We study lower bounds for the Riemann zeta function ζ(s){zeta(s)} along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of s ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of ζ(s){zeta(s)}.
我们研究了黎曼zeta函数ζ ( s ) {zeta(s)} 沿临界带右半部垂直算术级数的下界。我们证明，当 s 的虚部在给定区间内时，离散情况下获得的下界与连续情况下已知的下界重合，直至指数中的常数。我们的方法基于共振法的离散化，用于估计 ζ ( s ) {zeta(s)} 的极值。
{"title":"Discrete Ω-results for the Riemann zeta function","authors":"Paolo Minelli, Athanasios Sourmelidis","doi":"10.1515/forum-2023-0324","DOIUrl":"https://doi.org/10.1515/forum-2023-0324","url":null,"abstract":"We study lower bounds for the Riemann zeta function <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ζ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0324_eq_0200.png\"/> <jats:tex-math>{zeta(s)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of <jats:italic>s</jats:italic> ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ζ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0324_eq_0200.png\"/> <jats:tex-math>{zeta(s)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abdolnaser Bahlekeh, Fahimeh Sadat Fotouhi, Mohammad Amin Hamlehdari, Shokrollah Salarian
Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>𝔫</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0317_eq_0072.png"/> <jats:tex-math>{(S,{mathfrak{n}})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a commutative noetherian local ring and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ω</m:mi> <m:mo>∈</m:mo> <m:mi>𝔫</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0317_eq_0207.png"/> <jats:tex-math>{omegain{mathfrak{n}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective <jats:italic>S</jats:italic>-modules, such that their cokernels are annihilated by ω. It is shown that these categories, which will be denoted by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒫</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0317_eq_0342.png"/> <jats:tex-math>{{mathsf{Mon}}(omega,mathcal{P})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒢</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0317_eq_0341.png"/> <jats:tex-math>{{mathsf{Mon}}(omega,mathcal{G})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, are both Frobenius categories with the same projective objects. It is also proved that the stable category <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder accentunder="true"> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒫</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0317_eq_0277.png"/> <jats:tex-math>{underline{mathsf{Mon}}(omega,mathcal{P})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is triangle equivalent to the category of D-bran
{"title":"The stable category of monomorphisms between (Gorenstein) projective modules with applications","authors":"Abdolnaser Bahlekeh, Fahimeh Sadat Fotouhi, Mohammad Amin Hamlehdari, Shokrollah Salarian","doi":"10.1515/forum-2023-0317","DOIUrl":"https://doi.org/10.1515/forum-2023-0317","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>𝔫</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0317_eq_0072.png\"/> <jats:tex-math>{(S,{mathfrak{n}})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a commutative noetherian local ring and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ω</m:mi> <m:mo>∈</m:mo> <m:mi>𝔫</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0317_eq_0207.png\"/> <jats:tex-math>{omegain{mathfrak{n}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective <jats:italic>S</jats:italic>-modules, such that their cokernels are annihilated by ω. It is shown that these categories, which will be denoted by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"script\">𝒫</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0317_eq_0342.png\"/> <jats:tex-math>{{mathsf{Mon}}(omega,mathcal{P})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"script\">𝒢</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0317_eq_0341.png\"/> <jats:tex-math>{{mathsf{Mon}}(omega,mathcal{G})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, are both Frobenius categories with the same projective objects. It is also proved that the stable category <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:munder accentunder=\"true\"> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"script\">𝒫</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0317_eq_0277.png\"/> <jats:tex-math>{underline{mathsf{Mon}}(omega,mathcal{P})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is triangle equivalent to the category of D-bran","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"60 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
映射 f ： 如果 f ( 1 + x ) = 1 + f ( x ) {f(1+x)=1+f(x)} 和 f ( x y ) = f ( x ) f ( y ) {f(xy)=f(x)f(y)} 当 x , y∈ R {x,yin R} 时，则 f 是一个括号变形。 .我们要解决的问题是，是否每个 brachymorphism 都是可加的（即 f ( x + y ) = f ( x ) + f ( y ) {f(x+y)=f(x)+f(y)} ），结果表明，在许多情况下，包括下面的情况，答案都是肯定的： - R 是有限的（或者，更一般地说，R 是左或右阿蒂尼环）；- R 是交换环上任何 2 × 2 {2times 2} 矩阵环；- R 是恩格尔环；- R 的每个元素都是π-正则元素与中心元素之和（这适用于π-正则环、巴纳赫代数和幂级数环）； - R 是任意环上阶数大于 1 的全矩阵环； - R 是交换环 K 和 π-regular 单元 M 的单元环 K [ M ] {K[M]}； - R 是具有正特征的交换环 K 上的韦尔代数 𝖠 1 ( K ) {mathsf{A}_{1}(K)}; - f 是任意环上的幂函数 x ↦ x n {xmapsto x^{n}} ； - f 是任意环 R 上 n×n {ntimes n} 矩阵的行列式函数，n ≥ 3 {ngeq 3} ，在交换环上，如果 n > 3 {n>3} ，则 R 包含 n 个标量矩阵。 则 R 包含 n 个除数差不为零的标量矩阵。
A module over a ring <jats:italic>R</jats:italic> is <jats:italic>pure projective</jats:italic> provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module <jats:italic>M</jats:italic>, we consider <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2024-0031_eq_1297.png"/> <jats:tex-math>{operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which consists of direct summands of direct sums of copies of <jats:italic>M</jats:italic>. We are primarily interested in the case where <jats:italic>R</jats:italic> is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2024-0031_eq_1297.png"/> <jats:tex-math>{operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of <jats:italic>M</jats:italic> and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2024-0031_eq_0807.png"/> <jats:tex-math>{V^{*}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, of isomorphism classes of countably generated modules in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m
环 R 上的模块是纯投影模块，前提是它与有限呈现模块的直接和的直接和同构。我们开发了对交换诺特环上的纯投影模块进行分类的工具。特别是，对于一个固定的有限呈现模块 M，我们考虑 Add ( M ) {operatorname{Add}(M)} ，它由 M 的副本的直接和的直接和组成。我们主要关注 R 是一维局部域的情况，以及无扭（或 Cohen-Macaulay）模块。我们证明，即使在这种情况下，Add ( M ) {operatorname{Add}(M)} 也可以有大量模块不是有限生成模块的直和。我们的工作基于这样一个事实：这种无限生成的直接和都是由有限生成的数据决定的。也就是说，M 的内态环的幂幂/迹理想和有限生成的投影模块模都是由这些幂幂理想决定的。这样，我们就可以将研究有限生成模块的直接和分解与其完备性比较的经典理论扩展到无限生成的情况。我们研究了单元 V * ( M ) {V^{*}(M)} 的结构，即 Add ( M ) {operatorname{Add}(M)} 中由直接相加诱导的可数生成模块的同构类。我们证明了 V * ( M ) {V^{*}(M)} 是 V * ( M ⊗ R R ^ ) {V^{*}(Motimes_{R}widehat{R})} 的子单体，这让我们可以进行计算。 这样，我们就可以利用实例进行计算，并证明一些实现结果。
{"title":"Big pure projective modules over commutative noetherian rings: Comparison with the completion","authors":"Dolors Herbera, Pavel Příhoda, Roger Wiegand","doi":"10.1515/forum-2024-0031","DOIUrl":"https://doi.org/10.1515/forum-2024-0031","url":null,"abstract":"A module over a ring <jats:italic>R</jats:italic> is <jats:italic>pure projective</jats:italic> provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module <jats:italic>M</jats:italic>, we consider <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_1297.png\"/> <jats:tex-math>{operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which consists of direct summands of direct sums of copies of <jats:italic>M</jats:italic>. We are primarily interested in the case where <jats:italic>R</jats:italic> is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_1297.png\"/> <jats:tex-math>{operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of <jats:italic>M</jats:italic> and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_0807.png\"/> <jats:tex-math>{V^{*}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, of isomorphism classes of countably generated modules in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that, for any given q≥0{qgeq 0}, any Sasakian structure on a closed manifold M is approximated in the Cq{C^{{q}}}-norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the C0{C^{0}}-norm to a Cq{C^{{q}}}-approximation.
我们证明，对于任何给定的 q ≥ 0 {qgeq 0}，闭流形 M 上的任何萨萨基结构在 C q {C^{{q}} 中都是近似的。｝ -的结构近似。为了得到这个结果，我们还加强了 [J. Ross and R. Thomas, Weighted Sasakian spheres] 中给出的用投影诱导的球面凯勒形式对球面凯勒形式的逼近。Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom.88 2011, 1, 109-159] 中的 C 0 {C^{0}} -norm。 -C q {C^{{q}} 的近似。｝ -的近似。
{"title":"Any Sasakian structure is approximated by embeddings into spheres","authors":"Andrea Loi, Giovanni Placini","doi":"10.1515/forum-2023-0364","DOIUrl":"https://doi.org/10.1515/forum-2023-0364","url":null,"abstract":"We show that, for any given <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>q</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0364_eq_0372.png\"/> <jats:tex-math>{qgeq 0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, any Sasakian structure on a closed manifold <jats:italic>M</jats:italic> is approximated in the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0364_eq_0146.png\"/> <jats:tex-math>{C^{{q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mn>0</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0364_eq_0138.png\"/> <jats:tex-math>{C^{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0364_eq_0146.png\"/> <jats:tex-math>{C^{{q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-approximation.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"161 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang–Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota–Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota–Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota–Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight 0, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.
{"title":"Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators","authors":"Chengming Bai, Guilai Liu, Yunhe Sheng, Rong Tang","doi":"10.1515/forum-2023-0268","DOIUrl":"https://doi.org/10.1515/forum-2023-0268","url":null,"abstract":"We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang–Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota–Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota–Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota–Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight 0, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"14 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the weighted estimates for multilinear pseudo-differential operators were systematically studied in rearrangement invariant Banach and quasi-Banach spaces. These spaces contain the Lebesgue space, the classical Lorentz space and Marcinkiewicz space as typical examples. More precisely, the weighted boundedness and weighted modular estimates, including the weak endpoint case, were established for multilinear pseudo-differential operators and their commutators. As applications, we show that the above results also hold for the multilinear Fourier multipliers, multilinear square functions, and a class of multilinear Calderón–Zygmund operators.
{"title":"Quantitative weighted estimates for the multilinear pseudo-differential operators in function spaces","authors":"Jiawei Tan, Qingying Xue","doi":"10.1515/forum-2023-0454","DOIUrl":"https://doi.org/10.1515/forum-2023-0454","url":null,"abstract":"In this paper, the weighted estimates for multilinear pseudo-differential operators were systematically studied in rearrangement invariant Banach and quasi-Banach spaces. These spaces contain the Lebesgue space, the classical Lorentz space and Marcinkiewicz space as typical examples. More precisely, the weighted boundedness and weighted modular estimates, including the weak endpoint case, were established for multilinear pseudo-differential operators and their commutators. As applications, we show that the above results also hold for the multilinear Fourier multipliers, multilinear square functions, and a class of multilinear Calderón–Zygmund operators.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this short note is to prove that if G is a (homomorphic images of a) soluble periodic linear group and N is a locally nilpotent normal subgroup of G such that N and G/N{G/N} have no isomorphic G-chief factors, then two supplements to N in G are conjugate provided that they have the same intersection with N. This result follows from well-known theorems in the theory of Schunck classes (see [A. Ballester-Bolinches and L. M. Ezquerro, On conjugacy of supplements of normal subgroups of finite groups, Bull. Aust. Math. Soc. 89 2014, 2, 293–299]), and it appeared as the main theorem of [C. Parker and P. Rowley, A note on conjugacy of supplements in finite soluble groups, Bull. Lond. Math. Soc. 42 2010, 3, 417–419].
本短文旨在证明，如果 G 是可溶周期线性群的（同态图像），而 N 是 G 的局部零potent 正则子群，使得 N 和 G / N {G/N} 没有同构的 G 主因，那么 G 中 N 的两个补充群是共轭的，条件是它们与 N 有相同的交集。这一结果源于 Schunck 类理论中的著名定理（见 [A. Ballester-Bolinches and L. M. Ezquerro, On conjugacy of supplements to N in G, Bull.Ballester-Bolinches and L. M. Ezquerro, On conjugacy of supplements of normal subgroups of finite groups, Bull. Aust.Aust.Math.89 2014, 2, 293-299]），并作为主定理出现在[C.Parker and P. Rowley, A note on conjugacy of supplements in finite soluble groups, Bull.Lond.Math.42 2010, 3, 417-419].
{"title":"A note on conjugacy of supplements in soluble periodic linear groups","authors":"Marco Trombetti","doi":"10.1515/forum-2024-0102","DOIUrl":"https://doi.org/10.1515/forum-2024-0102","url":null,"abstract":"The aim of this short note is to prove that if <jats:italic>G</jats:italic> is a (homomorphic images of a) soluble periodic linear group and <jats:italic>N</jats:italic> is a locally nilpotent normal subgroup of <jats:italic>G</jats:italic> such that <jats:italic>N</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0102_eq_0133.png\"/> <jats:tex-math>{G/N}</jats:tex-math> </jats:alternatives> </jats:inline-formula> have no isomorphic <jats:italic>G</jats:italic>-chief factors, then two supplements to <jats:italic>N</jats:italic> in <jats:italic>G</jats:italic> are conjugate provided that they have the same intersection with <jats:italic>N</jats:italic>. This result follows from well-known theorems in the theory of Schunck classes (see [A. Ballester-Bolinches and L. M. Ezquerro, On conjugacy of supplements of normal subgroups of finite groups, Bull. Aust. Math. Soc. 89 2014, 2, 293–299]), and it appeared as the main theorem of [C. Parker and P. Rowley, A note on conjugacy of supplements in finite soluble groups, Bull. Lond. Math. Soc. 42 2010, 3, 417–419].","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"199 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let A be a superalgebra over a field F of characteristic zero. We prove tight relations between graded automorphisms, pseudoautomorphisms, superautomorphisms and K-gradings on A, where K is the Klein group. Moreover, we investigate the consequences of such connections within the theory of polynomial identities. In the second part we focus on the superalgebra UTn(F){UT_{n}(F)} of n×n{ntimes n} upper triangular matrices by completely classifying the graded-pseudo-super automorphism that one can define on it. Finally, we compute the ideals of identities of UTn(F){UT_{n}(F)} endowed with a graded or a pseudo automorphism, for any n, and the ideals of identities with superautomorphism in the cases n=2{n=2} and n=3{n=3}.
设 A 是特征为零的域 F 上的超代数。我们证明了 A 上的级数自变形、伪自变形、超自变形和 K 级数（其中 K 是克莱因群）之间的紧密关系。此外，我们还研究了这种联系在多项式同构理论中的后果。在第二部分中，我们将重点放在 n × n {ntimes n} 上三角矩阵的超代数 U T n ( F ) {UT_{n}(F)} 上，对可以在其上定义的分级伪超自变量进行完全分类。最后，我们计算了任意 n 的 U T n ( F ) {UT_{n}(F)} 带有分级或伪自形性的同调理想，以及 n = 2 {n=2} 和 n = 3 {n=3} 两种情况下带超同形性的同调理想。
{"title":"Gradings and graded linear maps on algebras","authors":"Antonio Ioppolo, Fabrizio Martino","doi":"10.1515/forum-2024-0098","DOIUrl":"https://doi.org/10.1515/forum-2024-0098","url":null,"abstract":"Let <jats:italic>A</jats:italic> be a superalgebra over a field <jats:italic>F</jats:italic> of characteristic zero. We prove tight relations between graded automorphisms, pseudoautomorphisms, superautomorphisms and <jats:italic>K</jats:italic>-gradings on <jats:italic>A</jats:italic>, where <jats:italic>K</jats:italic> is the Klein group. Moreover, we investigate the consequences of such connections within the theory of polynomial identities. In the second part we focus on the superalgebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>U</m:mi> <m:mo></m:mo> <m:msub> <m:mi>T</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0098_eq_0217.png\"/> <jats:tex-math>{UT_{n}(F)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0098_eq_0407.png\"/> <jats:tex-math>{ntimes n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> upper triangular matrices by completely classifying the graded-pseudo-super automorphism that one can define on it. Finally, we compute the ideals of identities of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>U</m:mi> <m:mo></m:mo> <m:msub> <m:mi>T</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0098_eq_0217.png\"/> <jats:tex-math>{UT_{n}(F)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> endowed with a graded or a pseudo automorphism, for any <jats:italic>n</jats:italic>, and the ideals of identities with superautomorphism in the cases <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0098_eq_0403.png\"/> <jats:tex-math>{n=2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0098_eq_0404.png\"/> <jats:tex-math>{n=3}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"40 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the quotient variety of the space of foliations on ℂℙ2{mathbb{CP}^{2}} of degree 2 up to change of coordinates. We find the intersection Betti numbers of this variety. As a corollary, we have that these intersection Betti numbers coincide with the intersection Betti numbers of the quotient variety of quartic plane curves. Finally, we give an explicit isomorphism between the space of foliations of degree 2 with different singular points, without invariant lines and the space of smooth quartic plane curves.
{"title":"GIT quotient of holomorphic foliations on ℂℙ2 of degree 2 and quartic plane curves","authors":"Claudia R. Alcántara, Juan Vásquez Aquino","doi":"10.1515/forum-2024-0043","DOIUrl":"https://doi.org/10.1515/forum-2024-0043","url":null,"abstract":"We study the quotient variety of the space of foliations on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ℂ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>ℙ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0043_eq_1268.png\"/> <jats:tex-math>{mathbb{CP}^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of degree 2 up to change of coordinates. We find the intersection Betti numbers of this variety. As a corollary, we have that these intersection Betti numbers coincide with the intersection Betti numbers of the quotient variety of quartic plane curves. Finally, we give an explicit isomorphism between the space of foliations of degree 2 with different singular points, without invariant lines and the space of smooth quartic plane curves.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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