Pub Date : 2024-09-26DOI: 10.1016/j.cam.2024.116291
The well-known high order stabilized codes (such as DUMKA and ROCK) have several drawbacks: numerically obtained stability polynomials (which do not have a closed analytic form), poor internal stability and convergence. RKC-type methods have much better computational properties. However, these types of methods currently have a second order maximum. In this paper, a family of third order stabilized methods with an explicit analytical solution of stability polynomials is presented. This was made possible by usage of two-step Runge–Kutta methods. A new code TSRKC3 is proposed, illustrated by several examples, and compared to existing programs.
{"title":"Third order two-step Runge–Kutta–Chebyshev methods","authors":"","doi":"10.1016/j.cam.2024.116291","DOIUrl":"10.1016/j.cam.2024.116291","url":null,"abstract":"<div><div>The well-known high order stabilized codes (such as DUMKA and ROCK) have several drawbacks: numerically obtained stability polynomials (which do not have a closed analytic form), poor internal stability and convergence. RKC-type methods have much better computational properties. However, these types of methods currently have a second order maximum. In this paper, a family of third order stabilized methods with an explicit analytical solution of stability polynomials is presented. This was made possible by usage of two-step Runge–Kutta methods. A new code TSRKC3 is proposed, illustrated by several examples, and compared to existing programs.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326350","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 : 2024-09-20DOI: 10.1016/j.cam.2024.116286
In this paper, we propose two numerical methods for the stochastic Helmholtz equation driven by white noise. We obtain the approximate stochastic problem by approximating the white noise with piecewise constant process, provide some regularity of its solution and the truncation error between the approximate stochastic problem and the original problem. The limitation on the wave number of the finite difference method (FDM) is analyzed and a stochastic finite difference (SFD) scheme is presented. The error analysis shows that the stochastic finite difference method is efficient with a certain convergence rate. Numerical experiments are provided to examine our theoretical results.
本文针对白噪声驱动的随机亥姆霍兹方程提出了两种数值方法。我们通过用片断常数过程逼近白噪声得到近似随机问题，给出了其解的一些规律性以及近似随机问题与原问题之间的截断误差。分析了有限差分法（FDM）对波数 k 的限制，并提出了一种随机有限差分法（SFD）方案。误差分析表明，随机有限差分法具有一定的收敛效率。我们还提供了数值实验来检验我们的理论结果。
{"title":"Finite difference methods for stochastic Helmholtz equation driven by white noise","authors":"","doi":"10.1016/j.cam.2024.116286","DOIUrl":"10.1016/j.cam.2024.116286","url":null,"abstract":"<div><div>In this paper, we propose two numerical methods for the stochastic Helmholtz equation driven by white noise. We obtain the approximate stochastic problem by approximating the white noise with piecewise constant process, provide some regularity of its solution and the truncation error between the approximate stochastic problem and the original problem. The limitation on the wave number <span><math><mi>k</mi></math></span> of the finite difference method (FDM) is analyzed and a stochastic finite difference (SFD) scheme is presented. The error analysis shows that the stochastic finite difference method is efficient with a certain convergence rate. Numerical experiments are provided to examine our theoretical results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142320199","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 : 2024-09-19DOI: 10.1016/j.cam.2024.116289
The presence of TV regularizer always induces an unsatisfactory staircase effect. To overcome the staircase while better sustaining edge information, this work proposes a novel model for Poisson noise removal. The model is based on non-convex mixed regularizers, which involves introducing a non-convex penalty into a composition of the total variation and the higher-order total variation. The iterative reweighted algorithm was used to convert the non-convex model into a convex one. The classic alternating direction method of multipliers was then employed to obtain approximate solutions of the model. When applying this model to degraded images contaminated by Poisson noise of medium to high intensity, its performance in noise suppression was tested. The regularizer was compared with others in the terms of the visual effect of the picture, time cost, and several commonly accepted quantitative indicators for evaluation, such as peak signal-to-noise ratio, feature similarity index and structural similarity index. Numerical experiments showed that the present model not only eliminates block artifacts but also retains sharp edges.
{"title":"Poisson noise removal based on non-convex hybrid regularizers","authors":"","doi":"10.1016/j.cam.2024.116289","DOIUrl":"10.1016/j.cam.2024.116289","url":null,"abstract":"<div><div>The presence of TV regularizer always induces an unsatisfactory staircase effect. To overcome the staircase while better sustaining edge information, this work proposes a novel model for Poisson noise removal. The model is based on non-convex mixed regularizers, which involves introducing a non-convex penalty into a composition of the total variation and the higher-order total variation. The iterative reweighted <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> algorithm was used to convert the non-convex model into a convex one. The classic alternating direction method of multipliers was then employed to obtain approximate solutions of the model. When applying this model to degraded images contaminated by Poisson noise of medium to high intensity, its performance in noise suppression was tested. The regularizer was compared with others in the terms of the visual effect of the picture, time cost, and several commonly accepted quantitative indicators for evaluation, such as peak signal-to-noise ratio, feature similarity index and structural similarity index. Numerical experiments showed that the present model not only eliminates block artifacts but also retains sharp edges.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326348","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 : 2024-09-18DOI: 10.1016/j.cam.2024.116271
The LFC control problem for a class of nonlinear power systems with time-varying delays is under study. Considering uncertainties arising from nonlinear issues such as the generation rate constraint (GRC) and governor dead band (GDB), as well as the high variability of renewable energy sources like wind power, the model is transformed into a discrete-time Takagi–Sugeno (T–S) fuzzy model with parameter uncertainty. By constructing a Lyapunov–Krasovskii functional, and employing difference inequalities and generalized cross-convex matrix inequalities, sufficient conditions for the asymptotic stability of power systems are provided. Based on the obtained conditions, a controller is designed to ensure the asymptotic stability of Multi-Area Power Systems (MAPS), with the performance index being . Finally, simulation results demonstrate the correctness and effectiveness of the theorem.
{"title":"Robust H∞ control for LFC of discrete T–S fuzzy MAPS with DFIG and time-varying delays","authors":"","doi":"10.1016/j.cam.2024.116271","DOIUrl":"10.1016/j.cam.2024.116271","url":null,"abstract":"<div><div>The <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> LFC control problem for a class of nonlinear power systems with time-varying delays is under study. Considering uncertainties arising from nonlinear issues such as the generation rate constraint (GRC) and governor dead band (GDB), as well as the high variability of renewable energy sources like wind power, the model is transformed into a discrete-time Takagi–Sugeno (T–S) fuzzy model with parameter uncertainty. By constructing a Lyapunov–Krasovskii functional, and employing difference inequalities and generalized cross-convex matrix inequalities, sufficient conditions for the asymptotic stability of power systems are provided. Based on the obtained conditions, a controller is designed to ensure the asymptotic stability of Multi-Area Power Systems (MAPS), with the performance index being <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span>. Finally, simulation results demonstrate the correctness and effectiveness of the theorem.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312507","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 : 2024-09-18DOI: 10.1016/j.cam.2024.116285
In this paper, we propose a numerical algorithm that combines the fading regularization method with the method of fundamental solutions (MFS) to solve a Cauchy problem associated with the biharmonic equation. We introduce a new stopping criterion for the iterative process and compare its performance with previous criteria. Numerical simulations using MFS validate the accuracy of this stopping criterion for both compatible and noisy data and demonstrate the convergence, stability, and efficiency of the proposed algorithm, as well as its ability to deblur noisy data.
{"title":"Fading regularization method for an inverse boundary value problem associated with the biharmonic equation","authors":"","doi":"10.1016/j.cam.2024.116285","DOIUrl":"10.1016/j.cam.2024.116285","url":null,"abstract":"<div><div>In this paper, we propose a numerical algorithm that combines the fading regularization method with the method of fundamental solutions (MFS) to solve a Cauchy problem associated with the biharmonic equation. We introduce a new stopping criterion for the iterative process and compare its performance with previous criteria. Numerical simulations using MFS validate the accuracy of this stopping criterion for both compatible and noisy data and demonstrate the convergence, stability, and efficiency of the proposed algorithm, as well as its ability to deblur noisy data.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-16DOI: 10.1016/j.cam.2024.116224
This paper focuses on the error analysis of a first-order, time-discrete scheme for solving the nonlinear Allen–Cahn equation. The discretization of the nonlinear potential is achieved through the EIEQ method, which employs an auxiliary variable to linearize the nonlinear double-well potential effectively. The energy stability of the scheme is demonstrated, along with its decoupled type implementation. Under a set of reasonable assumptions related to boundedness and continuity, an extensive error analysis is performed. This analysis results in the establishment of and error bounds for the numerical solution. Furthermore, a variety of numerical examples are conducted to illustrate the accuracy of the EIEQ scheme, highlighting its effectiveness in addressing complex dynamical systems governed by the Allen–Cahn equation.
{"title":"Error analysis of the explicit-invariant energy quadratization (EIEQ) numerical scheme for solving the Allen–Cahn equation","authors":"","doi":"10.1016/j.cam.2024.116224","DOIUrl":"10.1016/j.cam.2024.116224","url":null,"abstract":"<div><div>This paper focuses on the error analysis of a first-order, time-discrete scheme for solving the nonlinear Allen–Cahn equation. The discretization of the nonlinear potential is achieved through the EIEQ method, which employs an auxiliary variable to linearize the nonlinear double-well potential effectively. The energy stability of the scheme is demonstrated, along with its decoupled type implementation. Under a set of reasonable assumptions related to boundedness and continuity, an extensive error analysis is performed. This analysis results in the establishment of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> error bounds for the numerical solution. Furthermore, a variety of numerical examples are conducted to illustrate the accuracy of the EIEQ scheme, highlighting its effectiveness in addressing complex dynamical systems governed by the Allen–Cahn equation.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312508","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 : 2024-09-13DOI: 10.1016/j.cam.2024.116275
In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.
{"title":"Generalized Multiscale Finite Element Method for discrete network (graph) models","authors":"","doi":"10.1016/j.cam.2024.116275","DOIUrl":"10.1016/j.cam.2024.116275","url":null,"abstract":"<div><p>In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.</p></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142241449","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 : 2024-09-13DOI: 10.1016/j.cam.2024.116283
This paper constructs a linearized transformed virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical -fractional differential system derived from a smoothing transformation of variables , . By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in -norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.
本文为广义非线性耦合时间分数薛定谔方程构建了线性化变换 L1 虚拟元素方法。此类问题的解通常在开始时表现出奇异行为。为了避免这一缺陷，我们引入了一个由变量 t=s1/α, 0<α<1 的平滑变换导出的相同 s 分式微分方程系统。此外，我们还在不限制网格比的情况下，以 L2 规范推导出了所提出的完全离散方案的无条件最优误差边界。最后，对一组多边形网格进行了数值测试，以验证理论结果。
{"title":"Unconditional error analysis of the linearized transformed L1 virtual element method for nonlinear coupled time-fractional Schrödinger equations","authors":"","doi":"10.1016/j.cam.2024.116283","DOIUrl":"10.1016/j.cam.2024.116283","url":null,"abstract":"<div><div>This paper constructs a linearized transformed <span><math><mrow><mi>L</mi><mn>1</mn></mrow></math></span> virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical <span><math><mi>s</mi></math></span>-fractional differential system derived from a smoothing transformation of variables <span><math><mrow><mi>t</mi><mo>=</mo><msup><mrow><mi>s</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>α</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></math></span>. By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312509","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 : 2024-09-12DOI: 10.1016/j.cam.2024.116277
Interface problems, where distinct materials or physical domains meet, pose significant challenges in numerical simulations due to the discontinuities and sharp gradients across interfaces. Traditional finite element methods struggle to capture such behavior accurately. A new space transformed finite element method (ST-FEM) is developed for solving elliptic interface problems in . A homeomorphic stretching transformation is introduced to obtain an equivalent problem in the transformed domain which can be solved easily, and the solution can be projected back to original domain by the inverse transformation. Compared with the existing methods, this new scheme has capability of handling discontinuities across the interface. The proposed approach has advantages in circumventing interface approximation properties and reducing the degree of freedom. We initially develop ST-FEM for elliptic problems and subsequently expand upon this concept to address elliptic interface problems. We prove optimal a priori error estimates in the and norms, and quasi-optimal error estimate for the maximum norm. Finally, numerical experiments demonstrate the superior accuracy and convergence properties of the ST-FEM when compared to the standard finite element method. The interface is assumed to be a -sphere, nevertheless, our analysis can cover symmetric domains such as an ellipsoid or a cylinder.
{"title":"A new space transformed finite element method for elliptic interface problems in Rn","authors":"","doi":"10.1016/j.cam.2024.116277","DOIUrl":"10.1016/j.cam.2024.116277","url":null,"abstract":"<div><p>Interface problems, where distinct materials or physical domains meet, pose significant challenges in numerical simulations due to the discontinuities and sharp gradients across interfaces. Traditional finite element methods struggle to capture such behavior accurately. A new space transformed finite element method (ST-FEM) is developed for solving elliptic interface problems in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A homeomorphic stretching transformation is introduced to obtain an equivalent problem in the transformed domain which can be solved easily, and the solution can be projected back to original domain by the inverse transformation. Compared with the existing methods, this new scheme has capability of handling discontinuities across the interface. The proposed approach has advantages in circumventing interface approximation properties and reducing the degree of freedom. We initially develop ST-FEM for elliptic problems and subsequently expand upon this concept to address elliptic interface problems. We prove optimal a priori error estimates in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norms, and quasi-optimal error estimate for the maximum norm. Finally, numerical experiments demonstrate the superior accuracy and convergence properties of the ST-FEM when compared to the standard finite element method. The interface is assumed to be a <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-sphere, nevertheless, our analysis can cover symmetric domains such as an ellipsoid or a cylinder.</p></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142241453","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 : 2024-09-12DOI: 10.1016/j.cam.2024.116258
We establish the exact expressions for the hypergeometric function of a Hermitian matrix argument. This result allows for the eigenvalues of the matrix argument to occur with arbitrary multiplicities and can be used for numerical computation. These exact expressions are particularly important since they provide the key ingredient which allows many results which involve this function to be useful from a practical engineering perspective.
{"title":"Representation computation for the hypergeometric function of a Hermitian matrix argument","authors":"","doi":"10.1016/j.cam.2024.116258","DOIUrl":"10.1016/j.cam.2024.116258","url":null,"abstract":"<div><p>We establish the exact expressions for the hypergeometric function of a Hermitian matrix argument. This result allows for the eigenvalues of the matrix argument to occur with arbitrary multiplicities and can be used for numerical computation. These exact expressions are particularly important since they provide the key ingredient which allows many results which involve this function to be useful from a practical engineering perspective.</p></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142241451","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号