In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results.
{"title":"Hopf bifurcation analysis of a two‐delayed diffusive predator–prey model with spatial memory of prey","authors":"Hongyan Wang, Yunxian Dai, Shumin Zhou","doi":"10.1002/mma.10416","DOIUrl":"https://doi.org/10.1002/mma.10416","url":null,"abstract":"In this paper, we consider a diffusive predator–prey model with <jats:styled-content>spatial</jats:styled-content> memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Editorial for special issue “Current trends in Applied Mathematics”","authors":"Gerassimos Barbatis, Athanasios Yannacopoulos","doi":"10.1002/mma.10360","DOIUrl":"10.1002/mma.10360","url":null,"abstract":"","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ismail Onder, Aydin Secer, Muslum Ozisik, Mustafa Bayram
In this study, we obtained optical soliton solutions of the perturbed nonlinear Schrödinger–Hirota equation with generalized anti‐cubic law nonlinearity in the presence of spatio‐temporal dispersion. This equation models the propagation of optical pulses in fiber optic cables. Due to the anti‐cubic nonlinearity, perturbation, and spatio‐temporal dispersion present in the model, it provides more accurate results for high‐speed and long‐distance transmissions. Given the significant developments in the field of optics, studies on complex equations such as this model are of great importance. With the increase in real‐life applications, obtaining solutions to optical equations has become crucial. In this study, we used the improved F‐expansion method to derive the optical soliton solutions for the relevant model. This technique allows for obtaining various solutions through the Jacobi elliptic auxiliary functions it employs. The obtained solutions consist of trigonometric and hyperbolic functions. As a result of the application, 10 solutions were obtained, and 2D and 3D graphics of these solutions are included. These graphs illustrate the motion directions of optical solitons and the effect of the nonlinearity parameter and spatio‐temporal dispersion parameter on soliton behavior. No restrictions were encountered during the study. Finally, the originality of the study lies in the first application of this technique to the relevant model and in examining the effect of the parameters and on this model.
在这项研究中，我们获得了在存在时空色散的情况下，具有广义反立方律非线性的扰动非线性薛定谔-希罗塔方程的光孤子解。该方程模拟了光脉冲在光缆中的传播。由于模型中存在反立方非线性、扰动和时空色散，它能为高速和长距离传输提供更精确的结果。鉴于光学领域的重大发展，对该模型等复杂方程的研究具有重要意义。随着实际应用的增加，获得光学方程的解已变得至关重要。在这项研究中，我们使用改进的 F 展开法推导出了相关模型的光学孤子解。这种技术可以通过其采用的雅可比椭圆辅助函数获得各种解。获得的解包括三角函数和双曲函数。应用该技术后，共获得了 10 个解决方案，其中包括这些解决方案的二维和三维图形。这些图表说明了光学孤子的运动方向以及非线性参数和时空色散参数对孤子行为的影响。研究过程中没有遇到任何限制。最后，本研究的独创性在于首次将这一技术应用于相关模型，并研究了参数和对该模型的影响。
{"title":"Retrieval of the optical soliton solutions of the perturbed Schrödinger–Hirota equation with generalized anti‐cubic law nonlinearity having the spatio‐temporal dispersion","authors":"Ismail Onder, Aydin Secer, Muslum Ozisik, Mustafa Bayram","doi":"10.1002/mma.10429","DOIUrl":"https://doi.org/10.1002/mma.10429","url":null,"abstract":"In this study, we obtained optical soliton solutions of the perturbed nonlinear Schrödinger–Hirota equation with generalized anti‐cubic law nonlinearity in the presence of spatio‐temporal dispersion. This equation models the propagation of optical pulses in fiber optic cables. Due to the anti‐cubic nonlinearity, perturbation, and spatio‐temporal dispersion present in the model, it provides more accurate results for high‐speed and long‐distance transmissions. Given the significant developments in the field of optics, studies on complex equations such as this model are of great importance. With the increase in real‐life applications, obtaining solutions to optical equations has become crucial. In this study, we used the improved F‐expansion method to derive the optical soliton solutions for the relevant model. This technique allows for obtaining various solutions through the Jacobi elliptic auxiliary functions it employs. The obtained solutions consist of trigonometric and hyperbolic functions. As a result of the application, 10 solutions were obtained, and 2D and 3D graphics of these solutions are included. These graphs illustrate the motion directions of optical solitons and the effect of the nonlinearity parameter and spatio‐temporal dispersion parameter on soliton behavior. No restrictions were encountered during the study. Finally, the originality of the study lies in the first application of this technique to the relevant model and in examining the effect of the parameters and on this model.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a fourth‐order hyperbolic equation involving Kirchhoff type ‐Laplacian and superlinear source, subject to zero Navier boundary condition, where is an open bounded domain in with ; denotes the maximal existence time; and and are constants. For , using auxiliary function method and Sobolev inequality, we prove that there are only global solutions. For , we obtain the optimal classification of initial energy and Nehari energy, which guarantees the existence of blow‐up solutions and global solutions. In the critical case , we find out that the coefficients of the Kirchhoff term and the superlinear source play important role in separating out the property of weak solutions.
{"title":"Critical exponent for global solutions in a fourth‐order hyperbolic equation of p‐Kirchhoff type","authors":"Bingchen Liu, Jiaxin Dou","doi":"10.1002/mma.10438","DOIUrl":"https://doi.org/10.1002/mma.10438","url":null,"abstract":"We study a fourth‐order hyperbolic equation involving Kirchhoff type ‐Laplacian and superlinear source, subject to zero Navier boundary condition, <jats:disp-formula> </jats:disp-formula>where is an open bounded domain in with ; denotes the maximal existence time; and and are constants. For , using auxiliary function method and Sobolev inequality, we prove that there are only global solutions. For , we obtain the optimal classification of initial energy and Nehari energy, which guarantees the existence of blow‐up solutions and global solutions. In the critical case , we find out that the coefficients of the Kirchhoff term and the superlinear source play important role in separating out the property of weak solutions.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we discuss the existence of multiplicity of positive solutions to a new generalized Riemann–Liouville type fractional Fisher‐like equation on a semi‐infinite interval equipped with nonlocal multipoint boundary conditions involving Riemann–Liouville fractional derivative and integral operators. The existence of at least two positive solutions for the given problem is established by using the concept of complete continuity and iterative positive solutions. We show the existence of at least three positive solutions to the problem at hand by applying the generalized Leggett–Williams fixed‐point theorem due to Bai and Ge [Z. Bai, B. Ge, Existence of three positive solutions for some second‐order boundary value problems, Comput. Math. Appl. 48 (2014) 699‐70]. Illustrative examples are constructed to demonstrate the effectiveness of the main results. It has also been indicated in Section 5 that some new results appear as special cases by choosing the parameters involved in the given problem appropriately.
本文讨论了半无限区间上一个新的广义黎曼-黎奥维尔型分数费雪方程正解的多重性问题，该方程配备了涉及黎曼-黎奥维尔分数导数和积分算子的非局部多点边界条件。利用完全连续性和迭代正解的概念，确定了给定问题至少存在两个正解。我们应用白和葛的广义 Leggett-Williams 定点定理 [Z. Bai, B. Ge, Existence of the problem at hand] 证明了至少三个正解的存在。Bai, B. Ge, Existence of three positive solutions for some second-order boundary value problems, Comput.Math.48 (2014) 699-70]。为了证明主要结果的有效性，我们构建了一些示例。第 5 节还指出，通过适当选择给定问题所涉及的参数，一些新结果会作为特例出现。
{"title":"Existence results for the generalized Riemann–Liouville type fractional Fisher‐like equation on the half‐line","authors":"N. Nyamoradi, Bashir Ahmad","doi":"10.1002/mma.10398","DOIUrl":"https://doi.org/10.1002/mma.10398","url":null,"abstract":"In this paper, we discuss the existence of multiplicity of positive solutions to a new generalized Riemann–Liouville type fractional Fisher‐like equation on a semi‐infinite interval equipped with nonlocal multipoint boundary conditions involving Riemann–Liouville fractional derivative and integral operators. The existence of at least two positive solutions for the given problem is established by using the concept of complete continuity and iterative positive solutions. We show the existence of at least three positive solutions to the problem at hand by applying the generalized Leggett–Williams fixed‐point theorem due to Bai and Ge [Z. Bai, B. Ge, Existence of three positive solutions for some second‐order boundary value problems, Comput. Math. Appl. 48 (2014) 699‐70]. Illustrative examples are constructed to demonstrate the effectiveness of the main results. It has also been indicated in Section 5 that some new results appear as special cases by choosing the parameters involved in the given problem appropriately.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141921135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion.
{"title":"Besicovitch almost automorphic solutions in finite‐dimensional distributions to stochastic semilinear differential equations driven by both Brownian and fractional Brownian motions","authors":"Yongkun Li, Zhicong Bai","doi":"10.1002/mma.10403","DOIUrl":"https://doi.org/10.1002/mma.10403","url":null,"abstract":"In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141922964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper considers the initial‐boundary value problem for equation , in an N‐dimensional domain with a homogeneous Dirichlet condition. The fractional derivative is taken in the sense of Caputo. The main goal of the work is to solve the inverse problem of simultaneously determining two parameters: the order of the fractional derivative and the degree of the Laplace operator . A new formulation and solution method for this inverse problem are proposed. It is proved that in the new formulation the solution to the inverse problem exists and is unique for an arbitrary initial function from the class . Note that in previously known works, only the uniqueness of the solution to the inverse problem was proved and the initial function was required to be sufficiently smooth and non‐negative.
本文考虑的是 N 维域中具有同质 Dirichlet 条件的方程 , 的初始边界值问题。在 Caputo 的意义上取分数导数。工作的主要目标是解决同时确定两个参数的逆问题：分数导数的阶数和拉普拉斯算子的度数。针对这个逆问题提出了一种新的公式和求解方法。研究证明，在新的表述中，逆问题的解是存在的，并且对于.类中的任意初始函数都是唯一的。需要注意的是，在之前已知的工作中，只证明了逆问题解的唯一性，而且要求初始函数足够平滑且非负。
{"title":"Determining the order of time and spatial fractional derivatives","authors":"Ravshan Ashurov, Ilyoskhuja Sulaymonov","doi":"10.1002/mma.10393","DOIUrl":"https://doi.org/10.1002/mma.10393","url":null,"abstract":"The paper considers the initial‐boundary value problem for equation , in an N‐dimensional domain with a homogeneous Dirichlet condition. The fractional derivative is taken in the sense of Caputo. The main goal of the work is to solve the inverse problem of simultaneously determining two parameters: the order of the fractional derivative and the degree of the Laplace operator . A new formulation and solution method for this inverse problem are proposed. It is proved that in the new formulation the solution to the inverse problem exists and is unique for an arbitrary initial function from the class . Note that in previously known works, only the uniqueness of the solution to the inverse problem was proved and the initial function was required to be sufficiently smooth and non‐negative.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we generalized Mittag‐Leffler‐type functions , and , which correspond, respectively, to the familiar Lauricella hypergeometric functions , and of three variables. Initially, from the Mittag‐Leffler type function in the simplest form to the functions we are studying, necessary information about the development history, study, and importance of this and hypergeometric type functions will be introduced. Among the various properties and characteristics of these three‐variable Mittag‐Leffler‐type function , which we investigate in the article, include their relationships with other extensions and generalizations of the classical Mittag‐Leffler functions, their three‐dimensional convergence regions, their Euler‐type integral representations, their Laplace transforms, and their connections with the Riemann‐Liouville operators of fractional calculus. The link of three‐variable Mittag‐Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics.Therefore, we provide the systems of partial differential equations which are associated with them.
{"title":"Mittag‐Leffler type functions of three variables","authors":"A. Hasanov, Hilola Yuldashova","doi":"10.1002/mma.10401","DOIUrl":"https://doi.org/10.1002/mma.10401","url":null,"abstract":"In this article, we generalized Mittag‐Leffler‐type functions \u0000, and \u0000, which correspond, respectively, to the familiar Lauricella hypergeometric functions \u0000, and \u0000 of three variables. Initially, from the Mittag‐Leffler type function in the simplest form to the functions we are studying, necessary information about the development history, study, and importance of this and hypergeometric type functions will be introduced. Among the various properties and characteristics of these three‐variable Mittag‐Leffler‐type function \u0000, which we investigate in the article, include their relationships with other extensions and generalizations of the classical Mittag‐Leffler functions, their three‐dimensional convergence regions, their Euler‐type integral representations, their Laplace transforms, and their connections with the Riemann‐Liouville operators of fractional calculus. The link of three‐variable Mittag‐Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics.Therefore, we provide the systems of partial differential equations which are associated with them.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141921941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The study aims to introduce some geometric properties of dual Cartan numbers. Since a unit timelike Cartan number with lightlike vector part corresponds to a parabolic rotation around a lightlike axis, it has been shown that the unit dual Cartan number with lightlike dual vector also corresponds to a parabolic rotation and a translation transformation in the Cartan frame. Also, the angle between the directed pseudo‐null lines that tangent the same parabola is defined. Thus, the rotation and translation relationship between two directed pseudo‐null lines is obtained. Also, the E. Study transformation is given for the directed pseudo‐null lines.
本研究旨在介绍对偶卡当数的一些几何性质。由于具有类光向量部分的单位时间卡当数对应于绕类光轴线的抛物线旋转，因此证明了具有类光对偶向量的单位对偶卡当数也对应于在卡当框架中的抛物线旋转和平移变换。此外，还定义了与同一抛物线相切的有向伪零线之间的夹角。因此，可以得到两条有向伪零线之间的旋转和平移关系。此外，还给出了有向伪零线的 E. Study 变换。
{"title":"Dual Cartan numbers and pseudo‐null lines","authors":"İskender Öztürk","doi":"10.1002/mma.10323","DOIUrl":"https://doi.org/10.1002/mma.10323","url":null,"abstract":"The study aims to introduce some geometric properties of dual Cartan numbers. Since a unit timelike Cartan number with lightlike vector part corresponds to a parabolic rotation around a lightlike axis, it has been shown that the unit dual Cartan number with lightlike dual vector also corresponds to a parabolic rotation and a translation transformation in the Cartan frame. Also, the angle between the directed pseudo‐null lines that tangent the same parabola is defined. Thus, the rotation and translation relationship between two directed pseudo‐null lines is obtained. Also, the E. Study transformation is given for the directed pseudo‐null lines.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141929598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on the three‐dimensional (3D) incompressible anisotropic Boussinesq system while the velocity of fluid only involves horizontal dissipation and the temperature has a damping term. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space of perturbations near the hydrostatic equilibrium. Moreover, we take an effective approach to obtain the optimal decay rates for the global solution itself as well as its derivatives. In this paper, we aim to reveal the mechanism of how the temperature helps stabilize the fluid. Additionally, exploring the stability of perturbations near hydrostatic equilibrium may provide valuable insights into specific severe weather phenomena.
{"title":"Stability and optimal decay estimates for the 3D anisotropic Boussinesq equations","authors":"Wan‐Rong Yang, Meng‐Zhen Peng","doi":"10.1002/mma.10391","DOIUrl":"https://doi.org/10.1002/mma.10391","url":null,"abstract":"This paper focuses on the three‐dimensional (3D) incompressible anisotropic Boussinesq system while the velocity of fluid only involves horizontal dissipation and the temperature has a damping term. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space of perturbations near the hydrostatic equilibrium. Moreover, we take an effective approach to obtain the optimal decay rates for the global solution itself as well as its derivatives. In this paper, we aim to reveal the mechanism of how the temperature helps stabilize the fluid. Additionally, exploring the stability of perturbations near hydrostatic equilibrium may provide valuable insights into specific severe weather phenomena.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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