Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.10.012
Galina Filipuk
In this paper we consider several Hamiltonian functions for the second quasi-Painlevé equation. One of the features of these functions is that they give rise to the same final chart regular systems once using certain blowups and twists in the regularisation procedure. We also discuss what happens if we iterate the blowup process for these final chart systems. Using birational transformations between different Hamiltonian systems we show how to construct new Hamiltonian functions which give rise to the second quasi-Painlevé equation with shifted coefficients. We also give an explicit example of the Bäcklund transformation.
{"title":"Regularisation and iterated regularisation of Hamiltonian systems of the second quasi-Painlevé equation","authors":"Galina Filipuk","doi":"10.1016/j.apnum.2024.10.012","DOIUrl":"10.1016/j.apnum.2024.10.012","url":null,"abstract":"<div><div>In this paper we consider several Hamiltonian functions for the second quasi-Painlevé equation. One of the features of these functions is that they give rise to the same final chart regular systems once using certain blowups and twists in the regularisation procedure. We also discuss what happens if we iterate the blowup process for these final chart systems. Using birational transformations between different Hamiltonian systems we show how to construct new Hamiltonian functions which give rise to the second quasi-Painlevé equation with shifted coefficients. We also give an explicit example of the Bäcklund transformation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 290-300"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141564","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 : 2025-02-01DOI: 10.1016/j.apnum.2024.10.002
Nicoletta Del Buono , Flavia Esposito , Laura Selicato , Rafał Zdunek
Learning tasks are often based on penalized optimization problems in which a sparse solution is desired. This can lead to more interpretative results by identifying a smaller subset of important features or components and reducing the dimensionality of the data representation, as well. In this study, we propose a new method to solve a constrained Frobenius norm-based nonnegative low-rank approximation, and the tuning of the associated penalty hyperparameter, simultaneously. The penalty term added is a particular diversity measure that is more effective for sparseness purposes than other classical norm-based penalties (i.e., or norms). As it is well known, setting the hyperparameters of an algorithm is not an easy task. Our work drew on developing an optimization method and the corresponding algorithm that simultaneously solves the sparsity-constrained nonnegative approximation problem and optimizes its associated penalty hyperparameters. We test the proposed method by numerical experiments and show its promising results on several synthetic and real datasets.
{"title":"Penalty hyperparameter optimization with diversity measure for nonnegative low-rank approximation","authors":"Nicoletta Del Buono , Flavia Esposito , Laura Selicato , Rafał Zdunek","doi":"10.1016/j.apnum.2024.10.002","DOIUrl":"10.1016/j.apnum.2024.10.002","url":null,"abstract":"<div><div>Learning tasks are often based on penalized optimization problems in which a sparse solution is desired. This can lead to more interpretative results by identifying a smaller subset of important features or components and reducing the dimensionality of the data representation, as well. In this study, we propose a new method to solve a constrained Frobenius norm-based nonnegative low-rank approximation, and the tuning of the associated penalty hyperparameter, simultaneously. The penalty term added is a particular diversity measure that is more effective for sparseness purposes than other classical norm-based penalties (i.e., <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span> norms). As it is well known, setting the hyperparameters of an algorithm is not an easy task. Our work drew on developing an optimization method and the corresponding algorithm that simultaneously solves the sparsity-constrained nonnegative approximation problem and optimizes its associated penalty hyperparameters. We test the proposed method by numerical experiments and show its promising results on several synthetic and real datasets.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 189-204"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2023.09.010
Omar A. Alhelali , S.D. Georgiou , C. Koukouvinos , S. Stylianou
Designs for computer experiments constitute an important class of experimental designs. Computer experiments are used when the physical experiments are expensive or time-consuming and attracted a lot of attention in recent years. In this paper, we proposed a method for generating computer experiments with many factors and symmetric runs. These designs are suitable for computer experiments and are constructed using known sequences with zero autocorrelation function, such as T-sequences, Bases sequences, Normal sequences, and other. The results appear to be encouraging as the methodologies can transform the known sequences into designs for computer experiments without the need for a computer search. The generated designs have some favorable properties, including the symmetry in their runs which results in all the even orders effects being orthogonal to the main effects.
{"title":"Orthogonal designs for computer experiments constructed from sequences with zero autocorrelation","authors":"Omar A. Alhelali , S.D. Georgiou , C. Koukouvinos , S. Stylianou","doi":"10.1016/j.apnum.2023.09.010","DOIUrl":"10.1016/j.apnum.2023.09.010","url":null,"abstract":"<div><div>Designs for computer experiments constitute an important class of experimental designs<span>. Computer experiments are used when the physical experiments are expensive or time-consuming and attracted a lot of attention in recent years. In this paper, we proposed a method for generating computer experiments with many factors and symmetric runs. These designs are suitable for computer experiments and are constructed using known sequences with zero autocorrelation function<span>, such as T-sequences, Bases sequences, Normal sequences, and other. The results appear to be encouraging as the methodologies can transform the known sequences into designs for computer experiments without the need for a computer search. The generated designs have some favorable properties, including the symmetry in their runs which results in all the even orders effects being orthogonal to the main effects.</span></span></div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 22-31"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134976513","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 : 2025-02-01DOI: 10.1016/j.apnum.2024.01.021
Cristina Anton
We propose a systematic approach to construct a new family of stochastic symplectic schemes for the strong approximation of the solution of stochastic Hamiltonian systems. Our approach is based both on B-series and generating functions. The proposed schemes are a generalization of the implicit midpoint rule, they require derivatives of the Hamiltonian functions of at most order two, and are constructed by defining a generating function. We construct some schemes with strong convergence order one and a half, and we illustrate numerically their long term performance.
我们提出了一种系统的方法来构建新的随机交映方案系列，用于强逼近随机哈密顿系统的解。我们的方法基于 B 序列和生成函数。所提出的方案是隐式中点规则的广义化，它们要求哈密顿函数的导数最多为二阶，并通过定义一个生成函数来构建。我们构建了一些具有一阶半强收敛性的方案，并用数值说明了它们的长期性能。
{"title":"A new class of symplectic methods for stochastic Hamiltonian systems","authors":"Cristina Anton","doi":"10.1016/j.apnum.2024.01.021","DOIUrl":"10.1016/j.apnum.2024.01.021","url":null,"abstract":"<div><div>We propose a systematic approach to construct a new family of stochastic symplectic schemes for the strong approximation of the solution of stochastic Hamiltonian systems. Our approach is based both on B-series and generating functions. The proposed schemes are a generalization of the implicit midpoint rule, they require derivatives of the Hamiltonian functions of at most order two, and are constructed by defining a generating function. We construct some schemes with strong convergence order one and a half, and we illustrate numerically their long term performance.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 43-59"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657336","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 : 2025-02-01DOI: 10.1016/j.apnum.2024.01.017
Anas El Hachimi , Khalide Jbilou , Ahmed Ratnani
In this paper, we describe some Krylov subspace methods for computing eigentubes and eigenvectors (eigenslices) for large and sparse third-order tensors. This work provides projection methods for computing some of the largest (or smallest) eigentubes and eigenslices using the t-product. In particular, we use the tensor Arnoldi's approach for the non-hermitian case and the tensor Lanczos's approach for f-hermitian tensors. We also use the tensor block Arnoldi method to approximate the extreme eigentubes of a large tensor. Computed examples are given to illustrate the effectiveness of these methods.
本文介绍了一些计算大型稀疏三阶张量的特征管和特征向量（特征切片）的 Krylov 子空间方法。这项工作提供了使用 t-乘积计算一些最大（或最小）特征管和特征切片的投影方法。特别是，我们在非全米情况下使用了张量阿诺迪方法，在 f 全米张量情况下使用了张量兰克佐斯方法。我们还使用张量块阿诺迪方法来逼近大型张量的极值。我们给出了一些计算实例来说明这些方法的有效性。
{"title":"Krylov subspace methods for large multidimensional eigenvalue computation","authors":"Anas El Hachimi , Khalide Jbilou , Ahmed Ratnani","doi":"10.1016/j.apnum.2024.01.017","DOIUrl":"10.1016/j.apnum.2024.01.017","url":null,"abstract":"<div><div><span>In this paper, we describe some Krylov subspace methods for computing eigentubes and </span>eigenvectors (eigenslices) for large and sparse third-order tensors. This work provides projection methods for computing some of the largest (or smallest) eigentubes and eigenslices using the t-product. In particular, we use the tensor Arnoldi's approach for the non-hermitian case and the tensor Lanczos's approach for f-hermitian tensors. We also use the tensor block Arnoldi method to approximate the extreme eigentubes of a large tensor. Computed examples are given to illustrate the effectiveness of these methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 205-221"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657341","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 : 2025-02-01DOI: 10.1016/j.apnum.2024.10.016
M. Srati , A. Oulmelk , L. Afraites , A. Hadri , M.A. Zaky , A. Aldraiweesh , A.S. Hendy
In this study, we address an inverse problem in nonlinear time-fractional diffusion equations using a deep neural network. The challenge arises from the equation's nonlinear behavior, the involvement of time-based fractional Caputo derivatives, and the need to estimate parameters influenced by space or the solution of the fractional PDE. Our solution involves a fractional physics-informed neural network (FPINN). Initially, we use FPINN to solve a straightforward problem. Then, we apply FPINN to the inverse problem of estimating parameter and model non-linearity. For the inverse problem, we enhance our method by including the mean square error of final observations in the FPINN's cost function. This adjustment helps effectively in tackling the unique challenges of the time-fractional diffusion equation. Numerical tests involving regular and singular examples demonstrate the effectiveness of the physics-informed neural network approach in accurately recovering parameters. We reinforce this finding through a numerical comparison with alternative methods such as the alternating direction multiplier method (ADMM), the gradient descent, and the DeepONets (deep operator networks) method.
{"title":"An inverse problem of determining the parameters in diffusion equations by using fractional physics-informed neural networks","authors":"M. Srati , A. Oulmelk , L. Afraites , A. Hadri , M.A. Zaky , A. Aldraiweesh , A.S. Hendy","doi":"10.1016/j.apnum.2024.10.016","DOIUrl":"10.1016/j.apnum.2024.10.016","url":null,"abstract":"<div><div>In this study, we address an inverse problem in nonlinear time-fractional diffusion equations using a deep neural network. The challenge arises from the equation's nonlinear behavior, the involvement of time-based fractional Caputo derivatives, and the need to estimate parameters influenced by space or the solution of the fractional PDE. Our solution involves a fractional physics-informed neural network (FPINN). Initially, we use FPINN to solve a straightforward problem. Then, we apply FPINN to the inverse problem of estimating parameter and model non-linearity. For the inverse problem, we enhance our method by including the mean square error of final observations in the FPINN's cost function. This adjustment helps effectively in tackling the unique challenges of the time-fractional diffusion equation. Numerical tests involving regular and singular examples demonstrate the effectiveness of the physics-informed neural network approach in accurately recovering parameters. We reinforce this finding through a numerical comparison with alternative methods such as the alternating direction multiplier method (ADMM), the gradient descent, and the DeepONets (deep operator networks) method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 189-213"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141227","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 : 2025-02-01DOI: 10.1016/j.apnum.2024.10.009
Xiaohui Yu , Qingbiao Wu
Two efficient iteration methods are proposed for solving the absolute value equation which are the accelerated generalized SOR-like (AGSOR-like) iteration method and the preconditioned generalized SOR-like (PGSOR-like) iteration method. We prove the convergence of the two proposed iterative methods after applying some qualification conditions to the parameters involved. We also discuss the optimal values of the parameters involved in the two methods. Also, some numerical experiments demonstrate the practicability, robustness and high efficiency of the two new methods. In addition, applying the optimal parameter values obtained from theoretical analysis to the PGSOR-like method, it can give solutions with high accuracy after a small number of iterations, demonstrating significant advantages.
{"title":"Two efficient iteration methods for solving the absolute value equations","authors":"Xiaohui Yu , Qingbiao Wu","doi":"10.1016/j.apnum.2024.10.009","DOIUrl":"10.1016/j.apnum.2024.10.009","url":null,"abstract":"<div><div>Two efficient iteration methods are proposed for solving the absolute value equation which are the accelerated generalized SOR-like (AGSOR-like) iteration method and the preconditioned generalized SOR-like (PGSOR-like) iteration method. We prove the convergence of the two proposed iterative methods after applying some qualification conditions to the parameters involved. We also discuss the optimal values of the parameters involved in the two methods. Also, some numerical experiments demonstrate the practicability, robustness and high efficiency of the two new methods. In addition, applying the optimal parameter values obtained from theoretical analysis to the PGSOR-like method, it can give solutions with high accuracy after a small number of iterations, demonstrating significant advantages.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 148-159"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141569","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 : 2025-02-01DOI: 10.1016/j.apnum.2024.10.003
Mohamed Boujoudar , Abdelaziz Beljadid , Ahmed Taik
Modeling unsaturated flow through soils with water uptake by plant root has many applications in agriculture and water resources management. In this study, our aim is to develop efficient numerical techniques for solving the Richards equation with a sink term due to plant root water uptake. The Feddes model is used for water absorption by plant roots, and the van-Genuchten model is employed for capillary pressure. We introduce a numerical approach that combines the localized exponential radial basis function (EXP-RBF) method for space and the second-order backward differentiation formula (BDF2) for temporal discretization. The localized RBF methods eliminate the need for mesh generation and avoid ill-conditioning problems. This approach yields a sparse matrix for the global system, optimizing memory usage and computational time. The proposed implicit EXP-RBF techniques have advantages in terms of accuracy and computational efficiency thanks to the use of BDF2 and the localized RBF method. Modified Picards iteration method for the mixed form of the Richards equation is employed to linearize the system. Various numerical experiments are conducted to validate the proposed numerical model of infiltration with plant root water absorption. The obtained results conclusively demonstrate the effectiveness of the proposed numerical model in accurately predicting soil moisture dynamics under water uptake by plant roots. The proposed numerical techniques can be incorporated in the numerical models where unsaturated flows and water uptake by plant roots are involved such as in hydrology, agriculture, and water management.
{"title":"Implicit EXP-RBF techniques for modeling unsaturated flow through soils with water uptake by plant roots","authors":"Mohamed Boujoudar , Abdelaziz Beljadid , Ahmed Taik","doi":"10.1016/j.apnum.2024.10.003","DOIUrl":"10.1016/j.apnum.2024.10.003","url":null,"abstract":"<div><div>Modeling unsaturated flow through soils with water uptake by plant root has many applications in agriculture and water resources management. In this study, our aim is to develop efficient numerical techniques for solving the Richards equation with a sink term due to plant root water uptake. The Feddes model is used for water absorption by plant roots, and the van-Genuchten model is employed for capillary pressure. We introduce a numerical approach that combines the localized exponential radial basis function (EXP-RBF) method for space and the second-order backward differentiation formula (BDF2) for temporal discretization. The localized RBF methods eliminate the need for mesh generation and avoid ill-conditioning problems. This approach yields a sparse matrix for the global system, optimizing memory usage and computational time. The proposed implicit EXP-RBF techniques have advantages in terms of accuracy and computational efficiency thanks to the use of BDF2 and the localized RBF method. Modified Picards iteration method for the mixed form of the Richards equation is employed to linearize the system. Various numerical experiments are conducted to validate the proposed numerical model of infiltration with plant root water absorption. The obtained results conclusively demonstrate the effectiveness of the proposed numerical model in accurately predicting soil moisture dynamics under water uptake by plant roots. The proposed numerical techniques can be incorporated in the numerical models where unsaturated flows and water uptake by plant roots are involved such as in hydrology, agriculture, and water management.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 79-97"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143097230","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 : 2025-02-01DOI: 10.1016/j.apnum.2024.11.001
Sandra Carillo , Michel Chipot
The goal of this note is to study nonlinear parabolic problems nonlocal in time and space. We first establish the existence of a solution and its uniqueness in certain cases. Finally we consider its asymptotic behaviour.
{"title":"On some nonlocal in time and space parabolic problem","authors":"Sandra Carillo , Michel Chipot","doi":"10.1016/j.apnum.2024.11.001","DOIUrl":"10.1016/j.apnum.2024.11.001","url":null,"abstract":"<div><div>The goal of this note is to study nonlinear parabolic problems nonlocal in time and space. We first establish the existence of a solution and its uniqueness in certain cases. Finally we consider its asymptotic behaviour.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 314-322"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143097233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2023.11.019
Maryam Boubekraoui
Multilinear PageRank is a variant of the well-known PageRank model. With this model, web page ranking can be more accurate and efficient by taking into account higher-order connections between pages. The higher-order power method is commonly employed for computing the multilinear PageRank vector, since it is a natural extension of the traditional power method used in the PageRank algorithm. However, this method may not be efficient when the hyperlink tensor becomes large or the damping factor value fails to meet the necessary conditions for convergence. In this work, we propose a novel approach to efficiently computing the multilinear PageRank vector using tensor splitting and vector extrapolation methods.
{"title":"Extrapolated splitting methods for multilinear PageRank computations","authors":"Maryam Boubekraoui","doi":"10.1016/j.apnum.2023.11.019","DOIUrl":"10.1016/j.apnum.2023.11.019","url":null,"abstract":"<div><div>Multilinear PageRank is a variant of the well-known PageRank model. With this model, web page ranking can be more accurate and efficient by taking into account higher-order connections between pages. The higher-order power method is commonly employed for computing the multilinear PageRank vector, since it is a natural extension of the traditional power method used in the PageRank algorithm. However, this method may not be efficient when the hyperlink<span> tensor becomes large or the damping factor value fails to meet the necessary conditions for convergence. In this work, we propose a novel approach to efficiently computing the multilinear PageRank vector using tensor splitting and vector extrapolation methods.</span></div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 92-103"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507247","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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