Pub Date : 2024-11-26DOI: 10.1016/j.aim.2024.110051
Satoshi Naito , Daisuke Sagaki
The purpose of this paper is to prove a Pieri-type multiplication formula, conjectured by Lenart-Maeno, for quantum Grothendieck polynomials. This formula would enable us to compute explicitly the quantum product of two arbitrary (opposite) Schubert classes in the (small) quantum K-theory ring of the (full) flag manifold of type based on the fact that quantum Grothendieck polynomials represent (opposite) Schubert classes in .
{"title":"Pieri-type multiplication formula for quantum Grothendieck polynomials","authors":"Satoshi Naito , Daisuke Sagaki","doi":"10.1016/j.aim.2024.110051","DOIUrl":"10.1016/j.aim.2024.110051","url":null,"abstract":"<div><div>The purpose of this paper is to prove a Pieri-type multiplication formula, conjectured by Lenart-Maeno, for quantum Grothendieck polynomials. This formula would enable us to compute explicitly the quantum product of two arbitrary (opposite) Schubert classes in the (small) quantum <em>K</em>-theory ring <span><math><mi>Q</mi><mi>K</mi><mo>(</mo><mi>F</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> of the (full) flag manifold <span><math><mi>F</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> based on the fact that quantum Grothendieck polynomials represent (opposite) Schubert classes in <span><math><mi>Q</mi><mi>K</mi><mo>(</mo><mi>F</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110051"},"PeriodicalIF":1.5,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723200","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 : 2024-11-26DOI: 10.1016/j.aim.2024.110032
Yinhe Peng
We introduce an iteration of forcing notions satisfying the countable chain condition with minimal damage to a strong coloring. Applying this method, we prove that Martin's axiom is strictly stronger than its restriction to forcing notions satisfying the countable chain condition in all finite powers. Our method shows also the finer distinction, that Martin's axiom is strictly stronger than its restriction to forcing notions whose squares satisfy the countable chain condition.
{"title":"Distinguishing Martin's axiom from its restrictions","authors":"Yinhe Peng","doi":"10.1016/j.aim.2024.110032","DOIUrl":"10.1016/j.aim.2024.110032","url":null,"abstract":"<div><div>We introduce an iteration of forcing notions satisfying the countable chain condition with minimal damage to a strong coloring. Applying this method, we prove that Martin's axiom is strictly stronger than its restriction to forcing notions satisfying the countable chain condition in all finite powers. Our method shows also the finer distinction, that Martin's axiom is strictly stronger than its restriction to forcing notions whose squares satisfy the countable chain condition.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110032"},"PeriodicalIF":1.5,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723199","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 : 2024-11-26DOI: 10.1016/j.aim.2024.110036
Nir Lev , Anton Tselishchev
It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space for any . To the contrary, there do exist unconditional Schauder frames of translates in for every . The existence of such a system for , however, has remained an open problem. In this paper the problem is solved in the negative: we prove that none of the spaces , , admits an unconditional Schauder frame of translates.
{"title":"There are no unconditional Schauder frames of translates in Lp(R), 1 ⩽ p ⩽ 2","authors":"Nir Lev , Anton Tselishchev","doi":"10.1016/j.aim.2024.110036","DOIUrl":"10.1016/j.aim.2024.110036","url":null,"abstract":"<div><div>It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for any <span><math><mn>1</mn><mo>⩽</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span>. To the contrary, there do exist unconditional Schauder frames of translates in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for every <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span>. The existence of such a system for <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo>⩽</mo><mn>2</mn></math></span>, however, has remained an open problem. In this paper the problem is solved in the negative: we prove that none of the spaces <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, <span><math><mn>1</mn><mo>⩽</mo><mi>p</mi><mo>⩽</mo><mn>2</mn></math></span>, admits an unconditional Schauder frame of translates.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110036"},"PeriodicalIF":1.5,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723189","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 : 2024-11-26DOI: 10.1016/j.aim.2024.110050
Atsushi Kanazawa
The present article is concerned with mirror symmetry for generalized K3 surfaces, with particular emphasis on complex and Kähler rigid structures. Inspired by the works of Dolgachev, Aspinwall–Morrison and Huybrechts, we introduce a formulation of mirror symmetry for generalized K3 surfaces by using Mukai lattice polarizations. This approach solves issues in the conventional formulations of mirror symmetry for K3 surfaces. In particular, we provide a solution to the problem of mirror symmetry for singular K3 surfaces. Along the way, we investigate complex and Kähler rigid structures of generalized K3 surfaces.
{"title":"Mirror symmetry and rigid structures of generalized K3 surfaces","authors":"Atsushi Kanazawa","doi":"10.1016/j.aim.2024.110050","DOIUrl":"10.1016/j.aim.2024.110050","url":null,"abstract":"<div><div>The present article is concerned with mirror symmetry for generalized K3 surfaces, with particular emphasis on complex and Kähler rigid structures. Inspired by the works of Dolgachev, Aspinwall–Morrison and Huybrechts, we introduce a formulation of mirror symmetry for generalized K3 surfaces by using Mukai lattice polarizations. This approach solves issues in the conventional formulations of mirror symmetry for K3 surfaces. In particular, we provide a solution to the problem of mirror symmetry for singular K3 surfaces. Along the way, we investigate complex and Kähler rigid structures of generalized K3 surfaces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110050"},"PeriodicalIF":1.5,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723201","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 : 2024-11-22DOI: 10.1016/j.aim.2024.110020
Paweł Goldstein , Zofia Grochulska , Piotr Hajłasz
We prove that for any measurable mapping T into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals T outside a set of measure less than ε. We use this fact to prove that for any measurable mapping T into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals T almost everywhere.
我们利用这一事实证明，对于进入具有正行列式（无符号限制）的矩阵空间的任何可测映射 T，存在一个在度量小于 ε 的集合外导数等于 T 的差分同构。
{"title":"Constructing diffeomorphisms and homeomorphisms with prescribed derivative","authors":"Paweł Goldstein , Zofia Grochulska , Piotr Hajłasz","doi":"10.1016/j.aim.2024.110020","DOIUrl":"10.1016/j.aim.2024.110020","url":null,"abstract":"<div><div>We prove that for any measurable mapping <em>T</em> into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals <em>T</em> outside a set of measure less than <em>ε</em>. We use this fact to prove that for any measurable mapping <em>T</em> into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals <em>T</em> almost everywhere.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110020"},"PeriodicalIF":1.5,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704787","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 : 2024-11-22DOI: 10.1016/j.aim.2024.110033
François Bacher
Let be a Brody-hyperbolic singular holomorphic foliation on a compact complex manifold M. Suppose that has isolated singularities and that its Poincaré metric is complete. This is the case for a very large class of singularities, namely, non-degenerate and saddle-nodes in dimension 2. Let μ be an ergodic harmonic measure on . We show that the upper and lower local hyperbolic entropies of μ are leafwise constant almost everywhere. Moreover, we show that the entropy of μ is at least 2.
{"title":"Hyperbolic entropy for harmonic measures on singular holomorphic foliations","authors":"François Bacher","doi":"10.1016/j.aim.2024.110033","DOIUrl":"10.1016/j.aim.2024.110033","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi><mo>=</mo><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>L</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math></span> be a Brody-hyperbolic singular holomorphic foliation on a compact complex manifold <em>M</em>. Suppose that <span><math><mi>F</mi></math></span> has isolated singularities and that its Poincaré metric is complete. This is the case for a very large class of singularities, namely, non-degenerate and saddle-nodes in dimension 2. Let <em>μ</em> be an ergodic harmonic measure on <span><math><mi>F</mi></math></span>. We show that the upper and lower local hyperbolic entropies of <em>μ</em> are leafwise constant almost everywhere. Moreover, we show that the entropy of <em>μ</em> is at least 2.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110033"},"PeriodicalIF":1.5,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704786","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 : 2024-11-21DOI: 10.1016/j.aim.2024.110022
León Carvajales , Florian Stecker
We construct open domains of discontinuity for Anosov representations acting on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces. This generalizes work of Kapovich-Leeb-Porti on flag spaces. Our results complement those of Guéritaud-Guichard-Kassel-Wienhard, who constructed proper actions of Anosov representations. For Zariski dense Anosov representations with respect to a minimal parabolic subgroup acting on some symmetric spaces, we show that our construction describes the largest possible open domains of discontinuity.
{"title":"Anosov representations acting on homogeneous spaces: Domains of discontinuity","authors":"León Carvajales , Florian Stecker","doi":"10.1016/j.aim.2024.110022","DOIUrl":"10.1016/j.aim.2024.110022","url":null,"abstract":"<div><div>We construct open domains of discontinuity for Anosov representations acting on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces. This generalizes work of Kapovich-Leeb-Porti on flag spaces. Our results complement those of Guéritaud-Guichard-Kassel-Wienhard, who constructed proper actions of Anosov representations. For Zariski dense Anosov representations with respect to a minimal parabolic subgroup acting on some symmetric spaces, we show that our construction describes the largest possible open domains of discontinuity.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110022"},"PeriodicalIF":1.5,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698438","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 : 2024-11-21DOI: 10.1016/j.aim.2024.110027
Alejandro Poveda
We produce a model where every supercompact cardinal is -supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in [25] and provides a definitive answer to a question of Bagaria [7, p. 19]. This configuration is a consequence of a new axiom we introduce –called – which is showed to be compatible with Woodin's cardinals. We also answer a question of V. Gitman and G. Goldberg on the relationship between supercompactness and cardinal-preserving extendibility. As an incidental result, we prove a theorem suggesting that supercompactness is the strongest large-cardinal notion preserved by Radin forcing.
我们建立了一个模型，在这个模型中，每一个超紧密红心都是 C(1)超紧密的，都有不可访问的目标。这是对[25]中得到的主要身份脆性配置的重大改进，并为巴加利亚的一个问题[7，第 19 页]提供了明确的答案。这个配置是我们引入的一个新公理--称为 A--的结果，它被证明与伍丁的 I0 红心相容。我们还回答了 V. Gitman 和 G. Goldberg 提出的关于超紧密性与保全红心可扩展性之间关系的问题。作为一个附带结果，我们证明了一个定理，表明超紧密性是由拉丁强制所保留的最强的大红心概念。
{"title":"Axiom A and supercompactness","authors":"Alejandro Poveda","doi":"10.1016/j.aim.2024.110027","DOIUrl":"10.1016/j.aim.2024.110027","url":null,"abstract":"<div><div>We produce a model where every supercompact cardinal is <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in <span><span>[25]</span></span> and provides a definitive answer to a question of Bagaria <span><span>[7, p. 19]</span></span>. This configuration is a consequence of a new axiom we introduce –called <span><math><mi>A</mi></math></span>– which is showed to be compatible with Woodin's <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> cardinals. We also answer a question of V. Gitman and G. Goldberg on the relationship between supercompactness and cardinal-preserving extendibility. As an incidental result, we prove a theorem suggesting that supercompactness is the strongest large-cardinal notion preserved by Radin forcing.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110027"},"PeriodicalIF":1.5,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698437","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 : 2024-11-21DOI: 10.1016/j.aim.2024.110028
Jiayin Pan , Zhu Ye
We study the rigidity problems for open (complete and noncompact) n-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of M properly contains a Euclidean , then the first Betti number of M is at most ; moreover, if equality holds, then M is flat. Next, we study the geometry of the orbit , where acts on the universal cover . Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of . We also give the first example of a manifold M of and but with a varying orbit growth order.
我们研究了具有非负里奇曲率的开放（完整和非紧凑）n-形的刚性问题。我们证明，如果 M 的渐近锥恰当地包含欧几里得 Rk-1，那么 M 的第一个贝蒂数至多为 n-k；此外，如果相等成立，那么 M 是平坦的。接下来，我们研究轨道Γp˜的几何，其中Γ=π1(M,p) 作用于普遍盖（M˜,p˜）。在类似的渐近条件下，我们证明了Γp˜ 增长阶的几何刚性。我们还给出了第一个 Ric>0 且 π1(M)=Z，但轨道增长阶变化的流形 M 的例子。
{"title":"Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity","authors":"Jiayin Pan , Zhu Ye","doi":"10.1016/j.aim.2024.110028","DOIUrl":"10.1016/j.aim.2024.110028","url":null,"abstract":"<div><div>We study the rigidity problems for open (complete and noncompact) <em>n</em>-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of <em>M</em> properly contains a Euclidean <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, then the first Betti number of <em>M</em> is at most <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span>; moreover, if equality holds, then <em>M</em> is flat. Next, we study the geometry of the orbit <span><math><mi>Γ</mi><mover><mrow><mi>p</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>, where <span><math><mi>Γ</mi><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> acts on the universal cover <span><math><mo>(</mo><mover><mrow><mi>M</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>,</mo><mover><mrow><mi>p</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo></math></span>. Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of <span><math><mi>Γ</mi><mover><mrow><mi>p</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>. We also give the first example of a manifold <em>M</em> of <span><math><mrow><mi>Ric</mi></mrow><mo>></mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo><mo>=</mo><mi>Z</mi></math></span> but with a varying orbit growth order.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110028"},"PeriodicalIF":1.5,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698436","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 : 2024-11-20DOI: 10.1016/j.aim.2024.110019
Qiyue Chen , Travis Mandel , Fan Qin
Given a covering of a quiver (with potential), we show that the associated Bridgeland stability scattering diagrams are related by a restriction operation under the assumption of admitting a nice grading. We apply this to quivers with potential associated to marked surfaces. In combination with recent results of the second and third authors, our findings imply that the bracelets basis for a once-punctured closed surface coincides with the theta basis for the associated stability scattering diagram, and these stability scattering diagrams agree with the corresponding cluster scattering diagrams of Gross-Hacking-Keel-Kontsevich except in the case of the once-punctured torus.
{"title":"Stability scattering diagrams and quiver coverings","authors":"Qiyue Chen , Travis Mandel , Fan Qin","doi":"10.1016/j.aim.2024.110019","DOIUrl":"10.1016/j.aim.2024.110019","url":null,"abstract":"<div><div>Given a covering of a quiver (with potential), we show that the associated Bridgeland stability scattering diagrams are related by a restriction operation under the assumption of admitting a nice grading. We apply this to quivers with potential associated to marked surfaces. In combination with recent results of the second and third authors, our findings imply that the bracelets basis for a once-punctured closed surface coincides with the theta basis for the associated stability scattering diagram, and these stability scattering diagrams agree with the corresponding cluster scattering diagrams of Gross-Hacking-Keel-Kontsevich except in the case of the once-punctured torus.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110019"},"PeriodicalIF":1.5,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698441","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号