S. Butt, A. Akdemir, S. Aslan, I. Işcan, P. Agarwal
1COMSATS University Islamabad, Lahore Campus, Pakistan 2Department of Mathematics, Faculty of Arts and Sciences, Ağrı İbrahim Çeçen University, Ağrı, Turkey 3Institute of Graduate Studies, Ağrı İbrahim Çeçen University, Ağrı, Turkey 4Department of Mathematics, Faculty of Science and Arts, Giresun University, Giresun, Turkey 5International Center for Basic and Applied Sciences, Jaipur, India 6Department of Mathematics, Anand International College of Engineering, Jaipur, India
1COMSATS大学伊斯兰堡拉合尔校区2土耳其Ağrı İbrahim Çeçen大学Ağrı艺术与科学学院数学系3土耳其Ağrı İbrahim Çeçen大学Ağrı研究生院4土耳其吉雷松大学吉雷松科学与艺术学院数学系5印度斋浦尔国际基础与应用科学中心6印度斋浦尔阿南德国际工程学院数学系
{"title":"Some New Integral Inequalities via General Forms of Proportional Fractional Integral Operators","authors":"S. Butt, A. Akdemir, S. Aslan, I. Işcan, P. Agarwal","doi":"10.47443/cm.2021.0027","DOIUrl":"https://doi.org/10.47443/cm.2021.0027","url":null,"abstract":"1COMSATS University Islamabad, Lahore Campus, Pakistan 2Department of Mathematics, Faculty of Arts and Sciences, Ağrı İbrahim Çeçen University, Ağrı, Turkey 3Institute of Graduate Studies, Ağrı İbrahim Çeçen University, Ağrı, Turkey 4Department of Mathematics, Faculty of Science and Arts, Giresun University, Giresun, Turkey 5International Center for Basic and Applied Sciences, Jaipur, India 6Department of Mathematics, Anand International College of Engineering, Jaipur, India","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"21 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83724770","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}
Şerban Bărbuleanu, M. Mihăilescu, Denisa Stancu-Dumitru
Abstract Let Ω ⊂ R be an open and bounded set. Consider the eigenvalue problem of the Laplace operator with a drift term −∆u−x ·∇u = λu in Ω subject to the homogeneous Dirichlet boundary condition (u = 0 on ∂Ω). Denote by λ1(Ω) and λ2(Ω) the first two eigenvalues of the problem. We show that λ2(Ω)λ1(Ω) ≤ 1 + 4N−1. In particular, we complement a similar result obtained by Thompson [Stud. Appl. Math. 48 (1969) 281–283] for the classical eigenvalue problem of the Laplace operator.
{"title":"Estimates for the ratio of the first two eigenvalues of the Dirichlet-Laplace operator with\u0000a drift","authors":"Şerban Bărbuleanu, M. Mihăilescu, Denisa Stancu-Dumitru","doi":"10.47443/cm.2021.0043","DOIUrl":"https://doi.org/10.47443/cm.2021.0043","url":null,"abstract":"Abstract Let Ω ⊂ R be an open and bounded set. Consider the eigenvalue problem of the Laplace operator with a drift term −∆u−x ·∇u = λu in Ω subject to the homogeneous Dirichlet boundary condition (u = 0 on ∂Ω). Denote by λ1(Ω) and λ2(Ω) the first two eigenvalues of the problem. We show that λ2(Ω)λ1(Ω) ≤ 1 + 4N−1. In particular, we complement a similar result obtained by Thompson [Stud. Appl. Math. 48 (1969) 281–283] for the classical eigenvalue problem of the Laplace operator.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89157412","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}
A set S of vertices in a nontrivial connected graph G is a total dominating set if every vertex of G is adjacent to some vertex of S. The minimum cardinality of a total dominating set for G is the total domination number of G. A function h : V (G) → {0, 1} is a total dominating function of a graph G if σh(v) = ∑ u∈N(v) h(u) ≥ 1 for every vertex v of G. A total dominating function h of a nontrivial graph G is irregular if σh(u) 6= σh(v) for every two vertices u and v of G. While no graph possesses an irregular total dominating function, a graph G has an antiregular total dominating function h if there are exactly two vertices u and v of G such that σh(u) = σh(v). It is shown that for every integer n ≥ 3, there are exactly two non-isomorphic graphs of order n having an antiregular total dominating function. If h is a total dominating function of a graph G such that σh(v) is the same constant k for every vertex v of G, then h is a k-regular total dominating function of G. We present some results dealing with properties of regular total dominating functions of graphs. In particular, regular total dominating functions of trees are investigated. The only possible regular total dominating functions for a nontrivial tree are 1-regular total dominating functions. We characterize those trees having a 1-regular total dominating function. We also investigate k-regular total dominating functions of several well-known classes of regular graphs for various values of k.
{"title":"Total dominating functions of graphs: antiregularity versus regularity","authors":"Maria Talanda-Fisher, Ping Zhang","doi":"10.47443/cm.2020.0045","DOIUrl":"https://doi.org/10.47443/cm.2020.0045","url":null,"abstract":"A set S of vertices in a nontrivial connected graph G is a total dominating set if every vertex of G is adjacent to some vertex of S. The minimum cardinality of a total dominating set for G is the total domination number of G. A function h : V (G) → {0, 1} is a total dominating function of a graph G if σh(v) = ∑ u∈N(v) h(u) ≥ 1 for every vertex v of G. A total dominating function h of a nontrivial graph G is irregular if σh(u) 6= σh(v) for every two vertices u and v of G. While no graph possesses an irregular total dominating function, a graph G has an antiregular total dominating function h if there are exactly two vertices u and v of G such that σh(u) = σh(v). It is shown that for every integer n ≥ 3, there are exactly two non-isomorphic graphs of order n having an antiregular total dominating function. If h is a total dominating function of a graph G such that σh(v) is the same constant k for every vertex v of G, then h is a k-regular total dominating function of G. We present some results dealing with properties of regular total dominating functions of graphs. In particular, regular total dominating functions of trees are investigated. The only possible regular total dominating functions for a nontrivial tree are 1-regular total dominating functions. We characterize those trees having a 1-regular total dominating function. We also investigate k-regular total dominating functions of several well-known classes of regular graphs for various values of k.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84649158","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}
Fractional calculus, a branch of mathematics, is focused on the study and applications of the differential and integral operators of non-integer order. Although the fractional calculus is as old as the classical calculus, it has become one of the most developed areas of mathematics only in the last 40 years, not only because of the exponential growth of the number of publications in this area, but also due to its different applications and its overlapping with other areas of mathematics. This area has been developed intensively in recent years and it has found multiple applications in various fields. The classical results were basically extended in two fundamental directions: Riemann–Liouville fractional derivative and Caputo fractional derivative. As a result of the progress made in this area, numerous fractional (global) and generalized (local) operators have been appeared. These new operators give researchers the possibility to choose the one that suits best with the problem they investigate. Readers can consult the paper [2] where some reasons are given to justify the appearance of these new operators and where the applications and theoretical developments of these operators are discussed. These operators, developed by many mathematicians with a hardly specific formulation, include the Riemann–Liouville (RL), Weyl, Erdelyi-Kober and Hadamard integrals, and the fractional operators of Liouville and Katugampola. Many authors have even introduced new fractional operators generated from the general local differential operators. In this direction, a generalized local derivative was defined in [19], which generalizes both the conformable and non-conformable derivatives and that is the basis for the generalized integral operator proposed in [7], which contains as a particular case the fractional integral of Riemann– Liouville (see [31]). In fact, these new operators require a classification as they can cause confusion in researchers. Baleanu and Fernandez [3] gave a fairly complete classification of these fractional and generalized operators together with abundant information and references. For a more complete review, the readers are referred to Chapter 1 of [1], where a history of differential operators (both local and global) from Newton to Caputo is presented and where the qualitative differences between the operators are shown. Section 1.4 of [1] contains some conclusions that we want to highlight: “Therefore, we can conclude that the Riemann–Liouville and Caputo operators are not derivatives and, therefore, they are not fractional derivatives, but fractional operators. We agree with the result [27] that the local fractional operator is not a fractional derivative” (see p.24 in [1]). In this work, we present a new definition of the k-generalized fractional derivative of the Hilfer type, and we study its fundamental properties. We also present a particular case with a kernel defined in terms of the sigmoid function. The gamma function Γ (see [21
{"title":"Generalized fractional Hilfer integral and derivative","authors":"J. E. Valdés","doi":"10.47443/cm.2020.0036","DOIUrl":"https://doi.org/10.47443/cm.2020.0036","url":null,"abstract":"Fractional calculus, a branch of mathematics, is focused on the study and applications of the differential and integral operators of non-integer order. Although the fractional calculus is as old as the classical calculus, it has become one of the most developed areas of mathematics only in the last 40 years, not only because of the exponential growth of the number of publications in this area, but also due to its different applications and its overlapping with other areas of mathematics. This area has been developed intensively in recent years and it has found multiple applications in various fields. The classical results were basically extended in two fundamental directions: Riemann–Liouville fractional derivative and Caputo fractional derivative. As a result of the progress made in this area, numerous fractional (global) and generalized (local) operators have been appeared. These new operators give researchers the possibility to choose the one that suits best with the problem they investigate. Readers can consult the paper [2] where some reasons are given to justify the appearance of these new operators and where the applications and theoretical developments of these operators are discussed. These operators, developed by many mathematicians with a hardly specific formulation, include the Riemann–Liouville (RL), Weyl, Erdelyi-Kober and Hadamard integrals, and the fractional operators of Liouville and Katugampola. Many authors have even introduced new fractional operators generated from the general local differential operators. In this direction, a generalized local derivative was defined in [19], which generalizes both the conformable and non-conformable derivatives and that is the basis for the generalized integral operator proposed in [7], which contains as a particular case the fractional integral of Riemann– Liouville (see [31]). In fact, these new operators require a classification as they can cause confusion in researchers. Baleanu and Fernandez [3] gave a fairly complete classification of these fractional and generalized operators together with abundant information and references. For a more complete review, the readers are referred to Chapter 1 of [1], where a history of differential operators (both local and global) from Newton to Caputo is presented and where the qualitative differences between the operators are shown. Section 1.4 of [1] contains some conclusions that we want to highlight: “Therefore, we can conclude that the Riemann–Liouville and Caputo operators are not derivatives and, therefore, they are not fractional derivatives, but fractional operators. We agree with the result [27] that the local fractional operator is not a fractional derivative” (see p.24 in [1]). In this work, we present a new definition of the k-generalized fractional derivative of the Hilfer type, and we study its fundamental properties. We also present a particular case with a kernel defined in terms of the sigmoid function. The gamma function Γ (see [21","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79796383","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}
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers of the first kind we construct a number of expansions of functions in terms of inverse factorial series where the coefficients are special numbers. These results are used to prove/reprove the asymptotic expansion of some classical functions. We also prove a binomial formula involving inverse factorials.
{"title":"Stirling Numbers and Inverse Factorial Series","authors":"K. Boyadzhiev","doi":"10.47443/cm.2023.002","DOIUrl":"https://doi.org/10.47443/cm.2023.002","url":null,"abstract":"We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers of the first kind we construct a number of expansions of functions in terms of inverse factorial series where the coefficients are special numbers. These results are used to prove/reprove the asymptotic expansion of some classical functions. We also prove a binomial formula involving inverse factorials.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88696608","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}
In this paper, the convolution integrals (cid:82) t 0 ( t − s ) λ − 1 b ( s ) ds with hyper-singular kernels are considered, where λ ≤ 0 and either b is a smooth function or b belongs to L 1 ( R + ) . For such λ , these integrals diverge classically even for smooth b . These convolution integrals are defined in this paper for negative non-integer values of λ . Integral equations and inequalities are considered with the hyper-singular kernels ( t − s ) λ − 1 + for λ ≤ 0 , where t λ + := 0 for t < 0 . In particular, one is interested in the value λ = − 14 because it is important for the Navier-Stokes problem (NSP). Integral equations of the type b ( t ) = b 0 ( t ) + (cid:82) t 0 ( t − s ) λ − 1 b ( s ) ds , λ ≤ 0 , are also studied. The solution of these equations is investigated, and the existence and uniqueness of the solution is proved for λ = − 14 . The obtained results are applied to the analysis of the NSP in the space R 3 without boundaries. It is proved that the NSP is contradictory in the following sense: even if one assumes that v ( x, 0) > 0 , one proves that the solution v ( x, t ) to the NSP has the property v ( x, 0) = 0 , in general. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and it proves that the NSP does not have a solution, in general.
本文研究了具有超奇异核的卷积积分(cid:82) t 0 (t−s) λ−1 b (s) ds，其中λ≤0且b是光滑函数或b属于L 1 (R +)。对于这样的λ，这些积分即使在光滑b上也是发散的。本文对λ的负非整数值定义了这些卷积积分。考虑了λ≤0时具有超奇异核(t -s) λ−1 +的积分方程和不等式，其中t < 0时t λ + = 0。特别地，人们对λ = - 14的值感兴趣，因为它对Navier-Stokes问题(NSP)很重要。研究了b (t) = b 0 (t) + (cid:82) t 0 (t−s) λ−1 b (s) ds， λ≤0的积分方程。研究了这些方程的解，并证明了当λ =−14时解的存在唯一性。并将所得结果应用于无边界空间r3中NSP的分析。在以下意义上证明了NSP是矛盾的:即使假设v (x, 0) >，也证明了NSP的解v (x, t)一般具有v (x, 0) = 0的性质。这一悖论表明NSP不是对流体力学问题的正确描述，并证明NSP通常没有解。
{"title":"Theory of hyper-singular integrals and its application to the Navier-Stokes problem","authors":"A. Ramm","doi":"10.47443/cm.2020.0041","DOIUrl":"https://doi.org/10.47443/cm.2020.0041","url":null,"abstract":"In this paper, the convolution integrals (cid:82) t 0 ( t − s ) λ − 1 b ( s ) ds with hyper-singular kernels are considered, where λ ≤ 0 and either b is a smooth function or b belongs to L 1 ( R + ) . For such λ , these integrals diverge classically even for smooth b . These convolution integrals are defined in this paper for negative non-integer values of λ . Integral equations and inequalities are considered with the hyper-singular kernels ( t − s ) λ − 1 + for λ ≤ 0 , where t λ + := 0 for t < 0 . In particular, one is interested in the value λ = − 14 because it is important for the Navier-Stokes problem (NSP). Integral equations of the type b ( t ) = b 0 ( t ) + (cid:82) t 0 ( t − s ) λ − 1 b ( s ) ds , λ ≤ 0 , are also studied. The solution of these equations is investigated, and the existence and uniqueness of the solution is proved for λ = − 14 . The obtained results are applied to the analysis of the NSP in the space R 3 without boundaries. It is proved that the NSP is contradictory in the following sense: even if one assumes that v ( x, 0) > 0 , one proves that the solution v ( x, t ) to the NSP has the property v ( x, 0) = 0 , in general. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and it proves that the NSP does not have a solution, in general.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81803254","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}
An extension method for linear functionals is given. The proposed method provides extensions of a linear functional T defined on a subspace X of a vector space Y over a field K, by using a suitable isomorphism S : Y −→ Y that satisfies S (X) = X and TS = T. The extension Text : Xext −→ K is linear, and it is defined over a vector space Xext that contains X. Several illustrations are considered, including symmetric values, extension with respect to dilations, extended Cesàro summability of series, and extended multidimensional point values.
给出了线性泛函的一种扩展方法。该方法提供了扩展的线性泛函T定义在向量空间的子空间X Y /字段K,通过使用一个合适的同构S: Y−→Y满足S (X) = X和TS = T扩展文本:Xext−→K是线性的,它定义在向量空间Xext包含X考虑一些插图,包括对称值,扩展对相呼应,纬洛系列,可和性和多维延伸点值。
{"title":"Extended point values of distributions","authors":"R. Estrada","doi":"10.47443/cm.2020.0040","DOIUrl":"https://doi.org/10.47443/cm.2020.0040","url":null,"abstract":"An extension method for linear functionals is given. The proposed method provides extensions of a linear functional T defined on a subspace X of a vector space Y over a field K, by using a suitable isomorphism S : Y −→ Y that satisfies S (X) = X and TS = T. The extension Text : Xext −→ K is linear, and it is defined over a vector space Xext that contains X. Several illustrations are considered, including symmetric values, extension with respect to dilations, extended Cesàro summability of series, and extended multidimensional point values.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75228584","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}
For parameters $n,delta,B,C$, we obtain sharp asymptotic formula for number of $(n+lfloor n^deltarfloor)^2$ dimensional binary contingency tables with non-uniform margins $lfloor BCnrfloor$ and $lfloor Cnrfloor$. Furthermore, we compare our results with the classical textit{independent heuristic} and prove that the independent heuristic overestimates by a factor of $e^{Theta(n^{2delta})}$. Our comparison is based on the analysis of the textit{correlation ratio} and we obtain the explicit bound for the constant in $Theta$.
{"title":"Asymptotic enumeration of binary contingency tables and comparison with independence heuristic","authors":"Da Wu","doi":"10.47443/cm.2023.037","DOIUrl":"https://doi.org/10.47443/cm.2023.037","url":null,"abstract":"For parameters $n,delta,B,C$, we obtain sharp asymptotic formula for number of $(n+lfloor n^deltarfloor)^2$ dimensional binary contingency tables with non-uniform margins $lfloor BCnrfloor$ and $lfloor Cnrfloor$. Furthermore, we compare our results with the classical textit{independent heuristic} and prove that the independent heuristic overestimates by a factor of $e^{Theta(n^{2delta})}$. Our comparison is based on the analysis of the textit{correlation ratio} and we obtain the explicit bound for the constant in $Theta$.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"50 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83992027","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}
In this note, spectral conditions involving the eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues are derived for Hamiltonian properties of graphs.
本文给出了图的哈密顿性质中包含特征值、拉普拉斯特征值和无符号拉普拉斯特征值的谱条件。
{"title":"Combinations of some spectral invariants and Hamiltonian properties of graphs","authors":"Rao Li","doi":"10.47443/cm.2020.0013","DOIUrl":"https://doi.org/10.47443/cm.2020.0013","url":null,"abstract":"In this note, spectral conditions involving the eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues are derived for Hamiltonian properties of graphs.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"54 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77953656","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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