Nima Anari, Kuikui Liu, Shayan Oveis Gharan, Cynthia Vinzant
We give a self-contained proof of the strongest version of Mason’s conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.
{"title":"Log-concave polynomials III: Mason’s ultra-log-concavity conjecture for independent sets of matroids","authors":"Nima Anari, Kuikui Liu, Shayan Oveis Gharan, Cynthia Vinzant","doi":"10.1090/proc/16724","DOIUrl":"https://doi.org/10.1090/proc/16724","url":null,"abstract":"<p>We give a self-contained proof of the strongest version of Mason’s conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938833","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}
Abhishek Bharadwaj, Aprameyo Pal, Veekesh Kumar, R. Thangadurai
Let αalpha be a non-zero algebraic number. Let KK be the Galois closure of Q(α)mathbb {Q}(alpha ) with Galois group GG and Q¯bar {mathbb {Q}} be the algebraic closure of Qmathbb {Q}. In this article, among the other results, we prove the following. If f∈Q¯[G]fin bar {mathbb {Q}}[G] is a non-zero element of the group ring
让 α alpha 是一个非零代数数。让 K K 是 Q ( α ) mathbb {Q}(alpha ) 的伽罗瓦群 G G 的伽罗瓦封闭，Q ¯bar {mathbb {Q}} 是 Q mathbb {Q} 的代数封闭。在本文中，我们证明了以下结果。如果 f∈ Q ¯ [ G ] fin bar {mathbb {Q}}[G]是群环 Q ¯ [ G ] bar {mathbb {Q}}[G]的一个非零元素，并且 α alpha 是一个给定的代数数，使得 f ( α n ) f(alpha ^n) 对于无穷多个自然数 n n 是一个非零代数整数、那么 α alpha 是一个代数整数。这一结果概括了波利亚 [Rend. Circ Mat. Palermo, 40 (1915), pp.我们还证明了这一结果对于具有代数系数的有理函数的类似结果。受 B. de Smit [J. Number Theory 45 (1993), pp.为了证明这些结果，我们应用了 Corvaja 和 Zannier 的技术以及 Kulkarni 等人的结果 [Trans. Amer. Math. Soc. 371 (2019), pp.
{"title":"Sufficient conditions for a problem of Polya","authors":"Abhishek Bharadwaj, Aprameyo Pal, Veekesh Kumar, R. Thangadurai","doi":"10.1090/proc/16826","DOIUrl":"https://doi.org/10.1090/proc/16826","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding=\"application/x-tex\">alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a non-zero algebraic number. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Galois closure of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q left-parenthesis alpha right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}(alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with Galois group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar\"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">bar {mathbb {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the algebraic closure of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this article, among the other results, we prove the following. <italic>If <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f element-of ModifyingAbove double-struck upper Q With bar left-bracket upper G right-bracket\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">¯</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">fin bar {mathbb {Q}}[G]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a non-zero element of the group ring <inline-formula conten","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744661","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 prove that for any fixed unitary matrix UU, any abelian self-adjoint algebra of matrices that is invariant under conjugation by UU can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by UU. We use this result to analyse the structure of matrices AA for which A∗AA^*A commutes with AA∗AA^*, and to characterize matrices that are unitarily equivalent to weighted permutations.
我们证明，对于任何固定的单元矩阵 U U，任何在 U U 共轭下不变的矩阵无边自交代数都可以嵌入到一个在 U U 共轭下仍然不变的最大无边自交代数中。我们利用这一结果来分析 A ∗ A A^*A 与 A A ∗ AA^* 共轭的矩阵 A A 的结构，并描述与加权排列单元等价的矩阵的特征。
{"title":"Invariant embeddings and weighted permutations","authors":"M. Mastnak, H. Radjavi","doi":"10.1090/proc/16835","DOIUrl":"https://doi.org/10.1090/proc/16835","url":null,"abstract":"<p>We prove that for any fixed unitary matrix <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\"application/x-tex\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, any abelian self-adjoint algebra of matrices that is invariant under conjugation by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\"application/x-tex\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\"application/x-tex\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We use this result to analyse the structure of matrices <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Superscript asterisk Baseline upper A\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A^*A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> commutes with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A upper A Superscript asterisk\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">AA^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and to characterize matrices that are unitarily equivalent to weighted permutations.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151754","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 address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80’s. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger’s conjecture, we give simple proofs that two celebrated homological conjectures, namely the Huneke-Wiegand and the Auslander-Reiten problems, are true for the class of Ulrich modules.
{"title":"Tensor products and solutions to two homological conjectures for Ulrich modules","authors":"Cleto Miranda-Neto, Thyago Souza","doi":"10.1090/proc/16838","DOIUrl":"https://doi.org/10.1090/proc/16838","url":null,"abstract":"<p>We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80’s. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger’s conjecture, we give simple proofs that two celebrated homological conjectures, namely the Huneke-Wiegand and the Auslander-Reiten problems, are true for the class of Ulrich modules.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151927","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 1996 Stolz [Math. Ann. 304 (1996), pp. 785–800] conjectured that a string manifold with positive Ricci curvature has vanishing Witten genus. Here we prove this conjecture for toric string Fano manifolds and for string torus manifolds admitting invariant metrics of non-negative sectional curvature.
1996 年，Stolz [Math. Ann. 304 (1996), pp.在此，我们证明了环状弦法诺流形和允许非负截面曲率不变度量的弦环流形的这一猜想。
{"title":"On a conjecture of Stolz in the toric case","authors":"Michael Wiemeler","doi":"10.1090/proc/16823","DOIUrl":"https://doi.org/10.1090/proc/16823","url":null,"abstract":"<p>In 1996 Stolz [Math. Ann. 304 (1996), pp. 785–800] conjectured that a string manifold with positive Ricci curvature has vanishing Witten genus. Here we prove this conjecture for toric string Fano manifolds and for string torus manifolds admitting invariant metrics of non-negative sectional curvature.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516768","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, we consider kk-curvature equations σk(κ[Mu])=f(x,u,∇u)sigma _k(kappa [M_u])=f(x,u,nabla u) subject to (k+1)(k+1)-convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these kk-curvature equations.
本文考虑 k k -曲率方程 σ k ( κ [ M u ] ) = f ( x , u ,∇ u ) sigma _k(kappa [M_u])=f(x,u,nabla u) subject to ( k + 1 ) (k+1) -convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J 123 (2004)].123 (2004), pp.］通过使用 Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23] 的 Hessian 算子的关键凹不等式，我们推导出了这些 k k -曲率方程的半凸可纳解的 Pogorelov 估计值。
{"title":"Pogorelov estimates for semi-convex solutions of 𝑘-curvature equations","authors":"Xiaojuan Chen, Qiang Tu, Ni Xiang","doi":"10.1090/proc/16820","DOIUrl":"https://doi.org/10.1090/proc/16820","url":null,"abstract":"<p>In this paper, we consider <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-curvature equations <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma Subscript k Baseline left-parenthesis kappa left-bracket upper M Subscript u Baseline right-bracket right-parenthesis equals f left-parenthesis x comma u comma nabla u right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>κ</mml:mi> <mml:mo stretchy=\"false\">[</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>u</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">]</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant=\"normal\">∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">sigma _k(kappa [M_u])=f(x,u,nabla u)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> subject to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis k plus 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(k+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-curvature equations.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938981","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 note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that T(k)T(k)-local TR vanishes on connective Lnp,fL_n^{p,f}-acyclic E1mathbb {E}_1-rings for every 1≤k≤n1 leq k leq n and deduce consequences for connective Morava K-theory and the Thom spectra y(n)y(n). The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive ∞infty-categories which was recently established by Córdova Fedeli [Topological Hochschild homology of adic rings, Ph.D. thesis, University of Copenhagen, 2023].
在本注释中，我们建立了望远镜局部拓扑限制同调 TR 的消失结果。更准确地说，我们证明了 T ( k ) T(k) 局部 TR 在每 1 ≤ k ≤ n 1 leq k leq n 的连通 L n p , f L_n^{p,f} -acyclic E 1 mathbb {E}_1 -rings 上消失，并推导出连通莫拉瓦 K 理论和托姆谱 y ( n ) y(n) 的后果。证明依赖于 TR 与 K 理论上的曲线谱之间的关系，以及代数 K 理论保留了加性 ∞ infty - 类别的无限乘积这一事实，这一事实最近由科尔多瓦-费德利 (Córdova Fedeli) 建立[adic rings 的拓扑霍赫希尔德同源性，哥本哈根大学博士论文，2023 年]。
{"title":"A chromatic vanishing result for TR","authors":"Liam Keenan, Jonas McCandless","doi":"10.1090/proc/16840","DOIUrl":"https://doi.org/10.1090/proc/16840","url":null,"abstract":"<p>In this note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T left-parenthesis k right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-local TR vanishes on connective <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript n Superscript p comma f\"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">L_n^{p,f}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-acyclic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper E 1\"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"double-struck\">E</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">mathbb {E}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rings for every <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 less-than-or-equal-to k less-than-or-equal-to n\"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">1 leq k leq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and deduce consequences for connective Morava K-theory and the Thom spectra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"y left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">y(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal infinity\"> <mml:semantics> <mml:mi mathvariant=\"normal\">∞</mml:mi> <mml:annotation encoding=\"application/x-tex\">infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories which was recently established by Córdova Fedeli [<italic>Topological Hochschild homology of adic rings</italic>, Ph.D. thesis, University of Copenhagen, 2023].</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530999","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 investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of “close” abelian cubic number fields with class numbers as large as possible. We also give a first step toward an explicit lower bound for such extreme values of class numbers of abelian cubic fields.
{"title":"On abelian cubic fields with large class number","authors":"Jérémy Dousselin","doi":"10.1090/proc/16827","DOIUrl":"https://doi.org/10.1090/proc/16827","url":null,"abstract":"<p>We investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of “close” abelian cubic number fields with class numbers as large as possible. We also give a first step toward an explicit lower bound for such extreme values of class numbers of abelian cubic fields.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531000","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}
A new highly symmetrical model of the compact Lie algebra g2cmathfrak {g}^c_2 is provided as a twisted ring group for the group Z23mathbb {Z}_2^3 and the ring R⊕Rmathbb {R}oplus mathbb {R}. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in su(2)mathfrak {su}(2) and of the Gell-Mann matrices in su(3)mathfrak {su}(3). As a bonus, the split Lie algebra
本文提供了紧凑李代数 g 2 c mathfrak {g}^c_2 的一个新的高度对称模型，作为 Z 2 3 mathbb {Z}_2^3 群和环 R ⊕ R mathbb {R}mathbb {R}oplus mathbb {R} 的一个扭曲环群。这个模型是自足的，可以在没有关于根、八元数的推导或交叉积的知识的情况下使用。特别是，它提供了一个具有整数结构常量的正交基础，完全由半简单元素组成，是 s u ( 2 ) mathfrak {su}(2) 中的保利矩阵和 s u ( 3 ) mathfrak {su}(3) 中的盖尔-曼矩阵的广义化。作为奖励，分裂的李代数 g 2 （mathfrak {g}_2 ）也被视为一个扭曲的环群。
{"title":"The compact exceptional Lie algebra 𝔤^{𝔠}₂ as a twisted ring group","authors":"Cristina Draper","doi":"10.1090/proc/16821","DOIUrl":"https://doi.org/10.1090/proc/16821","url":null,"abstract":"<p>A new highly symmetrical model of the compact Lie algebra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German g 2 Superscript c\"> <mml:semantics> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\"fraktur\">g</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">mathfrak {g}^c_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is provided as a twisted ring group for the group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z 2 cubed\"> <mml:semantics> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">mathbb {Z}_2^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the ring <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R circled-plus double-struck upper R\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mo>⊕</mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {R}oplus mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German s German u left-parenthesis 2 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">s</mml:mi> <mml:mi mathvariant=\"fraktur\">u</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathfrak {su}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of the Gell-Mann matrices in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German s German u left-parenthesis 3 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">s</mml:mi> <mml:mi mathvariant=\"fraktur\">u</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathfrak {su}(3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a bonus, the split Lie algebra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German g","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744660","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 investigate relations between the pseudo-orbit-tracing property, topological stability and openness for tree-shifts. We prove that a tree-shift is of finite type if and only if it has the pseudo-orbit-tracing property which implies that the tree-shift is topologically stable and all shift maps are open. We also present an example of a tree-shift for which all shift maps are open but which is not of finite type. It also turns out that if a topologically stable tree-shift does not have isolated points then it is of finite type.
{"title":"On the stability and shadowing of tree-shifts of finite type","authors":"Dawid Bucki","doi":"10.1090/proc/16831","DOIUrl":"https://doi.org/10.1090/proc/16831","url":null,"abstract":"<p>We investigate relations between the pseudo-orbit-tracing property, topological stability and openness for tree-shifts. We prove that a tree-shift is of finite type if and only if it has the pseudo-orbit-tracing property which implies that the tree-shift is topologically stable and all shift maps are open. We also present an example of a tree-shift for which all shift maps are open but which is not of finite type. It also turns out that if a topologically stable tree-shift does not have isolated points then it is of finite type.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516765","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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