Pub Date : 2024-07-18DOI: 10.1134/s0965542524700532
M. I. Boldyrev
Abstract
Turbulent kinetic energy (TKE) is taken into account in the approximate HLLC Riemann solver. The Euler equations are supplemented with a hyperbolic equation for TKE, and turbulent pressure is taken into account in the momentum and energy balance equations. The Jacobian of this system of equations and its eigenvalues are found, which are used to modify the HLLC solver. The validity of TKE allowance in the modified HLLC Riemann solver is verified by solving Sod’s problem. It is shown that the scheme is unstable at high turbulent pressure if turbulence is ignored in the computation of characteristic velocities.
{"title":"Turbulent Kinetic Energy in an Approximate Riemann Solver","authors":"M. I. Boldyrev","doi":"10.1134/s0965542524700532","DOIUrl":"https://doi.org/10.1134/s0965542524700532","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Turbulent kinetic energy (TKE) is taken into account in the approximate HLLC Riemann solver. The Euler equations are supplemented with a hyperbolic equation for TKE, and turbulent pressure is taken into account in the momentum and energy balance equations. The Jacobian of this system of equations and its eigenvalues are found, which are used to modify the HLLC solver. The validity of TKE allowance in the modified HLLC Riemann solver is verified by solving Sod’s problem. It is shown that the scheme is unstable at high turbulent pressure if turbulence is ignored in the computation of characteristic velocities.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738282","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: 10.1134/s0965542524700520
A. Darya, N. Taghizadeh
Abstract
In this paper, we investigate some boundary value problems for the Cauchy–Riemann equations in the lens domain M. We apply the parqueting-reflection method for the domain to achieve the points of the complex plane. Then the Schwarz representation formula is constructed by the C-auchy–Pompeiu formula and an explicit solution for the Schwarz boundary value problem for the inhomogeneous Cauchy–Riemann equation on the domain is presented. We also discuss about the condition of solvability and by using the Schwarz boundary value problem, the homogeneous Ne-umann and the inhomogeneous Dirichlet boundary value problems are investigated.
{"title":"Three Boundary Value Problems for Complex Partial Differential Equations in the Lens Domain","authors":"A. Darya, N. Taghizadeh","doi":"10.1134/s0965542524700520","DOIUrl":"https://doi.org/10.1134/s0965542524700520","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper, we investigate some boundary value problems for the Cauchy–Riemann equations in the lens domain <i>M</i>. We apply the parqueting-reflection method for the domain to achieve the points of the complex plane. Then the Schwarz representation formula is constructed by the C-auchy–Pompeiu formula and an explicit solution for the Schwarz boundary value problem for the inhomogeneous Cauchy–Riemann equation on the domain is presented. We also discuss about the condition of solvability and by using the Schwarz boundary value problem, the homogeneous Ne-umann and the inhomogeneous Dirichlet boundary value problems are investigated.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738215","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: 10.1134/s0965542524700404
I. V. Voronin
Abstract
Asymptotic formulas are derived that admit a uniform estimate of the remainder for Toeplitz matrices of size (n) as (n to infty ) in the case when their symbol (a(t)) has the form (a(t) = (t - 2{{a}_{0}} + {{t}^{{ - 1}}}{{)}^{3}}). This result is a generalization of the result of Stukopin et al. (2021), who obtained similar asymptotic formulas for a seven-diagonal Toeplitz matrix with a similar symbol in the case ({{a}_{0}} = 1). The resulting formulas are of high computational efficiency and generalize the classical results of Parter and Widom on asymptotics of extreme eigenvalues.
{"title":"On Asymptotics of Eigenvalues of Seven-Diagonal Toeplitz Matrices","authors":"I. V. Voronin","doi":"10.1134/s0965542524700404","DOIUrl":"https://doi.org/10.1134/s0965542524700404","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Asymptotic formulas are derived that admit a uniform estimate of the remainder for Toeplitz matrices of size <span>(n)</span> as <span>(n to infty )</span> in the case when their symbol <span>(a(t))</span> has the form <span>(a(t) = (t - 2{{a}_{0}} + {{t}^{{ - 1}}}{{)}^{3}})</span>. This result is a generalization of the result of Stukopin et al. (2021), who obtained similar asymptotic formulas for a seven-diagonal Toeplitz matrix with a similar symbol in the case <span>({{a}_{0}} = 1)</span>. The resulting formulas are of high computational efficiency and generalize the classical results of Parter and Widom on asymptotics of extreme eigenvalues.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738202","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: 10.1134/s0965542524700465
M. Yu. Vatolkin
Abstract
A two-point (left( {n - 1,1} right))-type boundary value problem is investigated for the representation of eigenfunctions in the form of scalar series under the assumption that there is a functional (tilde {ell }), concentrated at one point, such that the first (n - 1) original boundary conditions and (tilde {ell }x = 1) turn into Cauchy conditions at this point. The eigenfunction of the boundary value problem under consideration, corresponding to the eigenvalue ({{lambda }_{ * }},) is presented by an expansion in powers of ({{lambda }_{ * }}.) The equation (Phi (lambda ) = 0,) where (Phi (lambda )) is the sum of the power series in (lambda ,) for finding the eigenvalues of the original problem is considered. Examples of calculating the first eigenvalue of some boundary value problems are given. Various estimates for the coefficients of such power series are obtained. A function of two variables (t) and (lambda ) is determined, and a partial differential equation with conditions for this function are obtained. The zeros of the “section” of this function coincide with the eigenvalues of the original boundary value problem, which can be used for their approximate calculation.
{"title":"On the Approximation of the First Eigenvalue of Some Boundary Value Problems","authors":"M. Yu. Vatolkin","doi":"10.1134/s0965542524700465","DOIUrl":"https://doi.org/10.1134/s0965542524700465","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A two-point <span>(left( {n - 1,1} right))</span>-type boundary value problem is investigated for the representation of eigenfunctions in the form of scalar series under the assumption that there is a functional <span>(tilde {ell })</span>, concentrated at one point, such that the first <span>(n - 1)</span> original boundary conditions and <span>(tilde {ell }x = 1)</span> turn into Cauchy conditions at this point. The eigenfunction of the boundary value problem under consideration, corresponding to the eigenvalue <span>({{lambda }_{ * }},)</span> is presented by an expansion in powers of <span>({{lambda }_{ * }}.)</span> The equation <span>(Phi (lambda ) = 0,)</span> where <span>(Phi (lambda ))</span> is the sum of the power series in <span>(lambda ,)</span> for finding the eigenvalues of the original problem is considered. Examples of calculating the first eigenvalue of some boundary value problems are given. Various estimates for the coefficients of such power series are obtained. A function of two variables <span>(t)</span> and <span>(lambda )</span> is determined, and a partial differential equation with conditions for this function are obtained. The zeros of the “section” of this function coincide with the eigenvalues of the original boundary value problem, which can be used for their approximate calculation.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738211","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: 10.1134/s0965542524700398
V. P. Varin
Abstract
Computations on a computer with a floating point arithmetic are always approximate. Conversely, computations with the rational arithmetic (in a computer algebra system, for example) are always absolutely exact and reproducible both on other computers and (theoretically) by hand. Consequently, these computations can be demonstrative in a sense that a proof obtained with their help is no different from a traditional one (computer assisted proof). However, usually such computations are impossible in a sufficiently complicated problem due to limitations on resources of memory and time. We propose a mechanism of rounding off rational numbers in computations with rational arithmetic, which solves this problem (of resources), i.e., computations can still be demonstrative but do not require unbounded resources. We give some examples of implementation of standard numerical algorithms with this arithmetic. The results have applications to analytical number theory.
{"title":"Rational Arithmetic with a Round-Off","authors":"V. P. Varin","doi":"10.1134/s0965542524700398","DOIUrl":"https://doi.org/10.1134/s0965542524700398","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Computations on a computer with a floating point arithmetic are always approximate. Conversely, computations with the rational arithmetic (in a computer algebra system, for example) are always absolutely exact and reproducible both on other computers and (theoretically) by hand. Consequently, these computations can be demonstrative in a sense that a proof obtained with their help is no different from a traditional one (computer assisted proof). However, usually such computations are impossible in a sufficiently complicated problem due to limitations on resources of memory and time. We propose a mechanism of rounding off rational numbers in computations with rational arithmetic, which solves this problem (of resources), i.e., computations can still be demonstrative but do not require unbounded resources. We give some examples of implementation of standard numerical algorithms with this arithmetic. The results have applications to analytical number theory.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738204","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: 10.1134/s096554252470043x
V. I. Elkin
Abstract
Symmetries of partial differential equations are examined by applying differential-geometric and algebraic methods of the theory of dynamical systems with control.
摘要 通过应用带控制的动力系统理论中的微分几何和代数方法，研究了偏微分方程的对称性。
{"title":"Symmetries and Decomposition of Systems of Partial Differential Equations and Control Systems with Distributed Parameters","authors":"V. I. Elkin","doi":"10.1134/s096554252470043x","DOIUrl":"https://doi.org/10.1134/s096554252470043x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Symmetries of partial differential equations are examined by applying differential-geometric and algebraic methods of the theory of dynamical systems with control.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738207","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: 10.1134/s0965542524700477
S. L. Skorokhodov, N. P. Kuzmina
Abstract
A new efficient analytical-numerical method is developed for solving a problem for the potential vorticity equation in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum. The method is used to analyze small perturbations of ocean currents of finite transverse scale with a general parabolic vertical profile of velocity. For the arising spectral non-self-adjoint problem, asymptotic expansions of the eigenfunctions and eigenvalues are constructed for small wave numbers (k) and the existence of a countable set of complex eigenvalues with an unboundedly decreasing imaginary part is shown. On the integration interval (z in [ - 1,1]), a system of three neighborhoods is introduced and a solution in each of them is constructed in the form of power series expansions, which are matched smoothly, so that the eigenfunctions and eigenvalues are efficiently calculated with high accuracy. For a varying wave number (k), the trajectories of complex eigenvalues are computed for various parameters of the problem and the existence of double eigenvalues is shown. The complex picture of instability developing in the simulated flow depending on physical parameters of the problem is briefly described.
摘要 开发了一种新的高效分析-数值方法，用于求解准地转近似的势涡度方程问题，并考虑了质量和动量的垂直扩散。该方法用于分析具有一般抛物线速度垂直剖面的有限横向尺度洋流的小扰动。对于所产生的谱非自交问题，构建了小波数 (k)的特征函数和特征值的渐近展开，并证明了存在一组虚部无限制递减的复特征值。在积分区间 (z 在 [ - 1,1]) 上，引入了一个由三个邻域组成的系统，并以幂级数展开的形式构建了每个邻域中的解，这些解平滑匹配，从而高效、高精度地计算出特征函数和特征值。对于变化的波数 (k)，计算了问题的各种参数的复特征值轨迹，并显示了双特征值的存在。简述了模拟流动中不稳定性发展的复杂情况，这取决于问题的物理参数。
{"title":"Analytical-Numerical Method for Solving the Spectral Problem in a Model of Geostrophic Ocean Currents","authors":"S. L. Skorokhodov, N. P. Kuzmina","doi":"10.1134/s0965542524700477","DOIUrl":"https://doi.org/10.1134/s0965542524700477","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A new efficient analytical-numerical method is developed for solving a problem for the potential vorticity equation in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum. The method is used to analyze small perturbations of ocean currents of finite transverse scale with a general parabolic vertical profile of velocity. For the arising spectral non-self-adjoint problem, asymptotic expansions of the eigenfunctions and eigenvalues are constructed for small wave numbers <span>(k)</span> and the existence of a countable set of complex eigenvalues with an unboundedly decreasing imaginary part is shown. On the integration interval <span>(z in [ - 1,1])</span>, a system of three neighborhoods is introduced and a solution in each of them is constructed in the form of power series expansions, which are matched smoothly, so that the eigenfunctions and eigenvalues are efficiently calculated with high accuracy. For a varying wave number <span>(k)</span>, the trajectories of complex eigenvalues are computed for various parameters of the problem and the existence of double eigenvalues is shown. The complex picture of instability developing in the simulated flow depending on physical parameters of the problem is briefly described.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738212","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: 10.1134/s0965542524700556
A. V. Kalinin, A. A. Tyukhtina, S. A. Malov
Abstract
Formulations of initial-boundary value problems for the system of Maxwell’s equations in various quasi-stationary approximations in homogeneous and inhomogeneous conducting media are considered. In the case of weakly inhomogeneous media, asymptotic expansions of solutions to the considered initial-boundary value problems in a parameter characterizing the degree of inhomogeneity of the medium are formulated and substantiated. It is shown that the construction of an asymptotic expansion for the quasi-stationary electromagnetic approximation leads to successively solving independent problems for the quasi-stationary electric and quasi-stationary magnetic approximations in a homogeneous medium. Conditions for the initial data providing the convergence of the asymptotic series are given.
{"title":"Problems of Determining Quasi-Stationary Electromagnetic Fields in Weakly Inhomogeneous Media","authors":"A. V. Kalinin, A. A. Tyukhtina, S. A. Malov","doi":"10.1134/s0965542524700556","DOIUrl":"https://doi.org/10.1134/s0965542524700556","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Formulations of initial-boundary value problems for the system of Maxwell’s equations in various quasi-stationary approximations in homogeneous and inhomogeneous conducting media are considered. In the case of weakly inhomogeneous media, asymptotic expansions of solutions to the considered initial-boundary value problems in a parameter characterizing the degree of inhomogeneity of the medium are formulated and substantiated. It is shown that the construction of an asymptotic expansion for the quasi-stationary electromagnetic approximation leads to successively solving independent problems for the quasi-stationary electric and quasi-stationary magnetic approximations in a homogeneous medium. Conditions for the initial data providing the convergence of the asymptotic series are given.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738285","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: 10.1134/s0965542524700568
E. V. Laskovets
Abstract
A mathematical model is proposed that describes the flow of a thin layer of liquid along an inclined unevenly heated substrate. The governing equations are the Navier–Stokes equations for a viscous incompressible liquid and relations representing generalized kinematic, dynamic, and energy conditions on the interface for the case of evaporation. The formulation is given in the two-dimensional case for large Reynolds numbers. The problem is solved within the framework of the long-wave approximation. A parametric analysis of the problem is carried out, and an evolutionary equation is derived to find the thickness of the liquid layer. An algorithm for a numerical solution is proposed for the problem of periodic flow of liquid along an inclined substrate. The influence of gravitational effects and the nature of heating of the solid substrate on the flow of the liquid layer is studied.
{"title":"Numerical Simulation of Convective Flows in a Thin Liquid Layer at Large Reynolds Numbers","authors":"E. V. Laskovets","doi":"10.1134/s0965542524700568","DOIUrl":"https://doi.org/10.1134/s0965542524700568","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A mathematical model is proposed that describes the flow of a thin layer of liquid along an inclined unevenly heated substrate. The governing equations are the Navier–Stokes equations for a viscous incompressible liquid and relations representing generalized kinematic, dynamic, and energy conditions on the interface for the case of evaporation. The formulation is given in the two-dimensional case for large Reynolds numbers. The problem is solved within the framework of the long-wave approximation. A parametric analysis of the problem is carried out, and an evolutionary equation is derived to find the thickness of the liquid layer. An algorithm for a numerical solution is proposed for the problem of periodic flow of liquid along an inclined substrate. The influence of gravitational effects and the nature of heating of the solid substrate on the flow of the liquid layer is studied.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738287","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: 10.1134/s0965542524700519
A. Liubavin, Mingkang Ni
Abstract
This article is considering the stability property of the solution with inner layer for singularly perturbed stationary equation with Neumann boundary conditions. The right-hand side is assumed to have discontinuity on some arbitrary curve (h(t)). Stability analysis is performed by obtaining the first non-zero coefficient of the series for eigenvalue and eigenfunction from the Sturm–Liouville problem. Theory of the asymptotic approximations is used in order to construct them.
{"title":"Application of Asymptotic Methods to the Question of Stability in Stationary Solution with Discontinuity on a Curve","authors":"A. Liubavin, Mingkang Ni","doi":"10.1134/s0965542524700519","DOIUrl":"https://doi.org/10.1134/s0965542524700519","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>This article is considering the stability property of the solution with inner layer for singularly perturbed stationary equation with Neumann boundary conditions. The right-hand side is assumed to have discontinuity on some arbitrary curve <span>(h(t))</span>. Stability analysis is performed by obtaining the first non-zero coefficient of the series for eigenvalue and eigenfunction from the Sturm–Liouville problem. Theory of the asymptotic approximations is used in order to construct them.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745846","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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