Pub Date : 2026-10-01Epub Date: 2026-02-10DOI: 10.1016/j.cam.2026.117422
Shashi Kant Mishra , Dheerendra Singh
In this paper, we consider a nonsmooth mathematical programming problem and establish the characterization of solution sets. We also give a normal cone condition for nonsmooth mathematical programming problems to obtain optimality conditions using Lagrange multipliers and tangential subdifferentials. We also provide some examples in support of our results.
{"title":"Characterizations of solution sets of nonsmooth mathematical programming problems","authors":"Shashi Kant Mishra , Dheerendra Singh","doi":"10.1016/j.cam.2026.117422","DOIUrl":"10.1016/j.cam.2026.117422","url":null,"abstract":"<div><div>In this paper, we consider a nonsmooth mathematical programming problem and establish the characterization of solution sets. We also give a normal cone condition for nonsmooth mathematical programming problems to obtain optimality conditions using Lagrange multipliers and tangential subdifferentials. We also provide some examples in support of our results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117422"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387119","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 : 2026-10-01Epub Date: 2026-02-19DOI: 10.1016/j.cam.2026.117459
Saba Asgarzadeh , M.R. Eslahchi
In this research, we introduce new classes of regularization matrices constructed using generalized operational matrices of the Caputo fractional derivative. Experimental results confirm the performance and effectiveness of the proposed method. In another part of this study, the obtained matrices are applied to classification tasks in machine learning. The results demonstrate improved classification accuracy and more effective data representation.
{"title":"Regularization via generalized operational matrices: Theory and applications in machine learning classification","authors":"Saba Asgarzadeh , M.R. Eslahchi","doi":"10.1016/j.cam.2026.117459","DOIUrl":"10.1016/j.cam.2026.117459","url":null,"abstract":"<div><div>In this research, we introduce new classes of regularization matrices constructed using generalized operational matrices of the Caputo fractional derivative. Experimental results confirm the performance and effectiveness of the proposed method. In another part of this study, the obtained matrices are applied to classification tasks in machine learning. The results demonstrate improved classification accuracy and more effective data representation.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117459"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387130","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 : 2026-10-01Epub Date: 2026-02-04DOI: 10.1016/j.nonrwa.2026.104616
A. Prabal, M. Devakar
In this paper, we present an analysis that establishes the existence and uniqueness of weak solution of the nonlinear system of partial differential equations governing the steady flow of an incompressible micropolar fluid flow through a homogeneous porous medium in a curved pipe. The Galerkin method along with a version of the Leray-Schauder principle has been used to prove the existence of a weak solution. It has been proved that there is a weak solution for sufficiently small values of curvature ratio (δ); furthermore, it has also been established that the solution is unique for sufficiently small values of Reynolds number (Re) and the micropolarity parameter (m). The regularity of the weak solution is also discussed in this paper; more importantly, if the cross-sectional area (Ω) is sufficiently smooth, specifically of class C3, then the weak solution becomes a classical solution.
{"title":"On the existence, uniqueness and regularity of solutions of micropolar fluid flow through porous medium in a curved pipe","authors":"A. Prabal, M. Devakar","doi":"10.1016/j.nonrwa.2026.104616","DOIUrl":"10.1016/j.nonrwa.2026.104616","url":null,"abstract":"<div><div>In this paper, we present an analysis that establishes the existence and uniqueness of weak solution of the nonlinear system of partial differential equations governing the steady flow of an incompressible micropolar fluid flow through a homogeneous porous medium in a curved pipe. The Galerkin method along with a version of the Leray-Schauder principle has been used to prove the existence of a weak solution. It has been proved that there is a weak solution for sufficiently small values of curvature ratio (<em>δ</em>); furthermore, it has also been established that the solution is unique for sufficiently small values of Reynolds number (<em>Re</em>) and the micropolarity parameter (<em>m</em>). The regularity of the weak solution is also discussed in this paper; more importantly, if the cross-sectional area (Ω) is sufficiently smooth, specifically of class <em>C</em><sup>3</sup>, then the weak solution becomes a classical solution.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104616"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146188913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2025-12-24DOI: 10.1016/j.nonrwa.2025.104560
Liyan Pang , Xiao Zhang
In this paper, the long time behavior for a two-species Lotka-Volterra reaction-diffusion system with strong competition in a periodic medium is concerned. We prove that under the compactly supported initial values, the solutions of Cauchy problem converge to a pair of diverging pulsating fronts. Further, we obtain a sufficient condition for solutions to converge to 1 with two different speeds to the left and right. Due to the spatial heterogeneity, the pulsating fronts depend on its direction and any pair of rightward and leftward wave speeds be asymmetrical. Therefore, our analysis mainly depends on constructing appropriate super- and subsolutions and using the comparison principle and asymptotic behavior of bistable pulsating fronts.
{"title":"Long time behavior for a Lotka-Volterra competition diffusion system in periodic medium","authors":"Liyan Pang , Xiao Zhang","doi":"10.1016/j.nonrwa.2025.104560","DOIUrl":"10.1016/j.nonrwa.2025.104560","url":null,"abstract":"<div><div>In this paper, the long time behavior for a two-species Lotka-Volterra reaction-diffusion system with strong competition in a periodic medium is concerned. We prove that under the compactly supported initial values, the solutions of Cauchy problem converge to a pair of diverging pulsating fronts. Further, we obtain a sufficient condition for solutions to converge to <strong><em>1</em></strong> with two different speeds to the left and right. Due to the spatial heterogeneity, the pulsating fronts depend on its direction and any pair of rightward and leftward wave speeds be asymmetrical. Therefore, our analysis mainly depends on constructing appropriate super- and subsolutions and using the comparison principle and asymptotic behavior of bistable pulsating fronts.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104560"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2025-12-22DOI: 10.1016/j.nonrwa.2025.104585
Fábio Natali
In this paper, we consider the problem of well-posedness and orbital stability of odd periodic traveling waves for the sine-Gordon equation. We first establish novel results concerning the local well-posedness in smoother periodic Sobolev spaces to guarantee the existence of a local time where the associated Cauchy problem has a unique solution with the zero mean property. Afterwards, we prove the orbital stability of odd periodic waves using a convenient index theorem applied to the constrained linearized operator defined in the Sobolev space with the zero mean property.
{"title":"Remarks on the orbital stability for the sine-Gordon equation","authors":"Fábio Natali","doi":"10.1016/j.nonrwa.2025.104585","DOIUrl":"10.1016/j.nonrwa.2025.104585","url":null,"abstract":"<div><div>In this paper, we consider the problem of well-posedness and orbital stability of odd periodic traveling waves for the sine-Gordon equation. We first establish novel results concerning the local well-posedness in smoother periodic Sobolev spaces to guarantee the existence of a local time where the associated Cauchy problem has a unique solution with the zero mean property. Afterwards, we prove the orbital stability of odd periodic waves using a convenient index theorem applied to the constrained linearized operator defined in the Sobolev space with the zero mean property.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104585"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-02-12DOI: 10.1016/j.cam.2026.117406
Pengyuan Liu , Zhaodong Xu , Zhiqiang Sheng
In this paper, we propose a positivity-preserving subspace method, termed PSNNW, which is based on neural networks formulated in the weak form for solving diffusion equations. The method employs a monotonic positivity-preserving nonlinear functions to transform the original equations into mathematically equivalent forms. The numerical solution of the transformed equation is subsequently computed using a subspace neural network method in the weak form designed for nonlinear problems. In this method, neural networks are employed to train and generate basis functions, which are then incorporated into iterative schemes, such as Picard iteration, to solve the problem within the Galerkin framework. Owing to the positivity-preserving transformation, the numerical solution of the original equation is guaranteed to remain positive. Numerical experiments demonstrate that the proposed method yields nonnegative solutions with high accuracy, confirming its simplicity and effectiveness in preserving positivity.
{"title":"A positivity-preserving subspace method based on neural networks for solving diffusion equations in the weak form","authors":"Pengyuan Liu , Zhaodong Xu , Zhiqiang Sheng","doi":"10.1016/j.cam.2026.117406","DOIUrl":"10.1016/j.cam.2026.117406","url":null,"abstract":"<div><div>In this paper, we propose a positivity-preserving subspace method, termed PSNNW, which is based on neural networks formulated in the weak form for solving diffusion equations. The method employs a monotonic positivity-preserving nonlinear functions to transform the original equations into mathematically equivalent forms. The numerical solution of the transformed equation is subsequently computed using a subspace neural network method in the weak form designed for nonlinear problems. In this method, neural networks are employed to train and generate basis functions, which are then incorporated into iterative schemes, such as Picard iteration, to solve the problem within the Galerkin framework. Owing to the positivity-preserving transformation, the numerical solution of the original equation is guaranteed to remain positive. Numerical experiments demonstrate that the proposed method yields nonnegative solutions with high accuracy, confirming its simplicity and effectiveness in preserving positivity.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117406"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387180","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}
In this paper, we present an efficient numerical method to address a thermodynamically consistent gas flow model in porous media involving compressible gas and deformable rock. The accurate modeling of gas flow in porous media often poses significant challenges due to their inherent nonlinearity, the coupling between gas and rock dynamics, and the need to preserve physical principles such as mass conservation, energy dissipation and molar density boundedness. The system is further complicated by the need to balance computational efficiency with the accuracy and stability of the numerical scheme. To tackle these challenges, we adopt a stabilization approach that is able to preserve the original energy dissipation while achieving linear energy-stable numerical schemes. We also prove the convergence of the adopted linear iterative method. At each time step, the stabilization parameter is adaptively updated using a simple and explicit formula to ensure compliance with the original energy dissipation law. The proposed method uses adaptive time stepping to improve computational efficiency while maintaining solution accuracy and boundedness. The adaptive time step size is calculated explicitly at each iteration, ensuring stability and allowing for efficient handling of highly dynamic scenarios. A mixed finite element method combined with an upwind scheme is employed as spatial discretization to ensure mass conservation and stability. Finally, we conduct a series of numerical experiments to validate the performance and robustness of the proposed numerical method.
{"title":"Bound-preserving adaptive time-stepping methods with energy stability for simulating compressible gas flow in poroelastic media","authors":"Huangxin Chen , Yuxiang Chen , Jisheng Kou , Shuyu Sun","doi":"10.1016/j.cam.2026.117552","DOIUrl":"10.1016/j.cam.2026.117552","url":null,"abstract":"<div><div>In this paper, we present an efficient numerical method to address a thermodynamically consistent gas flow model in porous media involving compressible gas and deformable rock. The accurate modeling of gas flow in porous media often poses significant challenges due to their inherent nonlinearity, the coupling between gas and rock dynamics, and the need to preserve physical principles such as mass conservation, energy dissipation and molar density boundedness. The system is further complicated by the need to balance computational efficiency with the accuracy and stability of the numerical scheme. To tackle these challenges, we adopt a stabilization approach that is able to preserve the original energy dissipation while achieving linear energy-stable numerical schemes. We also prove the convergence of the adopted linear iterative method. At each time step, the stabilization parameter is adaptively updated using a simple and explicit formula to ensure compliance with the original energy dissipation law. The proposed method uses adaptive time stepping to improve computational efficiency while maintaining solution accuracy and boundedness. The adaptive time step size is calculated explicitly at each iteration, ensuring stability and allowing for efficient handling of highly dynamic scenarios. A mixed finite element method combined with an upwind scheme is employed as spatial discretization to ensure mass conservation and stability. Finally, we conduct a series of numerical experiments to validate the performance and robustness of the proposed numerical method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117552"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386633","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 C1-continuity of calculated state curves is crucial for engineering problems subject to acceleration-dependent forces, such as hydrodynamic loads and control forces. Conventional variational integration schemes with prominent energy and momentum preserving properties are favored to calculate the dynamics of mechanical systems, however, can sometimes lose their efficacy due to a lack of continuity. In this work, a novel C1-continuous variational integration scheme is developed within a simple and general construction framework, ensuring the continuity of the generalized coordinates and their first derivative at the discrete-time points. This scheme is constructed by approximating generalized coordinates and velocities using Hermite polynomials within a certain time span with the action integral computed numerically. This framework greatly simplifies the derivation and implementation by avoiding the summation of discrete node variations, and it is also suitable for constructing other variational schemes based on Lagrangian polynomials of various orders. The algorithmic characteristics, including stability, dissipation, period elongation, and convergence order, are theoretically analyzed. The momentum-preserving and nearly energy-preserving properties are numerically demonstrated. Moreover, practical engineering problems subject to acceleration-dependent forces are investigated, which have well confirmed the feasibility of the proposed C1-continuous variational scheme in practical dynamic analyses.
{"title":"A new C1-continuous variational integration scheme for mechanical systems subjected to acceleration-dependent forces","authors":"Ping Zhou , Songhan Zhang , Hui Ren , Zheng Chen , Wei Fan","doi":"10.1016/j.cam.2026.117509","DOIUrl":"10.1016/j.cam.2026.117509","url":null,"abstract":"<div><div>The C<sup>1</sup>-continuity of calculated state curves is crucial for engineering problems subject to acceleration-dependent forces, such as hydrodynamic loads and control forces. Conventional variational integration schemes with prominent energy and momentum preserving properties are favored to calculate the dynamics of mechanical systems, however, can sometimes lose their efficacy due to a lack of continuity. In this work, a novel C<sup>1</sup>-continuous variational integration scheme is developed within a simple and general construction framework, ensuring the continuity of the generalized coordinates and their first derivative at the discrete-time points. This scheme is constructed by approximating generalized coordinates and velocities using Hermite polynomials within a certain time span with the action integral computed numerically. This framework greatly simplifies the derivation and implementation by avoiding the summation of discrete node variations, and it is also suitable for constructing other variational schemes based on Lagrangian polynomials of various orders. The algorithmic characteristics, including stability, dissipation, period elongation, and convergence order, are theoretically analyzed. The momentum-preserving and nearly energy-preserving properties are numerically demonstrated. Moreover, practical engineering problems subject to acceleration-dependent forces are investigated, which have well confirmed the feasibility of the proposed C<sup>1</sup>-continuous variational scheme in practical dynamic analyses.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117509"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386987","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 : 2026-10-01Epub Date: 2026-02-23DOI: 10.1016/j.cam.2026.117457
Yanfang Yang, Lu Xiao
In this paper, we study higher-order Multiscale Finite Element Method (MsFEM) to solve linear elasticity equations with oscillating coefficients. Compared to the Finite Element Method (FEM), MsFEM can obtain the characteristics of fine scale in coarse mesh by using carefully designed multiscale basis functions. By using higher-order basis functions, better accuracy can be achieved. To further improve accuracy, several techniques are considered: oversampling methods and oscillatory boundary conditions (OBCs) are used to prevent the influence of boundary conditions on the construction of multiscale basis functions; the Petrov-Galerkin method, with the trial functions being multiscale basis functions and the test functions being polynomial functions. Numerical examples are presented to demonstrate the efficiency of the proposed methods.
{"title":"Higher-order multiscale finite element method for linear elasticity equations with oscillating coefficients","authors":"Yanfang Yang, Lu Xiao","doi":"10.1016/j.cam.2026.117457","DOIUrl":"10.1016/j.cam.2026.117457","url":null,"abstract":"<div><div>In this paper, we study higher-order Multiscale Finite Element Method (MsFEM) to solve linear elasticity equations with oscillating coefficients. Compared to the Finite Element Method (FEM), MsFEM can obtain the characteristics of fine scale in coarse mesh by using carefully designed multiscale basis functions. By using higher-order basis functions, better accuracy can be achieved. To further improve accuracy, several techniques are considered: oversampling methods and oscillatory boundary conditions (OBCs) are used to prevent the influence of boundary conditions on the construction of multiscale basis functions; the Petrov-Galerkin method, with the trial functions being multiscale basis functions and the test functions being polynomial functions. Numerical examples are presented to demonstrate the efficiency of the proposed methods.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117457"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386989","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 : 2026-10-01Epub Date: 2026-02-24DOI: 10.1016/j.cam.2026.117542
Hasanen A. Hammad , Tarek Aboelenen
This study investigates the demanding problem of proving the existence of solutions for a tripled system that couples quantum integral equations with quadratic integral equations. To tackle the intrinsic nonlinear and noncompact features of the model, we employ Petryshyn’s fixed-point theorem, a significant generalization of Darbo’s theorem formulated within the framework of measures of noncompactness. Based on this approach, we derive rigorous and verifiable existence conditions applicable to a broad class of quantum integral systems. The theoretical findings are supported by a comprehensive illustrative example that confirms the validity of the proposed criteria. In addition, we develop a constructive collocation method founded on barycentric interpolation and Jackson quadrature for q-integrals, and we verify the required assumptions within the same example. Numerical experiments are finally presented to confirm the practical applicability of the existence results and to demonstrate the accuracy, stability, and robustness of the proposed discretization scheme.
{"title":"Applying fixed point techniques for solving tripled system of quantum integral equations with numerical results","authors":"Hasanen A. Hammad , Tarek Aboelenen","doi":"10.1016/j.cam.2026.117542","DOIUrl":"10.1016/j.cam.2026.117542","url":null,"abstract":"<div><div>This study investigates the demanding problem of proving the existence of solutions for a tripled system that couples quantum integral equations with quadratic integral equations. To tackle the intrinsic nonlinear and noncompact features of the model, we employ Petryshyn’s fixed-point theorem, a significant generalization of Darbo’s theorem formulated within the framework of measures of noncompactness. Based on this approach, we derive rigorous and verifiable existence conditions applicable to a broad class of quantum integral systems. The theoretical findings are supported by a comprehensive illustrative example that confirms the validity of the proposed criteria. In addition, we develop a constructive collocation method founded on barycentric interpolation and Jackson quadrature for q-integrals, and we verify the required assumptions within the same example. Numerical experiments are finally presented to confirm the practical applicability of the existence results and to demonstrate the accuracy, stability, and robustness of the proposed discretization scheme.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117542"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386995","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