Pub Date : 2025-12-10DOI: 10.1016/j.aml.2025.109848
Shifeng Geng, Pan Zhang
In this paper, we consider the stability problem of the 2D Boussinesq-MHD system with only fractional horizontal magnetic diffusion and thermal diffusivity. By employing the effects of magnetic field, and the decomposition of the horizontal average and oscillatory parts, we prove the global stability of the 2D Boussinesq-MHD system without velocity dissipation. And the result shows the magnetic field has a stabilizing effect on the fluid. Moreover, we obtain exponential decay of the solution in one direction.
{"title":"Stability of the 2D Boussinesq-MHD system with only fractional horizontal magnetic and thermal diffusion","authors":"Shifeng Geng, Pan Zhang","doi":"10.1016/j.aml.2025.109848","DOIUrl":"10.1016/j.aml.2025.109848","url":null,"abstract":"<div><div>In this paper, we consider the stability problem of the 2D Boussinesq-MHD system with only fractional horizontal magnetic diffusion and thermal diffusivity. By employing the effects of magnetic field, and the decomposition of the horizontal average and oscillatory parts, we prove the global stability of the 2D Boussinesq-MHD system without velocity dissipation. And the result shows the magnetic field has a stabilizing effect on the fluid. Moreover, we obtain exponential decay of the solution in one direction.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"175 ","pages":"Article 109848"},"PeriodicalIF":2.8,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731485","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 : 2025-12-10DOI: 10.1016/j.aml.2025.109847
To Fu Ma , Rodrigo N. Monteiro , Paulo N. Seminario-Huertas
In this paper, we consider a thermoelastic plate of Green–Lindsay type, characterized by two relaxation times and exhibiting finite-speed heat waves. The homogeneous problem was recently studied by Quintanilla et al. (2023). They pointed out that it was not known whether the domain of the semigroup generator is compactly embedded into the energy space. Nevertheless, through a detailed analysis, they established the well-posedness of the system and the exponential stability of its solution semigroup. Our aim is to investigate the asymptotic dynamics of the plate in the presence of a nonlinear foundation. We establish the existence of a finite dimensional global attractor with higher regularity.
本文考虑具有两个松弛时间且具有有限速度热波的Green-Lindsay型热弹性板。最近Quintanilla et al.（2023）研究了齐次问题。他们指出，尚不清楚半群发生器的域是否紧密嵌入到能量空间中。然而，通过详细的分析，他们建立了系统的适定性及其解半群的指数稳定性。我们的目的是研究在非线性基础存在下板的渐近动力学。建立了具有高正则性的有限维全局吸引子的存在性。
{"title":"Dynamics of a thermoelastic Green–Lindsay plate on a nonlinear foundation","authors":"To Fu Ma , Rodrigo N. Monteiro , Paulo N. Seminario-Huertas","doi":"10.1016/j.aml.2025.109847","DOIUrl":"10.1016/j.aml.2025.109847","url":null,"abstract":"<div><div>In this paper, we consider a thermoelastic plate of Green–Lindsay type, characterized by two relaxation times and exhibiting finite-speed heat waves. The homogeneous problem was recently studied by Quintanilla et al. (2023). They pointed out that it was not known whether the domain of the semigroup generator is compactly embedded into the energy space. Nevertheless, through a detailed analysis, they established the well-posedness of the system and the exponential stability of its solution semigroup. Our aim is to investigate the asymptotic dynamics of the plate in the presence of a nonlinear foundation. We establish the existence of a finite dimensional global attractor with higher regularity.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"175 ","pages":"Article 109847"},"PeriodicalIF":2.8,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731486","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 : 2025-12-08DOI: 10.1016/j.aml.2025.109845
Yaxiang Li , Jiangxing Wang
We propose a linearized hybridizable discontinuous Galerkin (HDG) method for solving the time-dependent nonlinear Schrödinger equation. By integrating the advantageous features of HDG spatial discretization with the temporal accuracy of a semi-implicit Crank–Nicolson scheme, the proposed method delivers both high-order accuracy and computational efficiency. A rigorous theoretical analysis establishes unconditional optimal error estimates for the numerical solution and its gradient without any restriction imposed between the time-step size and the spatial mesh size. Numerical examples are carried out to verify the theoretical results.
{"title":"Linearly implicit conservative HDG method for the nonlinear Schrödinger equation","authors":"Yaxiang Li , Jiangxing Wang","doi":"10.1016/j.aml.2025.109845","DOIUrl":"10.1016/j.aml.2025.109845","url":null,"abstract":"<div><div>We propose a linearized hybridizable discontinuous Galerkin (HDG) method for solving the time-dependent nonlinear Schrödinger equation. By integrating the advantageous features of HDG spatial discretization with the temporal accuracy of a semi-implicit Crank–Nicolson scheme, the proposed method delivers both high-order accuracy and computational efficiency. A rigorous theoretical analysis establishes unconditional optimal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> error estimates for the numerical solution and its gradient without any restriction imposed between the time-step size and the spatial mesh size. Numerical examples are carried out to verify the theoretical results.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"175 ","pages":"Article 109845"},"PeriodicalIF":2.8,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145705024","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 : 2025-12-08DOI: 10.1016/j.aml.2025.109846
Boxuan Zhao, Guotao Wang
This paper investigates the blow-up problem for the -Hessian equation with nonlinear gradient terms: where , are nonnegative constants with , is a smooth, bounded and strictly -convex domain, is a positive function and may be singular near . By the sub-supersolution method, we present the boundary behavior of large solutions to this problem. Our work essentially generalizes the relevant conclusions in Zhang and Feng (2018); Feng and Zhang (2020).
{"title":"Nonexistence and boundary behavior of solutions to the k-Hessian equation with nonlinear gradient terms","authors":"Boxuan Zhao, Guotao Wang","doi":"10.1016/j.aml.2025.109846","DOIUrl":"10.1016/j.aml.2025.109846","url":null,"abstract":"<div><div>This paper investigates the blow-up problem for the <span><math><mi>k</mi></math></span>-Hessian equation with nonlinear gradient terms: <span><math><mrow><msup><mrow><mrow><mo>(</mo><mi>γ</mi><mo>+</mo><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow><mrow><mi>k</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow></msup><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mrow><mo>(</mo><mo>ln</mo><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>β</mi></mrow></msup><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mi>z</mi><mo>∈</mo><mi>D</mi><mo>,</mo><mspace></mspace><mspace></mspace><mi>u</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>∂</mi><mi>D</mi></mrow></msub><mo>=</mo><mo>+</mo><mi>∞</mi><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi></mrow></math></span> are nonnegative constants with <span><math><mrow><mi>β</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>D</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> <span><math><mrow><mo>(</mo><mi>N</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></math></span> is a smooth, bounded and strictly <span><math><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-convex domain, <span><math><mrow><mi>h</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> is a positive function and may be singular near <span><math><mrow><mi>∂</mi><mi>D</mi></mrow></math></span>. By the sub-supersolution method, we present the boundary behavior of large solutions to this problem. Our work essentially generalizes the relevant conclusions in Zhang and Feng (2018); Feng and Zhang (2020).</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"175 ","pages":"Article 109846"},"PeriodicalIF":2.8,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145705030","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 : 2025-12-04DOI: 10.1016/j.aml.2025.109842
Yujia Zhang , Xin Wu , Zhaohai Ma
This study is devoted to proving the exponential stability of traveling wave solutions in a scalar age-structured model with spatial diffusion. By employing a comparison principle coupled with a weighted-energy approach, we demonstrate that traveling wave solutions are exponentially stable. This analytical conclusion is validated through numerical simulations.
{"title":"Exponential stability of traveling waves for a scalar age-structured equation","authors":"Yujia Zhang , Xin Wu , Zhaohai Ma","doi":"10.1016/j.aml.2025.109842","DOIUrl":"10.1016/j.aml.2025.109842","url":null,"abstract":"<div><div>This study is devoted to proving the exponential stability of traveling wave solutions in a scalar age-structured model with spatial diffusion. By employing a comparison principle coupled with a weighted-energy approach, we demonstrate that traveling wave solutions are exponentially stable. This analytical conclusion is validated through numerical simulations.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"175 ","pages":"Article 109842"},"PeriodicalIF":2.8,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145689357","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 : 2025-12-04DOI: 10.1016/j.aml.2025.109841
Henrik Garde
This short note modifies a reconstruction method by the author Garde (2020), for reconstructing piecewise constant conductivities in the Calderón problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.
{"title":"Reconstruction in the Calderón problem on a fixed partition from finite and partial boundary data","authors":"Henrik Garde","doi":"10.1016/j.aml.2025.109841","DOIUrl":"10.1016/j.aml.2025.109841","url":null,"abstract":"<div><div>This short note modifies a reconstruction method by the author Garde (2020), for reconstructing piecewise constant conductivities in the Calderón problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"175 ","pages":"Article 109841"},"PeriodicalIF":2.8,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145665711","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 : 2025-11-28DOI: 10.1016/j.aml.2025.109834
Paola Loreti, Daniela Sforza
In this paper, we address the question of estimating the energy decay of integrodifferential evolution equations with glassy memory. This class of memory kernel was not analyzed in previous studies. Moreover, a detailed analysis provides an explicit estimate of the connection between the kernel function’s decay constant and the energy’s decay constant.
{"title":"Energy decay for evolution equations with glassy type memory","authors":"Paola Loreti, Daniela Sforza","doi":"10.1016/j.aml.2025.109834","DOIUrl":"10.1016/j.aml.2025.109834","url":null,"abstract":"<div><div>In this paper, we address the question of estimating the energy decay of integrodifferential evolution equations with glassy memory. This class of memory kernel was not analyzed in previous studies. Moreover, a detailed analysis provides an explicit estimate of the connection between the kernel function’s decay constant and the energy’s decay constant.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"175 ","pages":"Article 109834"},"PeriodicalIF":2.8,"publicationDate":"2025-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145613965","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 : 2025-11-27DOI: 10.1016/j.aml.2025.109833
Chia-Yu Hsieh , Yongting Huang , Jiaqi Ren
We consider the Poisson–Nernst–Planck–Fourier system for the non-isothermal ionic transport. With the presence of permanent charges, the system admits nonconstant equilibria. In this paper, we prove the global well-posedness around nonconstant equilibria of the system.
{"title":"Global existence of solutions to the Poisson–Nernst–Planck–Fourier system near nonconstant equilibria","authors":"Chia-Yu Hsieh , Yongting Huang , Jiaqi Ren","doi":"10.1016/j.aml.2025.109833","DOIUrl":"10.1016/j.aml.2025.109833","url":null,"abstract":"<div><div>We consider the Poisson–Nernst–Planck–Fourier system for the non-isothermal ionic transport. With the presence of permanent charges, the system admits nonconstant equilibria. In this paper, we prove the global well-posedness around nonconstant equilibria of the system.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"174 ","pages":"Article 109833"},"PeriodicalIF":2.8,"publicationDate":"2025-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145608812","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 : 2025-11-25DOI: 10.1016/j.aml.2025.109832
Xudong Shang
In this paper, we consider the following Choquard equation involving the -Laplacian operator where , , and is the Riesz potential of order . By using the Ekeland variational principle and the implicit function theorem, we obtain the problem has a radial nodal solution for . For the case of , we employ the least energy radial nodal solution of pass to a limit procedure to obtain our result. This article extends some results of related literatures.
{"title":"Nodal solutions for a Choquard equation involving the p-Laplacian operator","authors":"Xudong Shang","doi":"10.1016/j.aml.2025.109832","DOIUrl":"10.1016/j.aml.2025.109832","url":null,"abstract":"<div><div>In this paper, we consider the following Choquard equation involving the <span><math><mi>p</mi></math></span>-Laplacian operator <span><span><span><math><mrow><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>u</mi><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>∗</mo><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi></mrow></msup><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mn>2</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>p</mi><mo>≤</mo><mi>q</mi><mo><</mo><mfrac><mrow><mi>p</mi><mrow><mo>(</mo><mi>N</mi><mo>+</mo><mi>α</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mi>p</mi><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>, and <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> is the Riesz potential of order <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mi>p</mi><mo>)</mo></mrow></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span>. By using the Ekeland variational principle and the implicit function theorem, we obtain the problem has a radial nodal solution for <span><math><mrow><mi>q</mi><mo>∈</mo><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mfrac><mrow><mi>p</mi><mrow><mo>(</mo><mi>N</mi><mo>+</mo><mi>α</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mi>p</mi><mo>)</mo></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></math></span>. For the case of <span><math><mrow><mi>q</mi><mo>=</mo><mi>p</mi></mrow></math></span>, we employ the least energy radial nodal solution of <span><math><mrow><mi>q</mi><mo>></mo><mi>p</mi></mrow></math></span> pass to a limit procedure <span><math><mrow><mi>q</mi><mo>→</mo><mi>p</mi></mrow></math></span> to obtain our result. This article extends some results of related literatures.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"174 ","pages":"Article 109832"},"PeriodicalIF":2.8,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145593474","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 : 2025-11-25DOI: 10.1016/j.aml.2025.109831
Yadong Zhong, Jingjing Ge, Yi Zhang
Through a direct semi-discretization procedure, we construct a discrete version of the Kuralay-II equation. By employing the Darboux transformation method, we derive multi-soliton solutions for the resulting discrete system. Finally, we also investigate the positon solution of the discrete equation and perform a comprehensive graphical analysis to illustrate its dynamic behavior.
{"title":"Integrable semi-discretization of the Kuralay-II equation and its positon solutions","authors":"Yadong Zhong, Jingjing Ge, Yi Zhang","doi":"10.1016/j.aml.2025.109831","DOIUrl":"10.1016/j.aml.2025.109831","url":null,"abstract":"<div><div>Through a direct semi-discretization procedure, we construct a discrete version of the Kuralay-II equation. By employing the Darboux transformation method, we derive multi-soliton solutions for the resulting discrete system. Finally, we also investigate the positon solution of the discrete equation and perform a comprehensive graphical analysis to illustrate its dynamic behavior.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"174 ","pages":"Article 109831"},"PeriodicalIF":2.8,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145592979","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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