We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmüller geodesic flow restricted to the hyperelliptic connected components of the strata of Abelian differentials, settling a question of [Far06]; This is the first description of the systole in any component of any stratum outside of genus two and three. We stress that all proofs and computations can be performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components and we provide a picture of that for small genera.
{"title":"Lengths spectrum of hyperelliptic components","authors":"Corentin Boissy, Erwan Lanneau","doi":"10.4171/jems/1150","DOIUrl":"https://doi.org/10.4171/jems/1150","url":null,"abstract":"We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmüller geodesic flow restricted to the hyperelliptic connected components of the strata of Abelian differentials, settling a question of [Far06]; This is the first description of the systole in any component of any stratum outside of genus two and three. We stress that all proofs and computations can be performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components and we provide a picture of that for small genera.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82064803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose $mathcal{E} to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $overline{mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $mathbb{R}$-divisors $overline{D}_X$ on the base curve $B$ over a number field, for each $X in E(k)otimes mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $overline{D}_Xcdot overline{D}_Y$, as a biquadratic form on $E(k)otimes mathbb{R}$. As a consequence, we have the following Bogomolov-type statement for the N'eron-Tate height functions on the fibers $E_t(overline{mathbb{Q}})$ of $mathcal{E}$ over $t in B(overline{mathbb{Q}})$: given points $P_1, ldots, P_m in E(k)$ with $mgeq 2$, there exist an infinite sequence $t_nin B(overline{mathbb{Q}})$ and small-height perturbations $P_{i,t_n}' in E_{t_n}(overline{mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set ${P_{1, t_n}', ldots, P_{m,t_n}'}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1, ldots, P_m$ are linearly dependent in $E(k)$. This gives a new proof of results of Masser and Zannier and of Barroero and Capuano and extends our earlier results. In the Appendix, we prove an equidistribution theorem for adelically metrized $mathbb{R}$-divisors on projective varieties (over a number field) using results of Moriwaki, extending the equidistribution theorem of Yuan.
假设$mathcal{E} to B$是定义在数域上的非等平凡椭圆曲面，对于光滑射影曲线$B$。设$k$表示函数场$overline{mathbb{Q}}(B)$, $E$表示相关的椭圆曲线$k$。在本文中，我们在一个数字字段上的基曲线$B$上为每个$X in E(k)otimes mathbb{R}$构造幂度量的$mathbb{R}$ -除数$overline{D}_X$。在$E(k)otimes mathbb{R}$上证明了Arakelov-Zhang交数$overline{D}_Xcdot overline{D}_Y$的双二次型的非简并性。因此，对于$mathcal{E}$ / $t in B(overline{mathbb{Q}})$的纤维$E_t(overline{mathbb{Q}})$上的nsamron - tate高度函数，我们有以下bogomolov型声明:给定点$P_1, ldots, P_m in E(k)$和$mgeq 2$，存在一个无限序列$t_nin B(overline{mathbb{Q}})$和专门化$P_{i,t_n}$的小高度扰动$P_{i,t_n}' in E_{t_n}(overline{mathbb{Q}})$，使得集合${P_{1, t_n}', ldots, P_{m,t_n}'}$对所有$n$满足至少两个独立的线性关系，当且仅当点$P_1, ldots, P_m$在$E(k)$中线性相关。这对Masser和Zannier以及Barroero和Capuano的结果给出了新的证明，并推广了我们之前的结果。在附录中，我们利用Moriwaki的结果，推广了Yuan的等分布定理，证明了(在数域上)射影变异上幂度量$mathbb{R}$ -因子的一个等分布定理。
{"title":"Elliptic surfaces and intersections of adelic $mathbb{R}$-divisors","authors":"Laura Demarco, Niki Myrto Mavraki","doi":"10.4171/jems/1354","DOIUrl":"https://doi.org/10.4171/jems/1354","url":null,"abstract":"Suppose $mathcal{E} to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $overline{mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $mathbb{R}$-divisors $overline{D}_X$ on the base curve $B$ over a number field, for each $X in E(k)otimes mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $overline{D}_Xcdot overline{D}_Y$, as a biquadratic form on $E(k)otimes mathbb{R}$. As a consequence, we have the following Bogomolov-type statement for the N'eron-Tate height functions on the fibers $E_t(overline{mathbb{Q}})$ of $mathcal{E}$ over $t in B(overline{mathbb{Q}})$: given points $P_1, ldots, P_m in E(k)$ with $mgeq 2$, there exist an infinite sequence $t_nin B(overline{mathbb{Q}})$ and small-height perturbations $P_{i,t_n}' in E_{t_n}(overline{mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set ${P_{1, t_n}', ldots, P_{m,t_n}'}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1, ldots, P_m$ are linearly dependent in $E(k)$. This gives a new proof of results of Masser and Zannier and of Barroero and Capuano and extends our earlier results. In the Appendix, we prove an equidistribution theorem for adelically metrized $mathbb{R}$-divisors on projective varieties (over a number field) using results of Moriwaki, extending the equidistribution theorem of Yuan.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89210119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define Ratner-Marklof-Strombergsson measures. These are probability measures supported on cut-and-project sets in R^d (d>1) which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project sets.
{"title":"Classification and statistics of cut-and-project sets","authors":"Ren'e Ruhr, Yotam Smilansky, B. Weiss","doi":"10.4171/jems/1338","DOIUrl":"https://doi.org/10.4171/jems/1338","url":null,"abstract":"We define Ratner-Marklof-Strombergsson measures. These are probability measures supported on cut-and-project sets in R^d (d>1) which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project sets.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76778091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(logL) for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E.
{"title":"Level spacing and Poisson statistics for continuum random Schrödinger operators","authors":"Adrian Dietlein, A. Elgart","doi":"10.4171/jems/1033","DOIUrl":"https://doi.org/10.4171/jems/1033","url":null,"abstract":"For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(logL) for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80305832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to [ {rm d}Z_t=sigma(Z_{t-}){rm d} X_t ]driven by a (symmetric) $alpha$-stable Levy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for $alphain(0,1)$. Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We proof integral tests for finiteness of path integrals under minimal assumptions.
{"title":"General path integrals and stable SDEs","authors":"S. Baguley, L. Doering, A. Kyprianou","doi":"10.4171/jems/1331","DOIUrl":"https://doi.org/10.4171/jems/1331","url":null,"abstract":"The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to [ \u0000{rm d}Z_t=sigma(Z_{t-}){rm d} X_t \u0000]driven by a (symmetric) $alpha$-stable Levy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for $alphain(0,1)$. Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We proof integral tests for finiteness of path integrals under minimal assumptions.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80326543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $D 0$ is an absolute positive constant independent of $D$. More generally, let $K$ be a bounded degree number field over $mathbb{Q}$ with the discriminant $D_K$ and the class number $h_K.$ We conjecture that a positive proportion of the ideal classes of $K$ contain a prime ideal with a norm less than $h_Klog(|D_K|)$.
{"title":"The least prime number represented by a binary quadratic form","authors":"Naser Talebizadeh Sardari","doi":"10.4171/jems/1031","DOIUrl":"https://doi.org/10.4171/jems/1031","url":null,"abstract":"Let $D 0$ is an absolute positive constant independent of $D$. More generally, let $K$ be a bounded degree number field over $mathbb{Q}$ with the discriminant $D_K$ and the class number $h_K.$ We conjecture that a positive proportion of the ideal classes of $K$ contain a prime ideal with a norm less than $h_Klog(|D_K|)$.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83852736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main result of this paper is that when $M_0$, $M_1$ are two simply connected spin manifolds of the same dimension $d geq 5$ which both admit a metric of positive scalar curvature, the spaces $mathcal{R}^+(M_0)$ and $mathcal{R}^+(M_1)$ of such metrics are homotopy equivalent. This supersedes a previous result of Chernysh and Walsh which gives the same conclusion when $M_0$ and $M_1$ are also spin cobordant. We also prove an analogous result for simply connected manifolds which do not admit a spin structure; we need to assume that $d neq 8$ in that case.
{"title":"On the homotopy type of the space of metrics of positive scalar curvature","authors":"Johannes Ebert, M. Wiemeler","doi":"10.4171/JEMS/1333","DOIUrl":"https://doi.org/10.4171/JEMS/1333","url":null,"abstract":"The main result of this paper is that when $M_0$, $M_1$ are two simply connected spin manifolds of the same dimension $d geq 5$ which both admit a metric of positive scalar curvature, the spaces $mathcal{R}^+(M_0)$ and $mathcal{R}^+(M_1)$ of such metrics are homotopy equivalent. This supersedes a previous result of Chernysh and Walsh which gives the same conclusion when $M_0$ and $M_1$ are also spin cobordant. \u0000We also prove an analogous result for simply connected manifolds which do not admit a spin structure; we need to assume that $d neq 8$ in that case.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90094742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $U_q'(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Lambda$ and $Lambda^infty$ introduced by the authors. We next define the reflections $mathcal{S}_k$ and $mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp. complete) duality datum, then $mathcal{S}_k(D)$ and $mathcal{S}_k^{-1}(D)$ are also strong (resp. complete ) duality data. We finally introduce the notion of affine cuspidal modules in $mathcal{C}_{mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.
{"title":"PBW theory for quantum affine algebras","authors":"M. Kashiwara, Myungho Kim, Se-jin Oh, E. Park","doi":"10.4171/jems/1323","DOIUrl":"https://doi.org/10.4171/jems/1323","url":null,"abstract":"Let $U_q'(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Lambda$ and $Lambda^infty$ introduced by the authors. We next define the reflections $mathcal{S}_k$ and $mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp. complete) duality datum, then $mathcal{S}_k(D)$ and $mathcal{S}_k^{-1}(D)$ are also strong (resp. complete ) duality data. We finally introduce the notion of affine cuspidal modules in $mathcal{C}_{mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89140468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
L. Manivel, M. Michałek, Leonid Monin, Tim Seynnaeve, Martin Vodivcka
We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.
{"title":"Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming","authors":"L. Manivel, M. Michałek, Leonid Monin, Tim Seynnaeve, Martin Vodivcka","doi":"10.4171/jems/1330","DOIUrl":"https://doi.org/10.4171/jems/1330","url":null,"abstract":"We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80232001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let S be a Gorenstein local ring and suppose that M is a finitely generated Cohen-Macaulay S-module of codimension c. Given a regular sequence f1, . . . , fc in the annihilator of M we set R = S/(f1, . . . , fc) and construct layered S-free and R-free resolutions of M . The construction inductively reduces the problem to the case of a Cohen-Macaulay module of codimension c 1 and leads to the inductive construction of a higher matrix factorization for M . In the case where M is a su ciently high R-syzygy of some module of finite projective dimension over S, the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP].
{"title":"Layered resolutions of Cohen–Macaulay modules","authors":"D. Eisenbud, I. Peeva","doi":"10.4171/jems/1024","DOIUrl":"https://doi.org/10.4171/jems/1024","url":null,"abstract":"Let S be a Gorenstein local ring and suppose that M is a finitely generated Cohen-Macaulay S-module of codimension c. Given a regular sequence f1, . . . , fc in the annihilator of M we set R = S/(f1, . . . , fc) and construct layered S-free and R-free resolutions of M . The construction inductively reduces the problem to the case of a Cohen-Macaulay module of codimension c 1 and leads to the inductive construction of a higher matrix factorization for M . In the case where M is a su ciently high R-syzygy of some module of finite projective dimension over S, the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP].","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91256786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：