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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 .

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.

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 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.

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.

Fractals are found in nature and play important roles in biological functions. However, it is challenging to controllably prepare biomolecule fractals. In this study, a series of Sierpiński triangles with global organizational chirality is successfully constructed by the coassembly of l-tryptophan and 1,3-bi(4-pyridyl)benzene molecules on Ag(111). The chirality is switched when replacing l-tryptophan by d-tryptophan. The fractal structures are characterized by low-temperature scanning tunneling microscopy at the single-molecule level. Density functional theory calculations reveal that intermolecular hydrogen bonds stabilize the Sierpiński triangles.

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.

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.

A novel virtual material layer model based on the fractal theory was proposed to predict the natural frequencies of carbon fiber reinforced plastic composite bolted joints. Rough contact surfaces of composite bolted joints are modeled with this new proposed approach. Numerical and experimental modal analyses were conducted to validate the effectiveness of the proposed model. A good consistence is noted between the numerical and experimental results. To demonstrate the necessity of accurately modeling the rough contact surfaces in the prediction of natural frequencies, virtual material layer model was compared with the widely used traditional model based on the Master-Slave contact algorithm and experiments, respectively. Results show that the proposed model has a better agreement with experiments than the widely used traditional model (the prediction accuracy is raised by 8.77% when the pre-tightening torque is 0.5 N∙m). Real contact area ratio A* of three different virtual material layers were calculated. Value of A* were discussed with dimensionless load P*, fractal dimension D and fractal roughness G. This work provides a new efficient way for accurately modeling the rough contact surfaces and predicting the natural frequencies of composite bolted joints, which can be used to help engineers in the dynamic design of composite materials.

In this paper, we consider the nonlinear discrete-time dynamic model proposed by Bischi and Baiardi (Chaos Solitons Fractals 79:145-156, 2015a). The model considers players with adaptive adjustment mechanisms towards the best reply and a form of inertia in adopting such mechanism. Moreover, we formulate an extension of the original model, where endogenous market size is considered. Through numerical simulations, we show that multiple attractors may exist in the presence of homogeneous agents and the emergence of non-synchronized trajectories both in the short (on-off intermittency) and long (global riddling) run. Therefore, the article highlights that strategic contexts exist in which the players’ knowledge of the market and the adoption of the best reply do not always allow the use of the representative agent’s rhetoric to describe the dynamics of the model.

The security of digital content during transmission and storage through insecure communication links and databases is a challenging issue in today's world. In this article, an encryption scheme based on fractal Tromino and Chebyshev polynomial-based generated chaotic matrix is presented. The scheme fulfills the most fundamental aspect of encryption that is diffusion and confusion. For confusion highly non-linear, pre-defined S-boxes are used. The proposed scheme has been tested using state-of-the-art key performance indicators including differential analysis, statistical analysis. Information entropy analysis, mean square error, and NIST-based randomness analysis. The encrypted images have the highest practically achievable entropy of 7.999 and the time analysis shows that the proposed system is suitable for real-time implementation. The rest of the results indicates that the proposed cryptosystem possesses high immunity toward various attacks. The security analysis compared with the existing scheme shows the strength of the suggested scheme.

Aimed at investigating the differentiation of acoustic emission (AE) signals and fractal precursory characteristics between strong, weak, and no bursting liability coals under uniaxial compression, as well as improving the accuracy of rockburst monitoring and early warning by AE techniques, we experimentally studied the evolution law and differences of AE ring count rate, energy rate, and correlation dimension between different loaded bursting liability coals by the YAW4306 electric mechanical test system and CTA-1 AE monitor. Our experimental results indicated that the AE count and energy of coal samples with different bursting liabilities showed a similar evolution law of “sharp increase-calm-sharp increase” before their main rupture. The active points of AE signals emitted from coal with strong, weak, and no bursting liability appeared at about 85∼90%, 75∼78%, and 51∼55% of the peak stress, respectively. The stronger the bursting liability of coal, the shorter the duration of main rupture and postpeak failure stage, and the greater the AE energy rate in the main rupture. The AE counts of different coals had obvious fractal characteristics, and the AE correlation dimension values of strong and weak bursting liability coal samples presented the phenomenon of “fluctuating rise to a peak value-sharp drop-continuous decrease,” which can be used as a precursory information of coal failure.

We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$. We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$.We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

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.

espanolSe propone una nueva tecnica de modelado fractal, adaptando el modelo de concentracion-numero (C-N, segun sus siglas en ingles) y el concepto de puntos de ruptura de umbral para interpretar las medidas de sondeo electrico vertical (VES, segun sus siglas en ingles) distribuidas a lo largode un perfil dado. En consecuencia, se propone un enfoque semicuantitativo para diferenciar facilmente entre diversas poblaciones de resistividad aparente, donde se podria construir una interpretacion semicuantitativa bidimensional (2D) y un analisis geologico preliminar. La nueva tecnica se aplico en un estudio de caso de la region del valle de Khanasser, norte de Siria, donde se interpretaron diferentes perfiles (LP1, LP2, LP3 yTP5). La idoneidad y viabilidad del enfoque propuesto se confirman y aprueban a traves de diferentes comparaciones entre las secciones transversales establecidas de multiples fractales y los modelos de interpretacion VES unidimensionales (1D) tradicionales. Se recomienda utilizar de forma rutinaria este nuevo enfoque fractal propuesto en la investigacion geoelectrica para interpretar las mediciones de VES distribuidas a lo largo de un perfil determinado. EnglishA new fractal modeling technique, adapting the concentration-number (C-N) model and the threshold break points concept is proposed to interpret vertical electrical sounding (VES) measurements distributed along a given profile. A new semi-quantitative approach is consequently proposed to readily differentiate between different apparent resistivity populations, where two-dimensional (2D) semi-quantitative interpretation and a preliminary geological analysis could be constructed. The newtechnique is applied and tested on a case study from the Khanasser Valley region, Northern Syria, where different selected profiles (LP1, LP2, LP3, and TP5) are interpreted. The suitability and feasibility of the proposed approach are confirmed and approved through different comparisons between the multi-fractal established cross-sections and the traditional one-dimensional (1D) VES interpretation models. It is recommended to routinely use this new proposed fractal approach in geoelectrical research for interpreting VES measurements distributed along a given profile

In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the $$(k+1)$$ ( k + 1 ) st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.

Fractality has been found in many areas and has been used to describe the internal features of time series. But is it possible to use fractal theory to improve the performance of hydrological models? This study aims at investigating the potential benefits of applying fractal theory in model calibration. A new criterion named the ratio of fractal dimensions (RD) is defined as the ratio of the fractal dimensions of simulated and observed streamflow series. To combine the advantages of fractal theory with classical criteria based on squared residuals, a multi-objective calibration strategy is designed. The selected classical criterion is the Nash–Sutcliffe efficiency (E). The E–RD strategy is tested in three study cases with different climates and geographies. The results reveal that, in most aspects, introducing RD into model calibration makes the simulation of streamflow components more reasonable. Also, pursuing a better RD during calibration leads to only a small decrease in E. We therefore recommend choosing the parameter set with the best E among the parameter sets with RD values of around 1.

A fractal tiling ( f -tiling) is a kind of rarely explored tiling by similar polygonal tiles which possesses self-similarity and the boundary of which is a fractal. Based on a tiling by similar isosceles right triangles, Dutch graphic artist M. C. Escher created an ingenious print Square Limit in which fish are uniformly reduced in size as they approach the boundaries of the tiling. In this article, we present four families of f -tilings and propose an easy-to-implement method to achieve similar Escher-like drawings. By systematically investigating the local star-shaped structure of f -tilings, we first enumerate four families of f -tilings admitted by kite-shaped or dart-shaped prototiles. Then, we establish a fast binning algorithm for visualising f -tilings. To facilitate the creation of Escher-like drawings on the reported f -tilings, we next introduce one-to-one mappings between the square, and kite and dart, respectively. This treatment allows a pre-designed square template to be deformed into all prototiles considered in the article. Finally, we specify some technical implementations and present a gallery of the resulting Escher-like drawings. The method established in this article is thus able to generate a great variety of exotic Escher-like drawings.

Knowledge of the behavior of highly compacted expansive clays, as an engineered barrier, in disposal of high-level nuclear waste (HLW) systems to prevent the pollution due to migration of radionuclide is extremely essential. The prominent properties of globally and widely used bentonites have been extensively studied during past two decades. In China, GaoMiaoZi (GMZ) bentonite is the first choice as a buffer or backfill material for deep geological repositories. This review article presents the recent progresses of knowledge on water retention properties, hydromechanical behavior, and fractal characteristics of GMZ bentonite-based materials, by reviewing 217 internationally published research articles. Firstly, the current literature regarding hydrogeochemical and mechanical characteristics of GMZ bentonite influenced by various saline solutions are critically summarized and reviewed. Then, the role of osmotic suction π alongside the application of surface fractal dimension Ds is presented from the standpoint of fractal theory. Finally, the strength characteristics of GMZ bentonites using fractal approach have been discussed. Furthermore, this study sheds light on gaps, opportunities, and further research for understanding and analyzing the long-term hydromechanical characteristics of the designed backfill material, from the standpoint of surface fractality of bentonites, and implications of sustainable buffer materials in the field of geoenvironmental engineering.

Compression shear failure of rock is a common failure mode in engineering, and there are often complicated seepage phenomena in its shear crack. Due to the anisotropic and irregular rough cross-section on both sides of the natural crack, the experimental study of rock seepage is difficult to operate. In order to study the microscopic seepage characteristics of rock cracks, granite crack specimens were prepared through shear experiments. The data was obtained by scanning the section with laser and imported into MATLAB software to reconstruct the three-dimensional section and crack opening model. In view of the difficulty of observing the microscopic characteristics of rock seepage flow, in this paper, the Reynolds equation of rock fracture flow considering the dynamic effect of boundary is derived, and under the condition of the definition of Reynolds equation theory, based on the advantage of fractal geometry for irregular cross-section, the relationship between flow velocity, permeability and simulation scale is studied by using COMSOL simulation software. The results show that the granite cross-section has a good statistical self-similarity feature, and the fractal matrix mesh covering the fracture surface can well express the feature of the fracture surface; The roughness of the fracture surface has a positive correlation with the fractal dimension. The larger the fractal dimension, the greater the roughness of the fracture surface, which verifies the reliability of the fractal dimension to describe the roughness of the section. There is a positive correlation between the scale-up of the simulation and the velocity of the micro-research points, and the permeability of the model; The feasibility of the fractal percolation model is investigated and the general suggestions for simulating scale are given. Some of the results of this study can be used for reference in the analysis of seepage characteristics of fractures in rough rock mass.

Clay minerals significantly alter the pore size distribution (PSD) of the gas hydrate-bearing sediments and sandstone reservoir rock by adding an intense amount of micropores to the existing intragranular pore space. Therefore, in the present study, the internal pore space of various clay groups is investigated by manually segmenting Scanning Electron Microscopy (SEM) images. We focused on kaolinite, smectite, chlorite, and dissolution holes and characterized their specific pore space using fractal geometry theory and parameters such as pore count, pore size distribution, area, perimeter, circularity, and density. Herein, the fractal properties of different clay groups and dissolution holes were extracted using the box counting technique and were introduced for each group. It was observed that the presence of clays complicates the original PSD of the reservoir by adding about 1.31-61.30 pores/100 μm2 with sizes in the range of 0.003-87.69 μm2. Meanwhile, dissolution holes complicate the pore space by adding 4.88-8.17 extra pores/100 μm2 with sizes in the range of 0.06-119.75 μm2. The fractal dimension ( D ) and lacunarity ( L ) values of the clays’ internal pore structure fell in the ranges of 1.51-1.85 and 0.18-0.99, respectively. Likewise, D and L of the dissolution holes were in the ranges of, respectively, 1.63-1.65 and 0.56-0.62. The obtained results of the present study lay the foundation for developing improved fractal models of the reservoir properties which would help to better understand the fluid flow, irreducible fluid saturation, and capillary pressure. These issues are of significant importance for reservoir quality and calculating the accurate amount of producible oil and gas.

This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section.

The research work is devoted to the control of materials structures described on the basis of fractal representations. The formation of fractal structures of materials is carried out due to positive feedback. The article describes the first stage of the work – statements of the problem of identification of materials structures based on fractal representations. Many recent studies indicate the fractal nature of material structures, while the mechanisms of positive feedbacks are based on the generation of fractal structures. At the present stage, the fundamental physicochemical laws governing the emergence and transformation of material structures are developed and presented in such a way that it is difficult to use them for the synthesis of structure control algorithms. In other words, they do not meet the requirements of control models – they do not reflect the dependence of the output actions on external factors. Therefore, it seems useful to go along the path of creating fractal models of structures that are identification structures of materials and then development of control actions, in particular on the parameters of positive feedback for predicting and changing the material structure in the required direction. This corresponds to the method of control algorithms synthesis with an assessment of the control object state and choice of controller gain coefficient. In this work, the statements of the problem of identification images of natural materials structures are formulated. Firstly, fractal models are built using well-known (standard) fractals, and secondly, fractal models are generated using the fractal structure formation mechanism. In the future, these statements of the problem will be used to develop methods and algorithms for materials structuresidentification in various industries, including mining and metallurgical industries.

PurposeThe purpose of this paper is the coupled nonlinear fractal Schrödinger system is defined by using fractal derivative, and its variational principle is constructed by the fractal semi-inverse method. The approximate analytical solution of the coupled nonlinear fractal Schrödinger system is obtained by the fractal variational iteration transform method based on the proposed variational theory and fractal two-scales transform method. Finally, an example illustrates the proposed method is efficient to deal with complex nonlinear fractal systems.Design/methodology/approachThe coupled nonlinear fractal Schrödinger system is described by using the fractal derivative, and its fractal variational principle is obtained by the fractal semi-inverse method. A novel approach is proposed to solve the fractal model based on the variational theory.FindingsThe fractal variational iteration transform method is an excellent method to solve the fractal differential equation system.Originality/valueThe author first presents the fractal variational iteration transform method to find the approximate analytical solution for fractal differential equation system. The example illustrates the accuracy and efficiency of the proposed approach.

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 study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. (Fractals 28:2040011 2020) in the frame of Chebyshev and Bielecki norms with time delay. The acquired results are obtained by using Banach fixed point theorems and the Picard operator (PO) method. Finally, a pertinent example of the results obtained is demonstrated.

A printed monopole in its planar and vertical configuration has been designed, fabricated and analyzed for microwave applications on low cost FR4 substrate material of thickness, h = 1.56 mm and relative permittivity, εr = 4.3. The designed planar monopole has been simulated and experimented to find its frequency response with coplanar waveguide feed to exhibit dual band characteristics with -10 dB reflection loss bandwidth of 45.078 % (i.e. 1.5:1 between 1.334 and 2.109 GHz) and 114.92 % (i.e. 3.7: 1 between 3.99 and 14.77 GHz). The vertical monopole using the same patch has also been simulated and experimented and -10 dB reflection loss bandwidth of 173.67% (14.2:1 between 0.925 and 13.125 GHz) has been obtained. The antenna finds many applications in microwave bands.

Purpose The purpose of this study is to develop a high-quality factor fractal inductor for wireless applications such as satellite, WLAN, Bluetooth, microwave, radar and cellular phone. Design/methodology/approach The Hilbert fractal curve is used in the implementation of the proposed inductor. In the proposed inductor, the metal width has split into multiple paths based on the skin depth of the metal. The simulations of the proposed inductor are performed in 180 nm CMOS technology using the Advanced Design System EM simulator. Findings The multipath technique reduces the skin effects and proximity effects, which, in turn, decreases the series resistance of the inductor and attains high-quality factor over conventional fractal inductor for the equal on-chip area. Research limitations/implications The width of the path has chosen higher than the skin depth of the metal for a required operating frequency. Due to cost constraints, the manufacturing of the proposed fractal inductor is limited to a single layer. Practical implications The proposed inductor will be useful for the implementation of critical building blocks of radio frequency integrated circuits and monolithic microwave integrated circuits such as low-noise amplifiers, voltage-controlled oscillators, mixers, filters and power amplifiers. Originality/value This paper presents for the first time the use of a multipath technique for the fractal inductors to enhance the quality factor.

In this paper, we have performed qualitative and quantitative analysis of LuMnO3 thin films surfaces, deposited by spin coating over Pt(111)/TiO2/SiO2/Si substrates, to evaluate their spatial patterns as a function of the film’s sintering temperature. Atomic force microscopy was employed to obtain topographic maps that were extensively analyzed via image processing techniques and mathematical tools. 3D (three-dimensional) topographical images revealed that films sintered at 650°C and 750°C presented the formation of smoother surfaces, while the film sintered at 850°C displayed a rougher surface with a root mean square roughness of ∼2.5 nm. On the other direction, the height distribution of the surface for all films has similar asymmetries and shape, although the film sintered using the highest temperature showed the lower density of rough peaks and a sharper peak shape. The advanced fractal parameters revealed that the film sintered at 850°C is dominated by low spatial frequencies, showing less spatial complexity, higher microtexture homogeneity, and uniform height distribution. These results suggest that the combination of stereometric and fractal parameters can be especially useful for identification of unique topographic spatial patterns in LuMnO3 thin films, helping in their implementation in technological applications, such as photovoltaic solar cells and information magnetic date storage and spintronic devices.

After defining the fractional Λ-derivative, having all the requirements for corresponding to a differential, the fractional Λ-strain is established. Contrary to the common strain, that has a local character, fractional strain access a non-local character, quite important for expressing deformations in non-homogeneous media with microcracks and inhomogeneities, that may change during deformation. The purpose of the present work is the establishement of the principles and laws of the non-linear Λ-fractional Elasticity. The Λ-fractional non-linear stress–strain relations are derived. The restriction into the linear fields is presented. Further, fractional deformation of a fractal bar is discussed. The Fractional deformations and fractional elastic problems are set up with the definition of stresses and displacements in the initial space. Further, the Λ-fractional analysis with its conjugate Λ-fractional space is presented, considering fractional derivatives of both sides in the bending of a cantilever beam under uniform continuously distributed loading.

This paper develops a numerical approach for solving two-dimensional fractal-fractional parabolic partial differential equations. The fractal-fractional derivative is defined in the Atangana-Rieman...

This paper presents the unique structure of a hybrid antenna with a blend of Peano–Gosper, Koch and Minkowski curves using a modified partial ground plane for multi-standard wireless applications. ...

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.

Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

A fractal-based approach is presented for the direct-stroke shielding analysis of high-voltage substations. The proposed method considers the effect of lightning polarity and the stochastic nature of the lightning attachment phenomenon. An application to a typical 69 kV substation reveals that shielding failures may occur even if the shielding design is realized with IEEE Std 998-2012 methods. This can be explained by the fact that the shielding design methods adopted by IEEE Std 998 do not consider the effects of lightning polarity and object height on striking distance. In addition, IEEE Std 998 methods adopt a deterministic approach on shielding failure analysis neglecting the branched and tortuous behavior of lightning discharges. Fractal-based simulation results show a considerable dispersion of striking distance and a significantly higher shielding failure probability for downward positive lightning than negative lightning. This work forms a discussion framework on shielding failure analysis following a stochastic approach; this approach may explain unexpected shielding failures, such as those reported in IEEE surveys, and can be used for the direct-stroke shielding design of mission-critical high-voltage substations.

This study illustrates the application of a geometric average integration of aeromagnetic, radiometric and satellite imagery data over a region prone to Cu‐bearing mineralization at Chahargonbad area in Kerman province of Iran. Processing aeromagnetic, radiometric and Advanced Spaceborne Thermal Emission and Reflection Radiometer satellite data can provide exploratory insights about favourable zones in association with porphyry‐type ore occurrences, which can be synthesized through a combination of knowledge‐ and data‐driven approaches as a geometric average and be represented in a mineral prospectivity map. The existence of known deposits in a prospect region can facilitate the investigation of significant exploratory footprints extracted from airborne data by calculating each indicator layer's weight by plotting a prediction–area curve accompanied by a concentration–area fractal curve. Among various indicators, the most important ones are determined based on derived weights from the prediction–area plots to be synthesized in a single Cu favorability map. To fulfil this aim, indicator layers from airborne geophysics (magmatic bodies, magnetic lineaments and potassium radiometry) and remote‐sensing data (alterations such as argillic, phyllic, propylitic and iron oxide along with geological lineaments) were prepared and evaluated using the known porphyry Cu mineralization by the simultaneous plot of the concentration–area fractal model and the prediction–area curve to attain the ore prediction rate and the relevant occupied area of each map for weight assignment of indicators. The geometric average prospectivity model was applied to synthesize the leading indicators, and the result was compared with a multi‐class index overlay map. This study's significance lies in improvement of the performance of the mineral prospectivity/potential mapping after running a geometric average by a higher ore prediction rate of 79%, which has occupied 21% of the area as potential zones for further mining investigations.

The paper delves into the plausibility of applying fractal, spectral, and nonlinear time series analyses for lung auscultation. The thirty-five sound signals of bronchial (BB) and pulmonary crackle (PC) analysed by fast Fourier transform and wavelet not only give the details of number, nature, and time of occurrence of the frequency components but also throw light onto the embedded air flow during breathing. Fractal dimension, phase portrait, and sample entropy help in divulging the greater randomness, antipersistent nature, and complexity of airflow dynamics in BB than PC. The potential of principal component analysis through the spectral feature extraction categorises BB, fine crackles, and coarse crackles. The phase portrait feature-based supervised classification proves to be better compared to the unsupervised machine learning technique. The present work elucidates phase portrait features as a better choice of classification, as it takes into consideration the temporal correlation between the data points of the time series signal, and thereby suggesting a novel surrogate method for the diagnosis in pulmonology. The study suggests the possible application of the techniques in the auscultation of coronavirus disease 2019 seriously affecting the respiratory system.

In protoplanetary discs and planetary atmospheres, dust grains coagulate to form fractal dust aggregates. The geometric cross-section of these aggregates is a crucial parameter characterizing aerodynamical friction, collision rates, and opacities. However, numerical measurements of the cross-section are often time-consuming as aggregates exhibit complex shapes. In this study, we derive a novel analytic expression for geometric cross-sections of fractal aggregates. If an aggregate consists of $N$ monomers of radius $R_0$, its geometric cross-section $G$ is expressed as \begin{equation} \frac{G}{N\pi R_0^2}=\frac{A}{1+(N-1)\tilde{\sigma}}, \nonumber \end{equation} where $\tilde{\sigma}$ is an overlapping efficiency, and $A$ is a numerical factor connecting the analytic expression to the small non-fractal cluster limit. The overlapping efficiency depends on the fractal dimension, fractal prefactor, and $N$ of the aggregate, and its analytic expression is derived as well. The analytic expressions successfully reproduce numerically measured cross-sections of aggregates. We also find that our expressions are compatible with the mean-field light scattering theory of aggregates in the geometrical optics limit. The analytic expressions greatly simplify an otherwise tedious calculation and will be useful in model calculations of fractal grain growth in protoplanetary discs and planetary atmospheres., Comment: 12 pages, 8 figures, 1 table; Accepted for publication in MNRAS

This paper builds upon two key principles behind the Bourgain–Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah–Logvinenko–Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on a fractal set. In addition to recovering the result of Bourgain–Dyatlov, we obtain analogous uniqueness results for denser fractals.

Fractal theory is an efficient method to characterize quantitatively tight rocks. Volcanic rocks are very promising reservoirs, and their pore structures have considerable influence on the occurrence and percolation mechanisms of oil and gas. Therefore, the research topics of pore structure and fractal characteristics are important for sustainable volcanic resource development. The pore structure and fractal characteristics of microscopic pores of volcanic reservoirs in the Changling Fault Depression, Songliao Basin, were studied by combining fractal theory with mercury intrusion porosimetry and low-temperature nitrogen adsorption technique. After quantitative characterization of the nanoscale pore size distribution (PSD), four fractal dimensions were calculated. Subsequently, the relationships between fractal dimensions and pore structure parameters were evaluated. Three types of PSD models of volcanic reservoirs in the study area were established. The relationships between fractal dimensions and PSD models, as well as the clustering characteristics of the fractal dimensions, were investigated. The study revealed that the volcanic reservoirs developed a certain proportion of nanoscale pores with morphological characteristics dominated by wedge-shaped pores and slit pores. D4 is relatively strongly negatively correlated with pore size, illustrating that it can better reflect pore structure compared to the other fractal dimensions. Additionally, D4 can reflect pore surface roughness. This study shows that D1 and D2 can distinguish PSD models well, but D3 and D4 are more effective for the identification of the Type II PSD model. The fractal dimensions were divided into four clusters via hierarchical cluster analysis, each corresponding well to a certain PSD model or PSD model combination.

Harvesting kinetic ambient energy from vibrations or impact loads to obtain electrical energy useful to supply electronic sensors is still a challenging issue, with a huge number of potential applications, from the industrial field to consumer goods. This work investigates four simple piezoelectric energy harvesters developed from fractal-inspired structures: these structures are square laminas with inner divisions that originate a fractal geometry, since it is obtained by the repetition of a base configuration. Starting from the description of the fractal-inspired structures, the work presents a computational modal analysis highlighting the peculiar wideband frequency response of these structures and thus their applicability as energy converters. Using commercial piezoelectric transducers, we built four energy harvesters and assessed their output voltage and output power performing dynamic tests on a vibrating table. The proposed structures exhibit a wideband frequency response and a good energy conversion, specifically at the fundamental eigenfrequency.

The pore structure of alkali-activated fly ash mortar (AAFAM) under different mix ratios and curing system was measured by mercury intrusion porosimetry (MIP) and scanning electron microscopy (SEM). Based on the fractal theory, the integral shape dimension of the pore surface of the AAFAM is obtained and discussed its relationship with total porosity, proportion of gel pores, transition pore, capillary pores and macropores, average pore diameter, median pore diameter, pore surface area, pore size distribution, and the influence of mass ratio of alkali activators, curing temperature, and curing age on pore structures. In addition, data regression analysis (DRA) is a statistical tool and applied to study the relationship between microscopic pore parameters and the fractal dimension of the AAFAM mixtures. The results show that the fractal dimension of the AAFAM is between 2.6 and 2.9, and the pore size distribution has a linear relationship with the fractal dimension, and the internal pore diameter of the AAFAM will gradually decrease and AAFAM’s matrix gradually becomes denser and the microstructure integrity is more stable as the curing age and temperature increases, and the mechanical strength will be higher. The AAFAM mixture composed of Na2SiO3/NAOH with a mass ratio of 1.0 and 100% fly ash is considered the most suitable mixture. AAFAM can make full use of industrial by-products (fly ash) and seems to be a suitable candidate for green environmental protection materials for engineering applications.

In the present study, we apply an analytical scheme to acquire wave solutions of a partial differential equation involving a local fractional derivative. The main idea of this scheme is to generalize the procedure of the well-known generalized exponential rational function technique. To test the method, we have considered the local fractional longitudinal wave equation in a magneto-electro-elastic circular rod (MEECR). The graphical representation of some derived solutions is also shown. The suggested technique is an efficient way to solve such type of differential equations with local fractional derivative. One of the valuable features of the suggested method is the possibility of using it in solving other similar equations with local fractional derivative.

The present work focuses on the three-dimensional mapping of fractal correlation dimensions and b-values of the Palghar (Maharashtra) and Pulichintala (Andhra Pradesh) regions of the Indian shield. The study is done using catalogues of 8766 Palghar earthquakes of ML0.4–4.7 and 965 Puclichintala earthquakes of ML ranging from −0.4 to 4.6. The b-values are estimated using the maximum likelihood approach, while fractal dimensions are modelled using the correlation dimension approach. The b-value is modelled to be 0.88 ± 0.02 for the Palghar sequence and 0.75 ± 0.04 for the Pulichintala sequence. The modelled b- and D2-values vary from 0.1 to 2.5 and 0.39 to 2.62, respectively, for the Palghar sequence, while they vary from 0.2 to 1.68, and 0.68 to 3.0, respectively, for the Pulichintala sequence. The modelled large b-value (~ 2.5) for the Palghar sequence in particular signifies the typical characteristic of swarm earthquakes. We examine the b-D2 correlations for both the swarm activity sequences and found a positive correlation (i.e. D2 = 1.085 + 0.0015b) for the Palghar sequence while a negative correlation (i.e. D2 = 0.99–0.0414b) for the Pulichintala sequence. The spatial distribution of D2 and b-values suggests that the seismicity in the Palghar region is occurring in the areas with lower to moderate b-values and moderate correlation dimensions, while the seismicity in the Pulichintala region is occurring in areas with moderate b-values and higher D2 values. Based on our results, we interpret the Palghar region as a high tectonic stress region, which can generate moderate earthquakes in future, while the Pulichintala region is a low stress region that can accumulate low levels of tectonic stress posing relatively less earthquake hazard.

The fractal geometry, anisotropy, discontinuity, and non-Darcy flow of tight reservoirs exert a significant effect on well production performance. In this study, the reservoir fractal geometry is represented by exponential functions on the basis of microseismic data, while the discontinuity of the fractures is presented as a nonequilibrium effect. The impact of the nonequilibrium effect and the low velocity non-Darcy flow on the temporal scale of the wellbore pressure is predicted herein. Results showed that the time scale analysis accurately simulates gas flow in a tight reservoir. The wellbore pressure gradually increases, whereas the pressure in the matrix lags when the nonequilibrium effect is considered. The wellbore pressure is affected in the early period by the nonequilibrium effect. However, at the later stage, the pressure in the matrix is mainly affected by the non-Darcy flow. When the non-Darcy flow is dominant, the pores without gas flowing through are better presented.

We study the dynamics of particles in cold electron plasma medium based on two dissimilar approaches: the fractional actionlike variational and fractal calculus approaches. In each case, the corres...

In this paper, we attain the multifractal Hewitt–Stromberg dimension functions of Moran measures associated with homogeneous Moran fractals and show that the multifractal Hewitt–Stromberg measures are mutually singular for which the multifractal functions do not necessarily coincide. In particular, we give a positive answer to questions posed in Attia and Selmi (J Geom Anal 31:825-862, 2021) and discuss some interesting examples.

We consider a 4-dimensional FRW Universe in the framework of fractal cosmology, where the space-time is assumed to be fractal, and the dimensionality of space changes with time. So, in fractal cosmology, the Universe and its matter components act in a fractal way over a full range of scales. Firstly, the modified Chaplygin gas and then the generalized cosmic Chaplygin gas are taken as the cosmic substratum. In these scenarios, the matter-antimatter asymmetry is investigated by two gravitational mechanisms: baryogenesis and leptogenesis. For gravitational baryogenesis, we find the ratio of baryon number density to entropy in both gases graphically. Also, for gravitational leptogenesis, we find the ratio of lepton number density to entropy in both gases graphically.