M. Tarasevich, I. Tsybulin, V. A. Onoprienko, D. Kulyamin, E. Volodin
Abstract Modern numerical models of the Earth system are complex and inherit its natural chaotic behaviour. The numerical results depend on various specifications of the simulation process, including computing systems, compilers, etc. Due to the chaotic behaviour, these minor differences lead to significant and unpredictable deviations. Therefore, some procedure verifying that simulation results describe the behaviour of the same physical system is of practical importance. The present paper proposes a statistical verification algorithm developed for the INM RAS Earth system model. Different ensemble generation techniques and statistical estimators are evaluated for verification suitability. The ability of the method to detect the deviations in the simulation results is demonstrated on a series of cases. Practical guidelines on how to choose the perturbation amplitude for the ensemble generation are provided for various verification cases.
{"title":"Ensemble-based statistical verification of INM RAS Earth system model","authors":"M. Tarasevich, I. Tsybulin, V. A. Onoprienko, D. Kulyamin, E. Volodin","doi":"10.1515/rnam-2023-0014","DOIUrl":"https://doi.org/10.1515/rnam-2023-0014","url":null,"abstract":"Abstract Modern numerical models of the Earth system are complex and inherit its natural chaotic behaviour. The numerical results depend on various specifications of the simulation process, including computing systems, compilers, etc. Due to the chaotic behaviour, these minor differences lead to significant and unpredictable deviations. Therefore, some procedure verifying that simulation results describe the behaviour of the same physical system is of practical importance. The present paper proposes a statistical verification algorithm developed for the INM RAS Earth system model. Different ensemble generation techniques and statistical estimators are evaluated for verification suitability. The ability of the method to detect the deviations in the simulation results is demonstrated on a series of cases. Practical guidelines on how to choose the perturbation amplitude for the ensemble generation are provided for various verification cases.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49593743","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}
S. Simakov, T. Gamilov, F. Liang, P. Chomakhidze, P. Kopylov
Abstract In the present work, we construct a model of coronary flow, which utilizes both CT scans of large coronary arteries and coronary CT perfusion. The model describes pulsatile flow in the patient’s network of coronary vessels and takes into account a number of physiological effects: myocardium contractions, stenoses, impairment of microvascular perfusion. The main novelty of this model is the new smooth boundary conditions that have not been used before in patient-specific simulations of coronary circulation. New boundary conditions use 0D lumped model approach and provide asymptotic convergence of the solution for the cases of one-to-one vascular connection and bifurcation with a very thin child vessel. The new boundary conditions make it possible to estimate the fractional flow margin more accurately. We also studied sensitivity of haemodynamic indices (fractional flow reserve, coronary flow reserve, instantaneous wave-free ratio) to the variations of microcirculation impairment. No substantial difference in sensitivity was observed between new model and original approach. The advantage of the presented approach is the availability of the required data in everyday clinical practice and, thus, improved personalization of the model.
{"title":"Validation of boundary conditions for coronary circulation model based on a lumped parameter approach","authors":"S. Simakov, T. Gamilov, F. Liang, P. Chomakhidze, P. Kopylov","doi":"10.1515/rnam-2023-0013","DOIUrl":"https://doi.org/10.1515/rnam-2023-0013","url":null,"abstract":"Abstract In the present work, we construct a model of coronary flow, which utilizes both CT scans of large coronary arteries and coronary CT perfusion. The model describes pulsatile flow in the patient’s network of coronary vessels and takes into account a number of physiological effects: myocardium contractions, stenoses, impairment of microvascular perfusion. The main novelty of this model is the new smooth boundary conditions that have not been used before in patient-specific simulations of coronary circulation. New boundary conditions use 0D lumped model approach and provide asymptotic convergence of the solution for the cases of one-to-one vascular connection and bifurcation with a very thin child vessel. The new boundary conditions make it possible to estimate the fractional flow margin more accurately. We also studied sensitivity of haemodynamic indices (fractional flow reserve, coronary flow reserve, instantaneous wave-free ratio) to the variations of microcirculation impairment. No substantial difference in sensitivity was observed between new model and original approach. The advantage of the presented approach is the availability of the required data in everyday clinical practice and, thus, improved personalization of the model.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44818146","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 : 2023-06-01DOI: 10.1515/rnam-2023-frontmatter3
{"title":"Frontmatter","authors":"","doi":"10.1515/rnam-2023-frontmatter3","DOIUrl":"https://doi.org/10.1515/rnam-2023-frontmatter3","url":null,"abstract":"","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136136003","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}
T. Tran, V. Shutyaev, H. S. Hoang, Shuai Li, Chinh Kien Nguyen, Hong Phong Nguyen, Thi Thanh Huong Duong
Abstract The present study promotes a new algorithm for estimating the water pollution propagation with the primary goal of providing more reliable and high quality estimates to decision makers. To date, the widely used variational method suffers from the large computational burden, which limits its application in practice. Moreover, this method, considering the initial state as a control variable, is very sensitive in specifying initial error, especially for unstable dynamical systems. The Neural Network Filter (NNF), proposed in the present paper, is aimed at overcoming these two drawbacks in the variational method: by its nature, the NNF is sequential (no batch large assimilation window used) and stable even for unstable dynamics, with the gain parameters as control variables. The NNF, developed in the present paper, is a Neural Network Filter (NNF) version of the Singular Evolutive Interpolated Kalman Filter (SEIKF). One of the new versions of this NNF is that it uses structure of the gain of SEIKF0 taken by the SEIKF at the first time moment of correction process. To deal with the uncertainty of the system parameters and of the noise covariance, the proposed Neural Network SEIKF0 named by NNSEIKF0 makes use of the covariance of a reduced rank iterated during assimilation process and of some pertinent gain parameters tuned adaptively to yield the minimum prediction error for the system output. The computational burden in implementation of the NNSEIKF0 is reduced drastically due to applying the optimization tool known as a simultaneous perturbation stochastic approximation (SPSA) algorithm, which requires only two integrations of the numerical model. No iterative loop is required at each assimilation instant as usually happens with the standard gradient descent optimization algorithms. Data assimilation experiment, carried out by the SEIKF0 and NNSEIKF0, is implemented for the Thanh Nhan Lake in Hanoi and the performance comparison between the NNSEIKF0 and SEIKF0 is given to show the high efficiency of the proposed NNSEIKF0.
{"title":"Neural networks singular evolutive interpolated Kalman filter and its application to data assimilation for 2D water pollution model","authors":"T. Tran, V. Shutyaev, H. S. Hoang, Shuai Li, Chinh Kien Nguyen, Hong Phong Nguyen, Thi Thanh Huong Duong","doi":"10.1515/rnam-2023-0015","DOIUrl":"https://doi.org/10.1515/rnam-2023-0015","url":null,"abstract":"Abstract The present study promotes a new algorithm for estimating the water pollution propagation with the primary goal of providing more reliable and high quality estimates to decision makers. To date, the widely used variational method suffers from the large computational burden, which limits its application in practice. Moreover, this method, considering the initial state as a control variable, is very sensitive in specifying initial error, especially for unstable dynamical systems. The Neural Network Filter (NNF), proposed in the present paper, is aimed at overcoming these two drawbacks in the variational method: by its nature, the NNF is sequential (no batch large assimilation window used) and stable even for unstable dynamics, with the gain parameters as control variables. The NNF, developed in the present paper, is a Neural Network Filter (NNF) version of the Singular Evolutive Interpolated Kalman Filter (SEIKF). One of the new versions of this NNF is that it uses structure of the gain of SEIKF0 taken by the SEIKF at the first time moment of correction process. To deal with the uncertainty of the system parameters and of the noise covariance, the proposed Neural Network SEIKF0 named by NNSEIKF0 makes use of the covariance of a reduced rank iterated during assimilation process and of some pertinent gain parameters tuned adaptively to yield the minimum prediction error for the system output. The computational burden in implementation of the NNSEIKF0 is reduced drastically due to applying the optimization tool known as a simultaneous perturbation stochastic approximation (SPSA) algorithm, which requires only two integrations of the numerical model. No iterative loop is required at each assimilation instant as usually happens with the standard gradient descent optimization algorithms. Data assimilation experiment, carried out by the SEIKF0 and NNSEIKF0, is implemented for the Thanh Nhan Lake in Hanoi and the performance comparison between the NNSEIKF0 and SEIKF0 is given to show the high efficiency of the proposed NNSEIKF0.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42973539","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 : 2023-03-01DOI: 10.48550/arXiv.2303.00421
P. Vabishchevich
Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.
{"title":"Operator-difference schemes on non-uniform grids for second-order evolutionary equations","authors":"P. Vabishchevich","doi":"10.48550/arXiv.2303.00421","DOIUrl":"https://doi.org/10.48550/arXiv.2303.00421","url":null,"abstract":"Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46700237","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}
Abstract Virus infection dynamics is governed by the processes on multiple scales: on the whole organism level, tissue level, and intracellular level. In this paper, we develop a multi-scale multi-compartment model of HIV infection in a simplified setting and the computational methods for numerical realization of the model. The multiscale model describes the processes from various scales and of different nature (cell motility, virus diffusion, intracellular virus replication). Intracellular replication model is based on a Markov chain with time-inhomogeneous propensities that depend on the extracellular level of virions. Reaction diffusion equations used to model free virion diffusion in the lymphoid tissue have moving sources, which are determined by the positions of the infected cells (immune cell motility model) and the rate of virion secretion from them (intracellular model). Immune cell motility model parameterizes the intercellular interaction forces, friction and the stochastic force of active cell motility. Together, this allows for a proper description of the intracellular stochasticity that propagates across multiple scales. A hybrid discrete-continuous stochastic-deterministic algorithm for simulation of the multiscale model based on the uniformization Monte Carlo method is implemented.
{"title":"Computational methods for multiscale modelling of virus infection dynamics","authors":"D. Grebennikov","doi":"10.1515/rnam-2023-0007","DOIUrl":"https://doi.org/10.1515/rnam-2023-0007","url":null,"abstract":"Abstract Virus infection dynamics is governed by the processes on multiple scales: on the whole organism level, tissue level, and intracellular level. In this paper, we develop a multi-scale multi-compartment model of HIV infection in a simplified setting and the computational methods for numerical realization of the model. The multiscale model describes the processes from various scales and of different nature (cell motility, virus diffusion, intracellular virus replication). Intracellular replication model is based on a Markov chain with time-inhomogeneous propensities that depend on the extracellular level of virions. Reaction diffusion equations used to model free virion diffusion in the lymphoid tissue have moving sources, which are determined by the positions of the infected cells (immune cell motility model) and the rate of virion secretion from them (intracellular model). Immune cell motility model parameterizes the intercellular interaction forces, friction and the stochastic force of active cell motility. Together, this allows for a proper description of the intracellular stochasticity that propagates across multiple scales. A hybrid discrete-continuous stochastic-deterministic algorithm for simulation of the multiscale model based on the uniformization Monte Carlo method is implemented.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49427510","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}
Abstract The kinetic formulation of the model problem of plasma waves excitation by a powerful short laser pulse is numerically studied for the first time. Kinetic and simplest hydrodynamic plasma models are also compared for the problem under consideration. It is shown that the considered hydrodynamic models do not provide good approximations to the solution to the Vlasov kinetic equation, namely, one leads to discontinuous solutions and the other has a significant qualitative distinction. At a low plasma temperature, the effect of non-isothermicity of the process is small, but it can lead to significant distortions of the solution during further heating. The results obtained here imply that the first two moments of the distribution function are not enough to describe the plasma hydrodynamics; higher-order moments should be used.
{"title":"Effect of electron temperature on formation of travelling waves in plasma: Kinetic and hydrodynamic models","authors":"E. V. Chizhonkov, A. A. Frolov","doi":"10.1515/rnam-2023-0006","DOIUrl":"https://doi.org/10.1515/rnam-2023-0006","url":null,"abstract":"Abstract The kinetic formulation of the model problem of plasma waves excitation by a powerful short laser pulse is numerically studied for the first time. Kinetic and simplest hydrodynamic plasma models are also compared for the problem under consideration. It is shown that the considered hydrodynamic models do not provide good approximations to the solution to the Vlasov kinetic equation, namely, one leads to discontinuous solutions and the other has a significant qualitative distinction. At a low plasma temperature, the effect of non-isothermicity of the process is small, but it can lead to significant distortions of the solution during further heating. The results obtained here imply that the first two moments of the distribution function are not enough to describe the plasma hydrodynamics; higher-order moments should be used.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46035987","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}
Abstract The paper is focused on the dependence of optimal disturbances of stable periodic solutions of time-delay systems on phases of such solutions. The results of numerical experiments with the well-known model of the dynamics of infection caused by lymphocytic choriomeningitis virus are presented and discussed. A new more efficient method for computing the optimal disturbances of periodic solutions is proposed and used.
{"title":"Dependence of optimal disturbances on periodic solution phases for time-delay systems","authors":"M. Y. Khristichenko, Y. Nechepurenko, G. Bocharov","doi":"10.1515/rnam-2023-0008","DOIUrl":"https://doi.org/10.1515/rnam-2023-0008","url":null,"abstract":"Abstract The paper is focused on the dependence of optimal disturbances of stable periodic solutions of time-delay systems on phases of such solutions. The results of numerical experiments with the well-known model of the dynamics of infection caused by lymphocytic choriomeningitis virus are presented and discussed. A new more efficient method for computing the optimal disturbances of periodic solutions is proposed and used.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41743164","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}
Abstract The article is devoted to the construction and study of a finite-difference scheme for a one-dimensional diffusion–convection equation with a fractional derivative with respect to the characteristic of the convection operator. It develops the previous results of the authors from [5, 6] in the following ways: the differential equation contains a fractional derivative of variable order along the characteristics of the convection operator and a quasi-linear diffusion operator; a new accuracy estimate is proved, which singles out the dependence of the accuracy of mesh scheme on the curvature of the characteristics.
{"title":"Finite difference scheme for a non-linear subdiffusion problem with a fractional derivative along the trajectory of motion","authors":"A. Lapin, V. Shaydurov, R. Yanbarisov","doi":"10.1515/rnam-2023-0003","DOIUrl":"https://doi.org/10.1515/rnam-2023-0003","url":null,"abstract":"Abstract The article is devoted to the construction and study of a finite-difference scheme for a one-dimensional diffusion–convection equation with a fractional derivative with respect to the characteristic of the convection operator. It develops the previous results of the authors from [5, 6] in the following ways: the differential equation contains a fractional derivative of variable order along the characteristics of the convection operator and a quasi-linear diffusion operator; a new accuracy estimate is proved, which singles out the dependence of the accuracy of mesh scheme on the curvature of the characteristics.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48692854","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}
Abstract We present a new approach for multidimensional surface–subsurface flow coupling. The method does not require mesh-to-canal refinement or alignment and is based on numerical characteristics of the intersection of one-dimensional hydraulic object beds and the surface faces of the 3D tetrahedral mesh. The method is validated using widely recognized tilted v-catchment with subsurface and Borden catchment benchmarks.
{"title":"Mesh-independent multidimensional coupling of surface and subsurface water flow models","authors":"K. Novikov","doi":"10.1515/rnam-2023-0004","DOIUrl":"https://doi.org/10.1515/rnam-2023-0004","url":null,"abstract":"Abstract We present a new approach for multidimensional surface–subsurface flow coupling. The method does not require mesh-to-canal refinement or alignment and is based on numerical characteristics of the intersection of one-dimensional hydraulic object beds and the surface faces of the 3D tetrahedral mesh. The method is validated using widely recognized tilted v-catchment with subsurface and Borden catchment benchmarks.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47999520","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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