Pub Date : 2020-01-10DOI: 10.1215/00127094-2021-0051
Samuel C. Edwards, H. Oh
Let $mathcal{M}=Gammabackslashmathbb{H}^{d+1}$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d/2$. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in $L^2(mathrm{T}^1(mathcal{M}))$, with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on $L^2(mathcal{M})$ due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varju on expanders, this implies uniform exponential mixing for congruence covers of $mathcal{M}$ when $Gamma$ is a thin subgroup of $mathrm{SO}^{circ}(d+1,1)$. Our result implies that, with respect to the Bowen-Margulis-Sullivan measure, the geodesic flow on $mathrm{T}^1(mathcal{M})$ is exponentially mixing, uniformly over congruence covers in the case when $Gamma$ is a thin subgroup.
{"title":"Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds","authors":"Samuel C. Edwards, H. Oh","doi":"10.1215/00127094-2021-0051","DOIUrl":"https://doi.org/10.1215/00127094-2021-0051","url":null,"abstract":"Let $mathcal{M}=Gammabackslashmathbb{H}^{d+1}$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d/2$. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in $L^2(mathrm{T}^1(mathcal{M}))$, with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on $L^2(mathcal{M})$ due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varju on expanders, this implies uniform exponential mixing for congruence covers of $mathcal{M}$ when $Gamma$ is a thin subgroup of $mathrm{SO}^{circ}(d+1,1)$. Our result implies that, with respect to the Bowen-Margulis-Sullivan measure, the geodesic flow on $mathrm{T}^1(mathcal{M})$ is exponentially mixing, uniformly over congruence covers in the case when $Gamma$ is a thin subgroup.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41742722","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}
Pub Date : 2020-01-03DOI: 10.1215/00127094-2021-0061
Anne Lonjou, Christian Urech
For each d we construct CAT(0) cube complexes on which Cremona groups rank d act by isometries. From these actions we deduce new and old group theoretical and dynamical results about Cremona groups. In particular, we study the dynamical behaviour of the irreducible components of exceptional loci, we prove regularization theorems, we find new constraints on the degree growth for non-regularizable birational transformations, and we show that the centralizer of certain birational transformations is small.
{"title":"Actions of Cremona groups on CAT(0) cube complexes","authors":"Anne Lonjou, Christian Urech","doi":"10.1215/00127094-2021-0061","DOIUrl":"https://doi.org/10.1215/00127094-2021-0061","url":null,"abstract":"For each d we construct CAT(0) cube complexes on which Cremona groups rank d act by isometries. From these actions we deduce new and old group theoretical and dynamical results about Cremona groups. In particular, we study the dynamical behaviour of the irreducible components of exceptional loci, we prove regularization theorems, we find new constraints on the degree growth for non-regularizable birational transformations, and we show that the centralizer of certain birational transformations is small.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44742457","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}
Pub Date : 2019-12-19DOI: 10.1215/00127094-2022-0052
Antoine Ducros, E. Hrushovski, F. Loeser
We explain how non-archimedean integrals considered by Chambert-Loir and Ducros naturally arise in asymptotics of families of complex integrals. To perform this analysis we work over a non-standard model of the field of complex numbers, which is endowed at the same time with an archimedean and a non-archimedean norm. Our main result states the existence of a natural morphism between bicomplexes of archimedean and non-archimedean forms which is compatible with integration.
{"title":"Non-Archimedean integrals as limits of complex integrals","authors":"Antoine Ducros, E. Hrushovski, F. Loeser","doi":"10.1215/00127094-2022-0052","DOIUrl":"https://doi.org/10.1215/00127094-2022-0052","url":null,"abstract":"We explain how non-archimedean integrals considered by Chambert-Loir and Ducros naturally arise in asymptotics of families of complex integrals. To perform this analysis we work over a non-standard model of the field of complex numbers, which is endowed at the same time with an archimedean and a non-archimedean norm. Our main result states the existence of a natural morphism between bicomplexes of archimedean and non-archimedean forms which is compatible with integration.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48578442","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}
Pub Date : 2019-12-19DOI: 10.1215/00127094-2021-0050
David Ellis, Noam Lifshitz
A family of permutations $mathcal{F} subset S_{n}$ is said to be $t$-intersecting if any two permutations in $mathcal{F}$ agree on at least $t$ points. It is said to be $(t-1)$-intersection-free if no two permutations in $mathcal{F}$ agree on exactly $t-1$ points. If $S,T subset {1,2,ldots,n}$ with $|S|=|T|$, and $pi: S to T$ is a bijection, the $pi$-star in $S_n$ is the family of all permutations in $S_n$ that agree with $pi$ on all of $S$. An $s$-star is a $pi$-star such that $pi$ is a bijection between sets of size $s$. Friedgut and Pilpel, and independently the first author, showed that if $mathcal{F} subset S_n$ is $t$-intersecting, and $n$ is sufficiently large depending on $t$, then $|mathcal{F}| leq (n-t)!$; this proved a conjecture of Deza and Frankl from 1977. Equality holds only if $mathcal{F}$ is a $t$-star. In this paper, we give a more `robust' proof of a strengthening of the Deza-Frankl conjecture, namely that if $n$ is sufficiently large depending on $t$, and $mathcal{F} subset S_n$ is $(t-1)$-intersection-free, then $|mathcal{F} leq (n-t)!$, with equality only if $mathcal{F}$ is a $t$-star. The main ingredient of our proof is a `junta approximation' result, namely, that any $(t-1)$-intersection-free family of permutations is essentially contained in a $t$-intersecting {em junta} (a `junta' being a union of a bounded number of $O(1)$-stars). The proof of our junta approximation result relies, in turn, on a weak regularity lemma for families of permutations, a combinatorial argument that `bootstraps' a weak notion of pseudorandomness into a stronger one, and finally a spectral argument for pairs of highly-pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature, and one being algebraic.
{"title":"Approximation by juntas in the symmetric group, and forbidden intersection problems","authors":"David Ellis, Noam Lifshitz","doi":"10.1215/00127094-2021-0050","DOIUrl":"https://doi.org/10.1215/00127094-2021-0050","url":null,"abstract":"A family of permutations $mathcal{F} subset S_{n}$ is said to be $t$-intersecting if any two permutations in $mathcal{F}$ agree on at least $t$ points. It is said to be $(t-1)$-intersection-free if no two permutations in $mathcal{F}$ agree on exactly $t-1$ points. If $S,T subset {1,2,ldots,n}$ with $|S|=|T|$, and $pi: S to T$ is a bijection, the $pi$-star in $S_n$ is the family of all permutations in $S_n$ that agree with $pi$ on all of $S$. An $s$-star is a $pi$-star such that $pi$ is a bijection between sets of size $s$. Friedgut and Pilpel, and independently the first author, showed that if $mathcal{F} subset S_n$ is $t$-intersecting, and $n$ is sufficiently large depending on $t$, then $|mathcal{F}| leq (n-t)!$; this proved a conjecture of Deza and Frankl from 1977. Equality holds only if $mathcal{F}$ is a $t$-star. \u0000In this paper, we give a more `robust' proof of a strengthening of the Deza-Frankl conjecture, namely that if $n$ is sufficiently large depending on $t$, and $mathcal{F} subset S_n$ is $(t-1)$-intersection-free, then $|mathcal{F} leq (n-t)!$, with equality only if $mathcal{F}$ is a $t$-star. The main ingredient of our proof is a `junta approximation' result, namely, that any $(t-1)$-intersection-free family of permutations is essentially contained in a $t$-intersecting {em junta} (a `junta' being a union of a bounded number of $O(1)$-stars). The proof of our junta approximation result relies, in turn, on a weak regularity lemma for families of permutations, a combinatorial argument that `bootstraps' a weak notion of pseudorandomness into a stronger one, and finally a spectral argument for pairs of highly-pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature, and one being algebraic.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45901484","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}
Pub Date : 2019-12-18DOI: 10.1215/00127094-2022-0025
Hamid Abban, M. Fedorchuk, I. Krylov
We introduce and study a new notion of stability for varieties fibered over curves, motivated by Koll'ar's stability for homogeneous polynomials with integral coefficients. We develop tools to study geometric properties of stable birational models of fibrations whose fibers are complete intersections in weighted projective spaces. As an application, we prove the existence of standard models of threefold degree one and two del Pezzo fibrations, settling a conjecture of Corti from 1996.
{"title":"Stability of fibrations over one-dimensional bases","authors":"Hamid Abban, M. Fedorchuk, I. Krylov","doi":"10.1215/00127094-2022-0025","DOIUrl":"https://doi.org/10.1215/00127094-2022-0025","url":null,"abstract":"We introduce and study a new notion of stability for varieties fibered over curves, motivated by Koll'ar's stability for homogeneous polynomials with integral coefficients. We develop tools to study geometric properties of stable birational models of fibrations whose fibers are complete intersections in weighted projective spaces. As an application, we prove the existence of standard models of threefold degree one and two del Pezzo fibrations, settling a conjecture of Corti from 1996.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42761931","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}
Pub Date : 2019-12-04DOI: 10.1215/00127094-2022-0063
Dawei Chen, Martin Möller, Adrien Sauvaget, A. Giacchetto, D. Lewanski
We describe a conjectural formula via intersection numbers for the Masur-Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to compute the top Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. An appendix proves that the intersection of this class with $psi$-classes can be computed by Eynard-Orantin topological recursion. As applications, we analyze numerical properties of Masur-Veech volumes, area Siegel-Veech constants and sums of Lyapunov exponents of the principal strata for fixed genus and varying number of zeros, which settles the corresponding conjectures due to Grivaux-Hubert, Fougeron, and elaborated in [the7]. We also describe conjectural formulas for area Siegel-Veech constants and sums of Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them for the principal strata.
{"title":"Masur–Veech volumes and intersection theory: The principal strata of quadratic differentials","authors":"Dawei Chen, Martin Möller, Adrien Sauvaget, A. Giacchetto, D. Lewanski","doi":"10.1215/00127094-2022-0063","DOIUrl":"https://doi.org/10.1215/00127094-2022-0063","url":null,"abstract":"We describe a conjectural formula via intersection numbers for the Masur-Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to compute the top Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. An appendix proves that the intersection of this class with $psi$-classes can be computed by Eynard-Orantin topological recursion. \u0000As applications, we analyze numerical properties of Masur-Veech volumes, area Siegel-Veech constants and sums of Lyapunov exponents of the principal strata for fixed genus and varying number of zeros, which settles the corresponding conjectures due to Grivaux-Hubert, Fougeron, and elaborated in [the7]. We also describe conjectural formulas for area Siegel-Veech constants and sums of Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them for the principal strata.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43751839","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}
Pub Date : 2019-11-20DOI: 10.1215/00127094-2023-0012
Antoine Song
We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian manifold and $Sigmasubset M$ be a closed embedded minimal hypersurface with area at most $A>0$ and with a singular set of Hausdorff dimension at most $n-7$. We show the following bounds: there is $C_A>0$ depending only on $n$, $g$, and $A$ so that $$sum_{i=0}^n b^i(Sigma) leq C_A big(1+index(Sigma)big) quad text{ if $3leq n+1leq 7$},$$ $$mathcal{H}^{n-7}big(Sing(Sigma)big) leq C_A big(1+index(Sigma)big)^{7/n} quad text{ if $n+1geq 8$},$$ where $b^i$ denote the Betti numbers over any field, $mathcal{H}^{n-7}$ is the $(n-7)$-dimensional Hausdorff measure and $Sing(Sigma)$ is the singular set of $Sigma$. In fact in dimension $n+1=3$, $C_A$ depends linearly on $A$. We list some open problems at the end of the paper.
{"title":"Morse index, Betti numbers, and singular set of bounded area minimal hypersurfaces","authors":"Antoine Song","doi":"10.1215/00127094-2023-0012","DOIUrl":"https://doi.org/10.1215/00127094-2023-0012","url":null,"abstract":"We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian manifold and $Sigmasubset M$ be a closed embedded minimal hypersurface with area at most $A>0$ and with a singular set of Hausdorff dimension at most $n-7$. We show the following bounds: there is $C_A>0$ depending only on $n$, $g$, and $A$ so that $$sum_{i=0}^n b^i(Sigma) leq C_A big(1+index(Sigma)big) quad text{ if $3leq n+1leq 7$},$$ $$mathcal{H}^{n-7}big(Sing(Sigma)big) leq C_A big(1+index(Sigma)big)^{7/n} quad text{ if $n+1geq 8$},$$ where $b^i$ denote the Betti numbers over any field, $mathcal{H}^{n-7}$ is the $(n-7)$-dimensional Hausdorff measure and $Sing(Sigma)$ is the singular set of $Sigma$. In fact in dimension $n+1=3$, $C_A$ depends linearly on $A$. We list some open problems at the end of the paper.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41453778","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}
Pub Date : 2019-11-14DOI: 10.1215/00127094-2022-0065
J. Nicaise, J. C. Ottem
We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. In particular, we show that very general quartic fivefolds and complete intersections of a quadric and a cubic in $mathbb P^6$ are stably irrational. An important new ingredient is the use of tropical degeneration techniques.
{"title":"Tropical degenerations and stable rationality","authors":"J. Nicaise, J. C. Ottem","doi":"10.1215/00127094-2022-0065","DOIUrl":"https://doi.org/10.1215/00127094-2022-0065","url":null,"abstract":"We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. In particular, we show that very general quartic fivefolds and complete intersections of a quadric and a cubic in $mathbb P^6$ are stably irrational. An important new ingredient is the use of tropical degeneration techniques.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47438376","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}
Pub Date : 2019-11-05DOI: 10.1215/00127094-2022-0050
Jacek Jendrej, M. Kowalczyk, A. Lawrie
This paper concerns classical nonlinear scalar field models on the real line. If the potential is a symmetric double-well, such a model admits static solutions called kinks and antikinks, which are perhaps the simplest examples of topological solitons. We study pure multi-kinks, which are solutions that converge in one infinite time direction to a superposition of a finite number of kinks and antikinks, without radiation. Our main result is a complete classification of all kink-antikink pairs in the strongly interacting regime, which means the speeds of the kinks tend asymptotically to zero. We show that up to translation there is only one such solution, and we give a precise description of the dynamics of the kink separation. We also establish the existence of strongly interacting $K$-multi-kinks, for any natural number $K$.
{"title":"Dynamics of strongly interacting kink-antikink pairs for scalar fields on a line","authors":"Jacek Jendrej, M. Kowalczyk, A. Lawrie","doi":"10.1215/00127094-2022-0050","DOIUrl":"https://doi.org/10.1215/00127094-2022-0050","url":null,"abstract":"This paper concerns classical nonlinear scalar field models on the real line. If the potential is a symmetric double-well, such a model admits static solutions called kinks and antikinks, which are perhaps the simplest examples of topological solitons. We study pure multi-kinks, which are solutions that converge in one infinite time direction to a superposition of a finite number of kinks and antikinks, without radiation. Our main result is a complete classification of all kink-antikink pairs in the strongly interacting regime, which means the speeds of the kinks tend asymptotically to zero. We show that up to translation there is only one such solution, and we give a precise description of the dynamics of the kink separation. We also establish the existence of strongly interacting $K$-multi-kinks, for any natural number $K$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44175377","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}
Pub Date : 2019-11-05DOI: 10.1215/00127094-2021-0027
Charles Ouyang, Andrea Tamburelli
We find a compactification of the $mathrm{SL}(3,mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.
{"title":"Limits of Blaschke metrics","authors":"Charles Ouyang, Andrea Tamburelli","doi":"10.1215/00127094-2021-0027","DOIUrl":"https://doi.org/10.1215/00127094-2021-0027","url":null,"abstract":"We find a compactification of the $mathrm{SL}(3,mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43220399","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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