Pub Date : 2026-04-01Epub Date: 2026-02-12DOI: 10.1016/j.aim.2026.110861
Qingchun Ji , Jun Yao
We formulate a division problem for a class of overdetermined systems introduced by L. Hörmander, and establish an effective divisibility criterion. In addition, we prove a coherence theorem which extends Nadel's coherence theorem from complex structures to elliptic systems of partial differential equations.
{"title":"Formally integrable structures II. Division problem","authors":"Qingchun Ji , Jun Yao","doi":"10.1016/j.aim.2026.110861","DOIUrl":"10.1016/j.aim.2026.110861","url":null,"abstract":"<div><div>We formulate a division problem for a class of overdetermined systems introduced by L. Hörmander, and establish an effective divisibility criterion. In addition, we prove a coherence theorem which extends Nadel's coherence theorem from complex structures to elliptic systems of partial differential equations.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110861"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174671","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 : 2026-04-01Epub Date: 2026-02-04DOI: 10.1016/j.aim.2026.110798
William Graham, Scott Joseph Larson
We study the torus-equivariant cohomology of weighted flag varieties, and prove a positivity property in the equivariant cohomology and Chow groups of weighted flag varieties, analogous to the non-weighted positivity proved in [23]. Our result strengthens and generalizes the positivity proved for weighted Grassmannians by Abe-Matsumura [1]. The positivity property is expressed in terms of weighted roots, which are used to describe weights of torus equivariant curves in weighted flag varieties. This provides a geometric interpretation of the parameters used in [1]. We approach weighted flag varieties from a uniform Lie-theoretic point of view, providing a more general definition than has appeared previously, and prove other general results about weighted flag varieties in this setting, including a Borel presentation of the equivariant cohomology. In addition, we generalize some results obtained for weighted Grassmannians or more generally type A ([1], [6]); in particular, we obtain descriptions of restrictions to fixed points, the GKM description of the cohomology, and a weighted Chevalley formula.
{"title":"Positivity in weighted flag varieties","authors":"William Graham, Scott Joseph Larson","doi":"10.1016/j.aim.2026.110798","DOIUrl":"10.1016/j.aim.2026.110798","url":null,"abstract":"<div><div>We study the torus-equivariant cohomology of weighted flag varieties, and prove a positivity property in the equivariant cohomology and Chow groups of weighted flag varieties, analogous to the non-weighted positivity proved in <span><span>[23]</span></span>. Our result strengthens and generalizes the positivity proved for weighted Grassmannians by Abe-Matsumura <span><span>[1]</span></span>. The positivity property is expressed in terms of weighted roots, which are used to describe weights of torus equivariant curves in weighted flag varieties. This provides a geometric interpretation of the parameters used in <span><span>[1]</span></span>. We approach weighted flag varieties from a uniform Lie-theoretic point of view, providing a more general definition than has appeared previously, and prove other general results about weighted flag varieties in this setting, including a Borel presentation of the equivariant cohomology. In addition, we generalize some results obtained for weighted Grassmannians or more generally type <em>A</em> (<span><span>[1]</span></span>, <span><span>[6]</span></span>); in particular, we obtain descriptions of restrictions to fixed points, the GKM description of the cohomology, and a weighted Chevalley formula.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110798"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174138","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 : 2026-04-01Epub Date: 2026-02-03DOI: 10.1016/j.aim.2026.110831
Frederik Benirschke, Carlos A. Serván
We correct a mistake in the proof of the main theorem of “Isometric embeddings of Teichmüller spaces are covering constructions.” Importantly, the results are unchanged.
{"title":"Erratum to “Isometric embeddings of Teichmüller spaces are covering constructions”","authors":"Frederik Benirschke, Carlos A. Serván","doi":"10.1016/j.aim.2026.110831","DOIUrl":"10.1016/j.aim.2026.110831","url":null,"abstract":"<div><div>We correct a mistake in the proof of the main theorem of “Isometric embeddings of Teichmüller spaces are covering constructions.” Importantly, the results are unchanged.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110831"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174442","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 : 2026-04-01Epub Date: 2026-01-23DOI: 10.1016/j.aim.2026.110806
Utsav Choudhury, Biman Roy
In this article we prove that any -connected smooth k-variety is -uniruled for any algebraically closed field k. We establish that if a non-empty open subscheme X of a smooth affine k-scheme is -weakly equivalent to , then as k-varieties for any field k of characteristic 0.
{"title":"A1-homotopy type of A2∖{(0,0)}","authors":"Utsav Choudhury, Biman Roy","doi":"10.1016/j.aim.2026.110806","DOIUrl":"10.1016/j.aim.2026.110806","url":null,"abstract":"<div><div>In this article we prove that any <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-connected smooth <em>k</em>-variety is <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-uniruled for any algebraically closed field <em>k</em>. We establish that if a non-empty open subscheme <em>X</em> of a smooth affine <em>k</em>-scheme is <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-weakly equivalent to <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>∖</mo><mrow><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>}</mo></mrow></math></span>, then <span><math><mi>X</mi><mo>≅</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>∖</mo><mrow><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>}</mo></mrow></math></span> as <em>k</em>-varieties for any field <em>k</em> of characteristic 0.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110806"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025720","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 : 2026-04-01Epub Date: 2026-02-13DOI: 10.1016/j.aim.2026.110845
Alexey Cheskidov , Zirong Zeng , Deng Zhang
The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with continuous energy profiles to both the 3D deterministic and stochastic incompressible Navier-Stokes equations. In the stochastic case the constructed solutions are probabilistically strong.
Our proof introduces a new backward convex integration scheme with delicate selections of initial relaxed solutions, backward time intervals, and energy profiles. Our initial relaxed solutions satisfy a new time-dependent frequency truncated NSE, different from the usual approximations as it decreases the large Reynolds error near the initial time, which plays a key role in the construction.
{"title":"Existence and non-uniqueness of weak solutions with continuous energy to the 3D deterministic and stochastic Navier-Stokes equations","authors":"Alexey Cheskidov , Zirong Zeng , Deng Zhang","doi":"10.1016/j.aim.2026.110845","DOIUrl":"10.1016/j.aim.2026.110845","url":null,"abstract":"<div><div>The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with continuous energy profiles to both the 3D deterministic and stochastic incompressible Navier-Stokes equations. In the stochastic case the constructed solutions are probabilistically strong.</div><div>Our proof introduces a new backward convex integration scheme with delicate selections of initial relaxed solutions, backward time intervals, and energy profiles. Our initial relaxed solutions satisfy a new time-dependent frequency truncated NSE, different from the usual approximations as it decreases the large Reynolds error near the initial time, which plays a key role in the construction.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110845"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174617","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 : 2026-04-01Epub Date: 2026-01-27DOI: 10.1016/j.aim.2026.110802
Xenia Flamm
The main result of this article is that Hitchin representations over real closed field extensions of correspond precisely to those representations of the fundamental group of a closed surface into that are conjugate to -positive representations, i.e. representations that admit an equivariant limit map from the set of fixed points in the boundary of the universal cover of the surface into the set of full flags in satisfying specific positivity properties. As the theorem treats general real closed fields, and not only the reals, the tools of analysis are not available. Instead, our proof is based on the Tarski–Seidenberg transfer principle and a multiplicative version of the Bonahon–Dreyer coordinates.
We use this result to prove that -positive representations form semi-algebraically connected components of the space of all representations, that consist entirely of injective and discrete representations, which are positively hyperbolic and weakly dynamics-preserving over . Furthermore, we show how to associate intersection geodesic currents to -positive representations, and conclude with applications to the Weyl chamber length compactification and to dual spaces of geodesic currents.
{"title":"Real spectrum compactification of Hitchin components, Weyl chamber valued lengths, and dual spaces","authors":"Xenia Flamm","doi":"10.1016/j.aim.2026.110802","DOIUrl":"10.1016/j.aim.2026.110802","url":null,"abstract":"<div><div>The main result of this article is that Hitchin representations over real closed field extensions <span><math><mi>F</mi></math></span> of <span><math><mi>R</mi></math></span> correspond precisely to those representations of the fundamental group of a closed surface into <span><math><mtext>PSL</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> that are conjugate to <span><math><mi>F</mi></math></span>-positive representations, i.e. representations that admit an equivariant limit map from the set of fixed points in the boundary of the universal cover of the surface into the set of full flags in <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> satisfying specific positivity properties. As the theorem treats general real closed fields, and not only the reals, the tools of analysis are not available. Instead, our proof is based on the Tarski–Seidenberg transfer principle and a multiplicative version of the Bonahon–Dreyer coordinates.</div><div>We use this result to prove that <span><math><mi>F</mi></math></span>-positive representations form semi-algebraically connected components of the space of all representations, that consist entirely of injective and discrete representations, which are positively hyperbolic and weakly dynamics-preserving over <span><math><mi>F</mi></math></span>. Furthermore, we show how to associate intersection geodesic currents to <span><math><mi>F</mi></math></span>-positive representations, and conclude with applications to the Weyl chamber length compactification and to dual spaces of geodesic currents.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110802"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080758","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 : 2026-04-01Epub Date: 2026-02-04DOI: 10.1016/j.aim.2026.110818
Marco Abate, Ian Short
We unify and advance a host of works on iterated function systems of holomorphic self-maps of hyperbolic Riemann surfaces. Our foremost result is a generalisation to left iterated function systems of an unpublished and little known theorem of Heins on iteration in the unit disc. Applications abound – to work of Benini et al. on transcendental dynamics, to the theory of hyperbolic steps of holomorphic maps, and to left semiconjugacy in the unit disc. We extend other work of Benini et al. and Ferreira on relatively compact left iterated function systems, and we prove a hyperbolic distance inequality for holomorphic maps that generalises a theorem of Bracci, Kraus, and Roth. Additionally, we strengthen results of the first author and Christodoulou on left iterated function systems, removing the need for Bloch domains, and we answer an open question from their work. Finally, we establish a version of the Heins theorem for right iterated functions systems, and we generalise theorems of Beardon and Kuznetsov on right iterated function systems in relatively compact semigroups of holomorphic maps.
我们统一并提出了关于双曲黎曼曲面全纯自映射的迭代函数系统的大量工作。我们最重要的结果是将Heins关于单位圆盘上迭代的一个尚未发表且鲜为人知的定理推广到左迭代函数系统。应用广泛- Benini等人在先验动力学上的工作，全纯映射的双曲阶理论，以及单位圆盘上的左半共轭。我们推广了Benini et al.和Ferreira在相对紧的左迭代函数系统上的其他工作，并证明了全纯映射的双曲距离不等式，推广了Bracci， Kraus和Roth的定理。此外，我们加强了第一作者和Christodoulou关于左迭代函数系统的结果，消除了对Bloch域的需要，并回答了他们工作中的一个开放问题。最后，我们建立了关于右迭代函数系统的Heins定理的一个版本，并推广了关于全纯映射的相对紧半群上的右迭代函数系统的Beardon定理和Kuznetsov定理。
{"title":"Iterated function systems of holomorphic maps","authors":"Marco Abate, Ian Short","doi":"10.1016/j.aim.2026.110818","DOIUrl":"10.1016/j.aim.2026.110818","url":null,"abstract":"<div><div>We unify and advance a host of works on iterated function systems of holomorphic self-maps of hyperbolic Riemann surfaces. Our foremost result is a generalisation to left iterated function systems of an unpublished and little known theorem of Heins on iteration in the unit disc. Applications abound – to work of Benini et al. on transcendental dynamics, to the theory of hyperbolic steps of holomorphic maps, and to left semiconjugacy in the unit disc. We extend other work of Benini et al. and Ferreira on relatively compact left iterated function systems, and we prove a hyperbolic distance inequality for holomorphic maps that generalises a theorem of Bracci, Kraus, and Roth. Additionally, we strengthen results of the first author and Christodoulou on left iterated function systems, removing the need for Bloch domains, and we answer an open question from their work. Finally, we establish a version of the Heins theorem for right iterated functions systems, and we generalise theorems of Beardon and Kuznetsov on right iterated function systems in relatively compact semigroups of holomorphic maps.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110818"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174624","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 : 2026-04-01Epub Date: 2026-01-23DOI: 10.1016/j.aim.2026.110789
Alexis Michelat , Andrea Mondino
We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area be uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay within a compact domain of the moduli space.
{"title":"Quantization of the Willmore energy in Riemannian manifolds","authors":"Alexis Michelat , Andrea Mondino","doi":"10.1016/j.aim.2026.110789","DOIUrl":"10.1016/j.aim.2026.110789","url":null,"abstract":"<div><div>We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area be uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay within a compact domain of the moduli space.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110789"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025723","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 : 2026-04-01Epub Date: 2026-02-02DOI: 10.1016/j.aim.2026.110828
Elena Cordero , Gianluca Giacchi , Edoardo Pucci , S. Ivan Trapasso
Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this end, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest.
As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms.
{"title":"Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement","authors":"Elena Cordero , Gianluca Giacchi , Edoardo Pucci , S. Ivan Trapasso","doi":"10.1016/j.aim.2026.110828","DOIUrl":"10.1016/j.aim.2026.110828","url":null,"abstract":"<div><div>Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this end, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest.</div><div>As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110828"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174566","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 : 2026-04-01Epub Date: 2026-01-29DOI: 10.1016/j.aim.2026.110823
Andrii Ilienko , Ilya Molchanov , Tommaso Visonà
We obtain a complete characterization of planar monotone σ-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or σ-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.
{"title":"Integer-valued valuations","authors":"Andrii Ilienko , Ilya Molchanov , Tommaso Visonà","doi":"10.1016/j.aim.2026.110823","DOIUrl":"10.1016/j.aim.2026.110823","url":null,"abstract":"<div><div>We obtain a complete characterization of planar monotone <em>σ</em>-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or <em>σ</em>-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110823"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080759","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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