Proceedings of the American Mathematical Society最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}