Pub Date : 2025-05-01Epub Date: 2025-02-24DOI: 10.1016/j.matpur.2025.103694
Kyoung-Seog Lee , Han-Bom Moon
We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the curve. It supports a prediction of the existence of a semiorthogonal decomposition of the derived category of the moduli space, expected by a motivic computation. As an application, we show that all Jacobian varieties, symmetric products of curves, and all principally polarized abelian varieties of dimension at most three, are Fano visitors. We also obtain similar results for motives.
{"title":"Derived categories of symmetric products and moduli spaces of vector bundles on a curve","authors":"Kyoung-Seog Lee , Han-Bom Moon","doi":"10.1016/j.matpur.2025.103694","DOIUrl":"10.1016/j.matpur.2025.103694","url":null,"abstract":"<div><div>We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the curve. It supports a prediction of the existence of a semiorthogonal decomposition of the derived category of the moduli space, expected by a motivic computation. As an application, we show that all Jacobian varieties, symmetric products of curves, and all principally polarized abelian varieties of dimension at most three, are Fano visitors. We also obtain similar results for motives.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"197 ","pages":"Article 103694"},"PeriodicalIF":2.1,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509395","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 : 2025-05-01Epub Date: 2025-02-24DOI: 10.1016/j.matpur.2025.103686
Huai-Dong Cao , Junming Xie
This is a sequel to our paper [24], in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete ancient solutions with WPIC, we are able to prove the 2-nonnegativity of the Ricci curvature and bound the curvature tensor Rm by . For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient expanding Ricci solitons with WPIC. Finally, motivated by the recent work [59], we improve our earlier results in [24] on 4D gradient shrinking Ricci solitons with half PIC or half WPIC, and also provide a characterization of complete gradient Kähler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.
这是我们论文[24]的续集，在[24]中，我们研究了具有半正（非负）各向同性曲率的四维梯度收缩Ricci孤子的几何。本文主要研究具有非负各向同性曲率（WPIC）或半非负各向同性曲率（half non - anisotropic curvature， WPIC）的四维梯度稳定Ricci孤子。特别是对于具有WPIC的4D完全古解，我们证明了Ricci曲率的2-非负性，并将曲率张量Rm限定为|Rm|≤R。对于具有WPIC的四维梯度稳定孤子，我们得到了一个分类结果。给出了具有半WPIC的四维梯度稳定Ricci孤子的部分分类。此外，我们还利用WPIC获得了4D完全梯度展开Ricci孤子的初步分类结果。最后，在最近工作[59]的激励下，我们改进了[24]中关于半PIC或半WPIC的4D梯度收缩Ricci孤子的早期结果，并在4维梯度Ricci收缩子中给出了复二维完全梯度Kähler-Ricci收缩子的表征。
{"title":"Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature","authors":"Huai-Dong Cao , Junming Xie","doi":"10.1016/j.matpur.2025.103686","DOIUrl":"10.1016/j.matpur.2025.103686","url":null,"abstract":"<div><div>This is a sequel to our paper <span><span>[24]</span></span>, in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete <em>ancient solutions</em> with WPIC, we are able to prove the 2-nonnegativity of the Ricci curvature and bound the curvature tensor <em>Rm</em> by <span><math><mo>|</mo><mi>R</mi><mi>m</mi><mo>|</mo><mo>≤</mo><mi>R</mi></math></span>. For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient <em>expanding Ricci solitons</em> with WPIC. Finally, motivated by the recent work <span><span>[59]</span></span>, we improve our earlier results in <span><span>[24]</span></span> on 4D gradient <em>shrinking Ricci solitons</em> with half PIC or half WPIC, and also provide a characterization of complete gradient Kähler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"197 ","pages":"Article 103686"},"PeriodicalIF":2.1,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511264","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 : 2025-04-01Epub Date: 2025-02-24DOI: 10.1016/j.matpur.2025.103688
Maarten V. de Hoop , Joonas Ilmavirta , Matti Lassas
Dix formulated the inverse problem of recovering an elastic body from the measurements of wave fronts of point sources. We geometrize this problem in the context of seismology, leading to the geometrical inverse problem of recovering a Finsler manifold from certain sphere data in a given open subset of the manifold. We solve this problem locally along any geodesic through the measurement set.
{"title":"Reconstruction along a geodesic from sphere data in Finsler geometry and anisotropic elasticity","authors":"Maarten V. de Hoop , Joonas Ilmavirta , Matti Lassas","doi":"10.1016/j.matpur.2025.103688","DOIUrl":"10.1016/j.matpur.2025.103688","url":null,"abstract":"<div><div>Dix formulated the inverse problem of recovering an elastic body from the measurements of wave fronts of point sources. We geometrize this problem in the context of seismology, leading to the geometrical inverse problem of recovering a Finsler manifold from certain sphere data in a given open subset of the manifold. We solve this problem locally along any geodesic through the measurement set.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"196 ","pages":"Article 103688"},"PeriodicalIF":2.1,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143507519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-01Epub Date: 2025-02-24DOI: 10.1016/j.matpur.2025.103693
Xiaojun Huang , Wanke Yin
Let M be a smooth pseudoconvex real hypersurface in with and let B be a subbundle of the CR tangent vector bundle of M. We prove that the commutator type and the Levi type associated with B are the same when either of them is less than 8. When the Levi type is eight or larger, we show that it is bounded from above by twice of the commutator type minus 8. Our results provide a partial solution to a generalized conjecture of D'Angelo.
{"title":"Commutator type and Levi type of a system of CR vector fields","authors":"Xiaojun Huang , Wanke Yin","doi":"10.1016/j.matpur.2025.103693","DOIUrl":"10.1016/j.matpur.2025.103693","url":null,"abstract":"<div><div>Let <em>M</em> be a smooth pseudoconvex real hypersurface in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> and let <em>B</em> be a subbundle of the CR tangent vector bundle of <em>M</em>. We prove that the commutator type and the Levi type associated with <em>B</em> are the same when either of them is less than 8. When the Levi type is eight or larger, we show that it is bounded from above by twice of the commutator type minus 8. Our results provide a partial solution to a generalized conjecture of D'Angelo.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"196 ","pages":"Article 103693"},"PeriodicalIF":2.1,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143507536","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 : 2025-04-01Epub Date: 2025-02-24DOI: 10.1016/j.matpur.2025.103690
Karl Kunisch , Gengsheng Wang , Huaiqiang Yu
A quantitative frequency-domain condition related to the exponential stabilizability for infinite-dimensional linear control systems is presented. It is proven that this condition is necessary and sufficient for the stabilizability of special systems, while it is a necessary condition for the stabilizability in general. Applications are provided.
{"title":"Frequency-domain criterion on the stabilizability for infinite-dimensional linear control systems","authors":"Karl Kunisch , Gengsheng Wang , Huaiqiang Yu","doi":"10.1016/j.matpur.2025.103690","DOIUrl":"10.1016/j.matpur.2025.103690","url":null,"abstract":"<div><div>A quantitative frequency-domain condition related to the exponential stabilizability for infinite-dimensional linear control systems is presented. It is proven that this condition is necessary and sufficient for the stabilizability of special systems, while it is a necessary condition for the stabilizability in general. Applications are provided.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"196 ","pages":"Article 103690"},"PeriodicalIF":2.1,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143507535","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 : 2025-04-01Epub Date: 2025-02-24DOI: 10.1016/j.matpur.2025.103692
Thomas Alazard , Jeremy L. Marzuola , Jian Wang
Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.
{"title":"Damping for fractional wave equations and applications to water waves","authors":"Thomas Alazard , Jeremy L. Marzuola , Jian Wang","doi":"10.1016/j.matpur.2025.103692","DOIUrl":"10.1016/j.matpur.2025.103692","url":null,"abstract":"<div><div>Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"196 ","pages":"Article 103692"},"PeriodicalIF":2.1,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529175","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 : 2025-03-01Epub Date: 2024-11-04DOI: 10.1016/j.matpur.2024.103620
Joaquin Moraga , Roberto Svaldi
Given a projective contraction and a log canonical pair such that is nef over a neighborhood of a closed point , one can define an invariant, the complexity of over , comparing the dimension of X and the relative Picard number of with the sum of the coefficients of those components of B intersecting the fiber over z. We prove that, in the hypotheses above, the complexity of the log pair over is non-negative and that when it is zero then is formally isomorphic to a morphism of toric varieties around . In particular, considering the case when π is the identity morphism, we get a geometric characterization of singularities that are formally isomorphic to toric singularities, thus resolving a conjecture due to Shokurov.
{"title":"A geometric characterization of toric singularities","authors":"Joaquin Moraga , Roberto Svaldi","doi":"10.1016/j.matpur.2024.103620","DOIUrl":"10.1016/j.matpur.2024.103620","url":null,"abstract":"<div><div>Given a projective contraction <span><math><mi>π</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Z</mi></math></span> and a log canonical pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> such that <span><math><mo>−</mo><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>+</mo><mi>B</mi><mo>)</mo></math></span> is nef over a neighborhood of a closed point <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span>, one can define an invariant, the complexity of <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> over <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span>, comparing the dimension of <em>X</em> and the relative Picard number of <span><math><mi>X</mi><mo>/</mo><mi>Z</mi></math></span> with the sum of the coefficients of those components of <em>B</em> intersecting the fiber over <em>z</em>. We prove that, in the hypotheses above, the complexity of the log pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> over <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span> is non-negative and that when it is zero then <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mo>⌊</mo><mi>B</mi><mo>⌋</mo><mo>)</mo><mo>→</mo><mi>Z</mi></math></span> is formally isomorphic to a morphism of toric varieties around <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span>. In particular, considering the case when <em>π</em> is the identity morphism, we get a geometric characterization of singularities that are formally isomorphic to toric singularities, thus resolving a conjecture due to Shokurov.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103620"},"PeriodicalIF":2.1,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155111","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 : 2025-03-01Epub Date: 2025-01-15DOI: 10.1016/j.matpur.2025.103671
Piotr B. Mucha , Maja Szlenk , Ewelina Zatorska
We analyze the pressureless Navier-Stokes system with nonlocal attraction–repulsion forces. Such systems appear in the context of models of collective behaviour. We prove the existence of weak solutions on the whole space in the case of density-dependent degenerate viscosity. For the nonlocal term it is assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. Under these assumptions, we derive the analog of the Bresch–Desjardins and Mellet–Vasseur estimates for the nonlocal system. In particular, we are able to adapt the approach of Vasseur and Yu [37], [36] to construct a weak solution.
{"title":"Construction of weak solutions to a model of pressureless viscous flow driven by nonlocal attraction–repulsion","authors":"Piotr B. Mucha , Maja Szlenk , Ewelina Zatorska","doi":"10.1016/j.matpur.2025.103671","DOIUrl":"10.1016/j.matpur.2025.103671","url":null,"abstract":"<div><div>We analyze the pressureless Navier-Stokes system with nonlocal attraction–repulsion forces. Such systems appear in the context of models of collective behaviour. We prove the existence of weak solutions on the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> in the case of density-dependent degenerate viscosity. For the nonlocal term it is assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. Under these assumptions, we derive the analog of the Bresch–Desjardins and Mellet–Vasseur estimates for the nonlocal system. In particular, we are able to adapt the approach of Vasseur and Yu <span><span>[37]</span></span>, <span><span>[36]</span></span> to construct a weak solution.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103671"},"PeriodicalIF":2.1,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155109","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 : 2025-03-01Epub Date: 2025-01-15DOI: 10.1016/j.matpur.2025.103669
Maria Gordina , Wei Qian , Yilin Wang
We use the SLEκ loop measure to construct a natural representation of the Virasoro algebra of central charge . In particular, we introduce a non-degenerate bilinear Hermitian form (and non positive-definite) using the SLE loop measure and show that the representation is indefinite unitary. Our proof relies on the infinitesimal conformal restriction property of the SLE loop measure.
{"title":"Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra","authors":"Maria Gordina , Wei Qian , Yilin Wang","doi":"10.1016/j.matpur.2025.103669","DOIUrl":"10.1016/j.matpur.2025.103669","url":null,"abstract":"<div><div>We use the SLE<sub><em>κ</em></sub> loop measure to construct a natural representation of the Virasoro algebra of central charge <span><math><mi>c</mi><mo>=</mo><mi>c</mi><mo>(</mo><mi>κ</mi><mo>)</mo><mo>≤</mo><mn>1</mn></math></span>. In particular, we introduce a non-degenerate bilinear Hermitian form (and non positive-definite) using the SLE loop measure and show that the representation is indefinite unitary. Our proof relies on the infinitesimal conformal restriction property of the SLE loop measure.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103669"},"PeriodicalIF":2.1,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01Epub Date: 2025-01-15DOI: 10.1016/j.matpur.2025.103657
Jean-Bernard Bru , Nathan Metraud
Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all times, for instance in the Schatten norm topology. This system presents remarkable ellipticity properties that turn out to be crucial for the study of the infinite-time limit of its solution, which is proven under relatively weak, albeit probably not necessary, hypotheses on the initial data. This system of differential equations is the elliptic counterpart of an hyperbolic flow applied to quantum field theory to diagonalize Hamiltonians that are quadratic in the bosonic field. In a similar way, this elliptic flow, in particular its asymptotics, has application in quantum field theory: it can be used to diagonalize Hamiltonians that are quadratic in the fermionic field while giving new explicit expressions and properties of these pivotal Hamiltonians of quantum field theory and quantum statistical mechanics.
{"title":"Non-linear operator-valued elliptic flows with application to quantum field theory","authors":"Jean-Bernard Bru , Nathan Metraud","doi":"10.1016/j.matpur.2025.103657","DOIUrl":"10.1016/j.matpur.2025.103657","url":null,"abstract":"<div><div>Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all times, for instance in the Schatten norm topology. This system presents remarkable ellipticity properties that turn out to be crucial for the study of the infinite-time limit of its solution, which is proven under relatively weak, albeit probably not necessary, hypotheses on the initial data. This system of differential equations is the elliptic counterpart of an hyperbolic flow applied to quantum field theory to diagonalize Hamiltonians that are quadratic in the bosonic field. In a similar way, this elliptic flow, in particular its asymptotics, has application in quantum field theory: it can be used to diagonalize Hamiltonians that are quadratic in the fermionic field while giving new explicit expressions and properties of these pivotal Hamiltonians of quantum field theory and quantum statistical mechanics.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103657"},"PeriodicalIF":2.1,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155106","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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