Pub Date : 2023-10-13DOI: 10.1142/s021919972350044x
Jun Hu, Lei Shi
In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra $R^Lambda(beta)$ associated to an {it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}in I$, where $Lambdain P^+$ and $betain Q_n^+$. As applications, we obtain some {it necessary and sufficient conditions} for the KLR idempotent $e(nu)$ (for any $nuin I^beta$) to be nonzero in the cyclotomic quiver Hecke algebra $R^Lambda(beta)$. We prove several level reduction results which decomposes $dim R^Lambda(beta)$ into a sum of some products of $dim R^{Lambda^i}(beta_i)$ with $Lambda=sum_iLambda^i$ and $beta=sum_{i}beta_i$, where $Lambda^iin P^+, beta^iin Q^+$ for each $i$. We construct some explicit monomial bases for the subspaces $e(widetilde{nu})R^Lambda(beta)e(mu)$ and $e(widetilde{nu})R^Lambda(beta)e(mu)$ of $R^Lambda(beta)$, where $muin I^beta$ is {it arbitrary} and $widetilde{nu}in I^beta$ is a certain specific $n$-tuple (see Section 4).Finally, we use our graded dimension formulae to provide some examples which show that $R^Lambda(n)$ is in general not graded free over its natural embedded subalgebra $R^Lambda(m)$ with $m
{"title":"Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras","authors":"Jun Hu, Lei Shi","doi":"10.1142/s021919972350044x","DOIUrl":"https://doi.org/10.1142/s021919972350044x","url":null,"abstract":"In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra $R^Lambda(beta)$ associated to an {it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}in I$, where $Lambdain P^+$ and $betain Q_n^+$. As applications, we obtain some {it necessary and sufficient conditions} for the KLR idempotent $e(nu)$ (for any $nuin I^beta$) to be nonzero in the cyclotomic quiver Hecke algebra $R^Lambda(beta)$. We prove several level reduction results which decomposes $dim R^Lambda(beta)$ into a sum of some products of $dim R^{Lambda^i}(beta_i)$ with $Lambda=sum_iLambda^i$ and $beta=sum_{i}beta_i$, where $Lambda^iin P^+, beta^iin Q^+$ for each $i$. We construct some explicit monomial bases for the subspaces $e(widetilde{nu})R^Lambda(beta)e(mu)$ and $e(widetilde{nu})R^Lambda(beta)e(mu)$ of $R^Lambda(beta)$, where $muin I^beta$ is {it arbitrary} and $widetilde{nu}in I^beta$ is a certain specific $n$-tuple (see Section 4).Finally, we use our graded dimension formulae to provide some examples which show that $R^Lambda(n)$ is in general not graded free over its natural embedded subalgebra $R^Lambda(m)$ with $m","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135854595","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}
Pub Date : 2023-10-13DOI: 10.1142/s0219199723500463
Jan-Henrik Metsch
We study half-spheres with small radii sitting on the boundary of a smooth bounded domain while meeting it orthogonally. Even though it is known that there exist families of CMC and Willmore type half-spheres near a nondegenerate critical point p of the domains boundaries mean curvature, it is unknown in both cases whether these provide a foliation of any deleted neighborhood of p. We prove that this is not guaranteed and establish a criterion in terms of the boundaries geometry that ensures or prevents the respective surfaces from providing such a foliation. This perhaps surprising phenomenon of conditional foliations is absent in the closely related Riemannian setting, where a foliation is guaranteed. We show how this unconditional foliation arises from symmetry considerations and how these fail to apply to the 'domain-setting'.
{"title":"On the Conditional Existence of Foliations by CMC and Willmore Type Half-Spheres","authors":"Jan-Henrik Metsch","doi":"10.1142/s0219199723500463","DOIUrl":"https://doi.org/10.1142/s0219199723500463","url":null,"abstract":"We study half-spheres with small radii sitting on the boundary of a smooth bounded domain while meeting it orthogonally. Even though it is known that there exist families of CMC and Willmore type half-spheres near a nondegenerate critical point p of the domains boundaries mean curvature, it is unknown in both cases whether these provide a foliation of any deleted neighborhood of p. We prove that this is not guaranteed and establish a criterion in terms of the boundaries geometry that ensures or prevents the respective surfaces from providing such a foliation. This perhaps surprising phenomenon of conditional foliations is absent in the closely related Riemannian setting, where a foliation is guaranteed. We show how this unconditional foliation arises from symmetry considerations and how these fail to apply to the 'domain-setting'.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135854922","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}
Pub Date : 2023-10-13DOI: 10.1142/s0219199723500475
Yang Liu, Xin Zhong
We study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a finite number of two-dimensional connected components. We derive the global existence and uniqueness of classical solutions provided that the initial total energy is suitably small. Our result generalizes the Cauchy problems of compressible Navier-Stokes equations with Coulomb force (J. Differential Equations 269: 8468--8508, 2020) and compressible MHD equations (SIAM J. Math. Anal. 45: 1356--1387, 2013) to the case of bounded domains although tackling many surface integrals caused by the slip boundary condition are complex. The main ingredient of this paper is to overcome the strong nonlinearity caused by Coulomb force, magnetic field, and rotation effect of micro-particles by applying piecewise-estimate method and delicate analysis based on the effective viscous fluxes involving velocity and micro-rotational velocity.
{"title":"Global classical solution for three-dimensional compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in bounded domains","authors":"Yang Liu, Xin Zhong","doi":"10.1142/s0219199723500475","DOIUrl":"https://doi.org/10.1142/s0219199723500475","url":null,"abstract":"We study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a finite number of two-dimensional connected components. We derive the global existence and uniqueness of classical solutions provided that the initial total energy is suitably small. Our result generalizes the Cauchy problems of compressible Navier-Stokes equations with Coulomb force (J. Differential Equations 269: 8468--8508, 2020) and compressible MHD equations (SIAM J. Math. Anal. 45: 1356--1387, 2013) to the case of bounded domains although tackling many surface integrals caused by the slip boundary condition are complex. The main ingredient of this paper is to overcome the strong nonlinearity caused by Coulomb force, magnetic field, and rotation effect of micro-particles by applying piecewise-estimate method and delicate analysis based on the effective viscous fluxes involving velocity and micro-rotational velocity.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135853578","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}
Pub Date : 2023-10-13DOI: 10.1142/s0219199723500505
Guilai Liu, Chengming Bai
The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang-Baxter equation and $mathcal O$-operators are studied.
{"title":"A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras","authors":"Guilai Liu, Chengming Bai","doi":"10.1142/s0219199723500505","DOIUrl":"https://doi.org/10.1142/s0219199723500505","url":null,"abstract":"The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang-Baxter equation and $mathcal O$-operators are studied.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135853272","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}
Pub Date : 2023-09-01DOI: 10.1142/s0219199723990018
Communications in Contemporary MathematicsVol. 25, No. 10, 2399001 (2023) No AccessAuthor index Volume 25 (2023)https://doi.org/10.1142/S0219199723990018Cited by:0 Previous AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Remember to check out the Most Cited Articles! Be inspired by these NEW Mathematics books for inspirations & latest information in your research area! FiguresReferencesRelatedDetails Recommended Vol. 25, No. 10 Metrics History PDF download
当代数学通讯卷。25，第10号，2399001(2023)没有访问作者索引卷25 (2023)https://doi.org/10.1142/S0219199723990018Cited by:0 Previous AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack citations推荐到图书馆分享分享在facebook上推特链接在redditemail记得检查最多被引用的文章!受到这些新的数学书籍的启发，在您的研究领域获得灵感和最新信息!FiguresReferencesRelatedDetails推荐Vol. 25, No. 10 Metrics History PDF下载
{"title":"Author index Volume 25 (2023)","authors":"","doi":"10.1142/s0219199723990018","DOIUrl":"https://doi.org/10.1142/s0219199723990018","url":null,"abstract":"Communications in Contemporary MathematicsVol. 25, No. 10, 2399001 (2023) No AccessAuthor index Volume 25 (2023)https://doi.org/10.1142/S0219199723990018Cited by:0 Previous AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Remember to check out the Most Cited Articles! Be inspired by these NEW Mathematics books for inspirations & latest information in your research area! FiguresReferencesRelatedDetails Recommended Vol. 25, No. 10 Metrics History PDF download","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135389411","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}
Pub Date : 2023-06-09DOI: 10.1142/s021919972350030x
Luccas Campos, Jason Murphy, T. Hoose
{"title":"Averaging for the 2d dispersion-managed NLS","authors":"Luccas Campos, Jason Murphy, T. Hoose","doi":"10.1142/s021919972350030x","DOIUrl":"https://doi.org/10.1142/s021919972350030x","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43284954","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}
Pub Date : 2023-06-09DOI: 10.1142/s0219199723500335
Yamin Wang
{"title":"Prescribed blow-up sets for sequences of solutions to a non-local Q-curvature equation in ℝ3","authors":"Yamin Wang","doi":"10.1142/s0219199723500335","DOIUrl":"https://doi.org/10.1142/s0219199723500335","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48423431","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}
Pub Date : 2023-05-23DOI: 10.1142/s0219199723500165
Pan Liu, Xin Yang Lu, Kunlun He
Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a [Formula: see text]-order (an)-isotropic total variation semi-norms [Formula: see text], [Formula: see text], defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative [Formula: see text], of such loss functions.
{"title":"Real order total variation with applications to the loss functions in learning schemes","authors":"Pan Liu, Xin Yang Lu, Kunlun He","doi":"10.1142/s0219199723500165","DOIUrl":"https://doi.org/10.1142/s0219199723500165","url":null,"abstract":"Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a [Formula: see text]-order (an)-isotropic total variation semi-norms [Formula: see text], [Formula: see text], defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative [Formula: see text], of such loss functions.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181131","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}
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