Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a6
Wansheng Wang, Mengli Mao, Zifeng Li
We develop a class of implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving parabolic integro-differential equations (PIDEs) with nonsmooth initial data, which describe several option pricing models in mathematical finance. Different from the usual IMEX RK methods, the proposed methods approximate the integral term explicitly by using an extrapolation operator based on the stage-values of RK methods, and we call them as IMEX stage-based interpolation RK (SBIRK) methods. It is shown that there exist arbitrarily high order IMEX SBIRK methods which are stable for abstract PIDEs under suitable time step restrictions. The consistency error and the global error bounds for this class of IMEX Runge-Kutta methods are derived for abstract PIDEs with nonsmooth initial data. The related higher time regularity analysis of the exact solution and stability estimates for IMEX SBIRK methods play key roles in deriving these error bounds. Numerical experiments for European options under jump-diffusion models and stochastic volatility model with jump verify and complement our theoretical results.
{"title":"IMEX variable step-size Runge-Kutta methods for parabolic integro-differential equations with nonsmooth initial data","authors":"Wansheng Wang, Mengli Mao, Zifeng Li","doi":"10.4310/cms.2024.v22.n6.a6","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a6","url":null,"abstract":"We develop a class of implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving parabolic integro-differential equations (PIDEs) with nonsmooth initial data, which describe several option pricing models in mathematical finance. Different from the usual IMEX RK methods, the proposed methods approximate the integral term explicitly by using an extrapolation operator based on the stage-values of RK methods, and we call them as IMEX stage-based interpolation RK (SBIRK) methods. It is shown that there exist arbitrarily high order IMEX SBIRK methods which are stable for abstract PIDEs under suitable time step restrictions. The consistency error and the global error bounds for this class of IMEX Runge-Kutta methods are derived for abstract PIDEs with nonsmooth initial data. The related higher time regularity analysis of the exact solution and stability estimates for IMEX SBIRK methods play key roles in deriving these error bounds. Numerical experiments for European options under jump-diffusion models and stochastic volatility model with jump verify and complement our theoretical results.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"130 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a2
Jean-Marc Hérard
We focus here on the relaxation process in a class of two-phase flow models, considering first gas-liquid flows, and then liquid-vapour mixtures. The whole analysis enables to exhibit a few conditions on the flow in order to guarantee the time decay of some variables. The former may depend on initial conditions but also on equations of state within each phase. The present analysis aims at providing some better understanding of inner processes, and it is also useful for numerical purposes, as emphasized in appendix B. It is a sequel of paper [J.M. Hérard and G. Jomée, ESAIM Proc. Surv., 72:19–40, 2023] where the sole pressure relaxation process in some multiphase flow models is investigated.
在此，我们重点研究一类两相流模型的弛豫过程，首先考虑气-液流动，然后考虑液-汽混合物。通过整个分析，我们可以对流动提出一些条件，以保证某些变量的时间衰减。前者可能取决于初始条件，但也取决于各相内的状态方程。正如附录 B 所强调的，本分析旨在提供对内在过程的更好理解，同时也有助于数值计算。这是论文[J.M. Hérard 和 G. Jomée，ESAIM Proc. Surv.，72:19-40，2023]的续篇，其中研究了一些多相流模型中的唯一压力松弛过程。
{"title":"A note on the relaxation process in a class of non-equilibrium two-phase flow models","authors":"Jean-Marc Hérard","doi":"10.4310/cms.2024.v22.n6.a2","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a2","url":null,"abstract":"We focus here on the relaxation process in a class of two-phase flow models, considering first gas-liquid flows, and then liquid-vapour mixtures. The whole analysis enables to exhibit a few conditions on the flow in order to guarantee the time decay of some variables. The former may depend on initial conditions but also on equations of state within each phase. The present analysis aims at providing some better understanding of inner processes, and it is also useful for numerical purposes, as emphasized in appendix B. It is a sequel of paper [J.M. Hérard and G. Jomée, ESAIM Proc. Surv., 72:19–40, 2023] where the sole pressure relaxation process in some multiphase flow models is investigated.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"31 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a4
Xueting Jin, Quansen Jiu
In this paper, we will apply the Fourier multiplier method to explore the stability for the 2D micropolar equations with partial dissipation near Couette flow. The difficulty will be encountered due to the facts that one order derivative of the microtation appears on the right term of velocity equations and that the velocity equations only have vertical dissipation. To overcome the difficulty, we will make use of a Fourier multiplier to grasp the enhanced dissipation created by the special structure $ypartial_x-nupartial_{y}^2$ and obtain some new and higher-order estimates of the solution in an elegant way. Also, a time-dependent elliptic operator $Lambda_t^b$ which commutes with linear part of the equations will be used to make our proof more clear.
{"title":"Stability for the 2D Micropolar equations with partial dissipation near Couette flow","authors":"Xueting Jin, Quansen Jiu","doi":"10.4310/cms.2024.v22.n6.a4","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a4","url":null,"abstract":"In this paper, we will apply the Fourier multiplier method to explore the stability for the 2D micropolar equations with partial dissipation near Couette flow. The difficulty will be encountered due to the facts that one order derivative of the microtation appears on the right term of velocity equations and that the velocity equations only have vertical dissipation. To overcome the difficulty, we will make use of a Fourier multiplier to grasp the enhanced dissipation created by the special structure $ypartial_x-nupartial_{y}^2$ and obtain some new and higher-order estimates of the solution in an elegant way. Also, a time-dependent elliptic operator $Lambda_t^b$ which commutes with linear part of the equations will be used to make our proof more clear.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"22 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a11
Xingyu Li, Marius Paicu, Arghir Zarnescu
We study the scaled anisotropic co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in dimension two. We prove the global well-posedness with small analytic data in a thin strip domain. Moreover, we justify the limit to a system involving the hydrostatic Navier-Stokes system with analytic data and prove the convergence.
{"title":"The hydrostatic limit of the Beris-Edwards system in dimension two","authors":"Xingyu Li, Marius Paicu, Arghir Zarnescu","doi":"10.4310/cms.2024.v22.n6.a11","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a11","url":null,"abstract":"We study the scaled anisotropic co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in dimension two. We prove the global well-posedness with small analytic data in a thin strip domain. Moreover, we justify the limit to a system involving the hydrostatic Navier-Stokes system with analytic data and prove the convergence.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"28 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a7
Paolo Antonelli, Boris Shakarov
We study a dissipative variant of the Gross-Pitaevskii equation with rotation. The model contains a nonlocal, nonlinear term that forces the conservation of $L^2$-norm of solutions. We are motivated by several physical experiments and numerical simulations studying the formation of vortices in Bose-Einstein condensates.We show local and global well-posedness of this model and investigate the asymptotic behavior of its solutions. In the linear case, the solution asymptotically tends to the eigenspace associated with the smallest eigenvalue in the decomposition of the initial datum. In the nonlinear case, we obtain weak convergence to a stationary state. Moreover, for initial energies in a specific range, we prove strong asymptotic stability of ground state solutions.
{"title":"Existence and large time behavior for a dissipative variant of the rotational NLS equation","authors":"Paolo Antonelli, Boris Shakarov","doi":"10.4310/cms.2024.v22.n6.a7","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a7","url":null,"abstract":"We study a dissipative variant of the Gross-Pitaevskii equation with rotation. The model contains a nonlocal, nonlinear term that forces the conservation of $L^2$-norm of solutions. We are motivated by several physical experiments and numerical simulations studying the formation of vortices in Bose-Einstein condensates.We show local and global well-posedness of this model and investigate the asymptotic behavior of its solutions. In the linear case, the solution asymptotically tends to the eigenspace associated with the smallest eigenvalue in the decomposition of the initial datum. In the nonlinear case, we obtain weak convergence to a stationary state. Moreover, for initial energies in a specific range, we prove strong asymptotic stability of ground state solutions.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"22 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a9
Jian Dong, Dingfang Li
This paper aims to propose a well-balanced positivity-preserving numerical scheme for the two-layer shallow water systems with arbitrary wet-dry fronts based on interface hydrostatic reconstructions (IHR). One key difficulty in solving the two-layer shallow water systems is the nonconservative product term which cannot be evaluated on the cell boundaries. Another difficulty is that the well-balanced property for the still water maybe missed when the computational domain has wet-dry fronts, especially, the wet-dry front is located at the discontinuous bottom topography. For the nonlinear stability of the numerical scheme, the positivity of the water height is vital. To this end, we discretize the nonconservative product term based on the IHR method, which is a particular choice of path-conservative methods. The intermediate bottom level used in the discretization of the bed source term of two layers is different. The nonconservative product term due to the momentum exchange between two layers is discretized using the intermediate interface water height. The resulting numerical scheme can preserve the positivity of two-layered heights and maintain the still water even when the computational domain has wet-dry fronts. The numerical scheme performs well in solving the complex problems, such as the Kelvin-Helmholtz instable problems. We demonstrate these properties of the current scheme through several classical problems of the two-layer shallow water systems with arbitrary wet-dry fronts.
本文旨在基于界面静水重构（IHR），为具有任意干湿前沿的两层浅水系统提出一种平衡良好的保正值数值方案。求解两层浅水系统的一个主要困难是单元边界上无法求值的非保守乘积项。另一个难点是，当计算域有干湿前沿时，尤其是干湿前沿位于不连续的底部地形时，可能会忽略静水的良好平衡特性。对于数值方案的非线性稳定性而言，水高的正向性至关重要。为此，我们采用 IHR 方法对非保守乘积项进行离散化处理，该方法是路径保守方法的一种特殊选择。两层床源项离散化所使用的中间底面是不同的。两层之间动量交换引起的非保守乘积项使用中间界面水高进行离散化。由此产生的数值方案可以保持两层高度的正向性，并在计算域出现干湿前沿时仍能保持静水。该数值方案在解决复杂问题（如开尔文-赫尔姆霍兹不稳定问题）时表现出色。我们通过几个具有任意干湿锋的两层浅水系统的经典问题来证明当前方案的这些特性。
{"title":"An efficient interface hydrostatic reconstruction for the two-layer shallow flows with arbitrary wet-dry fronts","authors":"Jian Dong, Dingfang Li","doi":"10.4310/cms.2024.v22.n6.a9","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a9","url":null,"abstract":"This paper aims to propose a well-balanced positivity-preserving numerical scheme for the two-layer shallow water systems with arbitrary wet-dry fronts based on interface hydrostatic reconstructions (IHR). One key difficulty in solving the two-layer shallow water systems is the nonconservative product term which cannot be evaluated on the cell boundaries. Another difficulty is that the well-balanced property for the still water maybe missed when the computational domain has wet-dry fronts, especially, the wet-dry front is located at the discontinuous bottom topography. For the nonlinear stability of the numerical scheme, the positivity of the water height is vital. To this end, we discretize the nonconservative product term based on the IHR method, which is a particular choice of path-conservative methods. The intermediate bottom level used in the discretization of the bed source term of two layers is different. The nonconservative product term due to the momentum exchange between two layers is discretized using the intermediate interface water height. The resulting numerical scheme can preserve the positivity of two-layered heights and maintain the still water even when the computational domain has wet-dry fronts. The numerical scheme performs well in solving the complex problems, such as the Kelvin-Helmholtz instable problems. We demonstrate these properties of the current scheme through several classical problems of the two-layer shallow water systems with arbitrary wet-dry fronts.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"15 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a10
Daniel Coble, Siming He
This paper explores the phenomena of enhanced dissipation in solutions to the passive scalar equations subject to time-dependent shear flows. The hypocoercivity functionals with carefully tuned time weights are applied in the analysis. We observe that as long as the critical points of the shear flow vary slowly, one can derive the sharp enhanced dissipation estimates, mirroring the ones obtained for the time-stationary case.
{"title":"A note on enhanced dissipation of time-dependent shear flows","authors":"Daniel Coble, Siming He","doi":"10.4310/cms.2024.v22.n6.a10","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a10","url":null,"abstract":"This paper explores the phenomena of enhanced dissipation in solutions to the passive scalar equations subject to time-dependent shear flows. The hypocoercivity functionals with carefully tuned time weights are applied in the analysis. We observe that as long as the critical points of the shear flow vary slowly, one can derive the sharp enhanced dissipation estimates, mirroring the ones obtained for the time-stationary case.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"13 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a8
Esteban G. Tabak, Giulio Trigila, Wenjun Zhao
A family of normalizing flows is introduced for selectively removing from a data set the variability attributable to a specific set of cofactors, while preserving the dependence on others. This is achieved by extending the barycenter problem of optimal transport theory to the newly introduced conditional barycenter problem. Rather than summarizing the data with a single probability distribution, as in the classical barycenter problem, the conditional barycenter is represented by a family of distributions labeled by the cofactors kept. The use of the conditional barycenter and its differences with the classical barycenter are illustrated on synthetic and real data addressing treatment effect estimation, super-resolution, anomaly detection and lightness transfer in image analysis.
{"title":"The conditional barycenter problem, its data-driven formulation and its solution through normalizing flows","authors":"Esteban G. Tabak, Giulio Trigila, Wenjun Zhao","doi":"10.4310/cms.2024.v22.n6.a8","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a8","url":null,"abstract":"A family of normalizing flows is introduced for selectively removing from a data set the variability attributable to a specific set of cofactors, while preserving the dependence on others. This is achieved by extending the barycenter problem of optimal transport theory to the newly introduced conditional barycenter problem. Rather than summarizing the data with a single probability distribution, as in the classical barycenter problem, the conditional barycenter is represented by a family of distributions labeled by the cofactors kept. The use of the conditional barycenter and its differences with the classical barycenter are illustrated on synthetic and real data addressing treatment effect estimation, super-resolution, anomaly detection and lightness transfer in image analysis.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a3
Yeping Li, Yujie Qian, Rong Yin
In this paper, we are concerned with the inflow problem on the half line $(0,+infty)$ for a one-dimensional compressible Navier-Stokes-Korteweg system, which is used to model compressible viscous fluids with internal capillarity, i.e., the liquid-vapor mixtures with phase interfaces. We first investigate that the asymptotic profile is a nonlinear wave: the superposition wave of a rarefaction wave and a boundary layer solution under the proper condition of the far fields and boundary values. The asymptotic stability on the nonlinear wave is shown under some conditions that the initial data are a small perturbation of the rarefaction wave and the strength of the stationary wave is small enough. The proofs are given by an elementary energy method.
{"title":"Asymptotic stability of nonlinear wave for an inflow problem to the compressible Navier-Stokes-Korteweg system","authors":"Yeping Li, Yujie Qian, Rong Yin","doi":"10.4310/cms.2024.v22.n6.a3","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a3","url":null,"abstract":"In this paper, we are concerned with the inflow problem on the half line $(0,+infty)$ for a one-dimensional compressible Navier-Stokes-Korteweg system, which is used to model compressible viscous fluids with internal capillarity, i.e., the liquid-vapor mixtures with phase interfaces. We first investigate that the asymptotic profile is a nonlinear wave: the superposition wave of a rarefaction wave and a boundary layer solution under the proper condition of the far fields and boundary values. The asymptotic stability on the nonlinear wave is shown under some conditions that the initial data are a small perturbation of the rarefaction wave and the strength of the stationary wave is small enough. The proofs are given by an elementary energy method.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a13
Michael Nestler, Axel Voigt
Approximating PDEs on surfaces by the diffuse interface approach allows us to use standard numerical tools to solve these problems. This makes it an attractive numerical approach. We extend this approach to vector-valued surface PDEs and explore their convergence properties. In contrast to the well-studied case of scalar-valued surface PDEs, the optimal order of convergence can only be achieved if certain higher-order relations between mesh size and interface width are fulfilled. This difference results from the increased coupling between the surface geometry and the PDE for vector-valued quantities defined on it.
{"title":"A diffuse interface approach for vector-valued PDEs on surfaces","authors":"Michael Nestler, Axel Voigt","doi":"10.4310/cms.2024.v22.n6.a13","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a13","url":null,"abstract":"Approximating PDEs on surfaces by the diffuse interface approach allows us to use standard numerical tools to solve these problems. This makes it an attractive numerical approach. We extend this approach to vector-valued surface PDEs and explore their convergence properties. In contrast to the well-studied case of scalar-valued surface PDEs, the optimal order of convergence can only be achieved if certain higher-order relations between mesh size and interface width are fulfilled. This difference results from the increased coupling between the surface geometry and the PDE for vector-valued quantities defined on it.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"5 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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