Pub Date : 2023-11-04DOI: 10.3390/fractalfract7110804
Naveen S., Parthiban V., Mohamed I. Abbas
This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω∈(1,2) for the RLC model. Based on Schaefer’s fixed point theorem and Banach’s contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples.
{"title":"Qualitative Analysis of RLC Circuit Described by Hilfer Derivative with Numerical Treatment Using the Lagrange Polynomial Method","authors":"Naveen S., Parthiban V., Mohamed I. Abbas","doi":"10.3390/fractalfract7110804","DOIUrl":"https://doi.org/10.3390/fractalfract7110804","url":null,"abstract":"This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω∈(1,2) for the RLC model. Based on Schaefer’s fixed point theorem and Banach’s contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"39 15","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135773648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-04DOI: 10.3390/fractalfract7110802
Xuefei Wang
In this paper, we conduct research on the fractal characteristics of the superposition of fractal surfaces from the view of fractal dimension. We give the upper bound of the lower and upper box dimensions of the graph of the sum of two bivariate continuous functions and calculate the exact values of them under some particular conditions. Further, it has been proven that the superposition of two continuous surfaces cannot keep the fractal dimensions invariable unless both of them are two-dimensional. A concrete example of a numerical experiment has been provided to verify our theoretical results. This study can be applied to the fractal analysis of metal fracture surfaces or computer image surfaces.
{"title":"An Investigation on Fractal Characteristics of the Superposition of Fractal Surfaces","authors":"Xuefei Wang","doi":"10.3390/fractalfract7110802","DOIUrl":"https://doi.org/10.3390/fractalfract7110802","url":null,"abstract":"In this paper, we conduct research on the fractal characteristics of the superposition of fractal surfaces from the view of fractal dimension. We give the upper bound of the lower and upper box dimensions of the graph of the sum of two bivariate continuous functions and calculate the exact values of them under some particular conditions. Further, it has been proven that the superposition of two continuous surfaces cannot keep the fractal dimensions invariable unless both of them are two-dimensional. A concrete example of a numerical experiment has been provided to verify our theoretical results. This study can be applied to the fractal analysis of metal fracture surfaces or computer image surfaces.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"39 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135773652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-04DOI: 10.3390/fractalfract7110803
Qian Zhang, Yanhui Dong, Shaoqing Tong
Pore structure features govern the capacity of gas storage and migration in shales and are highly dependent on the types of pores, i.e., interparticle (InterP) pores, intraparticle (IntraP) pores and organic matter (OM)-hosted pores. However, fractal features in terms of pore types and their respective contributions to permeability have been rarely addressed. On the basis of high-resolution imaging, fractal dimensions (Ds) have been determined from both pore size distributions and digital rock to quantify the heterogeneity in pore morphology and spatial textures. Overall, OM-hosted pores are smaller in size and more abundant in quantity, corresponding to a relatively high D, while IntraP pores are mainly isolated and scarce, translating into lower D values. Additionally, crack-like InterP pores with a moderate level of porosity and the D can play a pivotal role in shale seepage potential. A comparison of the estimated permeability among different pore types highlights that the contribution of interconnected OM pores to the overall permeability remains constrained unless they can link neighboring pore clusters, as commonly observed in organo-clay composites. Furthermore, the pore morphology and fractal features of shale rocks can exhibit noteworthy variations subjected to sedimentology, mineralogy, diagenesis and OM maturation.
{"title":"Pore-Type-Dependent Fractal Features of Shales and Implications on Permeability","authors":"Qian Zhang, Yanhui Dong, Shaoqing Tong","doi":"10.3390/fractalfract7110803","DOIUrl":"https://doi.org/10.3390/fractalfract7110803","url":null,"abstract":"Pore structure features govern the capacity of gas storage and migration in shales and are highly dependent on the types of pores, i.e., interparticle (InterP) pores, intraparticle (IntraP) pores and organic matter (OM)-hosted pores. However, fractal features in terms of pore types and their respective contributions to permeability have been rarely addressed. On the basis of high-resolution imaging, fractal dimensions (Ds) have been determined from both pore size distributions and digital rock to quantify the heterogeneity in pore morphology and spatial textures. Overall, OM-hosted pores are smaller in size and more abundant in quantity, corresponding to a relatively high D, while IntraP pores are mainly isolated and scarce, translating into lower D values. Additionally, crack-like InterP pores with a moderate level of porosity and the D can play a pivotal role in shale seepage potential. A comparison of the estimated permeability among different pore types highlights that the contribution of interconnected OM pores to the overall permeability remains constrained unless they can link neighboring pore clusters, as commonly observed in organo-clay composites. Furthermore, the pore morphology and fractal features of shale rocks can exhibit noteworthy variations subjected to sedimentology, mineralogy, diagenesis and OM maturation.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"39 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135773647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-02DOI: 10.3390/fractalfract7110799
Antonela Toma, Octavian Postavaru
Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems. We illustrate the theory with the aid of the damped oscillatory system.
{"title":"Fractional Complex Euler–Lagrange Equation: Nonconservative Systems","authors":"Antonela Toma, Octavian Postavaru","doi":"10.3390/fractalfract7110799","DOIUrl":"https://doi.org/10.3390/fractalfract7110799","url":null,"abstract":"Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems. We illustrate the theory with the aid of the damped oscillatory system.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"18 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135935122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present investigation aims to establish the existence and uniqueness of solutions for a system containing sequential fractional differential equations. Furthermore, boundary conditions that include the Riemann–Liouville fractional integral are taken into consideration. The existence of unknown functions, fractional derivatives, and fractional integrals at lower orders are necessary for the nonlinearity to exist. In order to provide proofs for the results presented in this study, the Leray–Schauder alternative and the Banach fixed-point theorem are utilised. Finally, examples are used to support the main results.
{"title":"Existence of Solutions for Coupled System of Sequential Liouville–Caputo-Type Fractional Integrodifferential Equations","authors":"Manigandan Murugesan, Subramanian Muthaiah, Rajarathinam Vadivel, Bundit Unyong","doi":"10.3390/fractalfract7110800","DOIUrl":"https://doi.org/10.3390/fractalfract7110800","url":null,"abstract":"The present investigation aims to establish the existence and uniqueness of solutions for a system containing sequential fractional differential equations. Furthermore, boundary conditions that include the Riemann–Liouville fractional integral are taken into consideration. The existence of unknown functions, fractional derivatives, and fractional integrals at lower orders are necessary for the nonlinearity to exist. In order to provide proofs for the results presented in this study, the Leray–Schauder alternative and the Banach fixed-point theorem are utilised. Finally, examples are used to support the main results.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"4 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135973606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-02DOI: 10.3390/fractalfract7110801
Elena N. Akimova, Murat A. Sultanov, Vladimir E. Misilov, Yerkebulan Nurlanuly
This paper is devoted to the development of a parallel algorithm for solving the inverse problem of identifying the space-dependent source term in the two-dimensional fractional diffusion equation. For solving the inverse problem, the regularized iterative conjugate gradient method is used. At each iteration of the method, we need to solve the auxilliary direct initial-boundary value problem. By using the finite difference scheme, this problem is reduced to solving a large system of a linear algebraic equation with a block-tridiagonal matrix at each time step. Solving the system takes almost the entire computation time. To solve this system, we construct and implement the direct parallel matrix sweep algorithm. We establish stability and correctness for this algorithm. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to study the performance of parallel implementations.
{"title":"Parallel Algorithm for Solving the Inverse Two-Dimensional Fractional Diffusion Problem of Identifying the Source Term","authors":"Elena N. Akimova, Murat A. Sultanov, Vladimir E. Misilov, Yerkebulan Nurlanuly","doi":"10.3390/fractalfract7110801","DOIUrl":"https://doi.org/10.3390/fractalfract7110801","url":null,"abstract":"This paper is devoted to the development of a parallel algorithm for solving the inverse problem of identifying the space-dependent source term in the two-dimensional fractional diffusion equation. For solving the inverse problem, the regularized iterative conjugate gradient method is used. At each iteration of the method, we need to solve the auxilliary direct initial-boundary value problem. By using the finite difference scheme, this problem is reduced to solving a large system of a linear algebraic equation with a block-tridiagonal matrix at each time step. Solving the system takes almost the entire computation time. To solve this system, we construct and implement the direct parallel matrix sweep algorithm. We establish stability and correctness for this algorithm. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to study the performance of parallel implementations.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"11 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135973023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.3390/fractalfract7110798
Tingting Guan, Lihong Zhang
In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove a new maximum principle with space-time fractional variable-order conformable derivatives and a generalized tempered fractional Laplace operator. Moreover, we discuss some results about comparison principles and properties of solutions for a family of space-time fractional variable-order conformable nonlinear differential equations with a generalized tempered fractional Laplace operator by maximum principle.
{"title":"Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator","authors":"Tingting Guan, Lihong Zhang","doi":"10.3390/fractalfract7110798","DOIUrl":"https://doi.org/10.3390/fractalfract7110798","url":null,"abstract":"In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove a new maximum principle with space-time fractional variable-order conformable derivatives and a generalized tempered fractional Laplace operator. Moreover, we discuss some results about comparison principles and properties of solutions for a family of space-time fractional variable-order conformable nonlinear differential equations with a generalized tempered fractional Laplace operator by maximum principle.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"9 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-31DOI: 10.3390/fractalfract7110796
Amnah E. Shammaky, Eslam M. Youssef, Mohamed A. Abdou, Mahmoud M. ElBorai, Wagdy G. ElSayed, Mai Taha
This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.
{"title":"A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space","authors":"Amnah E. Shammaky, Eslam M. Youssef, Mohamed A. Abdou, Mahmoud M. ElBorai, Wagdy G. ElSayed, Mai Taha","doi":"10.3390/fractalfract7110796","DOIUrl":"https://doi.org/10.3390/fractalfract7110796","url":null,"abstract":"This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135871346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-31DOI: 10.3390/fractalfract7110795
Fanqi Meng, Zhenglan Gu
With the advancement of information technology, the security of digital images has become increasingly important. To ensure the integrity of images, a novel color image-encryption algorithm based on extended DNA coding, Zig-Zag transform, and a fractional-order laser system is proposed in this paper. First, the dynamic characteristics of the fractional-order laser chaotic system (FLCS) were analyzed using a phase diagram and Lyapunov exponent spectra. The chaotic sequences generated by the system were used to design image-encryption algorithms. Second, a modified Zig-Zag confusing method was adopted to confuse the image. Finally, in the diffusion link, the DNA encoding scheme was extended to allow for a greater number of DNA encoding rules, increasing the randomness of the matrix and improving the security of the encryption scheme. The performance of the designed encryption algorithm is analyzed using key space, a histogram, information entropy, correlation coefficients, differential attack, and robustness analysis. The experimental results demonstrate that the algorithm can withstand multiple decryption methods and has strong encryption capability. The proposed novel color image-encryption scheme enables secure communication of digital images.
{"title":"A Color Image-Encryption Algorithm Using Extended DNA Coding and Zig-Zag Transform Based on a Fractional-Order Laser System","authors":"Fanqi Meng, Zhenglan Gu","doi":"10.3390/fractalfract7110795","DOIUrl":"https://doi.org/10.3390/fractalfract7110795","url":null,"abstract":"With the advancement of information technology, the security of digital images has become increasingly important. To ensure the integrity of images, a novel color image-encryption algorithm based on extended DNA coding, Zig-Zag transform, and a fractional-order laser system is proposed in this paper. First, the dynamic characteristics of the fractional-order laser chaotic system (FLCS) were analyzed using a phase diagram and Lyapunov exponent spectra. The chaotic sequences generated by the system were used to design image-encryption algorithms. Second, a modified Zig-Zag confusing method was adopted to confuse the image. Finally, in the diffusion link, the DNA encoding scheme was extended to allow for a greater number of DNA encoding rules, increasing the randomness of the matrix and improving the security of the encryption scheme. The performance of the designed encryption algorithm is analyzed using key space, a histogram, information entropy, correlation coefficients, differential attack, and robustness analysis. The experimental results demonstrate that the algorithm can withstand multiple decryption methods and has strong encryption capability. The proposed novel color image-encryption scheme enables secure communication of digital images.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"67 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135872139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-31DOI: 10.3390/fractalfract7110797
Sofia Ramzan, Muhammad Uzair Awan, Silvestru Sever Dragomir, Bandar Bin-Mohsin, Muhammad Aslam Noor
This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the s-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications of our main outcomes to special means of real numbers have been explored. Moreover, we have derived a new generic numerical scheme for solving non-linear equations, demonstrating an application of our main results in numerical analysis.
{"title":"Analysis and Applications of Some New Fractional Integral Inequalities","authors":"Sofia Ramzan, Muhammad Uzair Awan, Silvestru Sever Dragomir, Bandar Bin-Mohsin, Muhammad Aslam Noor","doi":"10.3390/fractalfract7110797","DOIUrl":"https://doi.org/10.3390/fractalfract7110797","url":null,"abstract":"This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the s-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications of our main outcomes to special means of real numbers have been explored. Moreover, we have derived a new generic numerical scheme for solving non-linear equations, demonstrating an application of our main results in numerical analysis.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135872475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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