In the present work, we examine and analyze an $hp$-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady fluid system on computational meshes consisting of polytopic elements on the boundary. This approach is based on the discontinuous Galerkin method, enriched by arbitrarily shaped elements techniques as has been introduced in [13]. In this framework, and employing extensions of trace, Markov-type, and $H^1/L^2$-type inverse estimates to arbitrary element shapes, we examine a stationary Stokes fluid system enabling the proof of the inf/sup condition and the $hp$- a priori error estimates, while we investigate the optimal convergence rates numerically. This approach recovers and integrates the flexibility and superiority of the discontinuous Galerkin methods for fluids whenever geometrical deformations are taking place by degenerating the edges, facets, of the polytopic elements only on the boundary, combined with the efficiency of the $hp$-version techniques based on arbitrarily shaped elements without requiring any mapping from a given reference frame.
{"title":"$hp$-Version Analysis for Arbitrarily Shaped Elements on the Boundary Discontinuous Galerkin Method for Stokes Systems","authors":"Efthymios N. Karatzas","doi":"10.4208/ijnam2024-1021","DOIUrl":"https://doi.org/10.4208/ijnam2024-1021","url":null,"abstract":"In the present work, we examine and analyze an $hp$-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady fluid system on\u0000computational meshes consisting of polytopic elements on the boundary. This approach is based\u0000on the discontinuous Galerkin method, enriched by arbitrarily shaped elements techniques as has\u0000been introduced in [13]. In this framework, and employing extensions of trace, Markov-type, and $H^1/L^2$-type inverse estimates to arbitrary element shapes, we examine a stationary Stokes fluid\u0000system enabling the proof of the inf/sup condition and the $hp$- a priori error estimates, while we\u0000investigate the optimal convergence rates numerically. This approach recovers and integrates the\u0000flexibility and superiority of the discontinuous Galerkin methods for fluids whenever geometrical\u0000deformations are taking place by degenerating the edges, facets, of the polytopic elements only\u0000on the boundary, combined with the efficiency of the $hp$-version techniques based on arbitrarily\u0000shaped elements without requiring any mapping from a given reference frame.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"71 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose a new method for the Darcy-Stokes equations based on the stabilizer-free weak Galerkin finite element method. In the proposed method, we remove the stabilizer term by increasing the degree of polynomial approximation space of the weak gradient operator. Compared with the classical weak Galerkin finite element method, it will not increase the size of global stiffness matrix. We show that the new algorithm not only has a simpler formula, but also reduces the computational complexity. Optimal order error estimates are established for the corresponding numerical approximation in various norms. Finally, we numerically illustrate the accuracy and convergence of this method.
{"title":"A Stabilizer-Free Weak Galerkin Finite Element Method for the Darcy-Stokes Equations","authors":"Kai He,Junjie Chen,Li Zhang, Maohua Ran","doi":"10.4208/ijnam2024-1018","DOIUrl":"https://doi.org/10.4208/ijnam2024-1018","url":null,"abstract":"In this paper, we propose a new method for the Darcy-Stokes equations based on\u0000the stabilizer-free weak Galerkin finite element method. In the proposed method, we remove the\u0000stabilizer term by increasing the degree of polynomial approximation space of the weak gradient\u0000operator. Compared with the classical weak Galerkin finite element method, it will not increase\u0000the size of global stiffness matrix. We show that the new algorithm not only has a simpler formula,\u0000but also reduces the computational complexity. Optimal order error estimates are established for\u0000the corresponding numerical approximation in various norms. Finally, we numerically illustrate\u0000the accuracy and convergence of this method.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"44 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove the existence of weak solution and the uniqueness of strong solution to a Voigt-regularization of the three-dimensional thermally coupled inviscid, resistive MHD equations. We also propose a fully discrete scheme for the considered problem, which is proven to be stable and convergent. All computational results support the theoretical analysis and demonstrate the effectiveness of the presented scheme.
{"title":"A Voigt-Regularization of the Thermally Coupled Inviscid, Resistive Magnetohydrodynamic","authors":"Xingwei Yang,Pengzhan Huang, Yinnian He","doi":"10.4208/ijnam2024-1019","DOIUrl":"https://doi.org/10.4208/ijnam2024-1019","url":null,"abstract":"In this paper, we prove the existence of weak solution and the uniqueness of strong\u0000solution to a Voigt-regularization of the three-dimensional thermally coupled inviscid, resistive\u0000MHD equations. We also propose a fully discrete scheme for the considered problem, which is\u0000proven to be stable and convergent. All computational results support the theoretical analysis\u0000and demonstrate the effectiveness of the presented scheme.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"31 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article develops and analyses a mixed virtual element scheme for the spatial discretization of linear parabolic integro-differential equations (PIDEs) combined with backward Euler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projection significantly helps in managing the integral terms, yielding optimal convergence of order $O(h^{k+1})$ for the two unknowns $p(x, t)$ and $sigma(x, t).$ In addition, a step-by-step analysis is proposed for the super convergence of the discrete solution of order $O(h^{k+2}).$ The fully discrete case has also been analyzed and discussed to achieve $O(tau)$ in time. Several computational experiments are discussed to validate the proposed schemes computational efficiency and support the theoretical conclusions.
{"title":"Mixed Virtual Element Method for Linear Parabolic Integro-Differential Equations","authors":"Meghana Suthar, Sangita Yadav","doi":"10.4208/ijnam2024-1020","DOIUrl":"https://doi.org/10.4208/ijnam2024-1020","url":null,"abstract":"This article develops and analyses a mixed virtual element scheme for the spatial\u0000discretization of linear parabolic integro-differential equations (PIDEs) combined with backward\u0000Euler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projection\u0000significantly helps in managing the integral terms, yielding optimal convergence of order $O(h^{k+1})$ for the two unknowns $p(x, t)$ and $sigma(x, t).$ In addition, a step-by-step analysis is proposed for the\u0000super convergence of the discrete solution of order $O(h^{k+2}).$ The fully discrete case has also been\u0000analyzed and discussed to achieve $O(tau)$ in time. Several computational experiments are discussed\u0000to validate the proposed schemes computational efficiency and support the theoretical conclusions.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"11 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we rigorously analyze an HIV-1 infection model with CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model with proving the positivity and boundedness of the solution. For this model, the basic reproduction number $R_0$ and the immune reproduction number $R_1$ are identified. Moreover, we have shown that the model has three equilibria, namely the infection-free equilibrium $E_0,$ the infectious equilibrium without immune response $E_1$ and the infectious equilibrium with immune response $E_2.$ By applying fluctuation lemma and Lyapunov functionals, we have demonstrated that the global stability of $E_0$ and $E_1$ are only related to $R_0$ and $R_1.$ The local stability of the third equilibrium is obtained under four situations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, some numerical simulations are carried out for illustrating the theoretical results.
{"title":"Dynamics Analysis of HIV-1 Infection Model with CTL Immune Response and Delays","authors":"Ting Guo, Fei Zhao","doi":"10.4208/ijnam2024-1022","DOIUrl":"https://doi.org/10.4208/ijnam2024-1022","url":null,"abstract":"In this paper, we rigorously analyze an HIV-1 infection model with CTL immune\u0000response and three time delays which represent the latent period, virus production period and\u0000immune response delay, respectively. We begin this model with proving the positivity and boundedness of the solution. For this model, the basic reproduction number $R_0$ and the immune\u0000reproduction number $R_1$ are identified. Moreover, we have shown that the model has three equilibria, namely the infection-free equilibrium $E_0,$ the infectious equilibrium without immune\u0000response $E_1$ and the infectious equilibrium with immune response $E_2.$ By applying fluctuation\u0000lemma and Lyapunov functionals, we have demonstrated that the global stability of $E_0$ and $E_1$ are only related to $R_0$ and $R_1.$ The local stability of the third equilibrium is obtained under four\u0000situations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, some\u0000numerical simulations are carried out for illustrating the theoretical results.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"44 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce a weak Galerkin finite element method for the dual-porosity-Stokes model. The dual-porosity-Stokes model couples the dual-porosity equations with the Stokes equations through four interface conditions. In this method, we define several weak Galerkin finite element spaces and weak differential operators. We provide the weak Galerkin scheme for the model, and establish the well-posedness of the numerical scheme. The optimal convergence orders of errors in the energy norm are derived. Finally, we verify the effectiveness of the numerical method with different weak Galerkin elements on different meshes.
{"title":"The Weak Galerkin Finite Element Method for the Dual-Porosity-Stokes Model","authors":"Lin Yang,Wei Mu,Hui Peng, Xiuli Wang","doi":"10.4208/ijnam2024-1023","DOIUrl":"https://doi.org/10.4208/ijnam2024-1023","url":null,"abstract":"In this paper, we introduce a weak Galerkin finite element method for the dual-porosity-Stokes model. The dual-porosity-Stokes model couples the dual-porosity equations with\u0000the Stokes equations through four interface conditions. In this method, we define several weak\u0000Galerkin finite element spaces and weak differential operators. We provide the weak Galerkin\u0000scheme for the model, and establish the well-posedness of the numerical scheme. The optimal\u0000convergence orders of errors in the energy norm are derived. Finally, we verify the effectiveness of\u0000the numerical method with different weak Galerkin elements on different meshes.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"140 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a direct method for efficiently solving three-dimensional elliptic interface problems featuring piecewise constant coefficients with a finite jump across the interface. A key advantage of our approach lies in its avoidance of augmented variables, distinguishing it from traditional methods. The computational framework relies on a finite difference scheme implemented on a uniform Cartesian grid system. By utilizing a seven-point Laplacian for grid points away from the interface, our method only requires coefficient modifications for grid points located near or on the interface. Numerical experiments validate our method’s effectiveness. Generally, it achieves second-order accuracy for both the solution and its gradient, measured in the maximum norm, particularly effective in scenarios with moderate coefficient jumps. Extending and building upon the recent work of [1] on 1D and 2D elliptic interfaces, our approach successfully introduces a simpler method for extension into three dimensions. Notably, our proposed method not only offers efficiency and accuracy but also enhances the simplicity of implementation, making it accessible to non-experts in the field.
{"title":"A Direct Method for Solving Three-Dimensional Elliptic Interface Problems","authors":"Kumudu Gamage,Yan Peng, Zhilin Li","doi":"10.4208/ijnam2024-1014","DOIUrl":"https://doi.org/10.4208/ijnam2024-1014","url":null,"abstract":"This paper presents a direct method for efficiently solving three-dimensional elliptic\u0000interface problems featuring piecewise constant coefficients with a finite jump across the interface.\u0000A key advantage of our approach lies in its avoidance of augmented variables, distinguishing\u0000it from traditional methods. The computational framework relies on a finite difference scheme\u0000implemented on a uniform Cartesian grid system. By utilizing a seven-point Laplacian for grid\u0000points away from the interface, our method only requires coefficient modifications for grid points\u0000located near or on the interface. Numerical experiments validate our method’s effectiveness.\u0000Generally, it achieves second-order accuracy for both the solution and its gradient, measured in\u0000the maximum norm, particularly effective in scenarios with moderate coefficient jumps. Extending\u0000and building upon the recent work of [1] on 1D and 2D elliptic interfaces, our approach successfully\u0000introduces a simpler method for extension into three dimensions. Notably, our proposed method\u0000not only offers efficiency and accuracy but also enhances the simplicity of implementation, making\u0000it accessible to non-experts in the field.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"19 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper develops novel fast numerical solvers for subdiffusion problems with spatial interfaces. These problems are modeled by partial differential equations that contain both fractional order and conventional first order time derivatives. The former is non-local and approximated by L1 and L2 discretizations along with fast evaluation algorithms based on approximation by sums of exponentials. This results in an effective treatment of the “long-tail” kernel of subdiffusion. The latter is local and hence conventional implicit Euler or Crank-Nicolson discretizations can be used. Finite volumes are utilized for spatial discretization based on consideration of local mass conservation. Interface conditions for mass and fractional fluxes are incorporated into these fast solvers. Computational complexity and implementation procedures are briefly discussed. Numerical experiments demonstrate accuracy and efficiency of these new fast solvers.
{"title":"Fast Numerical Solvers for Subdiffusion Problems with Spatial Interfaces","authors":"Boyang Yu,Yonghai Li, Jiangguo Liu","doi":"10.4208/ijnam2024-1017","DOIUrl":"https://doi.org/10.4208/ijnam2024-1017","url":null,"abstract":"This paper develops novel fast numerical solvers for subdiffusion problems with spatial interfaces. These problems are modeled by partial differential equations that contain both\u0000fractional order and conventional first order time derivatives. The former is non-local and approximated by L1 and L2 discretizations along with fast evaluation algorithms based on approximation\u0000by sums of exponentials. This results in an effective treatment of the “long-tail” kernel of subdiffusion. The latter is local and hence conventional implicit Euler or Crank-Nicolson discretizations\u0000can be used. Finite volumes are utilized for spatial discretization based on consideration of local\u0000mass conservation. Interface conditions for mass and fractional fluxes are incorporated into these\u0000fast solvers. Computational complexity and implementation procedures are briefly discussed.\u0000Numerical experiments demonstrate accuracy and efficiency of these new fast solvers.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"38 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jinwei Bai,Hongtao Liu,Xiaoming He,Wei Jiang, Yong Cao
This paper presents a recently developed full kinetic particle simulation code package, which is a two-dimensional highly integrated and universal framework for low-temperature plasma simulation on both Cartesian and axisymmetric coordinate systems. This code package is named CFIRM, since it is designed based on the continuous Galerkin immersed-finite-element (IFE) particle-in-cell (PIC) model with the polynomial-preserving-recovery (PPR) technique and the Monte-Carlo-collision (MCC) method. Both the traditional and implicit PIC methods were implemented in the package. Incorporating the advantages of all these methods together, the CFIRM code can adopt explicit or implicit PIC schemes to track the motion trajectory of charged particles and deal with the collisions between plasma and neutral gas. Additionally, it can conveniently handle complex interface problems on structured grids. The CFRIM code has excellent versatility in low-temperature plasma simulation and can easily extend to various particle processing modules, such as the variable weights and adaptive particle management algorithms which were incorporated into this code to reduce the memory utilization rate. The implementation for the main algorithms and the overall simulation framework of the CFIRM code package are rigorously described in details. Several simulations of the benchmark cases are carried out to validate the reliability and accuracy of the CFIRM code. Moreover, two typical low-temperature plasma engineering problems are simulated, including a hall thruster and a capacitively coupled plasma reactor, which demonstrates the applicability of this code package.
{"title":"CFIRM: An Integrated Code Package for the Low-Temperature Plasma Simulation on Structured Grids","authors":"Jinwei Bai,Hongtao Liu,Xiaoming He,Wei Jiang, Yong Cao","doi":"10.4208/ijnam2024-1015","DOIUrl":"https://doi.org/10.4208/ijnam2024-1015","url":null,"abstract":"This paper presents a recently developed full kinetic particle simulation code package, which is a two-dimensional highly integrated and universal framework for low-temperature\u0000plasma simulation on both Cartesian and axisymmetric coordinate systems. This code package\u0000is named CFIRM, since it is designed based on the continuous Galerkin immersed-finite-element\u0000(IFE) particle-in-cell (PIC) model with the polynomial-preserving-recovery (PPR) technique and\u0000the Monte-Carlo-collision (MCC) method. Both the traditional and implicit PIC methods were\u0000implemented in the package. Incorporating the advantages of all these methods together, the\u0000CFIRM code can adopt explicit or implicit PIC schemes to track the motion trajectory of charged\u0000particles and deal with the collisions between plasma and neutral gas. Additionally, it can conveniently handle complex interface problems on structured grids. The CFRIM code has excellent\u0000versatility in low-temperature plasma simulation and can easily extend to various particle processing modules, such as the variable weights and adaptive particle management algorithms which\u0000were incorporated into this code to reduce the memory utilization rate. The implementation for\u0000the main algorithms and the overall simulation framework of the CFIRM code package are rigorously described in details. Several simulations of the benchmark cases are carried out to validate\u0000the reliability and accuracy of the CFIRM code. Moreover, two typical low-temperature plasma\u0000engineering problems are simulated, including a hall thruster and a capacitively coupled plasma\u0000reactor, which demonstrates the applicability of this code package.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"49 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, a stationary 2d Navier-Stokes problem with nonlinear feedback forces field is considered in the stream-function formulation. We use the Continuous/Discontinuous Finite Element Method (CD-FEM), with interior penalty terms, to numerically solve the associated boundary-value problem. For the associated continuous and discrete problems, we prove the existence of weak solutions and establish the conditions for their uniqueness. Consistency, stability and convergence of the method are also shown analytically. To validate the numerical model regarding its applicability and robustness, several test cases are carried out.
{"title":"Continuous/Discontinuous Finite Element Approximation of a 2d Navier-Stokes Problem Arising in Fluid Confinement","authors":"Hermenegildo Borges De Oliveira, Nuno David Lopes","doi":"10.4208/ijnam2024-1013","DOIUrl":"https://doi.org/10.4208/ijnam2024-1013","url":null,"abstract":"In this work, a stationary 2d Navier-Stokes problem with nonlinear feedback forces\u0000field is considered in the stream-function formulation. We use the Continuous/Discontinuous Finite Element Method (CD-FEM), with interior penalty terms, to numerically solve the associated\u0000boundary-value problem. For the associated continuous and discrete problems, we prove the existence of weak solutions and establish the conditions for their uniqueness. Consistency, stability\u0000and convergence of the method are also shown analytically. To validate the numerical model\u0000regarding its applicability and robustness, several test cases are carried out.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"32 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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