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Parallel data processing is one of the specific infrastructure applications categorized as a service provided by cloud computing. In cloud computing environments, data-intensive applications increasingly use the parallel processing paradigm known as MapReduce. MapReduce is based on a strategy called "divide and conquer," which uses ordinary computers, also called "nodes," to do processing in parallel. This paper looks at how open multiprocessing (OpenMP), the best shared-memory parallel programming model for high-performance computing, can be used in the MapReduce application using proposed fractal network models. Two fractal network models are offered, and their work is compared with a well-known network model, the hypercube. The first fractal network model achieved an average speedup of 3.239 times while an efficiency ranged from 73-95%. In the second model of the network, the speedup got to 3.236 times while keeping an efficiency of 70-92%. Furthermore, the path-finding algorithm employed in the recommended fractal network models remarkably identified all paths and calculated the shortest and longest routes.

I investigate combinations of quasi-quanta expressions and how they yield alternatesolutions for expressions inMorphic Topology of Numeric Energy: A Fractal Morphism of Topological Counting Shows Real Differentiation of Numeric Energy. For Praising Jehovah, I do publish these mathematical gesturing forms from the infinity meaning of His word. Thanks mom! This quasi-quanta language package outlines methods for combining by topo- logical functor entanglement, symbolic, numeric-energy components. Methods, guidelines and algebraic rules for combining the quasi-quanta into the energy number equivalencies are also notated herein. The Quasi-Quanta Language Package is intended to show the symbolic pat- terns for configuring the quasi quanta symbology into the numeric energy ex- pressions. This should put to rest any doubt that Energy Numbers are indeed a real, logically configured phenomenon a priori to real or complex numbers, but optionally mappable to the real or complex plane. Pre-numeric energy symbol configurations offer a broad language of pat- tern detection and logical symbol operation delineated with particular solving methods herein. This hopefully provides a new way to looking at the branches of mathematics and their inter-operable analog functions. So, inevitably, we decompose the current perspective on numbers and prove a novel method for ordering and combining symbolic orientations.

The stock of shale gas in the Shahezi shale reservoir in the Changling fault depression, Songliao Basin, is believed to be worth exploring. In-depth study on the pore structure and fractal characterization of organic matter (OM) can help better understand the pore system of shale reservoir, which has implications for the exploration of lacustrine shale. To demonstrate the nanoscale pore structure and irregularity of the isolated OM, we collected a large number of samples and then conducted a series of laboratory experiments, such as X-ray diffraction, scanning electron microscopy, and [Formula: see text] and [Formula: see text] adsorption experiments to determine the pore structure parameters and reveal their heterogeneity according to Frenkel-Halsey-Hill theory. As suggested by the experimental results, the pore size distribution curve of the [Formula: see text] and [Formula: see text] exhibits similar pore peaks, whereas the trends of change in bulk shale and isolated OM samples are basically paralleled. The pore volume of the isolated OM ranges between 0.034 and [Formula: see text], which is approximately 0.90–3.06 times that of bulk shale samples. As for the fractal dimensions [Formula: see text] (2.594 on average) and [Formula: see text] (2.657 on average) of bulk shale, they are larger as compared with isolated OM, indicating that inorganic minerals can make a significant difference in the heterogeneity of shale pores. The fractal dimensions ([Formula: see text] and [Formula: see text]) of bulk shales indicate a close correlation with the parameters of pore structure, whereas there is no significant correlation observed between the dimensions of isolated OM and its parameters. In addition, thermal maturity and solid bitumen have only limited impact on the OM pore structure of isolated OM samples. Then, we conducted further research to reveal that the insoluble OM macerals derived from terrestrial higher plants can be used to explain the difference in pore structure and heterogeneity between isolated OM samples. Therefore, we arrived at the conclusion that the composition of macerals depends on the exact pore structure and fractal characteristics of isolated OM samples with similarity in thermal maturity.

The arrival behavior of elastic waves in a naturally fractured rock is studied based on numerical simulations. We use the discrete fracture network method to represent the distribution of a natural fracture system and employ the displacement discontinuity method to compute the propagation of elastic waves across individual fractures. We analyze macroscopic wavefield arrival properties collectively arising from the interaction between elastic waves and numerous fractures in the system. We show that the dimensionless angular frequency ῶ = ωZ/κ exerts a fundamental control on the arrival behavior of a plane wave traveling through the fractured rock, where ω, Z, and κ are the angular frequency, seismic impedance, and fracture stiffness, respectively. An asynchronous arrival phenomenon of the wave energy occurs and becomes more significant with an increased ῶ. Two regimes are identified according to the two-branch dependency of the fractal dimension D of the FFAW on ῶ, where the wave arrival behavior is within a non-fractal regime for ῶ smaller than the critical frequency ῶc ≈ 1.0, and enters the fractal regime for ῶ ≥ ῶc. The self-affine properties of the FFAW, i.e., the roughness exponent α and the correlation length lc, both linearly decrease as a function of the exponent ξ (with ῶ = 10ξ) in the fractal regime. Early breakthrough of wave transport occurs in regions with relatively low fracture density, while late-time arrival happens in regions of high fracture density., Rock Mechanics and Rock Engineering, 56 (4), ISSN:1434-453X, ISSN:0723-2632

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 the field of frontier research, information security has received a lot of interest, but in the field of information security algorithm, the introduction of decimals makes it impossible to bypass the topic of calculation accuracy. This article creatively proposes the definition and related proofs of double parameters fractal sorting matrix (DPFSM). As a new matrix classification with fractal properties, DPFSM contains self-similar structures in the ordering of both elements and sub-blocks in the matrix. These two self-similar structures are determined by two different parameters. To verify the theory, this paper presents a type of 2×2 DPFSM iterative generation method, as well as the theory, steps, and examples of the iteration. DPFSM is a space position transformation matrix, which has a better periodic law than a single parameter fractal sorting matrix (FSM). The proposal of DPFSM expands the fractal theory and solves the limitation of calculation accuracy on information security. The image encryption algorithm based on DPFSM is proposed, and the security analysis demonstrates the security. DPFSM has good application value in the field of information security.

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.

This article explores the emergence of Greenland as an Anthropocene island through anthropological fieldwork in and around the decommissioned Nalunaq goldmine in the south of the country. The article takes off from the idea that Anthropocene activities are characterized by the invention, movement, and marketing of seemingly mobile resource units that can be identified and invested in regardless of landscape specificities, and explores how the production of Greenlandic gold complicates this idea of extraction. In particular, the article discusses how Greenlandic post-colonial independence and ambitions for mining both go together and undermine each other, creating new dependencies and relationalities along the way. Through analyzing parts of Nalunaq’s political context, infrastructural challenges, the gold that came out, and eventual closure, the article presents Greenlandic gold mining as a set of partly congruous, partly contradictory practices and ideas. The article thus specifies an extractive project that both is and is not possible on the world’s biggest island, and brings this to bear on how we might understand the Anthropocene.

Background: To investigate the effect of single posterior mandibular implants on the survival, restorative status, pulpal, periapical and periodontal health of the adjacent teeth and surrounding trabecular bone structure. Material and Method: A total of 174 teeth adjacent to single-tooth implants and 174 teeth on the contralateral side were assessed. Survival and restorative, pulpal, periapical, and periodontal status of the teeth were recorded using numerical definitions. Fractal dimension (FD) of the trabecular bone on the mesial and distal sides of the implant and control teeth were measured on panoramic radiographs obtained after implant placement and 1 year after occlusal loading. Results: FD values measured around the teeth adjacent to the implants were significantly increased compared to the baseline (p < 0.001); a decrease was observed in the control group (p < 0.001). There was no statistically significant difference between the FD values in the mesial and distal regions of the control teeth (p = 0.982), and the teeth adjacent to the implants (p = 0.666). Change in the number of minimally restored teeth showed no difference between regions (p = 0.082), while the number of heavily restored teeth was significantly higher in the control region compared to the implant side (p = 0.032).Conclusions: The fractal dimension of the trabecular bone around the teeth adjacent to the implants increased after occlusal loading of the implants while decreased on the contralateral side. The survival, restorative, periapical, and periodontal conditions of the teeth adjacent to the implants changed minimally after occlusal loading., Amaç: Tek diş posterior mandibula implantlarının komşu dişlerin sağkalımı, restoratif durumu, pulpa, periapikal ve periodontal sağlığı ve çevreleyen trabeküler kemik yapısı üzerindeki etkisini araştırmaktırGereç ve Yöntemler: Tek diş implantlarına komşu 174 diş ve karşı tarafta 174 diş değerlendirildi. Dişlerin sağkalım ve restoratif, pulpal, periapikal ve periodontal durumları sayısal tanımlarla kaydedildi. İmplantın ve kontrol dişlerin mezial ve distal taraflarındaki trabeküler kemiğin fraktal boyutu (FD), implant yerleştirildikten sonra ve okluzal yüklemeden 1 yıl sonra elde edilen panoramik radyografilerde ölçüldü.Bulgular: İmplantlara komşu dişlerin çevresinde ölçülen FD değerleri başlangıca göre anlamlı olarak arttı (p

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.

Triaxial experiments were conducted on granular materials presenting uniform, graded and fractal particle distributions in order to investigate how the broadness of the distribution affects the phenomenon of strain localisation. The shear band thickness evolution is assessed by digital image correlation (DIC) using three cameras placed at different angles around a transparent triaxial cell. From the field of deformation, Gaussian distributions have made it possible to fit the data satisfactorily and determine the shear band width evolution. The latter exhibits a rapid decrease in the softening regime until a residual value is reached in all cases. In the conditions of the experiments, it is shown that the residual shear band thickness scales with the mean grain size and the ratio between the two increases with the broadness of the distribution. Samples with uniform distribution exhibit an average residual thickness of ∼ 10D50, samples with graded distribution exhibit an average residual thickness of ∼ 12·5D50 and samples with fractal distribution exhibit an average residual thickness of ∼ 17D50.

Funding: Economic and Social Research Council (Award number(s): [ES/V011383/1].). Drawing on both mathematical and anthropological understandings of fractality, this paper explores alternative perspectives of time as it relates to heroin addiction and poly-substance use in Scotland. The paper ethnographically illustrates temporalities which confound typical conceptualisations of linearity, and which can be better understood as fractal. Senses of linear time are disrupted for people who use heroin through intensive poly-substance use, an increasing trend in Scotland, as both time and memory become fragmented beyond coherence or re-assemblage. Distortedness and complexity being common descriptors applied to mathematical fractals, time shattered into uncountable and un-interpretable fragments similarly connotes fracture, dissonance and distortion. A meaningful engagement with fractal theory contains the potential to open up new vocabulary, imagery, and theoretical avenues with which to grasp complex and non-linear time experience. The aims of the paper are, therefore, twofold; to both provide a nuanced ethnographic exploration of substance use time, and to develop a reflexive analytical framework for temporal experience through fractals. Publisher PDF

Objectives: To evaluate the mandibular bone structure of patients with hyperlipidemia by fractal dimension (FD) analysis and panoramic radiomorphometric indices including mandibular cortical thickness measurement, panoramic mandibular index (PMI), mandibular cortical index (MCI) and to compare with the healthy group. Methods: In total, 60 panoramic radiographs were included, including panoramic radiographs of 30 individuals with hyperlipidemia and 30 individuals with systemically healthy. FD analysis in the mandibular condyle, angle of the mandible, the distal side of the second premolar and anterior to mental foramen, PMI, MCI, and mandibular cortical thickness measurements were evaluated on radiographs. Independent samples t-test was used for differences between healthy and hyperlipidemia groups with regard to age and PMI. Repeated measurement of variance with one within and one between factors in the comparison of four regions and two groups in terms of FD and cortical thickness measurements. Following this analysis, significant differences were detected by post-hoc Sidak test. Fisher-Freeman-Halton analysis was applied to determine the relationship between categorical variables. Results: FD values of the hyperlipidemic patients were found to be lower than the healthy group. Between the hyperlipidemic and healthy groups, there was a difference in the angle of the mandible FD values (p = 0.020). There were no differences in the cortical thickness measurements and PMI between the groups (p > 0.05). There was a difference in MCI values between the groups (p < 0.05). Conclusion: The trabecular structure of the angle of the mandible and the cortical bone structure of the mandible were found to be negatively affected by hyperlipidemia.

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.

Fractal is one of the important tools to produce efficient and well-suited antennas for navigation systems. A novel design of broadband fractal sausage Minkowski antenna is described in this research article. Sausage Minkowski is designed and simulated for global positioning system (GPS) application that relies on fractal geometry using computer simulation studio microwave (CST MW) software. The substrate sheet is Roger TMM4 has of 4.5 dielectric constant and 1.6 mm altitude. A small metallic patch over a large metallic ground layer is made of perfect electric conductor (PEC) material. The suggested fractal antenna can be used efficiently in the military applications which demand a narrow spectrum and a tiny antenna profile. By introducing four various levels of fractal iteration models and compared each one to the other. The main aim of this work is to increase antenna gain and reduce the antenna size, in addition to improving the semi-flat voltage standing wave ratio (VSWR) with respect to decreasing the resonance frequency is. For 10% decrease in the frequency for the 3rd iteration value led to a clear improvement in the antenna characteristics such as directivity, gain, and VSWR. The reflection coefficient and bandwidth remain in range.

Cyber-physical systems (CPSs) realize the automatic control of entities through computing systems and networks. Stability is an important factor in CPS for system upgrading and troubleshooting. Traditional analysis methods focus on simulation and formal analysis, which have two major limitations: first, the current state information of CPS is difficult to obtain; second, most CPS face the state space explosion problem. These problems can be avoided and a good analysis can be provided based on empirical data. The main work of this article is summarized as follows: first, a phase space reconstruction method is designed to divide the dataset into several subsequences with the same shape; second, we propose a stability analysis method based on correlation dimensions. Results indicate that the proposed approach can obtain a stable correlation dimension. CPS perform better if the correlation dimension is maintained within a certain range; otherwise, a destabilizing factor exists. The proposed stability analysis has less complexity and running time.

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.

The fractal and chaos are bound tightly, and their relevant researches are well-established. Few of them, however, concentrates on the research of the possibility of combining the fractal and the chaotic systems to generate multi-scroll chaotic attractors. This paper presents a novel non-equilibrium point chaotic system, exhibiting extremely rich and complex hidden behaviors including chaos, hyper-chaos, multi-scroll attractors, extreme multi-stability and initial offset-boosting. The proposed system is combined with fractal transformation respectively, and a new class of multi-scroll attractors, such as multi-ring attractors and separated-scroll attractors, is observed. Particularly, swallow-shaped attractors for the first time is found. Moreover, another efficient method to generate a different class of chaotic attractors uses parabola transformation and triangle transformation. Additionally, the spectrum entropy ( SE ) complexity is employed to discuss the complexity of the proposed system before and after fractal, resulting in a chaotic sequences with fractal transformation that has higher complexity. Finally, we develop a hardware platform to implement the presented attractors before and after fractal in a way to confirm the accuracy of the numerical simulations, providing a theoretical basis for the next application in image encryption.

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.

Due to the strong nonstationary characteristics of seismic signals, energy criteria-based methods are not robust for detecting moving targets, especially in data with low SNRs. To address this problem, we propose a new method for detecting ground moving target based on fractal dimension (FD) theory named FD-based support vector machine (FD-SVM). In this method, seismic signals are first measured by fractals, which can effectively extract seismic nonlinear features. These fractal features are then fed into an SVM to distinguish moving targets from noise. Two data sets are used to evaluate the proposed method. One is a set of seismic signals induced by wheeled and tracked vehicles. The other is a set of seismic signals generated by human footsteps. Experimental results demonstrate that the proposed FD-SVM algorithm achieves promising results on both data sets. Compared with the benchmark methods, the FD-SVM algorithm achieves a better precision rate, recall rate, and F1 score.

The environment around us naturally represents number of its components in fractal structures. Some fractal patterns are also artificially simulated using real life mathematical systems. In this paper, we use the fractal operator combined to the fractional operator with both exponential and Mittag-leffler laws to analyze and solve generalized three-dimensional systems related to real life phenomena. Numerical solutions are provided in each case and applications to some related systems are given. Numerical simulations show the existence of the models' initial three-dimensional structure followed by its self- replication in fractal structure mathematically produced. The whole dynamics are also impacted by the fractional part of the operator as the derivative order changes.

A decomposition and reconstruction theorem (DRT) is introduced to advance computation and provide physical understanding for the scattering from fractal surfaces (FPs). The profile of FP is decomposed into test profiles (TPs), and the scattering of FP is reconstructed using the scattering of TPs to enhance the computational efficiency. A new method that applies DRT on the extended boundary condition method combined with the truncated singular value decomposition technique (TEBCM) is presented and referred to as TEBCM-DRT. The method of TEBCM-DRT is employed to solve the scattering from realistic soil surfaces, and the results are validated against other scattering models. Moreover, the efficiency of TEBCM-DRT and its validity range are investigated. The result shows that TEBCM-DRT improves computational efficiency to 1.95 x 10⁵ and 7.35 x 10² times, respectively, compared to TEBCM and the conventional method of moments for a wide range of roughness. In addition, TEBCM-DRT indicates that the amplitude and direction of propagation of scattering modes are dependent on deterministic TPs. This relationship benefits for obtaining the accurate height of an arbitrary point on the profile from bistatic scattering coefficients.

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.

The particle morphology of granular materials comprises different characteristic scales, including particle shape and surface texture. Different methods have been proposed to characterise the morphology using three-dimensional parameters, among which is the fractal method. These methods, however, are applied either at the scale of particle shape or surface texture. A framework unifying the multi-scale morphology obtained from different measuring instruments could advance the current understanding to this topic, but is still lacking. This paper proposes a novel methodology to characterise the morphology of sand particles across different scales based on results from two previously adopted instruments with different measuring capabilities – an X-ray micro-computed tomography (μCT) and a high-resolution optical microscope equipped with an interferometer. The methodology is applied to sand-sized particles of a crushed granitic rock and a natural quarzitic sand (Fujian sand). By using spectrum analysis on data from both μCT and interferometer measurements, a single fractal dimension is found linking the spectrum of the two measurements for the crushed granitic rock. For Fujian sand, two self-affine patterns are observed, which serves as a separation between particle shape and surface texture, and also indicates that the fractal dimension obtained at larger scale may not be simply extended to small scales. The translation of surface measurements into numerically reconstructed particle morphology at particle shape and surface texture scale is demonstrated by using spherical harmonic expansion and power spectral density functions.

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.

Introduction. The development of new amplifiers and generators of the Ku- and K-bands (12…27 GHz) for use in onboard equipment is increasingly attracting research interest. Low-voltage multi-beam klystrons (LMBK) can be a promising element base for such devices. Serious problems are associated with the need to suppress parasitic modes of oscillations in NMLK operating in the centimeter and millimeter range. A possible solution is to use double-gap photonic-crystal resonators (DPCR) in LMBK. Another promising direction for improving the characteristics of such resonators is to use resonant segments of strip lines with fractal elements. In this case, the strip lines are placed on a dielectric substrate in the interaction space. Such resonators exhibit new properties that are useful for klystrons (an increase in characteristic impedance, suppression of the spectrum of unwanted frequencies, a reduction in mass and dimensions).Aim. Determination of an optimal set of electrodynamic and electronic parameters of double-gap photonic-crystal resonance systems with fractal elements "Minkowski Island" when operated as part of the LMBK resonator system, excited on π- and 2π-modes of oscillation.Materials and methods. To calculate the electrodynamic parameters of resonators, the method of finite differences in the time domain was used. The well-known Wessel-Berg method was used to calculate electronic parameters, such as the Ge / G0 electronic conductivity and the coupling coefficient M.Results. The main electrodynamic parameters of the resonator – Q-factor, resonant frequency and characteristic impedance – were investigated. The electronic parameters of the resonator, the coefficient of coupling with the electron beam, and the relative electronic conductivity for π- and 2π-modes of oscillations were calculated. In this case, three variants of the resonator with zero, first and second iterations of the fractal element were investigated. The amplitude-frequency characteristics of the resonator were investigated with a change in the pitch of the photonic crystal lattice. An estimation of the inhomogeneity of the high-frequency field in the interaction spaces of the resonator was carried out. Operational conditions were determined simultaneously for two types of oscillations without self-excitation.Conclusion. The results can find application in the development of resonator systems for klystron-type devices in the centimeter and millimeter ranges.

The current paper analyzed a fractal–fractional mathematical model of typhoid to study the dynamics of fever disease incorporating protection against infection. The considered model is investigated into fractional-order and fractal dimension in sense of Caputo derivative. The existence theory of the considered model is determined and the uniqueness of the solution by the approach of fixed point theorem. Furthermore, the Ulam–Hyres stability is derived for the considered system by perturbing the problem with a small factor. The non-integer order model is evaluated numerically by fractional Adams–Bashforth iterative techniques and compares the results of fractional order and fractal dimension with integer order.

In fractal structures, the box-counting dimension is used to characterize the un-evenness of geometric objects or spaces. In architecture it can be used as a measurement to investigate the complexity of a building’s geometric form or pattern. This study presents a developed approach using intersection testing on triangles and the octree structure, to calculate a fractal parameter for a building surface, derived from the geometric theory of the 3D box-counting fractal dimension. Using this method, we present a fractal parameter of the characteristic surface of a building, which can reflect the different space-filling capacity of the building due to the diversity of its geometric characteristics.

Cell studies play an important role in the basis of studies on cancer diagnosis and treatment. Reliable viability assays on cancer cell studies are essential for the development of effective drugs. Lens-free digital in-line holographic microscopy has become a powerful tool in the characterization and viability analysis of microparticles such as cancer cells due to its advantages such as high efficiency, low cost, and flexibility to integrate with other components. This study is designed to perform viability tests using fractal dimensions of alive and dead cancer cells based on digital holographic microscopy and machine learning. In the in-line holography configuration, a microscopy assembly consisting of inexpensive components was built using an LED source, and the images were reconstructed using computational methods. The standard US Air Force Resolution Target was used to evaluate the capability of our imaging setup then holograms of stained cancer cells were recorded. To characterize individual cells, 19 different rotational invariant fractal dimension values were extracted from the images as features. An artificial neural network technique was employed for the classification of fractal features extracted from cells. The artificial neural network was compared with four other machine learning techniques through five different classification performance measures. The empirical results indicated that artificial neural networks performed better than compared classification techniques with accuracies of 99.65%. The method proposed in this paper provides a new method for the study of cell viability which has the advantages of high accuracy and potential for laboratory application.

Duck down, as a natural keratin material, has been widely used as a filling material. The multilevel bifurcation structure of down has been observed and characterized through scanning electron microscopy. The structure is a complex fractal structure composed of four-level self-similar structures including five units, that is, the calamus, main barb, barb, barbule, and node or prong. The differential friction effect of the dynamic friction coefficients of the barb was reduced from 0.4 (dry state) to 0.23 (wet state), namely a decrease of 42.5%. The friction locking effect decreases due to the swelling of the fiber diameter. The down is zero gravity in water, and under the action of vibration and internal stress, down that has been subjected to friction or heat setting treatment can quickly return to its original shape in water. This shape memory mechanism was further confirmed, in which down after heat setting can restore its shape to the natural state by shaking it quickly and vigorously. This research provides inspiration to investigate more complicated functions of natural materials and encourages the creation of very intelligent synthetic polymers.

The need for telecommunications technology requires fast data transfer and increasing capacity. Therefore, telecommunications technology continues to be developed by creating transmitter and receiver devices that can work at Ultra-Wideband (UWB) frequencies. This study aims to design a coplanar Vivaldi antenna with a fractal structure so that the performance of the antenna radiation pattern can increase to apply UWB frequency. The conventional coplanar Vivaldi antenna, model-A, and Model-B fractal antennas are designed to see the performance of return loss values less than -10 dB, VSWR less than 2, and working frequency of 2-10 GHz. The fractal model is created with a circular repeating structure given at the edges of the antenna patch. From the simulation results, the main lobe value of the conventional Coplanar Vivaldi antenna was 5.21 dBi, the model-A fractal was 7.02 dBi, and the Model-B fractal was 7.84 dBi. The order of the best sidelobe performance at a frequency of 6 GHz is obtained for the model-A fractal of -9 dB. Beamwidth of 46 degrees and the best main lobe direction of 0 degrees is obtained for fractal model-B. By adding a fractal structure, the lobe magnitude/directivity, sidelobe level, beamwidth, and main lobe direction can be improved

This study presents a model application for the evaluation of Effective Stimulated Reservoir Volume (ESRV) in shale gas reservoirs. This current model is faster, cheaper, and readily available for estimating ESRV compared to previously published models. Key controlling parameters for efficient ESRV modeling, including geomechanical parameters and time, are considered for the model development. The model was validated for both single and multi-stage fractured reservoirs. For the single fractured reservoir, an ESRV of 3.07 × 106 ft3 was estimated against 3.99 × 106 ft3 of ESRV-FEM field data. Whereas, 7.00 × 109 ft3 ESRV was estimated from the multi-stage fractured reservoir against 7.90 × 109 ft3 of fractal-based model results. Stress dependence, time dependence, and permeability dependence of shale gas reservoirs are found to be essential parameters for the successful calculation of ESRV in reservoirs. An ESRV determined using this method can obtain the estimated ultimate recovery, propped volume, optimal fracture length, and spacing in fractured shale gas reservoirs.

Purpose The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

Fractal functions defined through iterated function systems provide a new technique for the approximation of functions. Non-self-referential bivariable fractal functions which approximate a given continuous function defined on a rectangle in $$\mathbb {R}^2$$ are developed herein. Moreover, by imposing suitable conditions on the scaling factors and on base functions, we study $$\mathcal {C}^r$$ -non-self-referential bivariable fractal functions.

For each of all 156 largest earthquakes (M ≥ 7.5, depth

This study presents a novel approach to design the single‐feed circularly polarized (CP) microstrip antenna by the periodic fractal parasitic structure (PFPS). By adding the PFPS along the radiated patch, a new independent orthogonal mode is generated. When the magnitudes of these two modes are the same and the time‐phase difference between them is 90°, the antenna can realise CP radiation. Parameter studies of the PFPS show how to design the CP microstrip antenna. The proposed CP microstrip antenna, designed for the satellite navigation system, has a 30 MHz 10‐dB impedance bandwidth and an 8 MHz 3‐dB axial ratio bandwidth in the centre frequency of 1.575 GHz. The advantage of this presented CP antenna is that these two modes for realising the CP radiation are independent of each other. This makes it more convenient to simplify the designing and debugging of the proposed CP antenna.