We study the geometry of spaces of fitrations on a Noetherian local domain. We introduce a metric $d_1$ on the space of saturated filtrations, inspired by the Darvas metric in complex geometry, such that it is a geodesic metric space. In the toric case, using Newton-Okounkov bodies, we identify the space of saturated monomial filtrations with a subspace of $L^1_mathrm{loc}$. We also consider several other topologies on such spaces and study the semi-continuity of the log canonical threshold function in the spirit of Koll'ar-Demailly. Moreover, there is a natural lattice structure on the space of saturated filtrations, which is a generalization of the classical result that the ideals of a ring form a lattice.
{"title":"On the geometry of spaces of filtrations on local rings","authors":"Lu Qi","doi":"arxiv-2409.01705","DOIUrl":"https://doi.org/arxiv-2409.01705","url":null,"abstract":"We study the geometry of spaces of fitrations on a Noetherian local domain.\u0000We introduce a metric $d_1$ on the space of saturated filtrations, inspired by\u0000the Darvas metric in complex geometry, such that it is a geodesic metric space.\u0000In the toric case, using Newton-Okounkov bodies, we identify the space of\u0000saturated monomial filtrations with a subspace of $L^1_mathrm{loc}$. We also\u0000consider several other topologies on such spaces and study the semi-continuity\u0000of the log canonical threshold function in the spirit of Koll'ar-Demailly.\u0000Moreover, there is a natural lattice structure on the space of saturated\u0000filtrations, which is a generalization of the classical result that the ideals\u0000of a ring form a lattice.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"139 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce some new graded Lie algebras associated with a Hom-Lie algebra. At first, we define the cup product bracket and its application to the deformation theory of Hom-Lie algebra morphisms. We observe an action of the well-known Hom-analogue of the Nijenhuis-Richardson graded Lie algebra on the cup product graded Lie algebra. Using the corresponding semidirect product, we define the Fr"{o}licher-Nijenhuis bracket and study its application to Nijenhuis operators. We show that the Nijenhuis-Richardson graded Lie algebra and the Fr"{o}licher-Nijenhuis algebra constitute a matched pair of graded Lie algebras. Finally, we define another graded Lie bracket, called the derived bracket that is useful to study Rota-Baxter operators on Hom-Lie algebras.
{"title":"Cup product, Frölicher-Nijenhuis bracket and the derived bracket associated to Hom-Lie algebras","authors":"Anusuiya Baishya, Apurba Das","doi":"arxiv-2409.01865","DOIUrl":"https://doi.org/arxiv-2409.01865","url":null,"abstract":"In this paper, we introduce some new graded Lie algebras associated with a\u0000Hom-Lie algebra. At first, we define the cup product bracket and its\u0000application to the deformation theory of Hom-Lie algebra morphisms. We observe\u0000an action of the well-known Hom-analogue of the Nijenhuis-Richardson graded Lie\u0000algebra on the cup product graded Lie algebra. Using the corresponding\u0000semidirect product, we define the Fr\"{o}licher-Nijenhuis bracket and study its\u0000application to Nijenhuis operators. We show that the Nijenhuis-Richardson\u0000graded Lie algebra and the Fr\"{o}licher-Nijenhuis algebra constitute a matched\u0000pair of graded Lie algebras. Finally, we define another graded Lie bracket,\u0000called the derived bracket that is useful to study Rota-Baxter operators on\u0000Hom-Lie algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $F$ be a field. Following the resolution of Milnor's conjecture relating the graded Witt ring of $F$ to its mod-2 Milnor $K$-theory, a major problem in the theory of symmetric bilinear forms is to understand, for any positive integer $n$, the low-dimensional part of $I^n(F)$, the $n$th power of the fundamental ideal in the Witt ring of $F$. In a 2004 paper, Karpenko used methods from the theory of algebraic cycles to show that if $mathfrak{b}$ is a non-zero anisotropic symmetric bilinear form of dimension $< 2^{n+1}$ representing an element of $I^n(F)$, then $mathfrak{b}$ has dimension $2^{n+1} - 2^i$ for some $1 leq i leq n$. When $i = n$, a classical result of Arason and Pfister says that $mathfrak{b}$ is similar to an $n$-fold Pfister form. At the next level, it has been conjectured that if $n geq 2$ and $i= n-1$, then $mathfrak{b}$ is isometric to the tensor product of an $(n-2)$-fold Pfister form and a $6$-dimensional form of trivial discriminant. This has only been shown to be true, however, when $n = 2$, or when $n = 3$ and $mathrm{char}(F) neq 2$ (another result of Pfister). In the present article, we prove the conjecture for all values of $n$ in the case where $mathrm{char}(F) =2$. In addition, we give a short and elementary proof of Karpenko's theorem in the characteristic-2 case, rendering it free from the use of subtle algebraic-geometric tools. Finally, we consider the question of whether additional dimension gaps can appear among the anisotropic forms of dimension $geq 2^{n+1}$ representing an element of $I^n(F)$. When $mathrm{char}(F) neq 2$, a result of Vishik asserts that there are no such gaps, but the situation seems to be less clear when $mathrm{char}(F) = 2$.
{"title":"On the holes in $I^n$ for symmetric bilinear forms in characteristic 2","authors":"Stephen Scully","doi":"arxiv-2409.02061","DOIUrl":"https://doi.org/arxiv-2409.02061","url":null,"abstract":"Let $F$ be a field. Following the resolution of Milnor's conjecture relating\u0000the graded Witt ring of $F$ to its mod-2 Milnor $K$-theory, a major problem in\u0000the theory of symmetric bilinear forms is to understand, for any positive\u0000integer $n$, the low-dimensional part of $I^n(F)$, the $n$th power of the\u0000fundamental ideal in the Witt ring of $F$. In a 2004 paper, Karpenko used\u0000methods from the theory of algebraic cycles to show that if $mathfrak{b}$ is a\u0000non-zero anisotropic symmetric bilinear form of dimension $< 2^{n+1}$\u0000representing an element of $I^n(F)$, then $mathfrak{b}$ has dimension $2^{n+1}\u0000- 2^i$ for some $1 leq i leq n$. When $i = n$, a classical result of Arason\u0000and Pfister says that $mathfrak{b}$ is similar to an $n$-fold Pfister form. At\u0000the next level, it has been conjectured that if $n geq 2$ and $i= n-1$, then\u0000$mathfrak{b}$ is isometric to the tensor product of an $(n-2)$-fold Pfister\u0000form and a $6$-dimensional form of trivial discriminant. This has only been\u0000shown to be true, however, when $n = 2$, or when $n = 3$ and $mathrm{char}(F)\u0000neq 2$ (another result of Pfister). In the present article, we prove the\u0000conjecture for all values of $n$ in the case where $mathrm{char}(F) =2$. In\u0000addition, we give a short and elementary proof of Karpenko's theorem in the\u0000characteristic-2 case, rendering it free from the use of subtle\u0000algebraic-geometric tools. Finally, we consider the question of whether\u0000additional dimension gaps can appear among the anisotropic forms of dimension\u0000$geq 2^{n+1}$ representing an element of $I^n(F)$. When $mathrm{char}(F) neq\u00002$, a result of Vishik asserts that there are no such gaps, but the situation\u0000seems to be less clear when $mathrm{char}(F) = 2$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A conjecture due to Y. Han asks whether that Hochschild homology groups of a finite dimensional algebra vanish for sufficiently large degrees would imply that the algebra is of finite global dimension. We investigate this conjecture from the viewpoint of recollements of derived categories. It is shown that for a recollement of unbounded derived categories of rings which extends downwards (or upwards) one step, Han's conjecture holds for the ring in the middle if and only if it holds for the two rings on the two sides and hence Han's conjecture is reduced to derived $2$-simple rings. Furthermore, this reduction result is applied to Han's conjecture for Morita contexts rings and exact contexts. Finally it is proved that Han's conjecture holds for skew-gentle algebras, category algebras of finite EI categories and Geiss-Leclerc-Schr"{o}er algebras associated to Cartan triples.
Y. Han 提出的一个猜想是，无限维代数的霍希契尔德同调群在足够大的度数下消失是否意味着该代数是有限全维的。我们从派生类的重组的角度研究了这一猜想。结果表明，对于向下（或向上）延伸一步的无界派生类环的重组，如果且只有当韩氏猜想对两边的两个环成立时，韩氏猜想才对中间的环成立，因此韩氏猜想被还原为派生的 2 美元简单环。此外，这一还原结果也适用于莫里塔上下文环和精确上下文的韩氏猜想。最后，证明了韩氏猜想对于与卡坦三元组相关联的斜温和代数、有限EI范畴的范畴代数和Geiss-Leclerc-Schr"{o}er代数是成立的。
{"title":"A recollement approach to Han's conjecture","authors":"Ren Wang, Xiaoxiao Xu, Jinbi Zhang, Guodong Zhou","doi":"arxiv-2409.00945","DOIUrl":"https://doi.org/arxiv-2409.00945","url":null,"abstract":"A conjecture due to Y. Han asks whether that Hochschild homology groups of a\u0000finite dimensional algebra vanish for sufficiently large degrees would imply\u0000that the algebra is of finite global dimension. We investigate this conjecture\u0000from the viewpoint of recollements of derived categories. It is shown that for\u0000a recollement of unbounded derived categories of rings which extends downwards\u0000(or upwards) one step, Han's conjecture holds for the ring in the middle if and\u0000only if it holds for the two rings on the two sides and hence Han's conjecture is reduced to derived $2$-simple rings. Furthermore, this\u0000reduction result is applied to Han's conjecture for Morita contexts rings and\u0000exact contexts. Finally it is proved that Han's conjecture holds for\u0000skew-gentle algebras, category algebras of finite EI categories and\u0000Geiss-Leclerc-Schr\"{o}er algebras associated to Cartan triples.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Clausen--Scholze introduced the notion of solid spectrum in their condensed mathematics program. We demonstrate that the solidification of algebraic $K$-theory recovers two known constructions: the semitopological $K$-theory of a real (associative) algebra and the topological (aka operator) $K$-theory of a real Banach algebra.
{"title":"(Semi)topological $K$-theory via solidification","authors":"Ko Aoki","doi":"arxiv-2409.01462","DOIUrl":"https://doi.org/arxiv-2409.01462","url":null,"abstract":"Clausen--Scholze introduced the notion of solid spectrum in their condensed\u0000mathematics program. We demonstrate that the solidification of algebraic\u0000$K$-theory recovers two known constructions: the semitopological $K$-theory of\u0000a real (associative) algebra and the topological (aka operator) $K$-theory of a\u0000real Banach algebra.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop the bialgebra theory for two classes of non-associative algebras: nearly associative algebras and $LR$-algebras. In particular, building on recent studies that reveal connections between these algebraic structures, we establish that nearly associative bialgebras and $LR$-bialgebras are, in fact, equivalent concepts. We also provide a characterization of these bialgebra classes based on the coproduct. Moreover, since the development of nearly associative bialgebras - and by extension, $LR$-bialgebras - requires the framework of nearly associative $L$-algebras, we introduce this class of non-associative algebras and explore their fundamental properties. Furthermore, we identify and characterize a special class of nearly associative bialgebras, the coboundary nearly associative bialgebras, which provides a natural framework for studying the Yang-Baxter equation (YBE) within this context.
{"title":"Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation","authors":"Elisabete Barreiro, Saïd Benayadi, Carla Rizzo","doi":"arxiv-2409.00390","DOIUrl":"https://doi.org/arxiv-2409.00390","url":null,"abstract":"We develop the bialgebra theory for two classes of non-associative algebras:\u0000nearly associative algebras and $LR$-algebras. In particular, building on\u0000recent studies that reveal connections between these algebraic structures, we\u0000establish that nearly associative bialgebras and $LR$-bialgebras are, in fact,\u0000equivalent concepts. We also provide a characterization of these bialgebra\u0000classes based on the coproduct. Moreover, since the development of nearly\u0000associative bialgebras - and by extension, $LR$-bialgebras - requires the\u0000framework of nearly associative $L$-algebras, we introduce this class of\u0000non-associative algebras and explore their fundamental properties. Furthermore,\u0000we identify and characterize a special class of nearly associative bialgebras,\u0000the coboundary nearly associative bialgebras, which provides a natural\u0000framework for studying the Yang-Baxter equation (YBE) within this context.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A quasi-twilled associative algebra is an associative algebra $mathbb{A}$ whose underlying vector space has a decomposition $mathbb{A} = A oplus B$ such that $B subset mathbb{A}$ is a subalgebra. In the first part of this paper, we give the Maurer-Cartan characterization and introduce the cohomology of a quasi-twilled associative algebra. In a quasi-twilled associative algebra $mathbb{A}$, a linear map $D: A rightarrow B$ is called a strong deformation map if $mathrm{Gr}(D) subset mathbb{A}$ is a subalgebra. Such a map generalizes associative algebra homomorphisms, derivations, crossed homomorphisms and the associative analogue of modified {sf r}-matrices. We introduce the cohomology of a strong deformation map $D$ unifying the cohomologies of all the operators mentioned above. We also define the governing algebra for the pair $(mathbb{A}, D)$ to study simultaneous deformations of both $mathbb{A}$ and $D$. On the other hand, a linear map $r: B rightarrow A$ is called a weak deformation map if $mathrm{Gr} (r) subset mathbb{A}$ is a subalgebra. Such a map generalizes relative Rota-Baxter operators of any weight, twisted Rota-Baxter operators, Reynolds operators, left-averaging operators and right-averaging operators. Here we define the cohomology and governing algebra of a weak deformation map $r$ (that unify the cohomologies of all the operators mentioned above) and also for the pair $(mathbb{A}, r)$ that govern simultaneous deformations.
{"title":"Quasi-twilled associative algebras, deformation maps and their governing algebras","authors":"Apurba Das, Ramkrishna Mandal","doi":"arxiv-2409.00443","DOIUrl":"https://doi.org/arxiv-2409.00443","url":null,"abstract":"A quasi-twilled associative algebra is an associative algebra $mathbb{A}$\u0000whose underlying vector space has a decomposition $mathbb{A} = A oplus B$\u0000such that $B subset mathbb{A}$ is a subalgebra. In the first part of this\u0000paper, we give the Maurer-Cartan characterization and introduce the cohomology\u0000of a quasi-twilled associative algebra. In a quasi-twilled associative algebra $mathbb{A}$, a linear map $D: A\u0000rightarrow B$ is called a strong deformation map if $mathrm{Gr}(D) subset\u0000mathbb{A}$ is a subalgebra. Such a map generalizes associative algebra\u0000homomorphisms, derivations, crossed homomorphisms and the associative analogue\u0000of modified {sf r}-matrices. We introduce the cohomology of a strong\u0000deformation map $D$ unifying the cohomologies of all the operators mentioned\u0000above. We also define the governing algebra for the pair $(mathbb{A}, D)$ to\u0000study simultaneous deformations of both $mathbb{A}$ and $D$. On the other hand, a linear map $r: B rightarrow A$ is called a weak\u0000deformation map if $mathrm{Gr} (r) subset mathbb{A}$ is a subalgebra. Such a\u0000map generalizes relative Rota-Baxter operators of any weight, twisted\u0000Rota-Baxter operators, Reynolds operators, left-averaging operators and\u0000right-averaging operators. Here we define the cohomology and governing algebra\u0000of a weak deformation map $r$ (that unify the cohomologies of all the operators\u0000mentioned above) and also for the pair $(mathbb{A}, r)$ that govern\u0000simultaneous deformations.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Leavitt path algebras, which are algebras associated to directed graphs, were first introduced about 20 years ago. They have strong connections to such topics as symbolic dynamics, operator algebras, non-commutative geometry, representation theory, and even chip firing. In this article we invite the reader to sneak a peek at these fascinating algebras and their interplay with several seemingly disparate parts of mathematics.
{"title":"Monoids, dynamics and Leavitt path algebras","authors":"Gene Abrams, Roozbeh Hazrat","doi":"arxiv-2409.00289","DOIUrl":"https://doi.org/arxiv-2409.00289","url":null,"abstract":"Leavitt path algebras, which are algebras associated to directed graphs, were\u0000first introduced about 20 years ago. They have strong connections to such\u0000topics as symbolic dynamics, operator algebras, non-commutative geometry,\u0000representation theory, and even chip firing. In this article we invite the\u0000reader to sneak a peek at these fascinating algebras and their interplay with\u0000several seemingly disparate parts of mathematics.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A multiplicatively idempotent rig (which we abbreviate to mirig) is a rig satisfying the equation r2 = r. We show that a free mirig on finitely many generators is finite and compute its size. This work was originally motivated by a collaborative effort on the decentralized social network Mastodon to compute the size of the free mirig on two generators.
{"title":"From free idempotent monoids to free multiplicatively idempotent rigs","authors":"Morgan Rogers","doi":"arxiv-2408.17440","DOIUrl":"https://doi.org/arxiv-2408.17440","url":null,"abstract":"A multiplicatively idempotent rig (which we abbreviate to mirig) is a rig\u0000satisfying the equation r2 = r. We show that a free mirig on finitely many\u0000generators is finite and compute its size. This work was originally motivated\u0000by a collaborative effort on the decentralized social network Mastodon to\u0000compute the size of the free mirig on two generators.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that all wild blocks of type $A$ Hecke algebras with quantum characteristic $e geqslant 3$ -- i.e. blocks of weight at least $2$ -- are strictly wild, with the possible exception of the weight $2$ Rouquier block for $e = 3$.
{"title":"Wild blocks of type $A$ Hecke algebras are strictly wild","authors":"Liron Speyer","doi":"arxiv-2408.16477","DOIUrl":"https://doi.org/arxiv-2408.16477","url":null,"abstract":"We prove that all wild blocks of type $A$ Hecke algebras with quantum\u0000characteristic $e geqslant 3$ -- i.e. blocks of weight at least $2$ -- are\u0000strictly wild, with the possible exception of the weight $2$ Rouquier block for\u0000$e = 3$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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