— We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures. Resume. — Cet article est une presentation rapide de la theorie spectrale et de la dynamique des varietes asymptotiquement hyperboliques a volume infini. Nous commencons par leur geometrie et quelques exemples, nous poursuivons en rappelant leur theorie spectrale, puis continuons sur des developpements recents de leur dynamique. Nous concluons par une discussion des resultats qui demontrent un rapport entre leurs mecaniques quantiques et classiques et enfin, nous offrons quelques idees et conjectures.
{"title":"On the spectral theory and dynamics of asymptotically hyperbolic manifolds","authors":"Julie Rowlett","doi":"10.5802/aif.2615","DOIUrl":"https://doi.org/10.5802/aif.2615","url":null,"abstract":"— We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures. Resume. — Cet article est une presentation rapide de la theorie spectrale et de la dynamique des varietes asymptotiquement hyperboliques a volume infini. Nous commencons par leur geometrie et quelques exemples, nous poursuivons en rappelant leur theorie spectrale, puis continuons sur des developpements recents de leur dynamique. Nous concluons par une discussion des resultats qui demontrent un rapport entre leurs mecaniques quantiques et classiques et enfin, nous offrons quelques idees et conjectures.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45064382","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 establish the existence and the uniqueness for the Boussinesq system in the whole 3D space in the critical space of continuous in time with values in the power 3 integrable in space functions for the velocity and square integrable in time with values in the power 3/2 integrable in space.
{"title":"Well-posedness for the Boussinesq system in critical spaces via maximal regularity","authors":"L. Brandolese, Sylvie Monniaux","doi":"10.5802/aif.3523","DOIUrl":"https://doi.org/10.5802/aif.3523","url":null,"abstract":"We establish the existence and the uniqueness for the Boussinesq system in the whole 3D space in the critical space of continuous in time with values in the power 3 integrable in space functions for the velocity and square integrable in time with values in the power 3/2 integrable in space.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46288172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study various trivializations of moment maps. First in the general framework of a reductive group $G$ acting on a smooth affine variety. We prove that the moment map is a locally trivial fibration over a regular locus of the center of the Lie algebra of $H$ a maximal compact subgroup of $G$. The construction relies on Kempf-Ness theory and Morse theory of the square norm of the moment map studied by Kirwan, Ness-Mumford and Sjamaar. Then we apply it together with ideas from Nakajima and Kronheimer to trivialize the hyperkaehler moment map for Nakajima quiver varieties. Notice this trivialization result about quiver varieties was known and used by experts such as Nakajima and Maffei but we could not locate a proof in the literature.
{"title":"Trivializations of moment maps","authors":"Mathieu Ballandras","doi":"10.5802/aif.3587","DOIUrl":"https://doi.org/10.5802/aif.3587","url":null,"abstract":"We study various trivializations of moment maps. First in the general framework of a reductive group $G$ acting on a smooth affine variety. We prove that the moment map is a locally trivial fibration over a regular locus of the center of the Lie algebra of $H$ a maximal compact subgroup of $G$. The construction relies on Kempf-Ness theory and Morse theory of the square norm of the moment map studied by Kirwan, Ness-Mumford and Sjamaar. Then we apply it together with ideas from Nakajima and Kronheimer to trivialize the hyperkaehler moment map for Nakajima quiver varieties. Notice this trivialization result about quiver varieties was known and used by experts such as Nakajima and Maffei but we could not locate a proof in the literature.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42188770","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}
M. Hitrik, R. Lascar, J. Sjoestrand, Maher Zerzeri
We study semiclassical Gevrey pseudodifferential operators, acting on exponentially weighted spaces of entire holomorphic functions. The symbols of such operators are Gevrey functions defined on suitable I-Lagrangian submanifolds of the complexified phase space, which are extended almost holomorphically in the same Gevrey class, or in some larger space, to complex neighborhoods of these submanifolds. Using almost holomorphic extensions, we obtain uniformly bounded realizations of such operators on a natural scale of exponentially weighted spaces of holomorphic functions for all Gevrey indices, with remainders that are optimally small, provided that the Gevrey index is $leq 2$.
{"title":"Semiclassical Gevrey operators in the complex domain","authors":"M. Hitrik, R. Lascar, J. Sjoestrand, Maher Zerzeri","doi":"10.5802/aif.3546","DOIUrl":"https://doi.org/10.5802/aif.3546","url":null,"abstract":"We study semiclassical Gevrey pseudodifferential operators, acting on exponentially weighted spaces of entire holomorphic functions. The symbols of such operators are Gevrey functions defined on suitable I-Lagrangian submanifolds of the complexified phase space, which are extended almost holomorphically in the same Gevrey class, or in some larger space, to complex neighborhoods of these submanifolds. Using almost holomorphic extensions, we obtain uniformly bounded realizations of such operators on a natural scale of exponentially weighted spaces of holomorphic functions for all Gevrey indices, with remainders that are optimally small, provided that the Gevrey index is $leq 2$.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71210748","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 IET be the group of bijections from $mathopen{[}0,1 mathclose{[}$ to itself that are continuous outside a finite set, right-continuous and piecewise translations. The abelianization homomorphism $f: text{IET} to A$, called SAF-homomorphism, was described by Arnoux-Fathi and Sah. The abelian group $A$ is the second exterior power of the reals over the rationals. For every subgroup $Gamma$ of $mathbb{R/Z}$ we define $text{IET}(Gamma)$ as the subgroup of $text{IET}$ consisting of all elements $f$ such that $f$ is continuous outside $Gamma$. Let $tilde{Gamma}$ be the preimage of $Gamma$ in $mathbb{R}$. We establish an isomorphism between the abelianization of $text{IET}(Gamma)$ and the second skew-symmetric power of $tilde{Gamma}$ over $mathbb{Z}$ denoted by ${}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma}$. This group often has non-trivial $2$-torsion, which is not detected by the SAF-homomorphism. We then define $text{IET}^{bowtie}$ the group of all interval exchange transformations with flips. Arnoux proved that this group is simple thus perfect. However for every subgroup $text{IET}^{bowtie}(Gamma)$ we establish an isomorphism between its abelianization and $langle lbrace a otimes a ~ [text{mod}~2] mid a in tilde{Gamma} rbrace rangle times langle lbrace ell wedge ell ~ [text{mod}~2] mid ell in tilde{Gamma} rbrace rangle$ which is a $2$-elementary abelian subgroup of $bigotimes^2_{mathbb{Z}} tilde{Gamma} / (2bigotimes^2_{mathbb{Z}} tilde{Gamma}) times {}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma} / (2 {}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma})$.
设IET为一组从$mathopen{[}0,1 mathclose{[}$到自身的双射，它们在有限集合外连续，右连续和分段平移。阿贝尔化同态$f: text{IET} to A$，称为saf同态，是由Arnoux-Fathi和Sah描述的。阿贝尔群$A$是实数对有理数的第二个外部幂。对于$mathbb{R/Z}$的每个子组$Gamma$，我们将$text{IET}(Gamma)$定义为$text{IET}$的子组，由所有元素$f$组成，使得$f$在$Gamma$之外连续。设$tilde{Gamma}$为$mathbb{R}$中的$Gamma$的原映像。建立了$text{IET}(Gamma)$的阿贝尔化与$tilde{Gamma}$ / $mathbb{Z}$的二阶偏对称幂(表示为${}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma}$)之间的同构关系。这个群通常具有不平凡的$2$ -扭转，这是saf同态检测不到的。然后我们定义$text{IET}^{bowtie}$所有带翻转的区间交换变换的集合。Arnoux证明了这个群体是简单的，因此是完美的。然而对于每一个子群$text{IET}^{bowtie}(Gamma)$，我们建立了它的阿贝尔化与$langle lbrace a otimes a ~ [text{mod}~2] mid a in tilde{Gamma} rbrace rangle times langle lbrace ell wedge ell ~ [text{mod}~2] mid ell in tilde{Gamma} rbrace rangle$之间的同构，这是$bigotimes^2_{mathbb{Z}} tilde{Gamma} / (2bigotimes^2_{mathbb{Z}} tilde{Gamma}) times {}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma} / (2 {}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma})$的一个$2$ -初等阿贝尔子群。
{"title":"Abelianization of some groups of interval exchanges","authors":"Octave Lacourte","doi":"10.5802/aif.3466","DOIUrl":"https://doi.org/10.5802/aif.3466","url":null,"abstract":"Let IET be the group of bijections from $mathopen{[}0,1 mathclose{[}$ to itself that are continuous outside a finite set, right-continuous and piecewise translations. The abelianization homomorphism $f: text{IET} to A$, called SAF-homomorphism, was described by Arnoux-Fathi and Sah. The abelian group $A$ is the second exterior power of the reals over the rationals. For every subgroup $Gamma$ of $mathbb{R/Z}$ we define $text{IET}(Gamma)$ as the subgroup of $text{IET}$ consisting of all elements $f$ such that $f$ is continuous outside $Gamma$. Let $tilde{Gamma}$ be the preimage of $Gamma$ in $mathbb{R}$. We establish an isomorphism between the abelianization of $text{IET}(Gamma)$ and the second skew-symmetric power of $tilde{Gamma}$ over $mathbb{Z}$ denoted by ${}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma}$. This group often has non-trivial $2$-torsion, which is not detected by the SAF-homomorphism. We then define $text{IET}^{bowtie}$ the group of all interval exchange transformations with flips. Arnoux proved that this group is simple thus perfect. However for every subgroup $text{IET}^{bowtie}(Gamma)$ we establish an isomorphism between its abelianization and $langle lbrace a otimes a ~ [text{mod}~2] mid a in tilde{Gamma} rbrace rangle times langle lbrace ell wedge ell ~ [text{mod}~2] mid ell in tilde{Gamma} rbrace rangle$ which is a $2$-elementary abelian subgroup of $bigotimes^2_{mathbb{Z}} tilde{Gamma} / (2bigotimes^2_{mathbb{Z}} tilde{Gamma}) times {}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma} / (2 {}^circleddash!!bigwedge^2_{mathbb{Z}} tilde{Gamma})$.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44773182","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 the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger-Gromov topology on the space of complete pointed riemannian manifolds.
{"title":"Prescription de courbure des feuilles des laminations : retour sur un théorème de Candel","authors":"Sébastien Alvarez, G. Smith","doi":"10.5802/aif.3476","DOIUrl":"https://doi.org/10.5802/aif.3476","url":null,"abstract":"In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger-Gromov topology on the space of complete pointed riemannian manifolds.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47319127","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 are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincare series). In case (B) we give a conjectural formula for the full mixed Poincare series.
{"title":"E-series of character varieties of non-orientable surfaces","authors":"E. Letellier, F. Rodriguez-Villegas","doi":"10.5802/aif.3540","DOIUrl":"https://doi.org/10.5802/aif.3540","url":null,"abstract":"In this paper we are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincare series). In case (B) we give a conjectural formula for the full mixed Poincare series.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42421423","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}
S. Bortz, J. Hoffman, S. Hofmann, J. García, Kaj Nystrom
We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension, and to Azzam and Schul in general) in the (classical) setting of Euclidean spaces with Hausdorff measure of integer dimension, where the approximating sets are Lipschitz graphs. Our result is a far reaching generalization of these results and we prove that coronizations imply big pieces squared is a generic property. In particular, our result applies, when suitably interpreted, in metric spaces having a fixed positive (perhaps non-integer) dimension, equipped with a Borel regular measure and with arbitrary approximating sets. As a novel application we highlight how to utilize this general setting in the context of parabolic uniform rectifiability.
{"title":"Coronizations and big pieces in metric spaces","authors":"S. Bortz, J. Hoffman, S. Hofmann, J. García, Kaj Nystrom","doi":"10.5802/aif.3518","DOIUrl":"https://doi.org/10.5802/aif.3518","url":null,"abstract":"We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension, and to Azzam and Schul in general) in the (classical) setting of Euclidean spaces with Hausdorff measure of integer dimension, where the approximating sets are Lipschitz graphs. Our result is a far reaching generalization of these results and we prove that coronizations imply big pieces squared is a generic property. In particular, our result applies, when suitably interpreted, in metric spaces having a fixed positive (perhaps non-integer) dimension, equipped with a Borel regular measure and with arbitrary approximating sets. As a novel application we highlight how to utilize this general setting in the context of parabolic uniform rectifiability.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46893382","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 vertex-transitive graph $mathcal{G}$ is called Local-to-Global rigid if there exists $R>0$ such that every other graph whose balls of radius $R$ are isometric to the balls of radius $R$ in $mathcal{G}$ is covered by $mathcal{G}$. An example of such a graph is given by the Bruhat-Tits building of $PSL_n(mathbb{K})$ with $ngeq 4$ and $mathbb{K}$ a non-Archimedean local field of characteristic zero.. In this paper we extend this rigidity property to a class of graphs quasi-isometric to the building including torsion-free lattices of $SL_n(mathbb{K})$. The demonstration is the occasion to prove a result on the local structure of the building. We show that if we fix a $PSL_n(mathbb{K})$-orbit in it, then a vertex is uniquely determined by the neighbouring vertices in this orbit.
{"title":"Local-to-Global-rigidity of lattices in SL n (𝕂)","authors":"Amandine Escalier","doi":"10.5802/aif.3490","DOIUrl":"https://doi.org/10.5802/aif.3490","url":null,"abstract":"A vertex-transitive graph $mathcal{G}$ is called Local-to-Global rigid if there exists $R>0$ such that every other graph whose balls of radius $R$ are isometric to the balls of radius $R$ in $mathcal{G}$ is covered by $mathcal{G}$. An example of such a graph is given by the Bruhat-Tits building of $PSL_n(mathbb{K})$ with $ngeq 4$ and $mathbb{K}$ a non-Archimedean local field of characteristic zero.. In this paper we extend this rigidity property to a class of graphs quasi-isometric to the building including torsion-free lattices of $SL_n(mathbb{K})$. The demonstration is the occasion to prove a result on the local structure of the building. We show that if we fix a $PSL_n(mathbb{K})$-orbit in it, then a vertex is uniquely determined by the neighbouring vertices in this orbit.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41745386","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 provide the first non-trivial examples of deformed G_2-instantons, originally called deformed Donaldson-Thomas connections. As a consequence, we see how deformed G_2-instantons can be used to distinguish between nearly parallel G_2-structures and isometric G_2-structures on 3-Sasakian 7-manifolds. Our examples give non-trivial deformed G_2-instantons with obstructed deformation theory and situations where the moduli space of deformed G_2-instantons has components of different dimensions. We finally study the relation between our examples and a Chern-Simons type functional which has deformed G_2-instantons as critical points.
{"title":"Examples of deformed G2-instantons/Donaldson–Thomas connections","authors":"Jason D. Lotay, Gonçalo Oliveira","doi":"10.5802/aif.3465","DOIUrl":"https://doi.org/10.5802/aif.3465","url":null,"abstract":"In this note, we provide the first non-trivial examples of deformed G_2-instantons, originally called deformed Donaldson-Thomas connections. As a consequence, we see how deformed G_2-instantons can be used to distinguish between nearly parallel G_2-structures and isometric G_2-structures on 3-Sasakian 7-manifolds. Our examples give non-trivial deformed G_2-instantons with obstructed deformation theory and situations where the moduli space of deformed G_2-instantons has components of different dimensions. We finally study the relation between our examples and a Chern-Simons type functional which has deformed G_2-instantons as critical points.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48745652","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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