Pub Date : 2026-01-01Epub Date: 2025-08-29DOI: 10.1016/j.apnum.2025.08.007
Min Zhang , Zimo Zhu , Qijia Zhai , Xiaoping Xie
This paper develops a hybridizable discontinuous Galerkin (HDG) finite element method of arbitrary order for the steady thermally coupled incompressible Magnetohydrodynamics (MHD) flow. The HDG scheme uses piecewise polynomials of degrees and respectively for the approximations of the velocity, the magnetic field, the pressure, the magnetic pseudo-pressure, and the temperature in the interior of elements, and uses piecewise polynomials of degree for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. Existence and uniqueness results for the discrete scheme are given and - optimal error estimates are derived for all the variables. Numerical experiments are provided to verify the obtained theoretical results.
{"title":"Robust globally divergence-free HDG finite element method for steady thermally coupled incompressible MHD flow","authors":"Min Zhang , Zimo Zhu , Qijia Zhai , Xiaoping Xie","doi":"10.1016/j.apnum.2025.08.007","DOIUrl":"10.1016/j.apnum.2025.08.007","url":null,"abstract":"<div><div>This paper develops a hybridizable discontinuous Galerkin (HDG) finite element method of arbitrary order for the steady thermally coupled incompressible Magnetohydrodynamics (MHD) flow. The HDG scheme uses piecewise polynomials of degrees <span><math><mrow><mi>k</mi><mo>(</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>)</mo><mo>,</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mi>k</mi></math></span> respectively for the approximations of the velocity, the magnetic field, the pressure, the magnetic pseudo-pressure, and the temperature in the interior of elements, and uses piecewise polynomials of degree <span><math><mi>k</mi></math></span> for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. Existence and uniqueness results for the discrete scheme are given and <span><math><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mi>k</mi></msup><mo>)</mo></mrow></math></span>- optimal error estimates are derived for all the variables. Numerical experiments are provided to verify the obtained theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 19-40"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144989796","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 : 2026-01-01Epub Date: 2025-09-10DOI: 10.1016/j.apnum.2025.09.003
Zhongdi Cen, Jian Huang, Aimin Xu
In this paper a singularly perturbed Riccati equation is considered. A hybrid difference method is used to approximate the singularly perturbed Riccati equation. A posteriori error analysis for the discretization method on an arbitrary mesh is given. The stability result of the differential operator used in a posteriori error analysis is obtained based on the properties of the exact solution and the numerical solution. A solution-adaptive algorithm based on a posteriori error estimation is designed to generate a posteriori mesh and the approximation solution. Numerical experiments verify that the method is second-order uniformly convergent with respect to small parameter and improves previous results.
{"title":"A numerical method on a posteriori mesh for a singularly perturbed Riccati equation","authors":"Zhongdi Cen, Jian Huang, Aimin Xu","doi":"10.1016/j.apnum.2025.09.003","DOIUrl":"10.1016/j.apnum.2025.09.003","url":null,"abstract":"<div><div>In this paper a singularly perturbed Riccati equation is considered. A hybrid difference method is used to approximate the singularly perturbed Riccati equation. A posteriori error analysis for the discretization method on an arbitrary mesh is given. The stability result of the differential operator used in a posteriori error analysis is obtained based on the properties of the exact solution and the numerical solution. A solution-adaptive algorithm based on a posteriori error estimation is designed to generate a posteriori mesh and the approximation solution. Numerical experiments verify that the method is second-order uniformly convergent with respect to small parameter and improves previous results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 86-95"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145046507","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 : 2026-01-01Epub Date: 2025-09-07DOI: 10.1016/j.apnum.2025.09.001
Yanping Chen , Haitao Leng
In this paper, we study an optimal control problem with point values of the state in the objective functional. The state and adjoint state are approximated by a hybridized discontinuous Galerkin (HDG) method, and the control is discretized by the variational discretization concept. With the help of the error estimates of Green’s function and Oswald interpolation, reliable and efficient a posteriori error estimates for the errors in the control, state and adjoint state variables are obtained. Several numerical examples are provided to show the performance of the obtained a posteriori error estimators.
{"title":"An adaptive HDG method for the pointwise tracking optimal control problem of elliptic equations","authors":"Yanping Chen , Haitao Leng","doi":"10.1016/j.apnum.2025.09.001","DOIUrl":"10.1016/j.apnum.2025.09.001","url":null,"abstract":"<div><div>In this paper, we study an optimal control problem with point values of the state in the objective functional. The state and adjoint state are approximated by a hybridized discontinuous Galerkin (HDG) method, and the control is discretized by the variational discretization concept. With the help of the error estimates of Green’s function and Oswald interpolation, reliable and efficient a posteriori error estimates for the errors in the control, state and adjoint state variables are obtained. Several numerical examples are provided to show the performance of the obtained a posteriori error estimators.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 73-85"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145046509","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 : 2026-01-01Epub Date: 2025-09-09DOI: 10.1016/j.apnum.2025.09.005
Chun-Hua Guo
Two iterative methods for solving the absolute value equations are recently proposed and analyzed in the paper by Yu and Wu (Appl. Numer. Math. 208 (2025) 148–159). We point out that the convergence analysis for both methods is incorrect and that the second method with “optimal” parameters is always slightly less efficient than the well-known generalized Newton method.
{"title":"Comments on: “Two efficient iteration methods for solving the absolute value equations”","authors":"Chun-Hua Guo","doi":"10.1016/j.apnum.2025.09.005","DOIUrl":"10.1016/j.apnum.2025.09.005","url":null,"abstract":"<div><div>Two iterative methods for solving the absolute value equations are recently proposed and analyzed in the paper by Yu and Wu (Appl. Numer. Math. 208 (2025) 148–159). We point out that the convergence analysis for both methods is incorrect and that the second method with “optimal” parameters is always slightly <em>less</em> efficient than the well-known generalized Newton method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 96-98"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145046590","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 : 2026-01-01Epub Date: 2025-09-03DOI: 10.1016/j.apnum.2025.08.011
Yan Xu, Boying Wu, Xiong Meng
In this paper, we analyze the local discontinuous Galerkin (LDG) method with generalized numerical fluxes to study the superconvergent properties of one-dimensional linearized KdV equations. Compared with traditional upwind and alternating fluxes, a slower error growth of the LDG solution using generalized numerical fluxes can be obtained for long time simulations. By establishing five energy identities and properties of correction functions with the appropriate numerical initial condition, we derive the supercloseness between the LDG solution and the interpolation function. The errors of the numerical fluxes as well as the cell averages achieve the th-order superconvergence. In addition, we prove that the superconvergent rates of the function and derivative values at the interior generalized Radau points are and , respectively. An extension to mixed boundary conditions is given, for which we present the generalized skew-symmetry property and propose an appropriate conservation property for the numerical initial condition. Numerical experiments are shown to demonstrate the theoretical results, including cases with other boundary conditions and nonlinear KdV equations.
{"title":"Superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linearized KdV equations","authors":"Yan Xu, Boying Wu, Xiong Meng","doi":"10.1016/j.apnum.2025.08.011","DOIUrl":"10.1016/j.apnum.2025.08.011","url":null,"abstract":"<div><div>In this paper, we analyze the local discontinuous Galerkin (LDG) method with generalized numerical fluxes to study the superconvergent properties of one-dimensional linearized KdV equations. Compared with traditional upwind and alternating fluxes, a slower error growth of the LDG solution using generalized numerical fluxes can be obtained for long time simulations. By establishing five energy identities and properties of correction functions with the appropriate numerical initial condition, we derive the supercloseness between the LDG solution and the interpolation function. The errors of the numerical fluxes as well as the cell averages achieve the <span><math><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>th-order superconvergence. In addition, we prove that the superconvergent rates of the function and derivative values at the interior generalized Radau points are <span><math><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span>, respectively. An extension to mixed boundary conditions is given, for which we present the generalized skew-symmetry property and propose an appropriate conservation property for the numerical initial condition. Numerical experiments are shown to demonstrate the theoretical results, including cases with other boundary conditions and nonlinear KdV equations.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 99-121"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145057114","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 : 2026-01-01Epub Date: 2025-09-15DOI: 10.1016/j.apnum.2025.09.007
Anh Ha Le , Quan M. Nguyen
We propose a Crank-Nicolson finite difference scheme to simulate a 2D perturbed soliton interaction under the framework of coupled (2+1)D nonlinear Schrödinger equations with saturable nonlinearity and nonlinear damping. We rigorously demonstrate that the proposed numerical scheme achieves a second-order convergence rate in both the discrete and norms, relative to the time step and spatial mesh size. We establish the boundedness of discrete energies to prove the existence and uniqueness of the solutions derived from the Crank-Nicolson scheme. The validity of the analysis is confirmed through numerical simulations that apply to the corresponding coupled (2+1)D saturable nonlinear Schrödinger equations with damping terms.
{"title":"A Crank-Nicolson finite difference scheme for coupled nonlinear Schrödinger equations with saturable nonlinearity and nonlinear damping","authors":"Anh Ha Le , Quan M. Nguyen","doi":"10.1016/j.apnum.2025.09.007","DOIUrl":"10.1016/j.apnum.2025.09.007","url":null,"abstract":"<div><div>We propose a Crank-Nicolson finite difference scheme to simulate a 2D perturbed soliton interaction under the framework of coupled (2+1)D nonlinear Schrödinger equations with saturable nonlinearity and nonlinear damping. We rigorously demonstrate that the proposed numerical scheme achieves a second-order convergence rate in both the discrete <span><math><msubsup><mi>H</mi><mn>0</mn><mn>1</mn></msubsup></math></span> and <span><math><msup><mi>L</mi><mn>2</mn></msup></math></span> norms, relative to the time step and spatial mesh size. We establish the boundedness of discrete energies to prove the existence and uniqueness of the solutions derived from the Crank-Nicolson scheme. The validity of the analysis is confirmed through numerical simulations that apply to the corresponding coupled (2+1)D saturable nonlinear Schrödinger equations with damping terms.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 219-238"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118204","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 : 2026-01-01Epub Date: 2025-08-26DOI: 10.1016/j.apnum.2025.08.008
Tao Guo , Yiqun Li , Wenlin Qiu
We investigate the numerical approximation to the Euler-Bernoulli (E-B) beams and plates with nonlinear nonlocal damping, which describes the damped mechanical behavior of beams and plates in real applications. We discretize the damping term by the composite Simpson’s rule and the six-point Simpson’s formula in the beam and plate problems, respectively, and then construct the fully discrete compact difference scheme for these problems. To account for the nonlinear-nonlocal term, we design several novel discrete norms to facilitate the error estimates of the damping term and the numerical scheme. The stability, convergence, and energy dissipation properties of the proposed scheme are proved, and numerical experiments are carried out to substantiate the theoretical findings.
{"title":"Compact difference method for Euler-Bernoulli beams and plates with nonlinear nonlocal damping","authors":"Tao Guo , Yiqun Li , Wenlin Qiu","doi":"10.1016/j.apnum.2025.08.008","DOIUrl":"10.1016/j.apnum.2025.08.008","url":null,"abstract":"<div><div>We investigate the numerical approximation to the Euler-Bernoulli (E-B) beams and plates with nonlinear nonlocal damping, which describes the damped mechanical behavior of beams and plates in real applications. We discretize the damping term by the composite Simpson’s rule and the six-point Simpson’s formula in the beam and plate problems, respectively, and then construct the fully discrete compact difference scheme for these problems. To account for the nonlinear-nonlocal term, we design several novel discrete norms to facilitate the error estimates of the damping term and the numerical scheme. The stability, convergence, and energy dissipation properties of the proposed scheme are proved, and numerical experiments are carried out to substantiate the theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 1-18"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926147","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 : 2026-01-01Epub Date: 2025-09-03DOI: 10.1016/j.apnum.2025.08.010
Yingdi Yi, Jijun Liu
The boundary impedance coefficient for an impenetrable obstacle represents its absorption ability for the incident waves, and consequently its indirect detection is of great importance in remote sensing, with the aim of detecting the property of obstacle boundary. We address an inverse acoustic scattering problem for a three-dimensional obstacle, focusing on the reconstruction of boundary impedance using the far-field pattern of the scattered wave corresponding to given incident plane waves. A two-point gradient method combined with the Kaczmarz type scheme is proposed to obtain satisfactory reconstruction. The iteration scheme is formulated by applying the adjoint operator for the forward scattering, based on the potential representation of the scattered wave. The convergence property of the iteration process is rigorously proved. To address the computational scheme for the surface potentials, we use an efficient numerical scheme tailored for three-dimensional geometries. Numerical experiments are presented to demonstrate the validity and robustness of our proposed approach.
{"title":"On the recovery of boundary impedance for 3-dimensional obstacle by acoustic wave scattering using modified Kaczmarz iteration algorithm","authors":"Yingdi Yi, Jijun Liu","doi":"10.1016/j.apnum.2025.08.010","DOIUrl":"10.1016/j.apnum.2025.08.010","url":null,"abstract":"<div><div>The boundary impedance coefficient for an impenetrable obstacle represents its absorption ability for the incident waves, and consequently its indirect detection is of great importance in remote sensing, with the aim of detecting the property of obstacle boundary. We address an inverse acoustic scattering problem for a three-dimensional obstacle, focusing on the reconstruction of boundary impedance using the far-field pattern of the scattered wave corresponding to given incident plane waves. A two-point gradient method combined with the Kaczmarz type scheme is proposed to obtain satisfactory reconstruction. The iteration scheme is formulated by applying the adjoint operator for the forward scattering, based on the potential representation of the scattered wave. The convergence property of the iteration process is rigorously proved. To address the computational scheme for the surface potentials, we use an efficient numerical scheme tailored for three-dimensional geometries. Numerical experiments are presented to demonstrate the validity and robustness of our proposed approach.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 53-72"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145046508","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 : 2026-01-01Epub Date: 2025-09-10DOI: 10.1016/j.apnum.2025.09.004
Maher Alwuthaynani , Muhammad Ahsan , Weidong Lei , Muhammad Abuzar , Masood Ahmad , Aditya Sharma
This study introduces a high-order Haar wavelet collocation method (HHWCM) as an enhanced version of the classical Haar wavelet collocation method (HWCM) for solving fifth-order ordinary differential equations (FoDEs) subject to simple, two-point, and integral boundary conditions. By incorporating a quasi-linearization strategy, the proposed method avoids Jacobian computations and achieves higher accuracy with faster convergence. The stability and convergence of the approach are rigorously analyzed. Numerical experiments on both linear and nonlinear FoDEs demonstrate that HHWCM significantly outperforms HWCM and other existing numerical methods in terms of precision, computational efficiency, and robustness across diverse problem settings.
{"title":"A high-order Haar wavelet approach to solve differential equations of fifth-order with simple, two-point and two-point integral conditions","authors":"Maher Alwuthaynani , Muhammad Ahsan , Weidong Lei , Muhammad Abuzar , Masood Ahmad , Aditya Sharma","doi":"10.1016/j.apnum.2025.09.004","DOIUrl":"10.1016/j.apnum.2025.09.004","url":null,"abstract":"<div><div>This study introduces a high-order Haar wavelet collocation method (HHWCM) as an enhanced version of the classical Haar wavelet collocation method (HWCM) for solving fifth-order ordinary differential equations (FoDEs) subject to simple, two-point, and integral boundary conditions. By incorporating a quasi-linearization strategy, the proposed method avoids Jacobian computations and achieves higher accuracy with faster convergence. The stability and convergence of the approach are rigorously analyzed. Numerical experiments on both linear and nonlinear FoDEs demonstrate that HHWCM significantly outperforms HWCM and other existing numerical methods in terms of precision, computational efficiency, and robustness across diverse problem settings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 122-144"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145059915","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 : 2026-01-01Epub Date: 2025-09-03DOI: 10.1016/j.apnum.2025.08.006
Lan Wang , Yiyang Luo , Meng Chen , Pengfei Zhu
An efficient fourth-order numerical scheme is developed for the Gross-Pitaevskii equation. The spatial direction is approximated by a fourth-order compact scheme and the temporal direction is discretized by a fourth-order splitting & composition method. This scheme not only preserves the symplectic structure and the discrete mass conservation law exactly but also maintains the discrete energy conservation law in some special case. Some numerical experiments confirm our theoretical expectation.
{"title":"A highly accurate symplectic-preserving scheme for Gross-Pitaevskii equation","authors":"Lan Wang , Yiyang Luo , Meng Chen , Pengfei Zhu","doi":"10.1016/j.apnum.2025.08.006","DOIUrl":"10.1016/j.apnum.2025.08.006","url":null,"abstract":"<div><div>An efficient fourth-order numerical scheme is developed for the Gross-Pitaevskii equation. The spatial direction is approximated by a fourth-order compact scheme and the temporal direction is discretized by a fourth-order splitting & composition method. This scheme not only preserves the symplectic structure and the discrete mass conservation law exactly but also maintains the discrete energy conservation law in some special case. Some numerical experiments confirm our theoretical expectation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 41-52"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145044374","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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