Pub Date : 2024-09-04DOI: 10.1186/s13661-024-01916-5
Chun Peng, Xiaoliang Li, Bo Du
In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.
{"title":"Periodic solution for neutral-type differential equation with piecewise impulses on time scales","authors":"Chun Peng, Xiaoliang Li, Bo Du","doi":"10.1186/s13661-024-01916-5","DOIUrl":"https://doi.org/10.1186/s13661-024-01916-5","url":null,"abstract":"In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197440","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 : 2024-09-04DOI: 10.1186/s13661-024-01914-7
Junping Li, Wanting Zhang
The main purpose of this paper is to consider the multiple birth properties for multi-type Markov branching processes. We first construct a new multi-dimensional Markov process based on the multi-type Markov branching process, which can reveal the multiple birth characteristics. Then the joint probability distribution of multiple birth of multi-type Markov branching process until any time t is obtained by using the new process. Furthermore, the probability distribution of multiple birth until the extinction of the process is also given.
本文的主要目的是考虑多类型马尔可夫分支过程的多重出生特性。首先，我们在多类型马尔可夫分支过程的基础上构建了一个新的多维马尔可夫过程，该过程可以揭示多生特性。然后，利用新过程求得多类型马尔可夫分支过程直到任意时间 t 的多次出生的联合概率分布。此外，还给出了该过程消亡前的多次出生概率分布。
{"title":"The multiple birth properties of multi-type Markov branching processes","authors":"Junping Li, Wanting Zhang","doi":"10.1186/s13661-024-01914-7","DOIUrl":"https://doi.org/10.1186/s13661-024-01914-7","url":null,"abstract":"The main purpose of this paper is to consider the multiple birth properties for multi-type Markov branching processes. We first construct a new multi-dimensional Markov process based on the multi-type Markov branching process, which can reveal the multiple birth characteristics. Then the joint probability distribution of multiple birth of multi-type Markov branching process until any time t is obtained by using the new process. Furthermore, the probability distribution of multiple birth until the extinction of the process is also given.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197438","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 : 2024-08-19DOI: 10.1186/s13661-024-01911-w
Xiumei Xing, Haiyan Wang, Shaoyong Lai
The second-order periodic system with p-Laplacian and unbounded time-dependent perturbation terms is investigated. Using the principle integral method, it is shown that under certain assumptions on the unbounded and periodic terms, all solutions to the equation possess boundedness.
{"title":"Boundedness of solutions to a second-order periodic system with p-Laplacian and unbounded perturbation terms","authors":"Xiumei Xing, Haiyan Wang, Shaoyong Lai","doi":"10.1186/s13661-024-01911-w","DOIUrl":"https://doi.org/10.1186/s13661-024-01911-w","url":null,"abstract":"The second-order periodic system with p-Laplacian and unbounded time-dependent perturbation terms is investigated. Using the principle integral method, it is shown that under certain assumptions on the unbounded and periodic terms, all solutions to the equation possess boundedness.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197315","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 : 2024-08-16DOI: 10.1186/s13661-024-01907-6
W. M. Abd-Elhameed, Y. H. Youssri, A. G. Atta
This study utilizes a spectral tau method to acquire an accurate numerical solution of the time-fractional diffusion equation. The central point of this approach is to use double basis functions in terms of certain Chebyshev polynomials, namely Chebyshev polynomials of the seventh-kind and their shifted ones. Some new formulas concerned with these polynomials are derived in this study. A rigorous error analysis of the proposed double expansion further corroborates our research. This analysis is based on establishing some inequalities regarding the selected basis functions. Several numerical examples validate the precision and effectiveness of the suggested method.
本研究利用谱 tau 方法获得时间分数扩散方程的精确数值解。该方法的核心是使用某些切比雪夫多项式（即七次切比雪夫多项式及其移位多项式）的双基函数。本研究得出了一些与这些多项式有关的新公式。对所提出的双重展开的严格误差分析进一步证实了我们的研究。该分析基于建立与所选基础函数相关的一些不等式。几个数值示例验证了所建议方法的精确性和有效性。
{"title":"Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials","authors":"W. M. Abd-Elhameed, Y. H. Youssri, A. G. Atta","doi":"10.1186/s13661-024-01907-6","DOIUrl":"https://doi.org/10.1186/s13661-024-01907-6","url":null,"abstract":"This study utilizes a spectral tau method to acquire an accurate numerical solution of the time-fractional diffusion equation. The central point of this approach is to use double basis functions in terms of certain Chebyshev polynomials, namely Chebyshev polynomials of the seventh-kind and their shifted ones. Some new formulas concerned with these polynomials are derived in this study. A rigorous error analysis of the proposed double expansion further corroborates our research. This analysis is based on establishing some inequalities regarding the selected basis functions. Several numerical examples validate the precision and effectiveness of the suggested method.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225221","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 : 2024-08-13DOI: 10.1186/s13661-024-01910-x
Yan Qiao, Fangqi Chen, Yukun An, Tao Lu
In this paper, a class of fractional Sturm–Liouville advection–dispersion equations with instantaneous and noninstantaneous impulses is considered, in particular, the nonlinearities discussed here include Caputo fractional derivatives. Since the nonlinear terms contain fractional derivatives, this problem does not directly have variational structure, we need to combine critical point theory and an iterative method to deal with such problems. Finally, the existence of at least one nontrivial solution is proved by the mountain pass theorem and the iterative method. At the same time, an example is given to illustrate the main result.
{"title":"New results on fractional advection–dispersion equations","authors":"Yan Qiao, Fangqi Chen, Yukun An, Tao Lu","doi":"10.1186/s13661-024-01910-x","DOIUrl":"https://doi.org/10.1186/s13661-024-01910-x","url":null,"abstract":"In this paper, a class of fractional Sturm–Liouville advection–dispersion equations with instantaneous and noninstantaneous impulses is considered, in particular, the nonlinearities discussed here include Caputo fractional derivatives. Since the nonlinear terms contain fractional derivatives, this problem does not directly have variational structure, we need to combine critical point theory and an iterative method to deal with such problems. Finally, the existence of at least one nontrivial solution is proved by the mountain pass theorem and the iterative method. At the same time, an example is given to illustrate the main result.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197441","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 : 2024-08-13DOI: 10.1186/s13661-024-01909-4
Abdelghani Lakhdari, Hüseyin Budak, Muhammad Uzair Awan, Badreddine Meftah
This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences.
本研究探索将米尔恩型不等式扩展到卡图甘波拉分式积分领域，旨在拓宽分式微积分的分析工具。通过引入一种新的积分特性，我们为具有扩展 s 凸一阶导数的函数建立了一系列米尔恩型不等式。随后，我们提出了一个配有图形表示的示例，以验证我们的理论发现。论文最后介绍了这些不等式的实际应用，展示了它们在数学和应用科学各个领域的潜在影响。
{"title":"Extension of Milne-type inequalities to Katugampola fractional integrals","authors":"Abdelghani Lakhdari, Hüseyin Budak, Muhammad Uzair Awan, Badreddine Meftah","doi":"10.1186/s13661-024-01909-4","DOIUrl":"https://doi.org/10.1186/s13661-024-01909-4","url":null,"abstract":"This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197442","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 : 2024-08-07DOI: 10.1186/s13661-024-01908-5
Qingfang Wang
This paper deals with the following nonlinear elliptic equation: $$ -Delta u=Q(|y'|,y'')u^{frac{N+2}{N-2}},,,u>0,,,text{in},{ mathbb{R}}^{N},,,uin D^{1,2}({mathbb{R}}^{N}), $$ where $(y',y'')in {mathbb{R}}^{2}times {mathbb{R}}^{N-2}$ , $Ngeq 5$ , $Q(|y'|,y'')$ is a bounded nonnegative function in $mathbb{R}^{2}times {mathbb{R}}^{N-2}$ . By using the local Pohozaev identities we prove a nondegeneracy result for the positive solutions constructed in (Peng et al. in J. Differ. Equ. 267:2503–2530, 2019).
本文涉及以下非线性椭圆方程：$$ -Delta u=Q(|y'|,y'')u^{frac{N+2}{N-2}},,,u>0,,,text{in},{ mathbb{R}}^{N},,,uin D^{1,2}({mathbb{R}}^{N}), $$ 其中$(y'、y'')in {mathbb{R}}^{2}times {mathbb{R}}^{N-2}$ , $Ngeq 5$ , $Q(|y'|,y'')$ 是 $mathbb{R}^{2}times {mathbb{R}}^{N-2}$ 中的有界非负函数。通过使用局部 Pohozaev 特性，我们证明了 (Peng et al. in J. Differ. Equ. 267:2503-2530, 2019) 中构建的正解的非退化结果。
{"title":"Nondegeneracy of the solutions for elliptic problem with critical exponent","authors":"Qingfang Wang","doi":"10.1186/s13661-024-01908-5","DOIUrl":"https://doi.org/10.1186/s13661-024-01908-5","url":null,"abstract":"This paper deals with the following nonlinear elliptic equation: $$ -Delta u=Q(|y'|,y'')u^{frac{N+2}{N-2}},,,u>0,,,text{in},{ mathbb{R}}^{N},,,uin D^{1,2}({mathbb{R}}^{N}), $$ where $(y',y'')in {mathbb{R}}^{2}times {mathbb{R}}^{N-2}$ , $Ngeq 5$ , $Q(|y'|,y'')$ is a bounded nonnegative function in $mathbb{R}^{2}times {mathbb{R}}^{N-2}$ . By using the local Pohozaev identities we prove a nondegeneracy result for the positive solutions constructed in (Peng et al. in J. Differ. Equ. 267:2503–2530, 2019).","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933522","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 : 2024-08-06DOI: 10.1186/s13661-024-01905-8
Jiqiang Jiang, Xuelin Sun
This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results.
{"title":"Existence of positive solutions for a class of p-Laplacian fractional differential equations with nonlocal boundary conditions","authors":"Jiqiang Jiang, Xuelin Sun","doi":"10.1186/s13661-024-01905-8","DOIUrl":"https://doi.org/10.1186/s13661-024-01905-8","url":null,"abstract":"This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933524","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 : 2024-08-06DOI: 10.1186/s13661-024-01906-7
Kuo-Chih Hung
We study the bifurcation curve and exact multiplicity of positive solutions in the space $C^{2}left ( (-L,L)right ) cap Cleft ( [-L,L]right ) $ for the Minkowski-curvature equation $$ left { textstylebegin{array}{l} -left ( dfrac{u^{prime }(x)}{sqrt{1-left ( {u^{prime }(x)}right ) ^{2}}}right ) ^{prime }=lambda f(u),text{ }-L< x< L, u(-L)=u(L)=0.end{array}displaystyle right . $$ where $lambda >0$ is a bifurcation parameter, $fin C[0,infty )cap C^{2}(0,infty )$ satisfies $f(u)>0$ for $u>0$ and f is either concave or geometrically concave on $(0,infty )$ . If f is a concave function, we prove that the bifurcation curve is monotone increasing on the $(lambda ,left Vert uright Vert _{infty })$ -plane. If f is a geometrically concave function, we prove that the bifurcation curve is either ⊂-shaped or monotone increasing on the $(lambda ,left Vert uright Vert _{infty })$ -plane under a mild condition. Some interesting applications are given.
我们研究了闵科夫斯基曲率方程 $$ C^{2}left ( (-L,L)right ) 空间中正解的分岔曲线和精确多重性。cap Cleft ( [-L,L]right ) $ 用于闵科夫斯基曲率方程 $$ left { textstylebegin{array}{l} -left ( dfrac{u^{prime }(x)}{sqrt{1-left ( {u^{prime }(x)}right )^{2}}}right )^{{prime }=lambda f(u),text{ }-L< x< L, u(-L)=u(L)=0.end{array}displaystyle right .$$ 其中 $lambda >0$ 是一个分岔参数，$fin C[0,infty )cap C^{2}(0,infty )$ 满足 $f(u)>0$ for $u>0$ 并且 f 在 $(0,infty )$ 上要么是凹函数要么是几何凹函数。如果 f 是凹函数，我们证明分岔曲线在 $(lambda ,left Vert uright Vert _{infty })$ 平面上是单调递增的。如果 f 是一个几何凹函数，我们证明在一个温和的条件下，分岔曲线在 $(lambda ,left Vert uright Vert _{infty })$ - 平面上要么是 ⊂ 形的，要么是单调递增的。文中给出了一些有趣的应用。
{"title":"Bifurcation curve for the Minkowski-curvature equation with concave or geometrically concave nonlinearity","authors":"Kuo-Chih Hung","doi":"10.1186/s13661-024-01906-7","DOIUrl":"https://doi.org/10.1186/s13661-024-01906-7","url":null,"abstract":"We study the bifurcation curve and exact multiplicity of positive solutions in the space $C^{2}left ( (-L,L)right ) cap Cleft ( [-L,L]right ) $ for the Minkowski-curvature equation $$ left { textstylebegin{array}{l} -left ( dfrac{u^{prime }(x)}{sqrt{1-left ( {u^{prime }(x)}right ) ^{2}}}right ) ^{prime }=lambda f(u),text{ }-L< x< L, u(-L)=u(L)=0.end{array}displaystyle right . $$ where $lambda >0$ is a bifurcation parameter, $fin C[0,infty )cap C^{2}(0,infty )$ satisfies $f(u)>0$ for $u>0$ and f is either concave or geometrically concave on $(0,infty )$ . If f is a concave function, we prove that the bifurcation curve is monotone increasing on the $(lambda ,left Vert uright Vert _{infty })$ -plane. If f is a geometrically concave function, we prove that the bifurcation curve is either ⊂-shaped or monotone increasing on the $(lambda ,left Vert uright Vert _{infty })$ -plane under a mild condition. Some interesting applications are given.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933523","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 : 2024-08-02DOI: 10.1186/s13661-024-01899-3
Suleman Alfalqi, Boumediene Boukhari, Ahmed Bchatnia, Abderrahmane Beniani
We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods. Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems.
{"title":"Advanced neural network approaches for coupled equations with fractional derivatives","authors":"Suleman Alfalqi, Boumediene Boukhari, Ahmed Bchatnia, Abderrahmane Beniani","doi":"10.1186/s13661-024-01899-3","DOIUrl":"https://doi.org/10.1186/s13661-024-01899-3","url":null,"abstract":"We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods. Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884359","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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