Journal of Pure and Applied Algebra最新文献

{"title":"On the tails of FI-modules","authors":"Peter Patzt,&nbsp;John D. Wiltshire-Gordon","doi":"10.1016/j.jpaa.2024.107741","DOIUrl":"10.1016/j.jpaa.2024.107741","url":null,"abstract":"<div><p>We study the end-behavior of integer-valued <span><math><mi>FI</mi></math></span>-modules. Our first result describes the high degrees of an <span><math><mi>FI</mi></math></span>-module in terms of newly defined tail invariants. Our main result provides an equivalence of categories between <span><math><mi>FI</mi></math></span>-tails and finitely supported modules for a new category that we call <span><math><mi>FJ</mi></math></span>. Objects of <span><math><mi>FJ</mi></math></span> are natural numbers, and morphisms are infinite series with summands drawn from certain modules of Lie brackets.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141278120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homotopy torsion theories","authors":"Sandra Mantovani ,&nbsp;Mariano Messora ,&nbsp;Enrico M. Vitale","doi":"10.1016/j.jpaa.2024.107742","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107742","url":null,"abstract":"<div><p>In the context of categories equipped with a structure of nullhomotopies, we introduce the notion of homotopy torsion theory. As special cases, we recover pretorsion theories as well as torsion theories in multi-pointed categories and in pre-pointed categories. Using the structure of nullhomotopies induced by the canonical string of adjunctions between a category <span><math><mi>A</mi></math></span> and the category <span><math><mrow><mi>Arr</mi></mrow><mo>(</mo><mi>A</mi><mo>)</mo></math></span> of arrows, we give a new proof of the correspondence between orthogonal factorization systems in <span><math><mi>A</mi></math></span> and homotopy torsion theories in <span><math><mrow><mi>Arr</mi></mrow><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, avoiding the request on the existence of pullbacks and pushouts in <span><math><mi>A</mi></math></span>. Moreover, such a correspondence is extended to weakly orthogonal factorization systems and weak homotopy torsion theories.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141313825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A semi-strictly generated closed structure on Gray-Cat","authors":"Adrian Miranda","doi":"10.1016/j.jpaa.2024.107740","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107740","url":null,"abstract":"<div><p>We show that the semi-strictly generated internal homs of <strong>Gray</strong>-categories <span><math><msub><mrow><mo>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>]</mo></mrow><mrow><mtext>ssg</mtext></mrow></msub></math></span> defined in <span>[19]</span> underlie a closed structure on the category <strong>Gray</strong>-<strong>Cat</strong> of <strong>Gray</strong>-categories and <strong>Gray</strong>-functors. The morphisms of <span><math><msub><mrow><mo>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>]</mo></mrow><mrow><mtext>ssg</mtext></mrow></msub></math></span> are composites of those trinatural transformations which satisfy the unit and composition conditions for pseudonatural transformations on the nose rather than up to an invertible 3-cell. Such trinatural transformations leverage three-dimensional strictification <span>[19]</span> while overcoming the challenges posed by failure of middle four interchange to hold in <strong>Gray</strong>-categories <span>[3]</span>. As a result we obtain a closed structure that is only partially monoidal with respect to <span>[8]</span>. As a corollary we obtain a slight strengthening of strictification results for braided monoidal bicategories <span>[13]</span>, which will be improved further in a forthcoming paper <span>[21]</span>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141249415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Toric Sylvester forms","authors":"Laurent Busé ,&nbsp;Carles Checa","doi":"10.1016/j.jpaa.2024.107739","DOIUrl":"10.1016/j.jpaa.2024.107739","url":null,"abstract":"<div><p>In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety <em>X</em> with respect to the irrelevant ideal of <em>X</em>. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on <em>X</em>. In particular, we prove that toric Sylvester forms yield bases of some graded components of <span><math><msup><mrow><mi>I</mi></mrow><mrow><mtext>sat</mtext></mrow></msup><mo>/</mo><mi>I</mi></math></span>, where <em>I</em> denotes an ideal generated by <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> generic forms, <em>n</em> is the dimension of <em>X</em> and <span><math><msup><mrow><mi>I</mi></mrow><mrow><mtext>sat</mtext></mrow></msup></math></span> is the saturation of <em>I</em> with respect to the irrelevant ideal of the Cox ring of <em>X</em>. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explore the computation of the toric residue of the product of two forms.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modular Virasoro vertex operator algebras with c=12","authors":"Chongying Dong ,&nbsp;Ching Hung Lam ,&nbsp;Li Ren","doi":"10.1016/j.jpaa.2024.107736","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107736","url":null,"abstract":"<div><p>Using a <span><math><mi>Z</mi><mo>[</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo></math></span>-form of Virasoro vertex operator algebra <span><math><mi>L</mi><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></math></span> with central charge <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, we obtain a modular vertex operator algebra over any field <span><math><mi>F</mi></math></span> of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141239691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Construction of even-level representations of SL2(o) with residue field of characteristic two","authors":"M. Hassain","doi":"10.1016/j.jpaa.2024.107737","DOIUrl":"10.1016/j.jpaa.2024.107737","url":null,"abstract":"<div><p>We construct the finite-dimensional continuous complex representations of <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> over compact discrete valuation rings of even residual characteristic, assuming the level is large enough compared to the ramification index, in the mixed characteristic case. We also prove that the complex group algebras of <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> over finite quotient rings of such compact discrete valuation rings depend on the characteristic of the ring. In particular, we prove that the group algebras <span><math><mi>C</mi><mo>[</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mi>Z</mi><mo>)</mo><mo>]</mo></math></span> and <span><math><mi>C</mi><mo>[</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo><mo>/</mo><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>)</mo><mo>]</mo></math></span> are not isomorphic for any <span><math><mi>r</mi><mo>≥</mo><mn>4</mn></math></span>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Differentiably simple rings and ring extensions defined by p-basis","authors":"Celia del Buey de Andrés ,&nbsp;Diego Sulca ,&nbsp;Orlando E. Villamayor","doi":"10.1016/j.jpaa.2024.107735","DOIUrl":"10.1016/j.jpaa.2024.107735","url":null,"abstract":"<div><p>We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit <em>p</em>-basis. These extensions are called <em>Galois extensions of exponent one</em>. For such an extension <span><math><mi>A</mi><mo>⊂</mo><mi>C</mi></math></span>, we introduce an <em>A</em>-scheme, called the <em>Yuan scheme</em>, which parametrizes subextensions <span><math><mi>A</mi><mo>⊂</mo><mi>B</mi><mo>⊂</mo><mi>C</mi></math></span> such that <span><math><mi>B</mi><mo>⊂</mo><mi>C</mi></math></span> is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"G-semisimple algebras","authors":"Rasool Hafezi ,&nbsp;Abdolnaser Bahlekeh","doi":"10.1016/j.jpaa.2024.107738","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107738","url":null,"abstract":"<div><p>Let Λ be an Artin algebra and <span><math><mrow><mi>mod</mi></mrow><mtext>-</mtext><mo>(</mo><munder><mrow><mrow><mi>Gprj</mi></mrow></mrow><mo>_</mo></munder><mtext>-</mtext><mi>Λ</mi><mo>)</mo></math></span> the category of finitely presented functors over the stable category <span><math><munder><mrow><mrow><mi>Gprj</mi></mrow></mrow><mo>_</mo></munder><mtext>-</mtext><mi>Λ</mi></math></span> of finitely generated Gorenstein projective Λ-modules. This paper deals with those algebras Λ in which <span><math><mrow><mi>mod</mi></mrow><mtext>-</mtext><mo>(</mo><munder><mrow><mrow><mi>Gprj</mi></mrow></mrow><mo>_</mo></munder><mtext>-</mtext><mi>Λ</mi><mo>)</mo></math></span> is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence (called representation equivalence in the terminology of Auslander), i.e. a full and dense functor that reflects isomorphisms, from the stable category of Gorenstein projective representations <span><math><munder><mrow><mrow><mi>Gprj</mi></mrow></mrow><mo>_</mo></munder><mo>(</mo><mi>Q</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></math></span> of a finite acyclic quiver <span><math><mi>Q</mi></math></span> to the category of representations <span><math><mrow><mi>rep</mi></mrow><mo>(</mo><mi>Q</mi><mo>,</mo><munder><mrow><mrow><mi>Gprj</mi></mrow></mrow><mo>_</mo></munder><mtext>-</mtext><mi>Λ</mi><mo>)</mo></math></span> over <span><math><munder><mrow><mrow><mi>Gprj</mi></mrow></mrow><mo>_</mo></munder><mtext>-</mtext><mi>Λ</mi></math></span>, provided Λ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra <span><math><mi>Λ</mi><mi>Q</mi></math></span> of the G-semisimple algebra Λ is <span><math><mi>CM</mi></math></span>-finite if and only if <span><math><mi>Q</mi></math></span> is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within <span><math><mrow><mi>Gprj</mi></mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>Λ</mi><mo>)</mo></math></span> of the linear quiver <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> over a G-semisimple algebra Λ. We also determine almost split sequences in <span><math><mrow><mi>Gprj</mi></mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>Λ</mi><mo>)</mo></math></span> with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver <span><math><mrow><mi>Gprj</mi></mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>Λ</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141263782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Power-closed ideals of polynomial and Laurent polynomial rings","authors":"Geir Agnarsson,&nbsp;Jim Lawrence","doi":"10.1016/j.jpaa.2024.107733","DOIUrl":"10.1016/j.jpaa.2024.107733","url":null,"abstract":"<div><p>We investigate the structure of power-closed ideals of the complex polynomial ring <span><math><mi>R</mi><mo>=</mo><mrow><mi>C</mi></mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>]</mo></math></span> and the Laurent polynomial ring <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>±</mo></mrow></msup><mo>=</mo><mrow><mi>C</mi></mrow><msup><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>]</mo></mrow><mrow><mo>±</mo></mrow></msup><mo>=</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mi>C</mi></mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>]</mo></math></span>, where <em>S</em> is the multiplicatively closed semigroup <span><math><mi>S</mi><mo>=</mo><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>]</mo></math></span>. Here, an ideal <em>I</em> is <em>power-closed</em> if <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>I</mi></math></span> implies <span><math><mi>f</mi><mo>(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>i</mi></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>)</mo><mo>∈</mo><mi>I</mi></math></span> for each natural number <em>i</em>. Important examples of such ideals are provided by the ideals of relations in Minkowski rings of convex polytopes. We investigate related closure and interior operators on the set of ideals of <em>R</em> and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>±</mo></mrow></msup></math></span> and we give a complete description of principal power-closed ideals and of radicals of general power-closed ideals of <em>R</em> and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>±</mo></mrow></msup></math></span>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141136971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Partial monoid actions on objects in categories with pullbacks and their globalizations","authors":"Mykola Khrypchenko ,&nbsp;Francisco Klock","doi":"10.1016/j.jpaa.2024.107734","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107734","url":null,"abstract":"<div><p>Let <em>M</em> be a monoid, <span><math><mi>C</mi></math></span> a category with pullbacks and <em>X</em> an object of <span><math><mi>C</mi></math></span>. We introduce the notion of a partial action <em>α</em> of <em>M</em> on <em>X</em> and study the globalization question for <em>α</em>. If <em>α</em> admits a reflection in the subcategory of global actions, then we reduce the problem to the verification that a certain diagram is a pullback in <span><math><mi>C</mi></math></span>. We then give a construction of such a reflection in terms of a colimit of a certain functor with values in <span><math><mi>C</mi></math></span>. We specify this construction to the case of categories admitting certain coproducts and coequalizers.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141090638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}