The Köthe–Bochner spaces Lρ(E) are the vector valued version of the scalar Köthe spaces Lρ, which generalize the Lebesgue spaces L , the Orlicz spaces and many other functional spaces. In the present paper we study the linear and continuous operators U : Lρ(E) → F , giving integral representations for them. These operators generate operators V : Lρ → L(E,F ) which we call “natural operators” and study here.
{"title":"Linear and continuous operators on Köthe–Bochner spaces","authors":"I. Chitescu, Razvan-Cornel Sfetcu","doi":"10.5186/AASFM.2019.4454","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4454","url":null,"abstract":"The Köthe–Bochner spaces Lρ(E) are the vector valued version of the scalar Köthe spaces Lρ, which generalize the Lebesgue spaces L , the Orlicz spaces and many other functional spaces. In the present paper we study the linear and continuous operators U : Lρ(E) → F , giving integral representations for them. These operators generate operators V : Lρ → L(E,F ) which we call “natural operators” and study here.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89390554","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 a nonlinear Robin problem driven by the p-Laplace differential operator and with a reaction term which depends also on the gradient (convection). Using a topological approach based on the Leray–Schauder alternative principle, we show that the problem has a positive smooth solution.
{"title":"Existence of positive solutions for nonlinear Robin problems with gradient dependence","authors":"Nikolaos S. Papageorgiou, Chao Zhang","doi":"10.5186/AASFM.2019.4437","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4437","url":null,"abstract":"We consider a nonlinear Robin problem driven by the p-Laplace differential operator and with a reaction term which depends also on the gradient (convection). Using a topological approach based on the Leray–Schauder alternative principle, we show that the problem has a positive smooth solution.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72510556","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 investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change the Segal entropy of the density of a normal, not necessarily normalised, state. Two cases are dealt with: a) no restriction on the map is imposed, b) the map represents a repeatable instrument in measurement theory which means that it is idempotent. Introduction In the paper, the question of invariance of Segal’s entropy under the action of a normal positive linear unital map is addressed in the case of a semifinite von Neumann algebra. The notion of Segal’s entropy was introduced by Segal in [9] for semifinite von Neumann algebras as a direct counterpart of von Neumann’s entropy defined for the full algebra B(H) of all bounded linear operators on a Hilbert space by means of the canonical trace. However, in the case of an arbitrary semifinite von Neumann algebra, where instead of the canonical trace we have a normal semifinite faithful trace, substantial differences between these two entropies arise. Perhaps the most fundamental one consists in the fact that while a normal state on B(H) is represented by a positive operator of trace one (the so-called ‘density matrix’), in the case of an arbitrary semifinite von Neumann algebra this ‘density matrix’ can be an unbounded operator. This prompted Segal to consider only the states whose ‘density matrices’ were in the algebra. In our analysis, we avoid this restriction as well as we allow the trace to be semifinite and not finite, the latter being also often assumed while dealing with Segal’s entropy. On the way to the main theorems, some auxiliary results about strict operator convexity or Jensen’s inequality for unbounded measurable operators are obtained which seem to be interesting and of some importance in their own right. 1. Preliminaries and notation Let M be a semifinite von Neumann algebra of operators acting on a Hilbert space H with a normal semifinite faithful trace τ , identity 1, and predual M∗. By M we shall denote the set of positive operators in M , and by M ∗ —the set of positive functionals in M∗. These functionals will be sometimes referred to as (nonnormalised) states. https://doi.org/10.5186/aasfm.2019.4439 2010 Mathematics Subject Classification: Primary 46L53; Secondary 81P45.
{"title":"Mappings preserving Segal's entropy in von Neumann algebras","authors":"A. Luczak, H. Podsędkowska","doi":"10.5186/AASFM.2019.4439","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4439","url":null,"abstract":"We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change the Segal entropy of the density of a normal, not necessarily normalised, state. Two cases are dealt with: a) no restriction on the map is imposed, b) the map represents a repeatable instrument in measurement theory which means that it is idempotent. Introduction In the paper, the question of invariance of Segal’s entropy under the action of a normal positive linear unital map is addressed in the case of a semifinite von Neumann algebra. The notion of Segal’s entropy was introduced by Segal in [9] for semifinite von Neumann algebras as a direct counterpart of von Neumann’s entropy defined for the full algebra B(H) of all bounded linear operators on a Hilbert space by means of the canonical trace. However, in the case of an arbitrary semifinite von Neumann algebra, where instead of the canonical trace we have a normal semifinite faithful trace, substantial differences between these two entropies arise. Perhaps the most fundamental one consists in the fact that while a normal state on B(H) is represented by a positive operator of trace one (the so-called ‘density matrix’), in the case of an arbitrary semifinite von Neumann algebra this ‘density matrix’ can be an unbounded operator. This prompted Segal to consider only the states whose ‘density matrices’ were in the algebra. In our analysis, we avoid this restriction as well as we allow the trace to be semifinite and not finite, the latter being also often assumed while dealing with Segal’s entropy. On the way to the main theorems, some auxiliary results about strict operator convexity or Jensen’s inequality for unbounded measurable operators are obtained which seem to be interesting and of some importance in their own right. 1. Preliminaries and notation Let M be a semifinite von Neumann algebra of operators acting on a Hilbert space H with a normal semifinite faithful trace τ , identity 1, and predual M∗. By M we shall denote the set of positive operators in M , and by M ∗ —the set of positive functionals in M∗. These functionals will be sometimes referred to as (nonnormalised) states. https://doi.org/10.5186/aasfm.2019.4439 2010 Mathematics Subject Classification: Primary 46L53; Secondary 81P45.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85138260","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}
Abstract. Due to Voronin’s universality theorem and Riemann–von Mangoldt formula, this paper concerns the problem of algebraic differential independence between the gamma function Γ and the function f(ζ), where ζ is the Riemann zeta function and f is a function with at least one zero-point. It is showed that Γ and f(ζ) cannot satisfy any nontrivial distinguished differential equation with meromorphic coefficients φ having Nevanlinna characteristic satisfying T (r, φ) = o(r) as r → ∞.
摘要利用Voronin的通用性定理和Riemann - von Mangoldt公式，研究了函数Γ与函数f(ζ)的代数微分无关性问题，其中ζ是Riemann ζ函数，f是至少有一个零点的函数。证明了Γ和f(ζ)不能满足任何亚纯系数φ具有满足T (r， φ) = o(r)为r→∞的Nevanlinna特征的非平凡微分方程。
{"title":"On algebraic differential equations of gamma function and Riemann zeta function","authors":"F. Lü","doi":"10.5186/AASFM.2019.4455","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4455","url":null,"abstract":"Abstract. Due to Voronin’s universality theorem and Riemann–von Mangoldt formula, this paper concerns the problem of algebraic differential independence between the gamma function Γ and the function f(ζ), where ζ is the Riemann zeta function and f is a function with at least one zero-point. It is showed that Γ and f(ζ) cannot satisfy any nontrivial distinguished differential equation with meromorphic coefficients φ having Nevanlinna characteristic satisfying T (r, φ) = o(r) as r → ∞.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78659252","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 we study generalized differentiability of functions on a class of fractals in Euclidean spaces. Such sets are not necessarily self-similar, but satisfy a weaker “scale-similar” property; in particular, they include the non self similar carpets introduced by Mackay–Tyson– Wildrick [12] but with different scale ratios. Specifically we identify certain geometric criteria for these fractals and, in the case that they have zero Lebesgue measure, we show that such fractals cannot support nonzero derivations in the sense of Weaver [16]. As a result (Theorem 26) such fractals cannot support Alberti representations and in particular, they cannot be Lipschitz differentiability spaces in the sense of Cheeger [3] and Keith [9]. 1. Motivation First order differentiable calculus has been extended from smooth manifolds to abstract metric spaces in many ways, by many authors. In this context, one important property of a metric space is the validity of Rademacher’s theorem, i.e. that Lipschitz functions are almost everywhere (a.e.) differentiable with respect to a choice of coordinates on that space. (For this reason, such spaces are known as Lipschitz differentiability spaces in the recent literature, e.g. [1, 2, 4] and said to have a measurable differentiable structure in earlier literature, e.g. [9, 11, 14].) The search for such a property naturally leads to questions of compatibility between a metric space and the choice of a Borel measure on that space. Even the case of Euclidean spaces has been addressed only recently. A result of De Phillipis and Rindler [5, Thm. 1.14] states that if Rademacher’s Theorem is true for a Radon measure μ on R, then μ must be absolutely continuous to m-dimensional Lebesgue measure. Here we address the case when μ is singular. As we will see, there is a large class of fractal sets, which we call Sierpiński-type fractals, for which Lipschitz functions do not even enjoy partial a.e. differentiability on the support of their natural measures. https://doi.org/10.5186/aasfm.2019.446
{"title":"Sierpinski-type fractals are differentiably trivial","authors":"E. Durand-Cartagena, Jasun Gong, J. Jaramillo","doi":"10.5186/AASFM.2019.4460","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4460","url":null,"abstract":"In this note we study generalized differentiability of functions on a class of fractals in Euclidean spaces. Such sets are not necessarily self-similar, but satisfy a weaker “scale-similar” property; in particular, they include the non self similar carpets introduced by Mackay–Tyson– Wildrick [12] but with different scale ratios. Specifically we identify certain geometric criteria for these fractals and, in the case that they have zero Lebesgue measure, we show that such fractals cannot support nonzero derivations in the sense of Weaver [16]. As a result (Theorem 26) such fractals cannot support Alberti representations and in particular, they cannot be Lipschitz differentiability spaces in the sense of Cheeger [3] and Keith [9]. 1. Motivation First order differentiable calculus has been extended from smooth manifolds to abstract metric spaces in many ways, by many authors. In this context, one important property of a metric space is the validity of Rademacher’s theorem, i.e. that Lipschitz functions are almost everywhere (a.e.) differentiable with respect to a choice of coordinates on that space. (For this reason, such spaces are known as Lipschitz differentiability spaces in the recent literature, e.g. [1, 2, 4] and said to have a measurable differentiable structure in earlier literature, e.g. [9, 11, 14].) The search for such a property naturally leads to questions of compatibility between a metric space and the choice of a Borel measure on that space. Even the case of Euclidean spaces has been addressed only recently. A result of De Phillipis and Rindler [5, Thm. 1.14] states that if Rademacher’s Theorem is true for a Radon measure μ on R, then μ must be absolutely continuous to m-dimensional Lebesgue measure. Here we address the case when μ is singular. As we will see, there is a large class of fractal sets, which we call Sierpiński-type fractals, for which Lipschitz functions do not even enjoy partial a.e. differentiability on the support of their natural measures. https://doi.org/10.5186/aasfm.2019.446","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86545762","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}
Abstract. We are concerned with the Neumann type boundary value problem to parabolic systems ∂tu− div(Dξf(x,Du)) = −Dug(x, u), where u is vector-valued, f satisfies a linear growth condition and ξ 7→ f(x, ξ) is convex. We prove that variational solutions of such systems can be approximated by variational solutions to ∂tu− div(Dξf(x,Du)) = −Dug(x, u) with p > 1. This can be interpreted both as a stability and existence result for general flows with linear growth.
{"title":"Existence for evolutionary Neumann problems with linear growth by stability results","authors":"Leah Schätzler","doi":"10.5186/AASFM.2019.4461","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4461","url":null,"abstract":"Abstract. We are concerned with the Neumann type boundary value problem to parabolic systems ∂tu− div(Dξf(x,Du)) = −Dug(x, u), where u is vector-valued, f satisfies a linear growth condition and ξ 7→ f(x, ξ) is convex. We prove that variational solutions of such systems can be approximated by variational solutions to ∂tu− div(Dξf(x,Du)) = −Dug(x, u) with p > 1. This can be interpreted both as a stability and existence result for general flows with linear growth.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77721990","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}
Circle packings have deep and well-established connections to conformal maps. Some methods for using circle packings to approximate quasiconformal maps have been studied, but they are not directly tied to the circle geometry. We present here a means to construct quasiconformal maps using Brooks’s parameterization of quadrilateral regions bounded by circles. The Brooks parameter acts as a sort of circle packing module, allowing us to directly affect the complex dilatation of our quasiconformal maps.
{"title":"Constructing quasiconformal maps using circle packings and Brooks's parameterization of quadrilaterals","authors":"G. Williams","doi":"10.5186/AASFM.2019.4445","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4445","url":null,"abstract":"Circle packings have deep and well-established connections to conformal maps. Some methods for using circle packings to approximate quasiconformal maps have been studied, but they are not directly tied to the circle geometry. We present here a means to construct quasiconformal maps using Brooks’s parameterization of quadrilateral regions bounded by circles. The Brooks parameter acts as a sort of circle packing module, allowing us to directly affect the complex dilatation of our quasiconformal maps.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81534295","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}
{"title":"On the cylindrical Green's function for representation theory and its applications","authors":"Lei Qiao","doi":"10.5186/AASFM.2019.4466","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4466","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77229202","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}
(1.2) −∆u + u = |u|u, x ∈ Ω, u ∈ H 0 (Ω), where 1 < p < 5. When Ω is a bounded domain, by applying the compactness of the embedding H 0 (Ω) →֒ L(Ω), 1 < p < 6, there is a positive solution of (1.2). If Ω is an unbounded domain, we can not obtain a solution for problem (1.2) by using Mountain Pass Theorem directly because the embedding H 0 (Ω) →֒ L(Ω), 1 < p < 6 is not compact. However, if Ω = R, Berestycki–Lions [3] proved that there is a radial positive solution of equation (1.2) by applying the compactness of the embedding H r (R ) →֒ L(R), 2 < p < 6, where H r (R) consists of the radially symmetric functions in H(R). By the Lions’s Concentration-Compactness Principle [13], there
(1.2)−∆u + u = | | u, x∈Ω,u∈H 0(Ω),1 < p < 5。当Ω是有界域时，通过应用嵌入H 0 (Ω)→ L(Ω)的紧性，1 < p < 6，存在(1.2)的正解。如果Ω是无界域，我们不能直接用山口定理得到问题(1.2)的解，因为嵌入H 0 (Ω)→ L(Ω)， 1 < p < 6是不紧的。然而，如果Ω = R, Berestycki-Lions[3]利用嵌入H R (R)→ L(R)， 2 < p < 6的紧性证明了方程(1.2)存在径向正解，其中H R (R)由H(R)中的径向对称函数组成。根据狮子会的集中-紧凑原则[13]，有
{"title":"Existence of positive solution for the nonlinear Kirchhoff type equations in the half space with a hole","authors":"Haiyang He, Xing Yi","doi":"10.5186/AASFM.2019.4462","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4462","url":null,"abstract":"(1.2) −∆u + u = |u|u, x ∈ Ω, u ∈ H 0 (Ω), where 1 < p < 5. When Ω is a bounded domain, by applying the compactness of the embedding H 0 (Ω) →֒ L(Ω), 1 < p < 6, there is a positive solution of (1.2). If Ω is an unbounded domain, we can not obtain a solution for problem (1.2) by using Mountain Pass Theorem directly because the embedding H 0 (Ω) →֒ L(Ω), 1 < p < 6 is not compact. However, if Ω = R, Berestycki–Lions [3] proved that there is a radial positive solution of equation (1.2) by applying the compactness of the embedding H r (R ) →֒ L(R), 2 < p < 6, where H r (R) consists of the radially symmetric functions in H(R). By the Lions’s Concentration-Compactness Principle [13], there","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86464085","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}
Abstract. We prove that every K-quasiconformal mapping w of the unit ball B ⊂ R, n ≥ 2 onto a C-Jordan domain Ω is Hölder continuous with constant α = 2 − n p , provided its weak Laplacian ∆w is in L(B) for some n/2 < p < n. In particular it is Hölder continuous for every 0 < α < 1 provided that ∆w ∈ L(B). Finally for p > n, we prove that w is Lipschitz continuous, a result, whose proof has been already sketched in [16] by the first author and Saksman. The paper contains the proofs of some results announced in [17].
摘要我们证明了单位球B∧R, n≥2在C-Jordan域Ω上的每一个k -拟共形映射w是Hölder连续的，且常数α = 2 - n p，只要它的弱拉普拉斯函数∆w在L(B)中，对于某些n/2 < p < n。特别是对于∆w∈L(B)，对于每一个0 < α < 1，它是Hölder连续的。最后，对于p > n，我们证明了w是Lipschitz连续的，这个结果的证明已经由第一作者和Saksman在[16]中勾画出来。本文包含[17]中公布的一些结果的证明。
{"title":"Quasiconformal mappings with controlled Laplacian and Hölder continuity","authors":"D. Kalaj, Arsen Zlaticanin","doi":"10.5186/AASFM.2019.4440","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4440","url":null,"abstract":"Abstract. We prove that every K-quasiconformal mapping w of the unit ball B ⊂ R, n ≥ 2 onto a C-Jordan domain Ω is Hölder continuous with constant α = 2 − n p , provided its weak Laplacian ∆w is in L(B) for some n/2 < p < n. In particular it is Hölder continuous for every 0 < α < 1 provided that ∆w ∈ L(B). Finally for p > n, we prove that w is Lipschitz continuous, a result, whose proof has been already sketched in [16] by the first author and Saksman. The paper contains the proofs of some results announced in [17].","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82138427","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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