We extend the LpLp theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of WW-random graphs.
{"title":"An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence","authors":"C. Borgs, J. Chayes, Henry Cohn, Yufei Zhao","doi":"10.1214/17-AOP1187","DOIUrl":"https://doi.org/10.1214/17-AOP1187","url":null,"abstract":"We extend the LpLp theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of WW-random graphs.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1187","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66060699","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}
A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.
{"title":"Mean field games with common noise","authors":"R. Carmona, F. Delarue, D. Lacker","doi":"10.1214/15-AOP1060","DOIUrl":"https://doi.org/10.1214/15-AOP1060","url":null,"abstract":"A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032342","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}
By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.
{"title":"Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients","authors":"Longjie Xie, Xicheng Zhang","doi":"10.1214/15-AOP1057","DOIUrl":"https://doi.org/10.1214/15-AOP1057","url":null,"abstract":"By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1057","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032574","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}
On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature ββ is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β>0β>0, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph.We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree dd, the Ising model has cutoff provided that β<κ/dβ<κ/d for some absolute constant κκ (a result which, up to the value of κκ, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1)O(1), just as when β=0β=0.Finally, the mixing time from almost every initial state is not more than a factor of 1+eβ1+eβ faster then the worst one (with eβ→0eβ→0 as β→0β→0), whereas the uniform starting state is at least 2−eβ2−eβ times faster.
{"title":"Universality of cutoff for the ising model","authors":"E. Lubetzky, A. Sly","doi":"10.1214/16-AOP1146","DOIUrl":"https://doi.org/10.1214/16-AOP1146","url":null,"abstract":"On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature ββ is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β>0β>0, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph.We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree dd, the Ising model has cutoff provided that β<κ/dβ<κ/d for some absolute constant κκ (a result which, up to the value of κκ, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1)O(1), just as when β=0β=0.Finally, the mixing time from almost every initial state is not more than a factor of 1+eβ1+eβ faster then the worst one (with eβ→0eβ→0 as β→0β→0), whereas the uniform starting state is at least 2−eβ2−eβ times faster.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1146","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047787","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 consider N×N random matrices of the form H=W+V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.
{"title":"Bulk universality for deformed Wigner matrices","authors":"J. Lee, Kevin Schnelli, Ben Stetler, H. Yau","doi":"10.1214/15-AOP1023","DOIUrl":"https://doi.org/10.1214/15-AOP1023","url":null,"abstract":"We consider N×N random matrices of the form H=W+V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031634","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}
Consider a discrete-time martingale {Xt}{Xt} taking values in a Hilbert space HH. We show that if for some L≥1L≥1, the bounds E[∥Xt+1−Xt∥2H|Xt]=1E[‖Xt+1−Xt‖H2|Xt]=1 and ∥Xt+1−Xt∥H≤L‖Xt+1−Xt‖H≤L are satisfied for all times t≥0t≥0, then there is a constant c=c(L)c=c(L) such that for 1≤R≤t√1≤R≤t, P(∥Xt−X0∥H≤R)≤cRt√. P(‖Xt−X0‖H≤R)≤cRt. Following Lee and Peres [Ann. Probab. 41 (2013) 3392–3419], this estimate has applications to small-ball estimates for random walks on vertex-transitive graphs: We show that for every infinite, connected, vertex-transitive graph GG with bounded degree, there is a constant CG>0CG>0 such that if {Zt}{Zt} is the simple random walk on GG, then for every e>0e>0 and t≥1/e2t≥1/e2, P(distG(Zt,Z0)≤et√)≤CGe, P(distG(Zt,Z0)≤et)≤CGe, where distGdistG denotes the graph distance in GG.
{"title":"A Gaussian upper bound for martingale small-ball probabilities","authors":"James R. Lee, Y. Peres, Charles K. Smart","doi":"10.1214/15-AOP1073","DOIUrl":"https://doi.org/10.1214/15-AOP1073","url":null,"abstract":"Consider a discrete-time martingale {Xt}{Xt} taking values in a Hilbert space HH. We show that if for some L≥1L≥1, the bounds E[∥Xt+1−Xt∥2H|Xt]=1E[‖Xt+1−Xt‖H2|Xt]=1 and ∥Xt+1−Xt∥H≤L‖Xt+1−Xt‖H≤L are satisfied for all times t≥0t≥0, then there is a constant c=c(L)c=c(L) such that for 1≤R≤t√1≤R≤t, \u0000 \u0000P(∥Xt−X0∥H≤R)≤cRt√. \u0000P(‖Xt−X0‖H≤R)≤cRt. \u0000Following Lee and Peres [Ann. Probab. 41 (2013) 3392–3419], this estimate has applications to small-ball estimates for random walks on vertex-transitive graphs: We show that for every infinite, connected, vertex-transitive graph GG with bounded degree, there is a constant CG>0CG>0 such that if {Zt}{Zt} is the simple random walk on GG, then for every e>0e>0 and t≥1/e2t≥1/e2, \u0000 \u0000P(distG(Zt,Z0)≤et√)≤CGe, \u0000P(distG(Zt,Z0)≤et)≤CGe, \u0000where distGdistG denotes the graph distance in GG.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1073","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032828","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 consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.
{"title":"Intermittency for branching random walk in Pareto environment","authors":"Marcel Ortgiese, Matthew I. Roberts","doi":"10.1214/15-AOP1021","DOIUrl":"https://doi.org/10.1214/15-AOP1021","url":null,"abstract":"We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031549","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}
This contribution establishes exact tail asymptotics of sup(s,t)∈E X(s,t) for a large class of nonhomogeneous Gaussian random fields X on a bounded convex set E⊂R2, with variance function that attains its maximum on a segment on E. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.
{"title":"Extremes of a class of nonhomogeneous Gaussian random fields","authors":"Krzysztof Dcebicki, E. Hashorva, L. Ji","doi":"10.1214/14-AOP994","DOIUrl":"https://doi.org/10.1214/14-AOP994","url":null,"abstract":"This contribution establishes exact tail asymptotics of sup(s,t)∈E X(s,t) for a large class of nonhomogeneous Gaussian random fields X on a bounded convex set E⊂R2, with variance function that attains its maximum on a segment on E. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP994","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66010589","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 M be a von Neumann algebra equipped with a faithful semifinite normal weight ϕ and N be a von Neumann subalgebra of M such that the restriction of ϕ to N is semifinite and such that N is invariant by the modular group of ϕ. Let E be the weight preserving conditional expectation from M onto N. As an application we show that there exists e0>0 such that for any free group Fn and any q≥4−e0, ∥Pt∥2→q≤1⇔t≥logq−1−−−−√, where (Pt) is the Poisson semigroup defined by the natural length function of Fn.
{"title":"A noncommutative martingale convexity inequality","authors":"'Eric Ricard, Quanhua Xu","doi":"10.1214/14-AOP990","DOIUrl":"https://doi.org/10.1214/14-AOP990","url":null,"abstract":"Let M be a von Neumann algebra equipped with a faithful semifinite normal weight ϕ and N be a von Neumann subalgebra of M such that the restriction of ϕ to N is semifinite and such that N is invariant by the modular group of ϕ. Let E be the weight preserving conditional expectation from M onto N. As an application we show that there exists e0>0 such that for any free group Fn and any q≥4−e0, \u0000∥Pt∥2→q≤1⇔t≥logq−1−−−−√, where (Pt) is the Poisson semigroup defined by the natural length function of Fn.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP990","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66010725","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 consider a class of kinetically constrained interacting particle systems on Zd which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of Zd, to each occupation variable ηx∈{0,1} a new value is proposed by tossing a (1−q)-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For d=1, the constraint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case, the process is the well-known East process. On Z2, the West or the South neighbor of the updating vertex must contain a vacancy, similarly, in higher dimensions. Despite of their apparent simplicity, in the limit q↘0 of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on Zd scales as the 1/d-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next, we compute the relaxation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multiscale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally, we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex x out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the l1-norm of x but also on its direction.
我们考虑了Zd上一类动力学约束的相互作用粒子系统，它们在描述玻璃动力学复杂行为的几种启发式定性和定量方法中起着关键作用。对于每个职业变量ηx∈{0,1}，在速率为1且独立于Zd的顶点之间，通过投掷(1−q)-硬币提出一个新值。如果当前配置满足某个局部约束，则建议的移动被接受，否则被拒绝。对于d=1，约束要求在更新顶点的左边顶点有一个空位。在本例中，该流程就是众所周知的East流程。在Z2上，更新顶点的西部或南部邻居必须包含一个空位，类似地，在更高的维度上。尽管它们表面上很简单，但在低空位密度的极限q d d 0下，对应于低温物理环境，这些过程具有相当复杂的动态行为，具有分层松弛时间尺度、异质性和普遍性。利用重整化群的思想，我们首先证明了Zd尺度上的弛豫时间是East过程弛豫时间的1/d-根，证实了大量数值模拟的结果。接下来，我们通过多尺度分析、容量方法和算法构建，仔细分析了微妙的能量熵竞争，计算了有限盒子中的松弛时间。我们的结果建立了动态异质性和对边界条件的戏剧性依赖。最后，我们证明了这些过程的一个相当强的各向异性性质:在原点(种子)孤立的顶点x上产生一个新的空位可能发生在(对数上)不同的时间尺度上，这不仅严重依赖于x的11范数，而且还依赖于它的方向。
{"title":"Relaxation to equilibrium of generalized East processes on $mathbb{Z}^{d}$: Renormalization group analysis and energy-entropy competition","authors":"P. Chleboun, A. Faggionato, F. Martinelli","doi":"10.1214/15-AOP1011","DOIUrl":"https://doi.org/10.1214/15-AOP1011","url":null,"abstract":"We consider a class of kinetically constrained interacting particle systems on Zd which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of Zd, to each occupation variable ηx∈{0,1} a new value is proposed by tossing a (1−q)-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For d=1, the constraint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case, the process is the well-known East process. On Z2, the West or the South neighbor of the updating vertex must contain a vacancy, similarly, in higher dimensions. Despite of their apparent simplicity, in the limit q↘0 of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on Zd scales as the 1/d-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next, we compute the relaxation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multiscale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally, we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex x out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the l1-norm of x but also on its direction.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031627","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}
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