Given a compact Riemannian manifold (M n , g) with boundary ∂M , we give an estimate for the quotient ∂M f dµ g M f dµ g , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [37], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a direct proof is given of the Faber-Krahn inequalities for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.
给定一个边界为∂M的紧黎曼流形(mn, g)，我们给出了商∂M f dµg M f dµg的估计，其中f是定义在M上的一个光滑正函数，它满足一些涉及标量拉普拉斯算子的不等式。利用[37]中建立的均值引理，我们给出了f的微分不等式，在某些曲率假设下，可以用贝塞尔函数来解释。作为我们的主要结果的一个应用，给出了Dirichlet和Robin Laplacian的Faber-Krahn不等式的一个直接证明。此外，对于除标量曲率外还包含贝塞尔函数正根的狄拉克算子的特征值，给出了一个新的估计。独立地，我们将函数上的罗宾拉普拉斯扩展到微分形式。证明了该自然推广定义了一个谱离散且由正实特征值组成的自伴随椭圆算子。特别地，我们描述了它的第一个特征值，并给出了它在贝塞尔函数中的下界。
{"title":"New eigenvalue estimates involving Bessel functions","authors":"F. Chami, N. Ginoux, Georges Habib","doi":"10.5565/PUBLMAT6522109","DOIUrl":"https://doi.org/10.5565/PUBLMAT6522109","url":null,"abstract":"Given a compact Riemannian manifold (M n , g) with boundary ∂M , we give an estimate for the quotient ∂M f dµ g M f dµ g , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [37], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a direct proof is given of the Faber-Krahn inequalities for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43743561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is well known that the set of values of a lower central word in a group $G$ need not be a subgroup. For a fixed lower central word $gamma_r$ and for $pge 5$, Guralnick showed that if $G$ is a finite $p$-group such that the verbal subgroup $gamma_r(G)$ is abelian and 2-generator, then $gamma_r(G)$ consists only of $gamma_r$-values. In this paper we extend this result, showing that the assumption that $gamma_r(G)$ is abelian can be dropped. Moreover, we show that the result remains true even if $p=3$. Finally, we prove that the analogous result for pro-$p$ groups is true.
{"title":"Lower central words in finite $p$-groups","authors":"Iker de las Heras, M. Morigi","doi":"10.5565/publmat6512107","DOIUrl":"https://doi.org/10.5565/publmat6512107","url":null,"abstract":"It is well known that the set of values of a lower central word in a group $G$ need not be a subgroup. For a fixed lower central word $gamma_r$ and for $pge 5$, Guralnick showed that if $G$ is a finite $p$-group such that the verbal subgroup $gamma_r(G)$ is abelian and 2-generator, then $gamma_r(G)$ consists only of $gamma_r$-values. In this paper we extend this result, showing that the assumption that $gamma_r(G)$ is abelian can be dropped. Moreover, we show that the result remains true even if $p=3$. Finally, we prove that the analogous result for pro-$p$ groups is true.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47488648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider $m in mathbb{N}$ and $beta in (1, m + 1]$. Assume that $ain mathbb{R}$ can be represented in base $beta$ using a development in series $a = sum^{infty}_{n = 1}x(n)beta^{-n}$ where the sequence $x = (x(n))_{n in mathbb{N}}$ take values in the alphabet $mathcal{A}_m := {0, ldots, m}$. The above expression is called the $beta$-expansion of $a$ and it is not necessarily unique. We are interested in sequences $x = (x(n))_{n in mathbb{N}} in mathcal{A}_m^mathbb{N}$ which are associated to all possible values $a$ which have a unique expansion. We denote the set of such $x$ (with some more technical restrictions) by $X_{m,beta} subsetmathcal{A}_m^mathbb{N}$. The space $X_{m, beta}$ is called the symmetric $beta$-shift associated to the pair $(m, beta)$. It is invariant by the shift map but in general it is not a subshift of finite type. Given a Holder continuous potential $A:X_{m, beta} tomathbb{R}$, we consider the Ruelle operator $mathcal{L}_A$ and we show the existence of a positive eigenfunction $psi_A$ and an eigenmeasure $rho_A$ for some appropriated values of $m$ and $beta$. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies $h(mu_{tA})_{t>1}$ converges, when $t toinfty$, to the maximal value among the set of all possible values of entropy of all $A$-maximizing probabilities.
考虑$minmathbb｛N｝$和$betain（1，m+1]$。假设$ainmath bb｛R｝$可以使用序列$a=sum^｛infty｝_｛N=1｝x（N）beta^｛-N｝$中的开发以$beta$为基表示，其中序列$x=（x（N{A}_m：=｛0，ldots，m｝$。上面的表达式被称为$a$的$beta$扩展，它不一定是唯一的。我们对序列$x=（x（n））_｛ninmathbb｛n｝｝inmathcal感兴趣{A}_m^mathb{N}$，其与具有唯一扩展的所有可能值$a$相关联。我们用$x_｛m，beta｝subet mathcal来表示这样的$x$的集合（具有一些更多的技术限制）{A}_m^mathb{N}$。空间$X_｛m，beta｝$被称为与对$（m，bita）$相关联的对称$beta$移位。它对移位映射是不变的，但通常它不是有限类型的子移位。给定Holder连续势$a:X_｛m，beta｝ to mathbb｛R｝$，我们考虑Ruelle算子$mathcal{L}_A$，并且我们证明了对于$m$和$beta$的一些适当值存在正本征函数$psi_a$和本征测度$rho_a$。我们还考虑了压力的一个变分原理。此外，我们证明了熵族$h（mu_{tA}）_{t>1}$在$ttoinfty$时收敛于所有$A$-最大化概率的熵的所有可能值的集合中的最大值。
{"title":"The Ruelle operator for symmetric $beta$-shifts","authors":"A. Lopes, V. Vargas","doi":"10.5565/PUBLMAT6422012","DOIUrl":"https://doi.org/10.5565/PUBLMAT6422012","url":null,"abstract":"Consider $m in mathbb{N}$ and $beta in (1, m + 1]$. Assume that $ain mathbb{R}$ can be represented in base $beta$ using a development in series $a = sum^{infty}_{n = 1}x(n)beta^{-n}$ where the sequence $x = (x(n))_{n in mathbb{N}}$ take values in the alphabet $mathcal{A}_m := {0, ldots, m}$. The above expression is called the $beta$-expansion of $a$ and it is not necessarily unique. We are interested in sequences $x = (x(n))_{n in mathbb{N}} in mathcal{A}_m^mathbb{N}$ which are associated to all possible values $a$ which have a unique expansion. We denote the set of such $x$ (with some more technical restrictions) by $X_{m,beta} subsetmathcal{A}_m^mathbb{N}$. The space $X_{m, beta}$ is called the symmetric $beta$-shift associated to the pair $(m, beta)$. It is invariant by the shift map but in general it is not a subshift of finite type. Given a Holder continuous potential $A:X_{m, beta} tomathbb{R}$, we consider the Ruelle operator $mathcal{L}_A$ and we show the existence of a positive eigenfunction $psi_A$ and an eigenmeasure $rho_A$ for some appropriated values of $m$ and $beta$. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies $h(mu_{tA})_{t>1}$ converges, when $t toinfty$, to the maximal value among the set of all possible values of entropy of all $A$-maximizing probabilities.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47141304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We answer several open questions and establish new results concerningdierential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If R is prime radical and δ is a derivation of R, then the dierential polynomial ring R[X; δ] is locally nilpotent. This answers an open question posed in [41]. The nil radical of a dierential polynomial ring R[X; δ] takes the form I[X; δ] for some ideal I of R, provided that the base field is infinite. This answers an open question posed in [30] for algebras over infinite fields. If R is a graded algebra generated in degree 1 over a field of characteristic zero and δ is a grading preserving derivation on R, then the Jacobson radical of R is δ-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp. Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent. The power series ring R[[X; σ; δ]] is well-defined whenever δ is a locally nilpotent σ-derivation; this answers a conjecture from [13], and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on δ.
{"title":"Five solved problems on radicals of Ore extensions","authors":"Be'eri Greenfeld, A. Smoktunowicz, M. Ziembowski","doi":"10.5565/PUBLMAT6321902","DOIUrl":"https://doi.org/10.5565/PUBLMAT6321902","url":null,"abstract":"We answer several open questions and establish new results concerningdierential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If R is prime radical and δ is a derivation of R, then the dierential polynomial ring R[X; δ] is locally nilpotent. This answers an open question posed in [41]. The nil radical of a dierential polynomial ring R[X; δ] takes the form I[X; δ] for some ideal I of R, provided that the base field is infinite. This answers an open question posed in [30] for algebras over infinite fields. If R is a graded algebra generated in degree 1 over a field of characteristic zero and δ is a grading preserving derivation on R, then the Jacobson radical of R is δ-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp. Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent. The power series ring R[[X; σ; δ]] is well-defined whenever δ is a locally nilpotent σ-derivation; this answers a conjecture from [13], and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on δ.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43464532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that, if D is a normal open subset of a Stein space X of puredimension such that D is locally Stein at every point of ∂D n Xsg, then, for every holomorphic vector bundle E over D and every discrete subset Ʌ of D Xsg whose set of accumulation points lies in ∂D Xsg, there is a holomorphic section of E over D with prescribed values on Ʌ. We apply this to the local Steinness problem and domains of holomorphy.
{"title":"An interpolation property of locally Stein sets","authors":"V. Vâjâitu","doi":"10.5565/PUBLMAT6321909","DOIUrl":"https://doi.org/10.5565/PUBLMAT6321909","url":null,"abstract":"We prove that, if D is a normal open subset of a Stein space X of puredimension such that D is locally Stein at every point of ∂D n Xsg, then, for every holomorphic vector bundle E over D and every discrete subset Ʌ of D Xsg whose set of accumulation points lies in ∂D Xsg, there is a holomorphic section of E over D with prescribed values on Ʌ. We apply this to the local Steinness problem and domains of holomorphy.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42653789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide an explicit technical framework for proving very general two-weight commutator estimates in arbitrary parameters. The aim is to both clarify existing literature, which often explicitly focuses on two parameters only, and to extend very recent results to the full generality of arbitrary parameters. More specifically, we study two-weight commutator estimates -- Bloom type estimates -- in the multi-parameter setting involving weighted product BMO and little BMO spaces, and their combinations.
{"title":"Two-weight commutator estimates: general multi-parameter framework","authors":"Emil Airta","doi":"10.5565/PUBLMAT6422013","DOIUrl":"https://doi.org/10.5565/PUBLMAT6422013","url":null,"abstract":"We provide an explicit technical framework for proving very general two-weight commutator estimates in arbitrary parameters. The aim is to both clarify existing literature, which often explicitly focuses on two parameters only, and to extend very recent results to the full generality of arbitrary parameters. More specifically, we study two-weight commutator estimates -- Bloom type estimates -- in the multi-parameter setting involving weighted product BMO and little BMO spaces, and their combinations.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":"12 1","pages":"681-729"},"PeriodicalIF":1.1,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72947371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning Euler systems for the multiplicative group over arbitrary number fields.
我们给出了罗伯特·科尔曼关于任意数域上的乘群的欧拉系统猜想的自然推广，并提供了有力的证据。
{"title":"On Euler systems for the multiplicative group over general number fields","authors":"D. Burns, Alexandre Daoud, Takamichi Sano, S. Seo","doi":"10.5565/publmat6712302","DOIUrl":"https://doi.org/10.5565/publmat6712302","url":null,"abstract":"We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning Euler systems for the multiplicative group over arbitrary number fields.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47079933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors form the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented along the exposition.
{"title":"Towards differentiation and integration between Hopf algebroids and Lie algebroids","authors":"A. Ardizzoni, Laiacbi El Kaoutit, P. Saracco","doi":"10.5565/PUBLMAT6712301","DOIUrl":"https://doi.org/10.5565/PUBLMAT6712301","url":null,"abstract":"In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors form the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented along the exposition.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44415372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that in cones having an isoperimetric property the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the $C^{1,1}$-metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].
{"title":"Isoperimetric cones and minimal solutions of partial overdetermined problems","authors":"F. Pacella, G. Tralli","doi":"10.5565/publmat6512102","DOIUrl":"https://doi.org/10.5565/publmat6512102","url":null,"abstract":"In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that in cones having an isoperimetric property the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the $C^{1,1}$-metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45307806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 study the existence of uniform a priori estimates for positive solutions to Navier problems of higher order Lane-Emden equations begin{equation*} (-Delta)^{m}u(x)=u^{p}(x), qquad ,, xinOmega end{equation*} for all large exponents $p$, where $Omegasubsetmathbb{R}^{n}$ is a star-shaped or strictly convex bounded domain with $C^{2m-2}$ boundary, $ngeq4$ and $2leq mleqfrac{n}{2}$. Our results extend those of previous authors for second order $m=1$ to general higher order cases $mgeq2$.
{"title":"Uniform a priori estimates for positive solutions of higher order Lane-Emden equations in $mathbb{R}^n$","authors":"Wei Dai, Thomas Duyckaerts","doi":"10.5565/publmat6512111","DOIUrl":"https://doi.org/10.5565/publmat6512111","url":null,"abstract":"In this paper, we study the existence of uniform a priori estimates for positive solutions to Navier problems of higher order Lane-Emden equations begin{equation*} \u0000(-Delta)^{m}u(x)=u^{p}(x), qquad ,, xinOmega end{equation*} for all large exponents $p$, where $Omegasubsetmathbb{R}^{n}$ is a star-shaped or strictly convex bounded domain with $C^{2m-2}$ boundary, $ngeq4$ and $2leq mleqfrac{n}{2}$. Our results extend those of previous authors for second order $m=1$ to general higher order cases $mgeq2$.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46826672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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