium-bearing sandstone were studied using mercury intrusion experiments and fractal theory, and the fractal dimension of the uranium-bearing sandstone was calculated. In addition, the effect of the fractal characteristics of the pore structure of the uranium-bearing sandstone on the uranium leaching kinetics was studied. Then, the kinetics was analyzed using a shrinking nuclear model, and it was determined that the rate of uranium leaching is mainly controlled by the diffusion reaction, and the dissolution rate constant (K) is linearly related to the pore specific surface fractal dimension (DS) and the pore volume fractal dimension (DV). Eventually, fractal kinetic models for predicting the in-situ leaching kinetics were established using the unreacted shrinking core model, and the linear relationship between the fractal dimension of the sample's pore structure and the dissolution rate during the leaching was fitted.

Structural health monitoring is an important research field being investigated around the globe. In recent years, meta-heuristics are being used to solve the complex inverse problem of structural damage assessment. In this work, a novel approach depending on a new meta-heuristic and effective objective function formulation is proposed. Firstly, by considering some research shortcomings, a triple modal-based objective function combination is employed to improve the precision of damage identification. Secondly, a new self-adaptive algorithm which combines the powerful features of the stochastic fractal search with improved mechanisms into one framework, is developed. Moreover, the concept of quasi-oppositional learning is utilized to improve the overall exploration in both initial and executive stages. The new algorithm, called the self- adaptive quasi-oppositional stochastic fractal search (SA-QSFS), is benchmarked using well-known benchmark functions and applied on the IASC-ASCE FE model for damage assessment. Various damage scenarios are studied using partial modal data and noisy conditions. The proposed technique demonstrates outstanding performance and can be recommended to solve continuous optimization problems.

Listeriosis is one of the zoonotic diseases affecting most parts of the Sub-Saharan countries. The infection is often transmitted by eating and it can also pass by respiratory and direct contact. In this paper, a listeriosis mathematical model is formulated involving fractal-fractional orders in both Caputo and Atangana-Baleanu derivatives. Moreover, future behaviors of the disease are investigated by considering the fractal-fractional operators that are very effective in modeling the real-life phenomena by virtue of their memory effect. The basic properties and steady states are also obtained. The threshold parameter for determining the spread of the disease is computed. Numerical results are presented for each fractal-fractional-order operator. The results obtained in the paper show that the numerical schemes are effective for predicting and analyzing complex phenomena. (C) 2019 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

In this work, we investigate deterministic mechanisms that generate and perpetuate flutter and fibrillation in the right atrium of the human heart using mathematical models and high-performance computational simulations. Using a GPU, the solutions of systems of as much as 360 000 cells of 29 variables each were distributed among the available processors to simulate a cell-to-cell electric behavior of heart tissue. Moreover, in a smaller array of 462 cells, we showed that fibrillation, fluttering, and fibrillation-to-fluttering can be modeled even by using the simplest excitable cell models including only a few variables as well as with realistic models of heart cells, simply by implementing aperiodicity and fractality to the connections network. Micro-reentries which may be identified with fibrillation of the systems can be simulated in silico experiments where emerge several self-perpetuating, colliding waves associated with non-periodic and mathematically calculated pseudo-EG.

As we know, the fractal theory is an effective method for irregular image processing and analysis, which is difficult to implement using other baseline methods. However, Extreme Learning Machine (ELM) algorithm in combination with fractal theory can provide a better solution than the single fractal theory in image processing, transformation, data prediction, and so on. The features including fractal dimension, area ratio, and average perimeter of hematite, magnetite, calcium silicate, calcium ferrite, and stomata in the microstructure of irregular pellet images are extracted by artificial intelligence (AI) technologies. An AL-based ELM model suitable for the features of pellet images was constructed, the ELM model was optimized by regularization, and the feature parameters of pellet images were predicted by improved ELM algorithm, the optimal number of neuron nodes in the hidden layer was obtained for every mineral. The results show that the efficiency of model training and prediction in the feature for irregular pellet images is highly comparable to baseline methods. The method can be used as a reference in the intelligent prediction, control, and optimization of pellet image quality.

Mathematical modeling of infectious diseases with non-integer order getting attentions from scientists and researchers day by day. It is obvious that classical models in epidemiology can only be described through a fixed order while models in fractional order derivative are of not fixed order. Having non fixed order the fractional derivative becomes more powerful in modeling real life problems. In the recent era, different novel concepts regarding fractional operators such as the exponential decay and the Mittag–Leffler kernel have been introduced which overcome the limitations of the previous fractional order derivatives. These new operators have been found effective in modeling problems arising in science and engineering. A more recent operator in fractional calculus was introduced that is known as fractal-fractional operator. In this study, we consider this novel approach and apply it to an epidemic model of dengue fever and explore their dynamics. We show some important analysis for the dengue epidemic model in the presence of this new operator. The uniqueness and existence results will be shown. We show the simulation results for the considered model with a novel numerical approach which is not yet considered by anyone for such epidemic model. We obtain results for fractal model when fractional order is one and will have fractional solution when fractal order is one and have when both are present. We show that the fractal-fractional approach is much suitable for an epidemic model rather than fractional operator.

Real time anomaly detection is important to performance and efficiency in many areas. This paper offers a complex method for detecting abnormal telecommunication traffic. The proposed method includes components based on entropy analysis, signature analysis and machine-learning detection of anomalies using fractal and recurrence analysis. Datasets containing realizations of normal traffic and realizations of attacks of different types were used as input data. Traffic containing the attack is considered to be abnormal. The multifractal and recurrence methods are briefly described. The results of multifractal and recurrence analysis of data-set traffic showed significant differences between normal and attacked realizations. Based on the obtained results, machine-learning algorithms are proposed. The classifiers random forest and neural network were used and showed good classification accuracy. The methods signature and entropy analysis are briefly described, and the results of their application showed a high degree of anomaly detection. The complex anomaly detection method combines entropy analysis, signature analysis, and machine-learning using the multifractal and recurrence analysis. The proposed complex method is tested on virtual network and is compared pervious methods. Simulation results showed that the proposed complex method has the highest percentage of detected attacks, the lowest percentage of undetected attacks and lost data.

To investigate the fractal dimension and its variation of intact and compacted loess, consolidated-drained (CD) triaxial tests were performed to obtain intact and compacted loess specimens at different stress-strain states, their pore-size distribution (PSD) curves were measured by using the mercury intrusion porosimetry (MIP) technique. Based on the PSDs, three models (namely, Menger sponge model, Neimark model and the model based on the law of thermodynamics) were used to determine the fractal dimensions of specimens. The results show that there have 3 or 4 fractal intervals (or fractal dimensions, i.e., Ds1, Ds2, Ds3, Ds4,) when using Menger sponge model, only Ds3 satisfies the definition of fractal dimension. Neimark model can be used to determine the fractal dimensions of inter-aggregate pores and intra-aggregate pores, respectively (i.e., DNmacro and DNmicro). Most DNmacro values are meaningless physically, DNmicro increases with the increase of axial strain during shearing, suggesting that the surfaces of pores become complex with the increase of axial strain. The model based on the law of thermodynamics is the most appropriate if only one fractal dimension (i.e., Df) is expected in the whole measurable range of pore diameter. Df increases linearly with the growth of confining stress during consolidation or axial strain during shearing for both loess soils, indicating that the roughness of the pore surfaces increases with the increase in stress or strain. Besides, the fractal dimensions of intact and compacted loess were compared. According to the variations of PSD and Df, the mechanism for the evolution of the loess soil pore structure is proposed. These findings are expected to provide new insight into establishment of the connection between microstructure and macro stress-strain state of loess.

Here, we use the modified Kudryashov’s algorithm and addendum to Kudryashov’s technique to obtain new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. Dark, bright, singular solitons and combo bright-singular solitons are retrieved. The modified Kudryashov’s algorithm provides dark, singular solitons, and combo bright-singular solitons, although the addendum to Kudryashov’s technique ensures bright and singular solitons. These solutions are practiced for three optional definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Also shown are graphic representations of the solutions obtained.

This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C 1 -rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form p i ( θ ) / q i ( θ ) , where p i ( θ ) and q i ( θ ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C 3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C 1 -RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.

In this paper, a rectangular fractal patch antenna for satellite communication is presented. The antenna is designed at frequency 20GHz. The antenna structure is depicted and inspected using 3-D finite difference time domain (FDTD) method and the results are compared with obtained by commercial software Ansys HFSS and experimental results. It is found that FDTD simulation results along with fabricated and HFSS simulations are agreed adequately. The return loss of the antenna operating is -33.14 dB and gain is 17dB. These results infer that the proposed antenna is well suitable for Satellite communication at Ku band and has large bandwidth.

In this study, we present the new generalized derivative and integral operators which are based on the newly constructed new generalized Caputo fractal–fractional derivatives (NGCFFDs). Based on these operators, a numerical method is developed to solve the fractal–fractional differential equations (FFDEs). We approximate the solution of the FFDEs as basis vectors of shifted Legendre polynomials (SLPs). We also extend the derivative operational matrix of SLPs to the generalized derivative operational matrix in the sense of NGCFFDs. The efficiency of the developed numerical method is tested by taking various test examples. We also compare the results of our proposed method with the methods existed in the literature In this paper, we specified the fractal–fractional differential operator of new generalized Caputo in three categories: (i) different values in ρ and fractal parameters, (ii) different values in fractional parameter while fractal and ρ parameters are fixed, and (iii) different values in fractal parameter controlling fractional and ρ parameters.

A general machine learning (ML) framework of surface wetting is proposed by considering a broad range of factors, including solid surface topography, solid surface chemistry, liquid properties, and environmental conditions. In particular, an XGBoost-based ML model is demonstrated for learning the surface wetting behaviors processed by a laser-based surface functionalization process, namely nanosecond laser-based high-throughput surface nanostructuring (nHSN). This is the first known attempt to apply machine learning to surface wetting by considering both surface topography and surface chemistry properties. Novel microscale and nanoscale topography parameters viz., roughness, fractal, entropy, feature periodicity are defined with suitable computer algorithms to comprehensively describe the surface topography. A novel set of surface chemistry parameters such as polarity, volume, and amount of functional groups are also used as the machine learning model input. Upon analyzing the importance of each parameter for the nHSN process, surface chemistry shows the greatest importance in determination of surface wettability, while surface morphology also plays a part in influencing the wettability.

The fractal property is one of the most important properties in complex networks. It describes the power law relationship between characteristics of the box and the box size. There are numerous research studies focusing on the fractal property in networks through different dimensions. In order to study the problems across various disciplines, fractal dimension and local dimension are proposed to study network and node properties respectively. In this review paper, various network covering algorithms, which form the basis for obtaining fractal dimension are being reviewed. The different dimensions used to describe the fractal property of networks and their applications are then discussed. Through these studies, we emphasize that the fractal property is an important tool for understanding network characteristics. In the last section, we give our conclusion and discuss possible future directions for fractal dimension research.

This work addresses the role of rigid inclusions in determining the elastic modulus of particle-filled colloidal networks by modifying an established fractal scaling model. The approach acknowledges the heterogeneous nature of stress distribution at length scales beyond the colloidal aggregates, while maintaining structural information at the level of individual clusters. This was achieved by introducing a scaling factor to account for system heterogeneity which contains intrinsic information about the network’s capacity to form load-bearing links. Rigid fillers bound to the network induce stress concentration, but additionally serve as junction zones which introduce additional load-bearing pathways. This gives rise to the observed increase in the modulus with filler volume fraction. The proposed relationship between the load-bearing network connectivity and scaling behavior may have additional implications on the fractal dimension determined by rheological methods. Further, this model accommodates an experimentally observed correlation between the scaling behavior of the modulus associated with the addition of fillers and that arising from increasing structurant concentration. The modified fractal model thus provides an alternative view of how fillers contribute to the small- and large-deformation mechanical behavior of filled colloidal gels in a manner consistent with experimental observations.

In this article, we propose the integrator circuit model by the fractal-fractional operator in which fractional-order has taken in the Atangana-Baleanu sense. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model posses at least one solution and via Banach fixed theorem, we guarantee that the proposed model has a unique solution. We derive the results for Ulam-Hyres stability by mean of non-linear functional analysis which shows that the proposed model is Ulam-Hyres stable under the new fractal-fractional derivative. We establish the numerical results of the model under consideration through Atanaga-Toufik method. We simulate the numerical results for different sets of fractional order and fractal dimension.

The generation method of three-dimensional fractal discrete fracture network (FDFN) based on multiplicative cascade process was developed. The complex multi-scale fracture system in shale after fracturing was characterized by coupling the artificial fracture model and the natural fracture model. Based on an assisted history matching (AHM) using multiple-proxy-based Markov chain Monte Carlo algorithm (MCMC), an embedded discrete fracture modeling (EDFM) incorporated with reservoir simulator was used to predict productivity of shale gas well. When using the natural fracture generation method, the distribution of natural fracture network can be controlled by fractal parameters, and the natural fracture network generated coupling with artificial fractures can characterize the complex system of different- scale fractures in shale after fracturing. The EDFM, with fewer grids and less computation time consumption, can characterize the attributes of natural fractures and artificial fractures flexibly, and simulate the details of mass transfer between matrix cells and fractures while reducing computation significantly. The combination of AMH and EDFM can lower the uncertainty of reservoir and fracture parameters, and realize effective inversion of key reservoir and fracture parameters and the productivity forecast of shale gas wells. Application demonstrates the results from the proposed productivity prediction model integrating FDFN, EDFM and AHM have high credibility.

Cough signal analysis for understanding the pathological condition has become important from the outset of the exigency posed by the epidemic COVID-19. The present work suggests a surrogate approach for the classification of cough signals - croup cough (CC) and pertussis (PT) – based on spectral, fractal, and nonlinear time-series techniques. The spectral analysis of CC reveals the presence of more frequency components in the short duration cough sound compared to PT. The musical nature of CC is unveiled not only through the spectral analysis but also through the phase portrait features – sample entropy (S), maximal Lyapunov exponent (L), and Hurst exponent (Hb). The modifications in the internal morphology of the respiratory tract, giving rise to more frequency components associated with the complex airflow dynamics, get staged through the higher fractal dimension of CC. Among the two supervised classification tools, cubic KNN (CKNN) and neural net pattern recognition (NNPR), used for classifying the CC and PT signals based on nonlinear time series parameters, NNPR is found better. Thus, the study opens the possibility of identification of pulmonary pathological conditions through cough sound signal analysis.

The Three Gorges Reservoir (TGR) in China is the largest hydroelectric project in the world, but the threat of sediment affecting ecological sustainability of the reservoir is a topic of concern. Sediment particle-size distribution (PSD) is informative in understanding sediment transport dynamics and biochemical functions. It is, therefore, important to quantitatively characterize the distribution of sediment particles. In the current study, fractal theory is applied to determine the PSD of suspended sediment in the TGR. The results show that the volumetric fractal dimension ( D v ) exhibits a significant seasonal difference (p D ( q ) – q , and multifractal singularity spectrum, f [ α ( q ) ] – α ( q ) , were calculated for each suspended sediment sample. Thereafter, the parameters, D(0), D(1), D(2), α(0), Δα(q), and Δ f [ α ( q ) ] , were determined to characterize the PSD. As a result, the coarser suspended sediment during the wet season is characterised by a more complex PSD pattern, with a wider range of particle sizes, greater heterogeneity, and greater homogeneity of distribution over the measurement interval. However, the multifractal structure of the PSD of suspended sediment is more complex during the dry season than during the wet season, with higher local dispersion and variability. The findings of the current study highlight that multifractal analysis provides important insight for understanding the PSD of suspended sediment in the TGR.

Determination and conceptualization of atomic structures of metallic glasses or amorphous alloys remain a grand challenge. Structural models proposed for bulk metallic glasses are still controversial owing to experimental difficulties in directly imaging the atom positions in three-dimensional structures. With the advanced atomic-resolution imaging, here we directly observed the atomic arrangements in atomically thin metallic glassy membranes obtained by vapor deposition. The atomic packing in the amorphous membrane is shown to have a fractal characteristic, with the fractal dimension depending on the atomic density. Locally, the atomic configuration for the metallic glass membrane is composed of many types of polygons with the bonding angles concentrated on 45°–55°. The fractal atomic structure is consistent with the analysis by the percolation theory, and may account for the enhanced relaxation dynamics and the easiness of glass transition as reported for the thin metallic glassy films or glassy surface.

This paper reports an alternative approach to the evaluation of infrared camouflage effectiveness via a multi-fractal method. By calculating multi-fractal spectra of the target region and the background regions in an infrared image, the spectrum shape features and the discrete Frechet distances among these spectra were used to analyze the camouflage effectiveness of the target qualitatively and quantitatively, and the correlation coefficients of the spectra were further used as the index of camouflage effectiveness. It was found that the camouflaged target had better camouflage effectiveness than the target without camouflage in the same one background, and the same one camouflaged target had different camouflage effectiveness in different backgrounds. On the whole, the target matching well with its background had high camouflage effectiveness value. This approach can expand the application of multi-fractal theory in infrared camouflage technology, which should be useful for the research of infrared camouflage materials, the design of camouflage patterns as well as the deployment of military equipment in battlefield.

This paper proposes a newly chaotic maps integrated stochastic fractal search (CMSFS) for solving the optimal distributed generation placement (ODGP) problem in radial distribution networks. The objective of the problem is to minimize the network real power loss satisfying the operational constraints of distributed generations (DGs) and the network. The proposed CMSFS approach is an improvement of the standard SFS approach by integrating chaotic maps into SFS to enhance its solution quality and convergence rate. The experimental results on the IEEE 33, 69, and 118-bus networks and the power loss reduction percentages with the integration of the optimal number of non-unity power factor DG are 99.21%, 99.43% and 92.36%, respectively. Moreover, the simulation results have also shown that the proposed CMSFS can provide better solution quality than many other methods for the considered scenarios. Therefore, the proposed CMSFS can be an effective alternative approach to the ODGP problems in RDNs.

We develop a new harmonic analysis of reproducing kernels. Our approach is with view to a number of recent applications: We explore both geometric and algorithmic consequences of our analysis. The list of applications includes: feature fitting in machine learning algorithms, dynamics in large networks, spline approximation, determinantal point processes, stochastic processes, and fractals.

Research on coal fragmentation can play an important role in understanding coal and gas outbursts. The study discussed in this paper explored the fragmentation of gas-containing coal particles using the drop-weight impact method. The effects of equilibrium gas pressures and type of adsorbate gas on particle size distributions and fragmentation energy were investigated in detail. We found that the Fractal particle size distribution model can most effectively describe the crushed coal particle sizes. The equilibrium pressure and type of gas can influence the Fractal distribution parameter. The crushing energy is composed of energy to create new surfaces and other forms of energy that are dissipated but the equilibrium gas pressure and type of adsorption gas can affect energy consumption and crushing efficiency. This research will be of guiding significance to the intensity evaluation and mechanism understanding of coal and gas outbursts.

An understanding of the particle transport characteristics in a branched network helps to predict the particle distribution and prevent undesired plugging in various engineering systems. Quantitative analysis of particle flow characteristics is challenging in that experiments are expensive and particle flow is difficult to detect without disturbing the flow. To overcome this difficulty, man-made fractal tree-like branched networks were built, and a coupled computational fluid dynamic and discrete element method model was applied. A series of numerical simulations was carried out to analyze the influence of fractal structure parameters of networks on the particle flow characteristics. The joint influence of inertial, shunt capacity and superposition from upstream branches on particle flow was investigated. The injection position at the inlet determined the particle velocity and its future flow path. The particle density ratio, particle size and bifurcation angle had a greater influence on the shunting of K2 branches than that in the K1 level and Nk22/Nk21 reached a maximum at 60°. Compared with a network with an even number of branches, there was a preferential branch when the branch number was odd. The preferential branch effect or asymmetry degree of the level (K2) branches had a more significant impact on particle shunting than that from the upstream branches (K1).

Gas-liquid mixing characteristics in fractal impeller stirred tank were experimentally investigated by measuring multi-scale entropy (MSE), relative power demand (RPD), local gas holdup, bubble size, and mass transfer coefficient (KLa). Results showed that fractal impeller could enhance 21.69% of MSE and 11.94% of RPD on the basis of pitched-blade impeller. Fractal impeller can enhance the local gas holdup, decrease the bubble size, reduce the dispersion coefficient (σ) and improve the gas-liquid dispersion performance. Meanwhile, the regression based on d 32 − for pitched-blade impeller and fractal impeller were obtained. In addition, fractal impeller can also increase 11.07% of volume mass transfer coefficient (KLa) compared with pitched-blade impeller in the gas-liquid mixing process. The results can provide theoretical guidance for the optimal design of gas-liquid stirred tank.

Density currents are caused by a difference in density, though low, of an entering fluid with the ambient fluid. This type of current is two-phased and found on riverbeds or in reservoirs behind dams, and is nonlinear in nature, complex, and sensitive to initial conditions. Fractal geometry is used as a powerful tool for studying complex natural phenomena. Using experimental studies and changes in inlet current conditions, the fractal and multi-fractal analyses of the interface between the density current and the ambient fluid were done. In addition, a search was made to find a possible connection between the nonlinear patterns. According to the results, with an increase in the inlet discharge and inlet density of the current the fractal dimension decreased. Further, the smaller the range of the singularity spectrum diagram was, i.e., the more it was less than 0.34, the lower the system's tendency was to be multi-fractal, and the system sensitive to large local changes. In the interface between the density current and the ambient fluid, using the fractal dimension-based Richardson number could improve experimental data by 12.4%. Moreover, with an increase in the Richardson number, the Reynolds number of the current decreased. Further, upon considering the fractal dimension, the Reynolds number improved by 23% and a good correlation with a coefficient of determination of 0.76.

Objectives Stenting of common femoral artery (CFA) bifurcation lesions is complex due to harmonious asymmetric geometry between the CFA and deep or superficial femoral artery. In order to ensure an optimal flow, the stents must be implanted according to the diameter of the CFA and the diameter of the daughter vessels. The aim of this study was to validate a mathematical formula for the CFA bifurcation in order to optimize the implantation of stents in this bifurcation with a fast and efficient method. Materials/Methods We retrospectively included all patients who underwent transcatheter aortic valve implantation and who had a healthy CFA bifurcation on the CT-scan. Diameters of the mother vessel (Dm), daughter vessels (Dl: larger vessel; Ds: smaller vessel), and the ratio between the two were calculated. According to the latter, the following linear model was proposed: [Dm = (Dm/Dl + Ds) × (Dl + Ds)] and was compared to the four existing models (Murray, HK, Flow, Finet). Finally, we calculated the relative error between the prediction of the four models and the measurements. Results Overall, 110 CFA bifurcations were included. Mean CFA diameter was 7.75 ± 1.67 mm. Mean Dl diameter was 5.79 ± 1.21 mm and mean Ds diameter was 5.23 ± 1.09 mm. A reduction of 25 % was seen between the CFA and the larger daughter-vessel. The mean ratio between the CFA and the daughter vessels was 0.71. Our model [Dm = 0.71× (Dl + Ds)] and the flow law were the most accurate (relative difference of 1.59 ± 11.97% and 1.01 ± 11.94%, respectively). However, Murray’s law had a statistically significant deviation from the real mother-vessel diameter (P Conclusions We developed a simple fractal ratio between CFA and daughter vessels, which could be easily used in daily practice during CFA percutaneous reconstruction to identify the optimal diameters of the diseased vessels.

The original approach of Hutchinson to fractals considers the defining equation as a fixed point problem, and then applies the Banach Contraction Principle. To do this, the Blaschke Completeness Theorem is essential. Avoiding Blaschke’s result, this note presents an alternative way to fractals via the Kuratowski noncompactness measure. Moreover, our technique extends the existence part of Hutchinson’s Theorem to condensing maps instead of contractions.

An investigation is done upon the lean H2-air flame under turbulent conditions to clarify the role of inherent instabilities and turbulence in self-acceleration of expanding flames. The result shows that, in weak turbulent flow fields, the two-stage self-acceleration feature still exists. As the turbulent intensity increases, the originally evident two-stage “transition-saturation” feature is weakened. When the Karlovitz number becomes greater than 1, the self-acceleration process no longer experiences the transition stage and performs a single-stage feature. These observations are correlated with the distribution and evolution of length scales in the cellular structure of the flamefront. A theoretical analysis is conducted to compare growth rates of disturbances induced by instability and turbulence, with consideration of their wavelength dependence. On this basis, interpretations are proposed on afore-mentioned phenomenal observations and response of quantitative parameters such as fractal excess and average cell wavelength.

System time offsets between Global Navigation Satellite Systems is a key issue in multi-system positioning. A high-precision time offsets prediction model is required in actual navigation applications. In this study, the fractal characteristics of time offsets in American and Russian navigation systems are directly confirmed through fractal dimension calculations. Then, the influence of the vertical scaling factor on the prediction accuracy of the fractal interpolation prediction model is confirmed. It is found that perturbation on the estimated vertical scaling factor causes a minimal attractor bias, which is positively correlated with the prediction error. After proving the existence of the estimated vertical scaling factor of the minimal attractor biases, an optimization method of the fractal interpolation prediction model is proposed, and the related algorithms are given. The numerical results show that the time offsets prediction errors using the new model are less than 1 ns for the American navigation system and close to 2 ns for the Russian navigation system. The prediction accuracy of this model has an improvement of at least 60% over the commonly used quadratic and grey models, and an improvement of more than 13% over the standard fractal interpolation prediction model.

This article creatively proposes a class of sorting matrices with fractal characteristics, named the fractal sorting matrix (FSM), and introduces its iterative calculation method. The FSM is irregular, self-similar and infinitely iterative. Notably, scrambling images or information based on this new cluster of matrices can effectively improve encryption algorithm security. Then, the article presents a new method of global pixel diffusion with two chaotic sequences, which offers good security and high encryption efficiency. Based on the FSM and global chaotic pixel diffusion, this paper constructs a more efficient and secure chaotic image encryption algorithm than other approaches. According to experimental comparison, the proposed algorithm is faster and has a higher pass rate associated with the local Shannon entropy. The data in the antidifferential attack test are closer to the theoretical values and smaller in data fluctuation, and the images obtained from the cropping and noise attacks are clearer. Therefore, the proposed algorithm shows better security and resistance to various attacks.

In this manuscript, a qualitative analysis of the mathematical model of novel coronavirus ( COVID -19) involving anew devised fractal-fractional operator in the Caputo sense having the fractional-order q and the fractal dimension p is considered. The concerned model is composed of eight compartments: susceptible, exposed, infected, super-spreaders, asymptomatic, hospitalized, recovery and fatality. When, choosing the fractal order one we obtain fractional order, and when choosing the fractional order one a fractal system is obtained. Considering both the operators together we present a model with fractal-fractional. Under the new derivative the existence and uniqueness of the solution for considered model are proved using Schaefer’s and Banach type fixed point approaches. Additionally, with the help of nonlinear functional analysis, the condition for Ulam’s type of stability of the solution to the considered model is established. For numerical simulation of proposed model, a fractional type of two-step Lagrange polynomial known as fractional Adams-Bashforth ( AB ) method is applied to simulate the results. At last, the results are tested with real data from COVID -19 outbreak in Wuhan City, Hubei Province of China from 4 January to 9 March 2020, taken from a source (Ndairou, 2020). The Numerical results are presented in terms of graphs for different fractional-order q and fractal dimensions p to describe the transmission dynamics of disease infection.

The purpose of this work is to analyze the dynamics of dengue fever in newly introduced operator known as fractal-fractional Atangana-Baleanu. Initially, we formulate a new dengue model with hospitalization class of notified infected cases and present the model basic results. Stability results for the model are shown when R 0 1 . We estimate the parameters for the model considering the infected cases occur in East Java Indonesia for the year 2018. We estimated that the basic reproduction number for this model is approximately equal to R 0 ≈ 1.8463 . We show the application of the fractal-fractional Atangana-Baleanu operator to the dengue model and present its numerical solution with a new method. We show the fractal-fractional model versus model fitting for different values of fractal and fractional orders and suggest that the fractal-fractional model provides better fitting to the infected cases of dengue infection. Some numerical results for different orders of fractal and fractional are shown for the validity and the applicability of the newly introduced operator.

Analysis of uterine contractions using electromyography signals is gaining importance due to its capability to measure the dynamics of uterus. Uterine electromyography (uEMG) provides information on the nature of uterine contractions non-invasively. In this study, the fluctuations in uEMG signals associated with Term pregnancies are analyzed. For this, Term uEMG signals corresponding to second (T1) and third (T2) trimesters are considered. The signals are subjected to Adaptive Fractal Analysis (AFA), wherein a global trend is obtained by using overlapping windows of three orders namely, 25%, 50% and 75%. The signals are detrended and the fluctuation function is estimated. Two Hurst exponent features computed at short range (Hs) and long range (Hl) are extracted and statistically analyzed. Results show that AFA is able to characterize variations in the fluctuations of Term delivery signals. The feature values are observed to vary significantly during different weeks of gestation. It is found that features of T2 signals are higher than that of T1 signals for all the considered overlaps, indicating that T2 signals possess smoother characteristics than T1 signals. Further, coefficient of variation is observed to be low, indicating that these features are able to handle the inter-subject variations in Term signals. Therefore, it appears that the proposed approach could aid in investigation of progressive changes in uterine contractions during Term pregnancies.

Ambulating while carrying a mission specific load is one of the most frequently executed occupational tasks for the military, especially for individuals in combat roles. Prolonged ambulation is a naturally dynamic and complex process, characterized by highly multi-dimensional interactions within the gait mechanics of the lower extremity. Recent wearable sensors studies, like inertial measurement unit (IMU)-related gait studies have demonstrated that machine learning (MLN) algorithms and fractal analysis can successfully discriminate between classes, such as movement patterns, injury, age and sex. This study attempts to classify fractal gait patterns of women and men using IMU-based signal data obtained from accelerometer, gyroscope and magnetometer during a 2 km loaded (20 kg) march. Random Forest (RF) MLN algorithm was used to generate a model that can measure the accuracy and identify the importance of IMU-based signal-related fractal variables. A total of 18 variables were calculated using 2 fractal methods, detrended fluctuation analysis (DFA) and wavelet transform-based power spectral density (PSD), from 3 IMU-based signals in their 3 axes (medial-lateral, vertical and anterior-posterior). A total of 33 healthy adults (17 men [26.7±5.9 years] and 16 women [25.2±4.5 years]) volunteered for this study. A 9-axis IMU sensor was attached to each participant at each of the following locations: feet, shanks, thighs and lumbar spine. An independent training-testing approach, called one-vs-one (i.e., variables from one IMU-based signal were trained and tested using another IMU-based signal) was applied to determine the classification accuracy (i.e., similarities between IMUs) and variable importance (score ranges: 0.0-1.0) measures. These values were then used to select the variables that best independently describe the rank in classification margin. The results from each IMU sensor placement based on the fractal values showed ‘moderate’ accuracy (50-75%), with the exception of two cases: the left shank yielded ‘good’ accuracy (80.1%) compared with the right shank, and the right thigh generated ‘poor’ accuracy (48.9%) compared with the left foot. No IMU location showed excellent accuracy (>90%). The results indicate that each IMU placement location has their own fractal patterns that are not similar to another IMU location in terms of sex classification. The analysis of the variable importance in the classification margin showed that most of the PSD resulted variables were classified as ‘most important’ compared with the DFA resulted variables. IMU sensors, and the associated analyses, could be used during military load carriage to evaluate changes in gait resulting from injury, fatigue or overtraining.

In this paper, a lattice structure whose mechanical properties can be systematically designed is presented using tailored sub-unit structures made from curved bars. First, the stiffness of circular-arc curved bars is theoretically studied, and the effects of geometric parameters of the curved bars on the mechanical properties are analyzed. Second, sub-unit structures are designed based on the fractal geometry, and periodic structures for lattice structures assembled from different sub-unit structures are precisely fabricated using 3D printing techniques. Third, mechanical properties of structures are mechanically measured and numerically simulated simultaneously, which present anisotropy of stiffness depending on the loading direction. These results show that the controllable stiffness of lattice structures along different directions can be designed by changing structure geometry and degrees of fractal. This study proposes a new method for the design of lattice structures and potential in versatile industrial and biomedical applications.

The statement of the problem of defects detection and recognition in aviation riveted joints is given. The solution is proposed to be implemented using a deep neural network with a recurrent LSTM layer. An example of solving the problem of detecting and recognizing a defect in rivets is considered. The accuracy of the result obtained in the simulation was 100% for two-class and 96.25% for four-class recognition by echo signals of an ultrasonic flaw detector. It is shown that it is possible to use the fractal properties of the non-destructive testing echo signal when detecting defects.

The surface of turbulent premixed flames is fractal within a finite range of scales and the fractal dimension and inner cutoff scale are key components of fractal turbulent combustion closures. In such closures, the estimate for the surface area is sensitive to the value of the inner fractal cutoff scale, whose modeling remains unclear and for which both experimental and numerical contradictory evidence exists. In this work, we analyze data from six direct numerical simulations of spherically expanding turbulent premixed flames at varying Reynolds and Karlovitz numbers. The flames propagate in decaying isotropic turbulence and fall in the flamelet regime. Past an initial transient, we find that the fractal dimension reaches an asymptotic value between 2.3 and 2.4 in good agreement with previous results at similar conditions. A minor dependence of the fractal dimension on the Reynolds and Karlovitz numbers is observed and explained by the relatively low values of the Reynolds number and narrow inertial and fractal ranges. The inner fractal cutoff scale Δ* is found to scale as Δ * / l ∼ Re λ − 1.14 , where l is the integral scale of turbulence and Reλ is the Reynolds number based on the Taylor micro-scale computed in the turbulence on the reactants’ side. The scaling is robust and insensitive to the Karlovitz number over the range of values considered in this study. An important implication is that the ratio Δ*/η grows with Reynolds number (η is the Kolmogorov scale), albeit at a rather slow rate that may explain the widespread observation that 4 ≤ Δ*/η ≤ 10. This suggests that Δ*, although smaller than λ, is not a dissipative length scale for the flame surface and scaled solely by η. Finally, a dissipative threshold scale that remains constant once normalized by η is identified.

A deformity of magnetic field discloses that different particle of ferrofluid exhibits different pattern to accumulate more closely with strong magnetic properties; this is because of the existence of a homogeneous ferromagnetic liquid. In this context, the Newtonian fluid (ordinary liquid) is magnetized by dispersing into porous media. A mathematical model for ferromagnetic fluid is developed by means of fractal–fractionalderivative based on the power law. A powerful technique of Laplace transform is invoked on the fractal-fractionalized MHD Newtonian fluid model embedded in porous medium based within power law kernel. The solution is investigated for velocity field by particularizing the format of mathematical special function so called Fox-H function H α , β + 1 1 , α F . The Fox-H function H α , β + 1 1 , α F is employed among solution for the elimination of gamma functions. In order to generate the physical insight of fractal-fractionalized ferrofluid, a novel behaviors of velocity field based on the power law has disclosed several similarities and differences through the embedded rheological parameters. In short, It is worth noting that there is no previous paper related to ferrofluid involved with fractal–fractional​ derivative based on the power law.

Local residual stress caused by impacts, machining and indentation results in a decrease in strength in most materials that fail in a brittle manner. The ratio of the critical crack size, c, and the fracture mirror size, r, also is affected by the existence of local residual stress. The global fracture toughness of non-R curve materials is not affected by the local residual stress. The fractal dimension of the fracture surface as characterized by the fractal dimensional increment, D*, is directly related to the square of the fracture toughness. This paper addresses the question of the effect of the local residual stress on the fractal dimension of the fracture surface. We derive a relationship between the fractal dimensional increment and the c/r ratio for materials fractured with and without local residual stress. We then compare the prediction with two cases of experimental results. We show the fractal dimension remains constant with the change in the c/r ratio for local residual stress conditions.

A short note on observations of dried patterns of a bi-dispersive colloidal drop on a relatively soft polymer-surface is presented here. The patterns have huge applications in the field of medical diagnosis, fertilization and drug delivery industry and improvement of coating/printing technology. The features in the dried drop involve self-organized open systems. The pattern analysis initiates the predicted mechanism of drying. Drops of liquid containing suspended particles are assumed to be drying on a horizontal semi-solid substrate of a polymer. Study of the fractal geometry of a pattern characterizes the system’s overall behavior and predicts the system’s basic wetting properties.

Laser surface texturing (LST) has shown immense promise in modifying the physical as well as interactive properties of the surfaces. Understanding the surface microtopography generated during LST provides key knowledge on developing desirable surfaces for different technologies. The work focuses on determining the range of frequency components present in the microtopography when processed at different parameters, such as beam diameter (BD), scanning speed (SS), and beam overlap (BO). Power spectral density (PSD) is utilized to evaluate the intensity of each spatial frequency and fitting models are applied to quantify their contribution to the overall microtopography. The experiments were performed on titanium alloy Ti6Al4V using a continuous watt fiber laser at constant power. The surface microtopography differs when processed under different conditions. The microtopography contains low and high spatial frequency components distributed along both the scan and overlap direction. The PSD analysis reveals the increase in high spatial frequency features when BD and SS are increased. Conversely, the growth of spatial features is observed with an increase in BO reducing the dominance of high spatial frequencies. With the increase in BD, the energy density decreases which reduces the growth of spatial features inducing increased contribution of high spatial frequency content noticeable by upward curvature in the PSD. An increase in SS causes rapid laser motion to affect the microtopography along the scan direction. The increase in BO leads to the enhanced overlapping between successive passes causing remelting and growth of previously formed microtopography and increasing the contribution of low spatial frequency content in the microtopography. The fitting model parameters from ABC at low spatial frequency and Fractal at high frequency provide the quantitative reasoning for the observed trend. Power spectral analysis reveals significant information about the surface microtopography formed during LST and accurate quantification of the PSD may help numerical models to fine-tune the surface features according to the desired functionality.

Many architectural systems are modelled by boundary value problems. Though there are many numerical and analytical methods to solve such problems, this paper suggests a simple but effective way to the third-order ordinary differential equations by the Taylor series technology. The governing equation is differentiated to obtain high order derivatives of a certain point, and solution is expanded into a Taylor series of the point, the boundary conditions are then used to identify the unknowns involved in the approximate solution. An example is given to elucidate the solution process, the maximal error is less 2%, and its accuracy can be further improved if a higher order series is solved. The approximate solution is compared with that by the variational iteration method, showing the present method is simpler and more straightforward. The comparison with the numerical solution culminates in its reliability and effectiveness. The method can be extended to fractal calculus.

This work proposes a new approach for mapping GPU threads onto a family of discrete embedded 2D fractals. A block-space map $\lambda: \mathbb{Z}_{\mathbb{E}}^{2} \mapsto \mathbb{Z}_{\mathbb{F}}^{2}$ is proposed, from Euclidean parallel space $\mathbb{E}$ to embedded fractal space $\mathbb{F}$, that maps in $\mathcal{O}(\log_2 \log_2(n))$ time and uses no more than $\mathcal{O}(n^\mathbb{H})$ threads with $\mathbb{H}$ being the Hausdorff dimension of the fractal, making it parallel space efficient. When compared to a bounding-box (BB) approach, $\lambda(\omega)$ offers a sub-exponential improvement in parallel space and a monotonically increasing speedup $n \ge n_0$. The Sierpinski gasket fractal is used as a particular case study and the experimental performance results show that $\lambda(\omega)$ reaches up to $9\times$ of speedup over the bounding-box approach. A tensor-core based implementation of $\lambda(\omega)$ is also proposed for modern GPUs, providing up to $\sim40\%$ of extra performance. The results obtained in this work show that doing efficient GPU thread mapping on fractal domains can significantly improve the performance of several applications that work with this type of geometry., Comment: 20 Pages. arXiv admin note: text overlap with arXiv:1706.04552

The hydrodynamics of floating and sinking particles mixing process in a stirred tank with four pitched-blade impellers and fractal impellers were investigated employing computational fluid dynamics (CFD) simulation. Axial solid concentration profile, solid concentration distribution, power consumption, trailing vortex, cloud height, water integrated velocity, just suspension speed (Njs) and just drawdown speed (Njd) in the stirred tank were predicted. Results showed that the mixing quality of floating and sinking particles increased with the increase of impeller speed. Smaller density difference between solid-liquid two phases and smaller particle diameter resulted in higher suspension degree. Fractal impeller can enhance water integrated velocity and improve suspension quality compared with four pitched-blade impeller under the constant power consumption, and increasingly so with fractal iteration number. Meanwhile, fractal impeller can reduce the size of trailing vortex and decrease the power consumption based on four pitched-blade impeller at the same impeller speed. In addition, the just suspension speed (Njs), just drawdown speed (Njd) and correlation equation for Njs and Njd in the three different impeller systems were obtained. Fractal impeller could decrease the Njs and Njd on the basis of four pitched-blade impeller, and decreasingly so with the fractal iteration number.