We report new hypergeometric constructions of rational approximations to Catalan's constant, $log2$, and $pi^2$, their connection with already known ones, and underlying `permutation group' structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.
{"title":"HYPERGEOMETRY INSPIRED BY IRRATIONALITY QUESTIONS","authors":"C. Krattenthaler, W. Zudilin","doi":"10.2206/kyushujm.73.189","DOIUrl":"https://doi.org/10.2206/kyushujm.73.189","url":null,"abstract":"We report new hypergeometric constructions of rational approximations to Catalan's constant, $log2$, and $pi^2$, their connection with already known ones, and underlying `permutation group' structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/kyushujm.73.189","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48528700","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 consider Mellin-Barnes integral representations of GKZ hypergeometric equations. We construct integration contours in an explicit way and show that suitable analytic continuations give rise to a basis of solutions.
{"title":"ON MELLIN-BARNES INTEGRAL REPRESENTATIONS FOR GKZ HYPERGEOMETRIC FUNCTIONS","authors":"Saiei-Jaeyeong Matsubara-Heo","doi":"10.2206/kyushujm.74.109","DOIUrl":"https://doi.org/10.2206/kyushujm.74.109","url":null,"abstract":"We consider Mellin-Barnes integral representations of GKZ hypergeometric equations. We construct integration contours in an explicit way and show that suitable analytic continuations give rise to a basis of solutions.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44410309","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}
Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the noncommutative polynomial algebra in two letters. These give a class of relations among multiple zeta values, which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.
{"title":"ROOTED TREE MAPS AND THE KAWASHIMA RELATIONS FOR MULTIPLE ZETA VALUES","authors":"Henrik Bachmann, Tatsushi Tanaka","doi":"10.2206/kyushujm.74.169","DOIUrl":"https://doi.org/10.2206/kyushujm.74.169","url":null,"abstract":"Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the noncommutative polynomial algebra in two letters. These give a class of relations among multiple zeta values, which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48124147","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, we establish the asymptotic formula for the second moment of the Riemann zeta-function twisted by a (3+ 1)-piece mollifier which is a generalization of the two-piece mollifier considered by Bui, Conrey and Young [Acta. Arith. 150(1) (2011), 35–64]. As an application, we obtain a lower bound for the proportion of critical zeros of the Riemann zeta-function.
{"title":"AN APPLICATION OF GENERALIZED MOLLIFIERS TO THE RIEMANN ZETA-FUNCTION","authors":"Keiju Sono","doi":"10.2206/KYUSHUJM.72.35","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.72.35","url":null,"abstract":"In this paper, we establish the asymptotic formula for the second moment of the Riemann zeta-function twisted by a (3+ 1)-piece mollifier which is a generalization of the two-piece mollifier considered by Bui, Conrey and Young [Acta. Arith. 150(1) (2011), 35–64]. As an application, we obtain a lower bound for the proportion of critical zeros of the Riemann zeta-function.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/KYUSHUJM.72.35","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555759","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}
The notion of absolute continuity for positive operators was studied by T. Ando, where parallel sums for such operators played an important role. On the other hand, a theory for parallel sums for densely defined positive self-adjoint operators (or more generally positive forms) was developed in our previous work. Based on this theory, we will investigate the notion of absolute continuity in such unbounded cases.
T. Ando研究了正算子的绝对连续性的概念，其中这类算子的并行和起了重要作用。另一方面，我们在以前的工作中发展了密集定义的正自伴随算子(或更一般的正形式)的平行和理论。基于这一理论，我们将研究这种无界情况下的绝对连续性的概念。
{"title":"ABSOLUTE CONTINUITY FOR UNBOUNDED POSITIVE SELF-ADJOINT OPERATORS","authors":"H. Kosaki","doi":"10.2206/KYUSHUJM.72.407","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.72.407","url":null,"abstract":"The notion of absolute continuity for positive operators was studied by T. Ando, where parallel sums for such operators played an important role. On the other hand, a theory for parallel sums for densely defined positive self-adjoint operators (or more generally positive forms) was developed in our previous work. Based on this theory, we will investigate the notion of absolute continuity in such unbounded cases.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/KYUSHUJM.72.407","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555791","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}
Summary: In the present article we study the even unimodular lattice which lies between the root lattice m · A n and the dual lattice ( m · A n ) # . Here m · A n is an orthogonal sum of m copies of the root lattice A n . In the course of the study the code over the ring A # n /A n arises in a natural way. We find that an intimate relationship between the even unimodular lattice containing m · A n as a sublattice and the error correcting code over the ring A # n /A n exists. As a consequence we could reconstruct sixteen non-isometric Niemeier lattices out of twenty-four non-isometric lattices by using the present approach.
摘要:本文研究了介于根格m·n和对偶格(m·n) #之间的偶单模格。这里m·n是根晶格n的m个拷贝的正交和。在研究过程中，环上的代码a# n / an自然出现。我们发现含有m·A·n作为子格的偶单模格与环A # n /A n上的纠错码之间存在密切关系。结果表明，利用本方法可以从24个非等距尼迈耶晶格中重构出16个非等距尼迈耶晶格。
{"title":"A DETAILED STUDY OF THE RELATIONSHIP BETWEEN SOME OF THE ROOT LATTICES AND THE CODING THEORY","authors":"M. Ozeki","doi":"10.2206/KYUSHUJM.72.123","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.72.123","url":null,"abstract":"Summary: In the present article we study the even unimodular lattice which lies between the root lattice m · A n and the dual lattice ( m · A n ) # . Here m · A n is an orthogonal sum of m copies of the root lattice A n . In the course of the study the code over the ring A # n /A n arises in a natural way. We find that an intimate relationship between the even unimodular lattice containing m · A n as a sublattice and the error correcting code over the ring A # n /A n exists. As a consequence we could reconstruct sixteen non-isometric Niemeier lattices out of twenty-four non-isometric lattices by using the present approach.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555003","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}
The first aim of this paper is to show the second main theorem for holomorphic maps from a compact Riemann surface into the complex projective space which is ramified over hypersurfaces in subgeneral position. We then use it to study the ramification over hypersurfaces of the generalized Gauss map of complete regular minimal surfaces in Rm with finite total curvature, sharing hypersurfaces in subgeneral position. The results generalize our previous results [Thai and Thoan, Vietnam J. Math. 2017, doi:10.1007/s10013-017-0259-6].
本文的第一个目的是证明从紧黎曼曲面到复射影空间的全纯映射的第二个主要定理。然后，我们利用它研究了有限总曲率Rm中完全正则极小曲面的广义高斯映射在超曲面上的分支，在次一般位置共享超曲面。结果概括了我们之前的结果[Thai and than, Vietnam J. Math. 2017, doi:10.1007/s10013-017-0259-6]。
{"title":"RAMIFICATION OVER HYPERSURFACES LOCATED IN SUBGENERAL POSITION OF THE GAUSS MAP OF COMPLETE MINIMAL SURFACES WITH FINITE TOTAL CURVATURE","authors":"D. D. Thai, Pham Duc Thoan","doi":"10.2206/KYUSHUJM.72.253","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.72.253","url":null,"abstract":"The first aim of this paper is to show the second main theorem for holomorphic maps from a compact Riemann surface into the complex projective space which is ramified over hypersurfaces in subgeneral position. We then use it to study the ramification over hypersurfaces of the generalized Gauss map of complete regular minimal surfaces in Rm with finite total curvature, sharing hypersurfaces in subgeneral position. The results generalize our previous results [Thai and Thoan, Vietnam J. Math. 2017, doi:10.1007/s10013-017-0259-6].","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/KYUSHUJM.72.253","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555328","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}
Ricci solitons are the self-similar solutions to the Ricci flow, which play an important role in understanding the singularity dilations of the Ricci flow. In this paper, we investigate eigenvalues of the Dirichlet problem of a drifting Laplacian on some important complete Ricci solitons: the product shrinking Ricci soliton, cigar soliton, and so on. Since eigenvalues are invariant of isometries, we can give the estimates for the eigenvalues of a drifting Laplacian on the rotationally invariant shrinking solitons. In addition, we also obtain a sharp upper bound of the kth eigenvalue of the a drifting Laplacian on the product Ricci soliton in the sense of order k.
{"title":"ESTIMATES FOR THE EIGENVALUES OF THE DRIFTING LAPLACIAN ON SOME COMPLETE RICCI SOLITONS","authors":"Lingzhong Zeng","doi":"10.2206/KYUSHUJM.72.143","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.72.143","url":null,"abstract":"Ricci solitons are the self-similar solutions to the Ricci flow, which play an important role in understanding the singularity dilations of the Ricci flow. In this paper, we investigate eigenvalues of the Dirichlet problem of a drifting Laplacian on some important complete Ricci solitons: the product shrinking Ricci soliton, cigar soliton, and so on. Since eigenvalues are invariant of isometries, we can give the estimates for the eigenvalues of a drifting Laplacian on the rotationally invariant shrinking solitons. In addition, we also obtain a sharp upper bound of the kth eigenvalue of the a drifting Laplacian on the product Ricci soliton in the sense of order k.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/KYUSHUJM.72.143","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555055","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}
Let G be a finite group and A(G) the Burnside ring of G. The family of rings A(H), where H ranges over the set of all proper subgroups of G, yields the inverse limit L(G) and a canonical homomorphism from A(G) to L(G) which is called the restriction map. Let Q(G) be the cokernel of this homomorphism. It is known that Q(G) is a finite abelian group and is isomorphic to the cartesian product of Q(G/N (p)), where p runs over the set of primes dividing the order of G and N (p) stands for the smallest normal subgroup of G such that the order of G/N (p) is a power of p. Therefore, it is important to investigate Q(G) for G of prime power order. In this paper we develop a way to compute Q(G) for cartesian products G of two cyclic p-groups.
{"title":"COKERNELS OF HOMOMORPHISMS FROM BURNSIDE RINGS TO INVERSE LIMITS II: G = Cpm × Cpn","authors":"M. Morimoto, Masafumi Sugimura","doi":"10.2206/KYUSHUJM.72.95","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.72.95","url":null,"abstract":"Let G be a finite group and A(G) the Burnside ring of G. The family of rings A(H), where H ranges over the set of all proper subgroups of G, yields the inverse limit L(G) and a canonical homomorphism from A(G) to L(G) which is called the restriction map. Let Q(G) be the cokernel of this homomorphism. It is known that Q(G) is a finite abelian group and is isomorphic to the cartesian product of Q(G/N (p)), where p runs over the set of primes dividing the order of G and N (p) stands for the smallest normal subgroup of G such that the order of G/N (p) is a power of p. Therefore, it is important to investigate Q(G) for G of prime power order. In this paper we develop a way to compute Q(G) for cartesian products G of two cyclic p-groups.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555428","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. Takeuchi showed that the negative logarithm of the Fubini–Study boundary distance function of pseudoconvex domains in the complex projective space CPn , n ∈ N, is strictly plurisubharmonic and solved the Levi problem for CPn . His estimate from below of the Levi form is nowadays called the ‘Takeuchi’s inequality.’ In this paper, we give the ‘Takeuchi’s equality,’ i.e. an explicit representation of the Levi form of the negative logarithm of the Fubini–Study distance to complex submanifolds in CPn . 0. Introduction Let D (CPn , n ∈ N, be a pseudoconvex domain and denote by δ∂D(P) the Fubini– Study distance from P ∈ D to the boundary ∂D of D. Takeuchi [21] found that the strict subharmonicity of the function −log tan−1|z| on C {0} leads the strict plurisubharmonicity of the function −log δ∂D on D, and solved the Levi problem for CPn . The inequality i∂∂̄(−log δ∂D)≥ 3ωF S on D is nowadays called the ‘Takeuchi’s inequality’ (cf. [1, 7, 9, 20, 22]). Recently, many mathematicians have been interested in the following problem: ‘Is there a smooth closed Levi-flat real hypersurface in CPn if n ≥ 2?’, where a real hypersurface M ⊂ CPn is said to be Levi-flat if its complement CPn M is locally pseudoconvex or equivalently locally Stein. When n ≥ 3, Lins Neto [11] proved the non-existence in the real analytic case, and Siu [19] proved it in the smooth case. When n = 2, the non-existence problem is still open even in the real analytic case. Then Takeuchi’s inequality is one of the key points to approach the non-existence problem or related topics. His paper [21] is frequently cited even now although he wrote it over 50 years ago (for example, see Adachi [1], Adachi and Brinkschulte [2], Brinkschulte [5], Brunella [6], Fu and Shaw [8], Harrington and Shaw [10], Ohsawa [16, 17], and Ohsawa and Sibony [18]). It follows from Takeuchi’s theorem that if S is a complex hypersurface in CPn and if δS denotes the Fubini–Study distance to S, then the function−log δS is strictly plurisubharmonic 2010 Mathematics Subject Classification: Primary 32E40, 32C25.
A. Takeuchi证明了复射影空间CPn (n∈n)中伪凸域的Fubini-Study边界距离函数的负对数是严格多次调和的，并解决了CPn的Levi问题。他对李维形式的估计现在被称为“竹内不等式”。在本文中，我们给出了“Takeuchi等式”，即CPn中复子流形的Fubini-Study距离的负对数的Levi形式的显式表示。0. 设D(CPn, n∈n)为伪凸域，用δ∂D(P)表示富比尼-研究从P∈D到D的边界∂D的距离。Takeuchi[21]发现函数- log tan - 1|z|在C {0}上的严格亚调和性导致函数- log δ∂D在D上的严格多亚调和性，并解决了CPn的Levi问题。不等式i∂∂(−log δ∂D)≥3ω fs on D现在被称为“Takeuchi不等式”(参见[1,7,9,20,22])。近年来，许多数学家对以下问题很感兴趣:当n≥2时，CPn中是否存在光滑闭合的列维平坦实超曲面?，其中实超曲面M∧CPn如果其补集CPn M是局部伪凸或等价的局部斯坦因，则称其为列维平坦。当n≥3时，Lins Neto[11]证明了实解析情况下的不存在性，Siu[19]证明了光滑情况下的不存在性。当n = 2时，即使在实际解析情况下，不存在性问题仍然是开放的。因此，竹内不等式是研究不存在问题或相关课题的关键之一。他的论文[21]虽然写于50多年前，但至今仍被频繁引用(如:Adachi[1]、Adachi and Brinkschulte[2]、Brinkschulte[5]、Brunella[6]、Fu and Shaw[8]、Harrington and Shaw[10]、Ohsawa[16, 17]、Ohsawa and Sibony[18])。由Takeuchi定理可知，如果S是CPn中的一个复超曲面，且δS表示到S的Fubini-Study距离，则函数- log δS是严格的多次谐波。
{"title":"Takeuchi’s equality for the levi form of the fubini–study distance to complex submanifolds in complex projective spaces","authors":"Kazuko Matsumoto","doi":"10.2206/KYUSHUJM.72.107","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.72.107","url":null,"abstract":"A. Takeuchi showed that the negative logarithm of the Fubini–Study boundary distance function of pseudoconvex domains in the complex projective space CPn , n ∈ N, is strictly plurisubharmonic and solved the Levi problem for CPn . His estimate from below of the Levi form is nowadays called the ‘Takeuchi’s inequality.’ In this paper, we give the ‘Takeuchi’s equality,’ i.e. an explicit representation of the Levi form of the negative logarithm of the Fubini–Study distance to complex submanifolds in CPn . 0. Introduction Let D (CPn , n ∈ N, be a pseudoconvex domain and denote by δ∂D(P) the Fubini– Study distance from P ∈ D to the boundary ∂D of D. Takeuchi [21] found that the strict subharmonicity of the function −log tan−1|z| on C {0} leads the strict plurisubharmonicity of the function −log δ∂D on D, and solved the Levi problem for CPn . The inequality i∂∂̄(−log δ∂D)≥ 3ωF S on D is nowadays called the ‘Takeuchi’s inequality’ (cf. [1, 7, 9, 20, 22]). Recently, many mathematicians have been interested in the following problem: ‘Is there a smooth closed Levi-flat real hypersurface in CPn if n ≥ 2?’, where a real hypersurface M ⊂ CPn is said to be Levi-flat if its complement CPn M is locally pseudoconvex or equivalently locally Stein. When n ≥ 3, Lins Neto [11] proved the non-existence in the real analytic case, and Siu [19] proved it in the smooth case. When n = 2, the non-existence problem is still open even in the real analytic case. Then Takeuchi’s inequality is one of the key points to approach the non-existence problem or related topics. His paper [21] is frequently cited even now although he wrote it over 50 years ago (for example, see Adachi [1], Adachi and Brinkschulte [2], Brinkschulte [5], Brunella [6], Fu and Shaw [8], Harrington and Shaw [10], Ohsawa [16, 17], and Ohsawa and Sibony [18]). It follows from Takeuchi’s theorem that if S is a complex hypersurface in CPn and if δS denotes the Fubini–Study distance to S, then the function−log δS is strictly plurisubharmonic 2010 Mathematics Subject Classification: Primary 32E40, 32C25.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68554978","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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