Fractal Approach Research Articles

ARTIFICIAL INTELLIGENCE TO DEAL WITH THE POST COVID-19 FRACTAL DYNAMICS LINKED WITH ECONOMY

Data-based studies have always provided useful insights of the research problem, provided that correct statistical modeling and inference strategies are adopted. For bi-directional studies or longitudinal datasets, it is always difficult to analyze the dependence of variables and their impact on the key features. With the advancement in the fields of data science and the applied mathematical modeling, these difficulties are well resolved. The machine learning algorithms can help to streamline the data-based studies in a robust manner. During this research, the COVID-19 data are analyzed with the aid of machine learning classification tools to identify the predictors, directly influenced by the pandemic. The impact of COVID-19 on the world’s economy can be better interpreted with the aid of data-based research. The data linked to the unemployment rates, during the frequent waves of COVID-19, are extracted from different sources and analyzed during this research for better forecasting measures. The nonlinear dynamical analysis can be visualized with the aid of the 2D fractal pattern generation approach. Thus the current research is an attempt to connect the outcomes of the classification analysis to the 2D fractal generation for better visual interpretation of lengthy large datasets.

A fractal power-law approach for investigating the interactions between reservoir microstructure and gas migration

As one of the most significant clean energy sources, the migration characteristics and extraction processes of coalbed methane (CBM) have been extensively studied. The structural distribution and evolutionary characteristics of reservoir fractures, as the main conduit for gas migration, significantly affect the permeability and gas production. However, few models have been able to quantitatively and accurately explore reservoir micro–macro interactions under coupled thermal-fluid–solid effects. This work develops a new highly coupled model based on the widely adopted power-law function to quantify reservoir thermal conduction effect, gas pressure evolution, reservoir deformation, in situ stress, the adsorption–desorption effect, and reservoir microstructure evolution. Three parameters are adopted to quantitatively characterize the reservoir structure: (1) fracture power index αf (to characterize the fracture density), (2) fracture length ratio rf (to characterize the fracture size), and (3) the maximum fracture length l. The results demonstrate that the fractal network is a special kind of network in the power-law length distribution. The proposed power-law seepage model is able to accurately characterize the evolution of reservoir microstructure and the impact of microevolution on extraction under multi-field coupling effects, compared to the traditional power-law model. The proposed model can provide a good theoretical and practical support for the study of CBM migration and extraction.

ARTIFICIAL INTELLIGENCE-DRIVEN APPROACH TO IDENTIFY AND RECOMMEND THE WINNER IN A TIED EVENT IN SPORTS SURVEILLANCE

The proliferation of fractal artificial intelligence (AI)-based decision-making has propelled advances in intelligent computing techniques. Fractal AI-driven decision-making approaches are used to solve a variety of real-world complex problems, especially in uncertain sports surveillance situations. To this end, we present a framework for deciding the winner in a tied sporting event. As a case study, a tied cricket match was investigated, and the issue was addressed with a systematic state-of-the-art approach by considering the team strength in terms of the player score, team score at different intervals, and total team scores (TTSs). The TTSs of teams were compared to recommend the winner. We believe that the proposed idea will help to identify the winner in a tied match, supporting intelligent surveillance systems. In addition, this approach can potentially address many existing issues and future challenges regarding critical decision-making processes in sports. Furthermore, we posit that this work will open new avenues for researchers in fractal AI.

YANG TRANSFORM FOR THE HOMOTOPY PERTURBATION METHOD: PROMISE FOR FRACTAL-FRACTIONAL MODELS

This study presents the modified form of the homotopy perturbation method (HPM), and the Yang transform is adopted to simplify the solving process for the Kuramoto–Sivashinsky (KS) problem with fractal derivatives. This scheme is established by combining the two-scale fractal scheme and Yang transform, which is very helpful to evaluate the approximate solution of the fractal KS problem. Initially, we transfer the fractal problem into its partners using the two-scale fractal approach, and then we use the Yang transform ([Formula: see text]T) to obtain the recurrent relation. Second, the HPM is then introduced to deal with the nonlinear elements of the fractal model. The numerical example demonstrates how the suggested technique is incredibly straightforward and precise for nonlinear fractal models. In addition, the graphical error of the proposed fractal model is compared with the calculated results of our suggested approach and the exact results. This graphical error displays the strength and authenticity of our proposed scheme.

A Fractional Integral and Fractal Dimension-Based Deep Learning Approach for Pavement Crack Detection in Transportation Service Management

With artificial intelligence prevailing in intelligent transportation system, pavement crack detection with deep learning has aroused wide attentions in both academia and transportation sector. Nevertheless, it still remains a challenge to accomplish crack detection due to the complexity in pavement background. Motivated by latest advents in computer vision research, a fractional integral-based filtering method is advocated to remove pavement noise, and a fractal dimension estimation method has also emerged to present shape feature at pixel level, with the multi-scale feature architecture. Therefore, we try to propose a deep learning method, integrating fractional integral with fractal dimension, for crack detection in transportation service management. Firstly, the crack image is taken as input in the bottom-up architecture to extract fractal dimension on multi-scale levels, and a per-level feature unit is built to incorporate maps to make context information flow. Secondly, after fed into a convolutional filter for dimension resizing, all the resized feature maps are next fused at each level to comprise a group network. Finally, extensive experiments are executed on different crack datasets, exhibiting that the proposed method surpasses existing cutting-edge ones in terms of both generalizability and accuracy, with the benefits from not only fractional integral filtering, but also multi-scale fractal dimension features.

Fractal signature as a rotational modulation and stellar noise classifier based on the active kepler stars

In this study, we report on the analysis of 701 stars in a solar vicinity defined in three categories namely subsolar, solar, and supersolar with rotation periods between 1 and 70 days, based on rotational modulation signatures inferred from time series from the Kepler mission’s Public Archives. In our analysis, we performed an initial selection based on the rotation period and position in the period–H diagram, where H denotes the Hurst exponent extracted from fractal analysis. To refine our analysis, we applied a fractal approach known as the R/S method, taking into account the fluctuations of the features associated with photometric modulation at different time intervals and the fractality traces that are present in the time series of our sample. In this sense, we computed the so-called Hurst exponent for the referred stars and found that it can provide a strong discriminant of rotational modulation and background noise behavior, going beyond what can be achieved with solely the rotation period itself. Furthermore, our results emphasize that the rotation period of stars is scaled by the exponent H which increases following the increase in the rotation period. Finally, our approach suggests that the referred exponent may be a powerful rotational modulation and noise classifier.

The application of NDT techniques to examination of thermally-induced cracking patterns of brick powder-Portland cement matrix

The effect of an elevated temperature on the properties of thermally induced cracking patterns (CPs) of the brick powder-Portland cement system (CM-BP) was analyzed by NDT techniques. Various contents of brick powder (BP), ranging from 0 to 20%, as a partial replacement for cement, were evaluated. CM-BP was subjected to a cyclically increasing thermal load in the range of 105–500 °C. To evaluate the cracking patterns the image segmentation procedure was implemented. Parameters such as the total crack area (TCA), crack density (CD), and average crack width (CW) were used to characterize the CPs. To evaluate the CPs morphology and its complexity, a fractal approach was implemented by introducing the CPs fractal dimension (FDB). Additionally, after each thermal loading cycle, the CM-BP was subjected to the basic mechanical tests and the ultrasonic pulse velocity test (UPV). The results obtained allowed describing the process of CM-BP thermal degradation, and the FDB was found to be the most statistically significant parameter in the CPs quantitative description. In this paper, a damage assessment index (DAI) based on FDB and UPV was proposed to evaluate the degree of CM-BP thermal degradation. DAI was compared with the relative strength loss (RSL) and DAI was found to be more sensitive to the thermally induced chemo-physical transformation of the CM-BP microstructure.

A Fractal Approach to Nonlinear Topographical Features of Healthy and Keratoconus Corneas Pre- and Post-Operation of Intracorneal Implants

Fractal dimension (FD) together with advances in imaging technologies has provided an increasing application of digital images to interpret biological phenomena. In ophthalmology, topography-based images are increasingly used in common practices of clinical settings. They provide detailed information about corneal surfaces. Few-micron alterations of the corneal geometry to the elevation and curvature cause a highly multifocal surface, change the corneal optical power up to several diopters, and therefore adversely affect the individual’s vision. Keratoconus (KCN) is a corneal disease characterized by a local alteration of the corneal anatomical and mechanical features. The formation of cone-shaped regions accompanied by thinning and weakening of the cornea are the major manifestations of KCN. The implantation of tiny arc-like polymeric sections, known as intracorneal implants, is considered to be effective in restoring the corneal curvature. This study investigated the FD nature of healthy corneas (n = 7) and compared it to the corresponding values before and after intracorneal implant surgery in KCN patients (n = 7). The generalized Hurst exponent, Higuchi, and Katz FDs were computed for topography-based parameters of corneal surfaces: front elevation (ELE-front), back elevation (ELE-back), and corneal curvature (CURV). The Katz FD showed better discriminating ability for the diseased group. It could reveal a significant difference between the healthy corneas and both pre- and post-implantation topographies (p < 0.001). Moreover, the Katz dimension varied between the topographic features of KCN patients before and after the treatment (p < 0.036). We propose to describe the curvature feature of corneal topography as a “strange attractor” with a self-similar (i.e., fractal) structure according to the Katz algorithm.

Review and prospects of metamaterials used to control elastic waves and vibrations

Acoustic metamaterials, artificial composite structures with exotic material properties used to control elastic waves, have become a new frontier in physics, materials science, engineering and chemistry. In this paper, the research progress and development prospect of acoustic metamaterials are reviewed. Related studies on passive acoustic metamaterials and active acoustic metamaterials are introduced and compared. Additionally, we discuss approaches to material structure design, including topology optimization approaches, as well as bio-inspired and fractal geometry-based approaches to structure design. Finally, we summarize and look forward to the prospects and directions of acoustic metamaterial research. With the development of additive manufacturing technology, the research potential of acoustic metamaterials is huge.

Surface and Structure of Phosphatidylcholine Membranes Reconstructed with CoFe2O4 Nanoparticles.

Structural changes in phosphatidylcholine lipid membranes caused by the introduction of insoluble CoFe2O4 nanoparticles (NPs) are analyzed. Changes in nuclear magnetic resonance spectrum, infrared spectrum, and ionic conductivity of membranes are observed with the addition of NPs. The presence of NPs in membranes is proved by atomic force and magnetic force microscopy. Structural changes in the membranes in the vicinity of the lipid C-O bonds caused by NPs are observed by Scanning near-field optical microscopy. Analysis of nuclear magnetic resonance (NMR) spectra allowed us to identify the affected atomic groups in the membrane surface layers. Conductivity measurements of the bilayer membranes were performed in DC as well as in time-resolved modes. Hydrophobic NPs stimulate surface distortion and creation of pores, which depending on NP concentration leads to an increase in the ionic conductivity of membranes. Concentration dependence demonstrating percolation threshold was analyzed in the frame of the fractal theory approach.

Fractal Analysis for Fatigue Crack Growth Rate Response of Engineering Structures with Complex Geometry

A growing fatigue crack in metallic materials and structures exhibits multifractal features that inherit signatures of the crack growth rate behavior of the material. This study exploits the recently established multifractal fatigue crack growth model to quantify the characteristic fatigue crack growth rate response of the AISI 410 martensitic stainless steel using an L-shaped bell crank structure. The objective is to demonstrate that the fatigue crack growth rate response of the material could be established by quantifying the fractality of the growing crack. The fractal approach avoids the need of the crack geometry factor when calculating the crack tip driving force. The fractal analysis of the crack image employs the box-counting algorithm to determine the fractal dimension along the edge of the crack length. The analysis is confined to the power law crack growth rate stage (Paris crack growth regime). Results show that the fatigue crack growth path in the bell crank structure is dictated by the Mode I (opening) component of the crack loading. The distribution of fractal-based fatigue crack growth rate data is within the 99% confidence limit of the median crack growth response by the Paris equation. Thus, the model could be employed for prediction of the fatigue crack growth response of engineering structures where the crack geometry factor is not readily available.

Dynamic studies of copper adsorption on mesoporous alginate beads using an integrated approach of fractal‐like kinetic reaction and diffusion modeling

AbstractThe kinetics of Cu(II) adsorption on calcium alginate (AL) beads was elucidated using reaction‐based and diffusion models. In the present work, the fractal psuedo first‐order kinetic model has been used in combination with the Vermeulen diffusion model. The Cu(II) adsorption parameters were optimized by batch adsorption studies. The AL beads were characterized using field emission scanning electron microscope (FESEM), energy‐dispersive X‐ray (EDAX), and Brunauer‐Emmett‐Teller (BET) surface area analysis. BET studies indicated beads to be mesoporous. The combined studies using fractal reaction and diffusion models helped to trace the path of mass transport of the Cu(II) ions from the bulk of the solution onto the surface of the bead and into the porous mass of the beads. The use of the fractal approach enabled us to determine the heterogeneity of the bead surface and hence gave a better insight into the understanding of the sorption phenomenon. The adsorption rate constants calculated using the equations for the “Fractal” and “Classical” adsorption kinetic models were drastically different, thus pointing toward the explicit need to model the data using diffusion kinetic models attributing to surface heterogeneity. Such combined studies may result in a better understanding of the adsorption of Cu(II) on the AL beads.

Some new aspects of fractal superconductivity

The purpose of this paper is to develop, in fractal dimension, the semi-quantitative extension of the Ginzburg-Landau theory of superconductivity. Our theoretical analysis is based on the notion of the non-standard Lagrangian approach which, based on recent studies, offer new insights in the theory of nonlinear differential equations and quantum mechanics. Besides, the fractal dimensional approach was based on the concept of the product-like fractal measure introduced by Li and Ostoja-Starzewski in order to study anisotropic media. The extended Ginzburg-Landau equation in fractal dimension was derived and several features have been revealed in particular for fractal dimension α=1/2. In particular, the theory is characterized by discrete effective mass and effective coherent length which is independent of the temperature and it don't diverge at critical temperature. This outcome is motivating since it allows manufacturing temperature-independent superconductors with a high current density without the need of refrigeration or explicit substances. The effective discrete coherent length was found to decrease with the energy levels and such a shrinking is detected in several superconductors including YBCO, Zr-doped (GdY) BCO tapes and SQUIDs. We have also studied the Abrikosov's anisotropic vortex lattice. It was found that the extended fractal theory is deformed due to the effective potential of theory which holds an inverse square part and the associated Schrödinger equation is analogous to the biconfluent Heun's differential equation. We have obtained the solution in terms of the Hermite polynomials and the energy equation was found to be altered considerably. However, it was observed that a transition between a type-II and a type-I superconductors takes place if only the non-standard parameter is ξ0≈0.99orξ0≈0.13where both numerical values result on a Ginzburg-Landau parameter κ≈1/2 although the theory is characterized by a deformed periodic lattice structure due to its anisotropic feature and non-standard Lagrangian structure. Further points are discussed and analyzed.

Interval type-3 fuzzy fractal approach in sound speaker quality control evaluation

This article is presenting a proposal for the mathematical definition of interval type-3 fuzzy sets, as a basis for later constructing all the theory needed for building interval type-3 fuzzy systems for different areas of application. It is well known that the most used form of fuzzy logic is the one called type-1, which later evolved into interval and general type-2. Recently, we have seen that type-2 fuzzy has outperformed type-1 in many real problems when there is high level of uncertainty, dynamic situations or nonlinearity. For these reasons it is worthwhile to explore the new area of type-3 fuzzy logic, and in this paper, we are putting forward some mathematical definitions of type-3 fuzzy sets and systems, as a starting point for later proposing their applications. In addition, we illustrate the potential of interval type-3 in a decision-making application in quality control. The aim of this study is to show the advantages of type-3 in a quality control problem that can be viewed as a decision-making process under uncertainty that estimates product quality Speaker manufacturing requires strict quality control and a combination of type-3 with fractal theory is able to outperform previous works with type-2 and type-1 in quality control. Simulation results of the proposed approach are compared with the results provided by human experts in sound speaker quality control to validate the effectiveness of the type-3 fuzzy fractal method in handling the uncertainty of this problem.

Comparison between Fractal and Statistical Approaches to Model Size Effects in VHCF

Size effects concern the anomalous scaling of relevant mechanical properties of materials and structures over a sufficiently wide dimensional range. In the last few years, thanks to technological advances, such effects have been experimentally detected also in the very high cycle fatigue (VHCF) tests. Research groups at Politecnico di Torino are very active in this field, observing size effects on fatigue strength, fatigue life and fatigue limit up to the VHCF regime for different metal alloys. In addition, different theoretical models have been put forward to explain these effects. In the present paper, two of them are introduced, respectively based on fractal geometry and statistical concepts. Furthermore, a comparison between the models and experimental results is provided. Both models are able to predict the decrement in the fatigue life and in the conventional fatigue limit.

Self-similar behavior of the neutron fracture functions

In this paper, we have employed Fractal formalism to calculate the Fracture Functions of the Leading neutron produced in [Formula: see text] collisions. The Fractal concept describes the self-similar behavior of the proton structure at Leading neutron production of semi-inclusive Deep Inelastic Scattering at the low values of the fractional momentum variable [Formula: see text]. The Fracture Functions (FFs) parameterized the non-perturbative part of the fragmentation process at the initial scale of Q[Formula: see text]. In this analysis, we benefit from the Leading neutron (Ln) experimental data published by H1 Collaboration and in order to estimate the uncertainty of neutron FFs and corresponding observables, we used the Hessian method. As a consequence of this Next-to-Leading order QCD analysis, we achieve a nice agreement between the prediction of our model for neutron FFs and experimental data. It seems that the Fractal approach based on the self-similar behavior of these conditional parton distribution functions can nicely describe the experimental data at low [Formula: see text].

Gauge fields renormalization groups and thermofractals

The perturbative approach to QCD has shown to be limited, and the difficulties to obtain accurate calculations in the low-energy region seems to be insurmountable. A recent approach uses the fractal structures of Yang-Mills Field Theory to circumvent those difficulties, allowing for the determination of an analytic expression for the running coupling. The results obtained are in agreement with several experimental findings, and explain many of the observed phenomena at high-energy collisions. In this work, we address some of the conceptual aspects of the fractal approach, which are expressed in terms of the renormalization group equation and the self-energy corrections to the parton mass. We associate these concepts with the origins of the fractal structure in the quantum field theory.

Solving constrained portfolio optimization model using stochastic fractal search approach

PurposeOptimum utilization of investments has always been considered one of the most crucial aspects of capital markets. Investment into various securities is the subject of portfolio optimization intent to maximize return at minimum risk. In this series, a population-based evolutionary approach, stochastic fractal search (SFS), is derived from the natural growth phenomenon. This study aims to develop portfolio selection model using SFS approach to construct an efficient portfolio by optimizing the Sharpe ratio with risk budgeting constraints.Design/methodology/approachThis paper proposes a constrained portfolio optimization model using the SFS approach with risk-budgeting constraints. SFS is an evolutionary method inspired by the natural growth process which has been modeled using the fractal theory. Experimental analysis has been conducted to determine the effectiveness of the proposed model by making comparisons with state-of-the-art from domain such as genetic algorithm, particle swarm optimization, simulated annealing and differential evolution. The real datasets of the Indian stock exchanges and datasets of global stock exchanges such as Nikkei 225, DAX 100, FTSE 100, Hang Seng31 and S&P 100 have been taken in the study.FindingsThe study confirms the better performance of the SFS model among its peers. Also, statistical analysis has been done using SPSS 20 to confirm the hypothesis developed in the experimental analysis.Originality/valueIn the recent past, researchers have already proposed a significant number of models to solve portfolio selection problems using the meta-heuristic approach. However, this is the first attempt to apply the SFS optimization approach to the problem.

A Dual Fractal Approach to Thermal–Hydrological–Mechanical Interactions of Unconventional Reservoir

A Dual Fractal Approach to Thermal–Hydrological–Mechanical Interactions of Unconventional Reservoir

On the test of novel constitutive relation of capacitor for electrical circuit analysis: a fractal calculus-based approach

PurposeThe purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor.Design/methodology/approachFor ceteris paribus, the constitutive relations of the resistor and inductor have been reformulated by following the novel constitutive relation of capacitor. The responses of RL, RC, LC and RLC circuits defined on the fractal set described by these definitions have been derived by means of the fractal calculus and fractal Laplace transformation. A comparative Hamiltonian formalism-based analysis has been performed where the circuits described by the conventional and the formerly proposed revisited constitutive relations have also been considered.FindingsThis study has found that the novel constitutive relations give unreasonable results unlike the conventional ones. Like such previous revisited constitutive relations, an odd Hamiltonian has been obtained. On the other hand, the conventional constitutive relations give a reasonable Hamiltonian.Originality/valueTo the best of the author’s knowledge, for the first time, the analysis of fractal set defined electrical circuits by means of unconventional constitutive relations has been performed where the deficiency of the tested capacitive constitutive relation has been pointed out.