Ben Hayes, David Jekel, Srivatsav Kunnawalkam Elayavalli
We obtain a new proof of Shlyakhtenko's result which states that if $G$ is a sofic, finitely presented group with vanishing first $ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede–Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also use the works of Shlyakhtenko and Shalom to give a short proof that the von Neumann algebras of sofic groups with Property (T) are strongly 1 bounded, which is a special case of another result by the authors.
我们得到了Shlyakhtenko结果的一个新的证明，即如果$G$是一个有限呈现的群，且第一个$ well ^2$-Betti数消失，则$L(G)$是强1有界的。我们对这一结果的证明适应并简化了荣格的技术论证，该论证表明在捕捉关系的矩阵的Fuglede-Kadison行列式的一定条件下强1有界性。我们的证明还包括Jung的一个关键思想，它涉及对微状态空间覆盖数的迭代估计。我们还利用Shlyakhtenko和Shalom的工作给出了一个简短的证明，证明了具有性质(T)的群的von Neumann代数是强有界的，这是作者另一个结果的特例。
{"title":"Vanishing first cohomology and strong 1-boundedness for von Neumann algebras","authors":"Ben Hayes, David Jekel, Srivatsav Kunnawalkam Elayavalli","doi":"10.4171/jncg/530","DOIUrl":"https://doi.org/10.4171/jncg/530","url":null,"abstract":"We obtain a new proof of Shlyakhtenko's result which states that if $G$ is a sofic, finitely presented group with vanishing first $ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede–Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also use the works of Shlyakhtenko and Shalom to give a short proof that the von Neumann algebras of sofic groups with Property (T) are strongly 1 bounded, which is a special case of another result by the authors.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"152 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Filippo Ambrosio, Giovanna Carnovale, Francesco Esposito, Lewis Topley
Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by $C^times$-equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First, we give a complete description of the cases in which the finite $W$-algebra is a universal filtered quantization of the slice, building on the work of Lehn–Namikawa–Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type, we prove that the finite $W$-algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally, we apply this result to give a presentation of the subregular finite $W$-algebra of type $mathsf{B}$ as a quotient of a shifted Yangian.
{"title":"Universal filtered quantizations of nilpotent Slodowy slices","authors":"Filippo Ambrosio, Giovanna Carnovale, Francesco Esposito, Lewis Topley","doi":"10.4171/jncg/544","DOIUrl":"https://doi.org/10.4171/jncg/544","url":null,"abstract":"Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by $C^times$-equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First, we give a complete description of the cases in which the finite $W$-algebra is a universal filtered quantization of the slice, building on the work of Lehn–Namikawa–Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type, we prove that the finite $W$-algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally, we apply this result to give a presentation of the subregular finite $W$-algebra of type $mathsf{B}$ as a quotient of a shifted Yangian.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"27 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136233439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones–Wenzl projections in Temperley–Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C$^*$-algebras that are $KK$-equivalent to $mathbb C$ and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz–Pimsner algebras under monoidal equivalence of quantum symmetry groups.
{"title":"Subproduct systems with quantum group symmetry","authors":"Erik Habbestad, Sergey Neshveyev","doi":"10.4171/jncg/523","DOIUrl":"https://doi.org/10.4171/jncg/523","url":null,"abstract":"We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones–Wenzl projections in Temperley–Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C$^*$-algebras that are $KK$-equivalent to $mathbb C$ and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz–Pimsner algebras under monoidal equivalence of quantum symmetry groups.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"BME-30 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The goal of this article is to explicitly compute the Hochschild (co)homology of the Fomin–Kirillov algebra on three generators over a field of characteristic different from $2$ and $3$. We also obtain the cyclic (co)homology of the Fomin–Kirillov algebra in case the characteristic of the field is zero. Moreover, we compute the algebra structure of the Hochschild cohomology.
{"title":"Hochschild and cyclic (co)homology of the Fomin–Kirillov algebra on $3$ generators","authors":"Estanislao Herscovich, Ziling Li","doi":"10.4171/jncg/525","DOIUrl":"https://doi.org/10.4171/jncg/525","url":null,"abstract":"The goal of this article is to explicitly compute the Hochschild (co)homology of the Fomin–Kirillov algebra on three generators over a field of characteristic different from $2$ and $3$. We also obtain the cyclic (co)homology of the Fomin–Kirillov algebra in case the characteristic of the field is zero. Moreover, we compute the algebra structure of the Hochschild cohomology.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135729103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^*$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$-theory classes. We apply the higher Kazhdan projections to establish a relation between $ell_2$-Betti numbers of a group and surjectivity of different Baum–Connes type assembly maps.
我们在群C^*$-代数和Roe代数上引入了矩阵代数中Kazhdan投影的高维类似。这些投影是在酉表示中带系数的上同调的框架中构造的，在某些情况下产生了非平凡的K -理论类。利用高哈兹丹投影，建立了群的$ well _2$-Betti数与不同Baum-Connes型集合映射的满性之间的关系。
{"title":"Higher Kazhdan projections, $ell_2$-Betti numbers and Baum–Connes conjectures","authors":"Kang Li, Piotr W. Nowak, Sanaz Pooya","doi":"10.4171/jncg/529","DOIUrl":"https://doi.org/10.4171/jncg/529","url":null,"abstract":"We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^*$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$-theory classes. We apply the higher Kazhdan projections to establish a relation between $ell_2$-Betti numbers of a group and surjectivity of different Baum–Connes type assembly maps.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, first, we introduce the notion of post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and the fact that there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman–Larson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions to the Yang–Baxter equation. Then, we introduce the notion of relative Rota–Baxter operator on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota–Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota–Baxter operator also induces a post-Hopf algebra. Finally, we show that relative Rota–Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota–Baxter operators give solutions to the Yang–Baxter equation in certain cocommutative Hopf algebras.
{"title":"Post-Hopf algebras, relative Rota--Baxter operators and solutions to the Yang--Baxter equation","authors":"Yunnan Li, Yunhe Sheng, Rong Tang","doi":"10.4171/jncg/537","DOIUrl":"https://doi.org/10.4171/jncg/537","url":null,"abstract":"In this paper, first, we introduce the notion of post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and the fact that there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman–Larson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions to the Yang–Baxter equation. Then, we introduce the notion of relative Rota–Baxter operator on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota–Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota–Baxter operator also induces a post-Hopf algebra. Finally, we show that relative Rota–Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota–Baxter operators give solutions to the Yang–Baxter equation in certain cocommutative Hopf algebras.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136184409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian $Delta=-(X_1^2+cdots+X_m^2)$ on a compact connected Lie group $G$ if $p$ is large enough, more precisely under the (sharp) condition $p > frac{d}{alpha}$, where $d$ is the local dimension of $(G,X)$ and where $0 < alpha leq 1$. We also provide locally compact variants of this result and generalizations for real second-order subelliptic operators. We also introduce a compact spectral triple (= noncommutative manifold) canonically associated to each subelliptic Laplacian on a compact group. In addition, we show that its spectral dimension is equal to the local dimension of $(G,X)$. Finally, we prove that the Connes spectral pseudo-metric allows us to recover the Carnot–Carathéodory distance.
{"title":"Sobolev algebras on Lie groups and noncommutative geometry","authors":"Cédric Arhancet","doi":"10.4171/jncg/532","DOIUrl":"https://doi.org/10.4171/jncg/532","url":null,"abstract":"We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian $Delta=-(X_1^2+cdots+X_m^2)$ on a compact connected Lie group $G$ if $p$ is large enough, more precisely under the (sharp) condition $p > frac{d}{alpha}$, where $d$ is the local dimension of $(G,X)$ and where $0 < alpha leq 1$. We also provide locally compact variants of this result and generalizations for real second-order subelliptic operators. We also introduce a compact spectral triple (= noncommutative manifold) canonically associated to each subelliptic Laplacian on a compact group. In addition, we show that its spectral dimension is equal to the local dimension of $(G,X)$. Finally, we prove that the Connes spectral pseudo-metric allows us to recover the Carnot–Carathéodory distance.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"238 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136184412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The equivariant coarse Baum–Connes conjecture interpolates between the Baum–Connes conjecture for a discrete group and the coarse Baum–Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group $Gamma$ acts properly and isometrically on a discrete metric space $X$ with bounded geometry, not necessarily cocompact. We show that if the quotient space $X/Gamma$ admits a coarse embedding into Hilbert space and $Gamma$ is amenable, and that the $Gamma$-orbits in $X$ are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum–Connes conjecture holds for $(X,Gamma)$. Along the way, we prove a $K$-theoretic amenability statement for the $Gamma$-space $X$ under the same assumptions as above; namely, the canonical quotient map from the maximal equivariant Roe algebra of $X$ to the reduced equivariant Roe algebra of $X$ induces an isomorphism on $K$-theory.
{"title":"The equivariant coarse Baum–Connes conjecture for metric spaces with proper group actions","authors":"Jintao Deng, Benyin Fu, Qin Wang","doi":"10.4171/jncg/519","DOIUrl":"https://doi.org/10.4171/jncg/519","url":null,"abstract":"The equivariant coarse Baum–Connes conjecture interpolates between the Baum–Connes conjecture for a discrete group and the coarse Baum–Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group $Gamma$ acts properly and isometrically on a discrete metric space $X$ with bounded geometry, not necessarily cocompact. We show that if the quotient space $X/Gamma$ admits a coarse embedding into Hilbert space and $Gamma$ is amenable, and that the $Gamma$-orbits in $X$ are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum–Connes conjecture holds for $(X,Gamma)$. Along the way, we prove a $K$-theoretic amenability statement for the $Gamma$-space $X$ under the same assumptions as above; namely, the canonical quotient map from the maximal equivariant Roe algebra of $X$ to the reduced equivariant Roe algebra of $X$ induces an isomorphism on $K$-theory.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135917830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that an action $rho:Ato M(C_0(mathbb{G})otimes A)$ of a locally compact quantum group on a $C^*$-algebra has a universal equivariant compactification and prove a number of other category-theoretic results on $mathbb{G}$-equivariant compactifications: that the categories compactifications of $rho$ and $A$, respectively, are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When $mathbb{G}$ is regular, coamenable we also show that the forgetful functor from unital $mathbb{G}$-$C^$-algebras to unital $C^$-algebras creates finite limits and is comonadic and that the monomorphisms in the former category are injective.
{"title":"Non-commutative ambits and equivariant compactifications","authors":"Alexandru Chirvasitu","doi":"10.4171/jncg/536","DOIUrl":"https://doi.org/10.4171/jncg/536","url":null,"abstract":"We prove that an action $rho:Ato M(C_0(mathbb{G})otimes A)$ of a locally compact quantum group on a $C^*$-algebra has a universal equivariant compactification and prove a number of other category-theoretic results on $mathbb{G}$-equivariant compactifications: that the categories compactifications of $rho$ and $A$, respectively, are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When $mathbb{G}$ is regular, coamenable we also show that the forgetful functor from unital $mathbb{G}$-$C^$-algebras to unital $C^$-algebras creates finite limits and is comonadic and that the monomorphisms in the former category are injective.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136064062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a manifold with corners $X$, we associate to it the corner structure simplicial complex $Sigma_X$. Its reduced K-homology is isomorphic to the K-theory of the $C^*$-algebra $mathcal{K}_b(X)$ of b-compact operators on $X$. Moreover, the homology of $Sigma_X$ is isomorphic to the conormal homology of $X$. In this note, we construct for an arbitrary abstract finite simplicial complex $Sigma$ a manifold with corners $X$ such that $Sigma_XcongSigma$. As a consequence, the homology and K-homology which occur for finite simplicial complexes also occur as conormal homology of manifolds with corners and as K-theory of their b-compact operators. In particular, these groups can contain torsion.
{"title":"Conormal homology of manifolds with corners","authors":"Thomas Schick, Mario Velasquez","doi":"10.4171/jncg/520","DOIUrl":"https://doi.org/10.4171/jncg/520","url":null,"abstract":"Given a manifold with corners $X$, we associate to it the corner structure simplicial complex $Sigma_X$. Its reduced K-homology is isomorphic to the K-theory of the $C^*$-algebra $mathcal{K}_b(X)$ of b-compact operators on $X$. Moreover, the homology of $Sigma_X$ is isomorphic to the conormal homology of $X$. In this note, we construct for an arbitrary abstract finite simplicial complex $Sigma$ a manifold with corners $X$ such that $Sigma_XcongSigma$. As a consequence, the homology and K-homology which occur for finite simplicial complexes also occur as conormal homology of manifolds with corners and as K-theory of their b-compact operators. In particular, these groups can contain torsion.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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