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":"28 1","pages":""},"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}
<p>In this paper, we make the first attempt to <italic>figure out</italic> the differences on Hölder regularity in time of solutions and conserved physical quantities between the ideal electron magnetohydrodynamic equations concerning Hall term and the incompressible Euler equations involving convection term. It is shown that the regularity in time of magnetic field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript t Superscript StartFraction alpha Over 2 EndFraction"> <mml:semantics> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mi>α</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">C_{t}^{frac {alpha }2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provided it belongs to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript t Superscript normal infinity Baseline upper C Subscript x Superscript alpha"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">L_{t}^{infty } C_{x}^{alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">alpha >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, its energy is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript t Superscript StartFraction 2 alpha Over 2 minus alpha EndFraction"> <mml:semantics> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">C_{t}^{frac {2alpha }{2-alpha }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as long as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:
在本文中，我们首次尝试找出涉及霍尔项的理想电子磁流体动力学方程与涉及对流项的不可压缩欧拉方程在解的时间霍尔德正则性和守恒物理量上的差异。研究表明，磁场 B B 的时间正则性为 C t α 2 C_{t}^{frac {alpha }2} ，条件是它属于 L t ∞ C x α L_{t}^{infty }。C_{x}^{alpha } for any α > 0 alpha >0 , its energy is C t 2 α 2 - α C_{t}^{frac {2alpha }{2-alpha }} as long as B B belongs to L t ∞ B ˙ 3 , ∞ α L_{t}^{infty }dot {B}^{alpha }_{3,infty } for any 0 > α > 1 0>alpha >；1 ，其磁螺旋度为 C t 2 α + 1 2 - α C_{t}^{frac {2alpha +1}{2-alpha }}，假设 B B 属于 L t ∞ B ˙ 3 ，∞ α L_{t}^{infty }。dot {B}^{alpha }_{3,infty } for any 0 > α > 1 2 0>alpha >frac 12 , 这与经典的不可压缩欧拉方程完全不同。
{"title":"Hölder regularity of solutions and physical quantities for the ideal electron magnetohydrodynamic equations","authors":"Yanqing Wang, Jitao Liu, Guoliang He","doi":"10.1090/proc/16829","DOIUrl":"https://doi.org/10.1090/proc/16829","url":null,"abstract":"<p>In this paper, we make the first attempt to <italic>figure out</italic> the differences on Hölder regularity in time of solutions and conserved physical quantities between the ideal electron magnetohydrodynamic equations concerning Hall term and the incompressible Euler equations involving convection term. It is shown that the regularity in time of magnetic field <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=\"application/x-tex\">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Subscript t Superscript StartFraction alpha Over 2 EndFraction\"> <mml:semantics> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mi>α</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">C_{t}^{frac {alpha }2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provided it belongs to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript t Superscript normal infinity Baseline upper C Subscript x Superscript alpha\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L_{t}^{infty } C_{x}^{alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">alpha >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, its energy is <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Subscript t Superscript StartFraction 2 alpha Over 2 minus alpha EndFraction\"> <mml:semantics> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">C_{t}^{frac {2alpha }{2-alpha }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as long as <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\"> <mml:semantics> <mml:","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"192 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516769","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 purpose of the present article is to provide an upper bound of the number of the negative eigenvalues of a generalized Schrödinger operator defined on a finite compact metric tree.
本文旨在提供定义在有限紧凑度量树上的广义薛定谔算子的负特征值个数的上限。
{"title":"On the number of the negative eigenvalues on a finite compact metric tree","authors":"Mohammed El Aïdi","doi":"10.1090/proc/16822","DOIUrl":"https://doi.org/10.1090/proc/16822","url":null,"abstract":"<p>The purpose of the present article is to provide an upper bound of the number of the negative eigenvalues of a generalized Schrödinger operator defined on a finite compact metric tree.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"119 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938831","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}
Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action.
{"title":"Categorifying equivariant monoids","authors":"Daniel Graves","doi":"10.1090/proc/16832","DOIUrl":"https://doi.org/10.1090/proc/16832","url":null,"abstract":"<p>Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"16 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516639","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 establish the local-in-time existence of weak solutions to the kinetic Cucker–Smale model with singular communication weights ϕ(x)=|x|−αphi (x) = |x|^{-alpha } with α∈(0,d)alpha in (0,d). In the case α∈(0,d−1]alpha in (0, d-1], we also provide the uniqueness of weak solutions extending the work of Carrillo et al [MMCS, ESAIM Proc. Surveys, vol. 47, EDP Sci., Les Ulis, 2014, pp. 17–35] where the existence and uniqueness of weak solutions are studied for α∈(0,d−1)alpha in (0,d-1).
我们建立了具有奇异通信权重 ϕ ( x ) = | x | - α phi (x) = |x|^{-alpha } 且 α ∈ ( 0 , d ) alpha in (0,d) 的动力学 Cucker-Smale 模型弱解的局部时间内存在性。在 α∈ ( 0 , d - 1 ] 的情况下 (0, d-1] . 我们还提供了弱解的唯一性，扩展了 Carrillo 等人的工作[MMCS, ESAIM Proc. Surveys, vol. 47, EDP Sci., Les Ulis, 2014, pp.
{"title":"On weak solutions to the kinetic Cucker–Smale model with singular communication weights","authors":"Young-Pil Choi, Jinwook Jung","doi":"10.1090/proc/16837","DOIUrl":"https://doi.org/10.1090/proc/16837","url":null,"abstract":"<p>We establish the local-in-time existence of weak solutions to the kinetic Cucker–Smale model with singular communication weights <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"phi left-parenthesis x right-parenthesis equals StartAbsoluteValue x EndAbsoluteValue Superscript negative alpha\"> <mml:semantics> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">phi (x) = |x|^{-alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha element-of left-parenthesis 0 comma d right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">alpha in (0,d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the case <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha element-of left-parenthesis 0 comma d minus 1 right-bracket\"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</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\">alpha in (0, d-1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we also provide the uniqueness of weak solutions extending the work of Carrillo et al [MMCS, ESAIM Proc. Surveys, vol. 47, EDP Sci., Les Ulis, 2014, pp. 17–35] where the existence and uniqueness of weak solutions are studied for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha element-of left-parenthesis 0 comma d minus 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</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\">alpha in (0,d-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"2 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516767","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}
This paper studies the combinatorial Calabi flow for circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. By using a Lyapunov function, we show that the flow exists for all time and converges exponentially fast to a circle pattern metric with prescribed attainable curvatures. This provides an algorithm to search for the desired circle patterns.
{"title":"Combinatorial Calabi flow on surfaces of finite topological type","authors":"Shengyu Li, Qianghua Luo, Yaping Xu","doi":"10.1090/proc/16839","DOIUrl":"https://doi.org/10.1090/proc/16839","url":null,"abstract":"<p>This paper studies the combinatorial Calabi flow for circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. By using a Lyapunov function, we show that the flow exists for all time and converges exponentially fast to a circle pattern metric with prescribed attainable curvatures. This provides an algorithm to search for the desired circle patterns.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"25 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780243","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 prove a vanishing theorem concerning the periods of cuspidal automorphic sheaves for GLm+noperatorname {GL}_{m+n} along the Levi subgroup GLm×GLnoperatorname {GL}_{m}times operatorname {GL}_{n} for m≠nm neq n.
在本文中，我们证明了一个关于 GL m + n operatorname {GL}_{m+n} 沿着 Levi 子群 GL m × GL n operatorname {GL}_{m} operatorname {GL}_{n} 的 m ≠ n m neq n 的 cuspidal 自动形剪周期的消失定理。
{"title":"Vanishing linear periods of cuspidal automorphic sheaves for 𝐺𝐿_{𝑚+𝑛}","authors":"Fang Shi","doi":"10.1090/proc/16836","DOIUrl":"https://doi.org/10.1090/proc/16836","url":null,"abstract":"<p>In this paper, we prove a vanishing theorem concerning the periods of cuspidal automorphic sheaves for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L Subscript m plus n\"> <mml:semantics> <mml:msub> <mml:mi>GL</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=\"application/x-tex\">operatorname {GL}_{m+n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> along the Levi subgroup <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L Subscript m Baseline times upper G upper L Subscript n\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>GL</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>GL</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">operatorname {GL}_{m}times operatorname {GL}_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m not-equals n\"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≠</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">m neq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"24 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059895","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 coarse maps between countable metric spaces of bounded geometry induce natural transformations of sufficiently good endofunctors of C∗C^{*}-algebras and prove that this correspondence is invariant with respect to coarse homotopies.
我们证明，有界几何的可数度量空间之间的粗映射会诱发 C ∗ C^{*} 的足够好的内函数的自然变换，并证明这种对应关系在粗同调方面是不变的。 -代数，并证明这种对应关系在粗同调方面是不变的。
{"title":"Roe functors preserve homotopies","authors":"Georgii Makeev","doi":"10.1090/proc/16824","DOIUrl":"https://doi.org/10.1090/proc/16824","url":null,"abstract":"<p>We show that coarse maps between countable metric spaces of bounded geometry induce natural transformations of sufficiently good endofunctors of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">C^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras and prove that this correspondence is invariant with respect to coarse homotopies.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"17 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516766","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 the Gauduchon metric g0g_0 of a compact locally conformally product manifold (M,c,D)(M,c,D) of dimension greater than 22 is adapted, in the sense that the Lee form of DD with respect to g0g_0 vanishes on the DD-flat distribution of MM. We also characterize adapted metrics as critical points of a natural functional defined on the conformal class.
我们证明，维度大于 2 2 的紧凑局部保角积流形 ( M , c , D ) (M,c,D) 的高都松度量 g 0 g_0 是自适应的，即 D D 关于 g 0 g_0 的李形式在 M M 的 D D 平面分布上消失。我们还将适配度量描述为定义在共形类上的自然函数的临界点。
{"title":"Adapted metrics on locally conformally product manifolds","authors":"Andrei Moroianu, Mihaela Pilca","doi":"10.1090/proc/16706","DOIUrl":"https://doi.org/10.1090/proc/16706","url":null,"abstract":"<p>We show that the Gauduchon metric <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g 0\"> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a compact locally conformally product manifold <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M comma c comma upper D right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M,c,D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension greater than <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=\"application/x-tex\">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is adapted, in the sense that the Lee form of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g 0\"> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanishes on the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-flat distribution of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also characterize adapted metrics as critical points of a natural functional defined on the conformal class.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"118 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938843","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}
G. Bessa, Vicent Gimeno i Garcia, Leandro Pessoa, Alberto Setti
In this short note we show that Doyle-Grigor’yan criterion for non-parabolicity is not necessary in dimension greater than or equal to four. This gives an negative answer to Problem # 1 of Grigor’yan [Bull. Amer. Math. Soc. (N.S) 36 (1999). pp. 135–249] in this dimensional range.
在这篇短文中，我们证明了在大于或等于四维时，多伊尔-格里高利的非抛物线判据是不必要的。这给出了格里高利问题 #1 [Bull. Amer. Math. Soc. (N.S.) 36 (1999). pp.
{"title":"On Doyle-Grigor’yan criterion for non-parabolicity","authors":"G. Bessa, Vicent Gimeno i Garcia, Leandro Pessoa, Alberto Setti","doi":"10.1090/proc/16804","DOIUrl":"https://doi.org/10.1090/proc/16804","url":null,"abstract":"<p>In this short note we show that Doyle-Grigor’yan criterion for non-parabolicity is not necessary in dimension greater than or equal to four. This gives an negative answer to Problem # 1 of Grigor’yan [Bull. Amer. Math. Soc. (N.S) 36 (1999). pp. 135–249] in this dimensional range.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"94 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938849","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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