Pub Date : 2022-01-01DOI: 10.4310/ajm.2022.v26.n3.a1
Shuanghe Fan, S. Yau, Huaiqing Zuo
{"title":"Non-existence of negative weight derivations of the local $1textrm{-st}$ Hessian algebras of singularities","authors":"Shuanghe Fan, S. Yau, Huaiqing Zuo","doi":"10.4310/ajm.2022.v26.n3.a1","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n3.a1","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-28DOI: 10.4310/ajm.2022.v26.n6.a3
Gabriel J. H. Khan, Jun Zhang
The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pair of torsion-free conjugate connections. In recent work, the authors considered parametrized probability distributions as partially-flat statistical manifolds admitting torsion and showed that there is a complex to symplectic duality on the tangent bundles of such manifolds, based on the dualistic geometry of the underlying manifold. In this paper, we explore this correspondence further in the context of Hessian manifolds, in which case the conjugate connections are both curvature- and torsion-free, and the associated dual pair of spaces are K"ahler manifolds. We focus on several key examples and their geometric features. In particular, we show that the moduli space of univariate normal distributions gives rise to a correspondence between the Siegel half-space and the Siegel-Jacobi space, which are spaces that appear in the context of automorphic forms.
{"title":"A hall of statistical mirrors","authors":"Gabriel J. H. Khan, Jun Zhang","doi":"10.4310/ajm.2022.v26.n6.a3","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n6.a3","url":null,"abstract":"The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pair of torsion-free conjugate connections. In recent work, the authors considered parametrized probability distributions as partially-flat statistical manifolds admitting torsion and showed that there is a complex to symplectic duality on the tangent bundles of such manifolds, based on the dualistic geometry of the underlying manifold. In this paper, we explore this correspondence further in the context of Hessian manifolds, in which case the conjugate connections are both curvature- and torsion-free, and the associated dual pair of spaces are K\"ahler manifolds. We focus on several key examples and their geometric features. In particular, we show that the moduli space of univariate normal distributions gives rise to a correspondence between the Siegel half-space and the Siegel-Jacobi space, which are spaces that appear in the context of automorphic forms.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48645034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-28DOI: 10.4310/ajm.2022.v26.n2.a4
J. Hoppe
Three classes of new, algebraic, zero-mean-curvature hypersurfaces in pseudo-Euclidean spaces are given.
给出了伪欧氏空间中三类新的代数零平均曲率超曲面。
{"title":"Exact algebraic m(em)brane solutions","authors":"J. Hoppe","doi":"10.4310/ajm.2022.v26.n2.a4","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n2.a4","url":null,"abstract":"Three classes of new, algebraic, zero-mean-curvature hypersurfaces in pseudo-Euclidean spaces are given.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46649538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-20DOI: 10.4310/ajm.2022.v26.n5.a2
Yuji Hirota, Tosiaki Kori
We shall give a twisted Dirac structure on the space of irreducible connections on a SU(n)-bundle over a three-manifold, and give a family of twisted Dirac structures on the space of irreducible connections on the trivial SU(n)-bundle over a four-manifold. The twist is described by the Cartan 3-form on the space of connections. It vanishes over the subspace of flat connections. So the spaces of flat connections are endowed with ( non-twisted ) Dirac structures. The Dirac structure on the space of flat connections over the three-manifold is obtained as the boundary restriction of a corresponding Dirac structure over the four-manifold. We discuss also the action of the group of gauge transformations over these Dirac structures.
{"title":"Dirac structures on the space of connections","authors":"Yuji Hirota, Tosiaki Kori","doi":"10.4310/ajm.2022.v26.n5.a2","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n5.a2","url":null,"abstract":"We shall give a twisted Dirac structure on the space of irreducible connections on a SU(n)-bundle over a three-manifold, and give a family of twisted Dirac structures on the space of irreducible connections on the trivial SU(n)-bundle over a four-manifold. The twist is described by the Cartan 3-form on the space of connections. It vanishes over the subspace of flat connections. So the spaces of flat connections are endowed with ( non-twisted ) Dirac structures. The Dirac structure on the space of flat connections over the three-manifold is obtained as the boundary restriction of a corresponding Dirac structure over the four-manifold. We discuss also the action of the group of gauge transformations over these Dirac structures.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46025127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-13DOI: 10.4310/ajm.2022.v26.n4.a3
J. Lauret
Let g be a G-invariant Einstein metric on a compact homogeneous space M=G/K. We use a formula for the Lichnerowicz Laplacian of g at G-invariant TT-tensors to study the stability type of g as a critical point of the scalar curvature function. The case when g is naturally reductive is studied in special detail.
{"title":"On the stability of homogeneous Einstein manifolds","authors":"J. Lauret","doi":"10.4310/ajm.2022.v26.n4.a3","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n4.a3","url":null,"abstract":"Let g be a G-invariant Einstein metric on a compact homogeneous space M=G/K. We use a formula for the Lichnerowicz Laplacian of g at G-invariant TT-tensors to study the stability type of g as a critical point of the scalar curvature function. The case when g is naturally reductive is studied in special detail.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47883581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-03DOI: 10.4310/AJM.2022.v26.n1.a4
Vincenzo Antonelli, G. Casnati, Ozhan Genc
We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. We determine a lower bound for the quantum number of a non-ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_Xge 2$ or $i_X=1$, $mathrm{Pic}(X)$ is cyclic and $X$ is ordinary. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non-ordinary instanton bundles on $mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension.
{"title":"Even and odd instanton bundles on Fano threefolds","authors":"Vincenzo Antonelli, G. Casnati, Ozhan Genc","doi":"10.4310/AJM.2022.v26.n1.a4","DOIUrl":"https://doi.org/10.4310/AJM.2022.v26.n1.a4","url":null,"abstract":"We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. We determine a lower bound for the quantum number of a non-ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_Xge 2$ or $i_X=1$, $mathrm{Pic}(X)$ is cyclic and $X$ is ordinary. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non-ordinary instanton bundles on $mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43879888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-20DOI: 10.4310/ajm.2021.v25.n3.a5
Nilanjan Bag, Rupam Barman
Abstract. The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by L-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the 6 and 8 order moments of the generalized quadratic Gauss sums weighted by L-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.
{"title":"Higher order moments of generalized quadratic Gauss sums weighted by $L$-functions","authors":"Nilanjan Bag, Rupam Barman","doi":"10.4310/ajm.2021.v25.n3.a5","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n3.a5","url":null,"abstract":"Abstract. The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by L-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the 6 and 8 order moments of the generalized quadratic Gauss sums weighted by L-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46412782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.4310/ajm.2021.v25.n6.a1
Shuliang Bai, Linyuan Lu, S. Yau
A graph is called Ricci-flat if its Ricci curvatures vanish on all edges, here the definition of Ricci curvature on graphs was given by Lin-Lu-Yau [5]. The authors in [4] and [2] obtained a complete characterization for all Ricci-flat graphs with girth at least five. In this paper, we completely determined all Ricci-flat graphs with maximum degree at most 4. keywords: Ricci curvature, Ricci-flat graph, maximum degree
如果图的Ricci曲率在所有边上都消失，则称之为Ricci平坦图，这里图上Ricci曲率的定义是由Lin Lu Yau[5]给出的。作者在[4]和[2]中获得了所有周长至少为5的Ricci平面图的完整刻画。在本文中，我们完全确定了所有最大度为4的Ricci平面图。关键词：Ricci曲率，Ricci平面图，最大度
{"title":"Ricci-flat graphs with maximum degree at most $4$","authors":"Shuliang Bai, Linyuan Lu, S. Yau","doi":"10.4310/ajm.2021.v25.n6.a1","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n6.a1","url":null,"abstract":"A graph is called Ricci-flat if its Ricci curvatures vanish on all edges, here the definition of Ricci curvature on graphs was given by Lin-Lu-Yau [5]. The authors in [4] and [2] obtained a complete characterization for all Ricci-flat graphs with girth at least five. In this paper, we completely determined all Ricci-flat graphs with maximum degree at most 4. keywords: Ricci curvature, Ricci-flat graph, maximum degree","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49142479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.4310/ajm.2021.v25.n1.a3
Chuanhao Wei
{"title":"Fibration of $operatorname{log}$-general type space over quasi-abelian varieties","authors":"Chuanhao Wei","doi":"10.4310/ajm.2021.v25.n1.a3","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n1.a3","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70391859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.4310/ajm.2021.v25.n5.a1
Eduardo Casas-Alvero
. We present new results regarding which Puiseux coefficients the analytic type of a complex irreducible plane curve singularity depends on.
． 给出了复不可约平面曲线奇异点解析型所依赖的普塞系数的新结果。
{"title":"On the analytic classification of irreducible plane curve singularities","authors":"Eduardo Casas-Alvero","doi":"10.4310/ajm.2021.v25.n5.a1","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n5.a1","url":null,"abstract":". We present new results regarding which Puiseux coefficients the analytic type of a complex irreducible plane curve singularity depends on.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：