We introduce the Gelfand-Kirillov base for a numerical semigroup ring over the prime field of characteristic pp, where pp is prime, and show its existence for the semigroup ring of the ordinary cusp by establishing a growth recurrence with respect to Frobenius.
我们介绍了特征为 p p 的素域上的数值半群环的格尔芬-基里洛夫基，其中 p p 是素数，并通过建立关于弗罗贝纽斯的增长递推关系，证明了普通尖顶半群环的格尔芬-基里洛夫基的存在性。
{"title":"The growth recurrence and Gelfand-Kirillov base of the ordinary cusp","authors":"Alan Dills, Florian Enescu","doi":"10.1090/proc/16913","DOIUrl":"https://doi.org/10.1090/proc/16913","url":null,"abstract":"<p>We introduce the Gelfand-Kirillov base for a numerical semigroup ring over the prime field of characteristic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is prime, and show its existence for the semigroup ring of the ordinary cusp by establishing a growth recurrence with respect to Frobenius.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"84 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945112","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>We consider the existence of invariant manifolds to evolution equations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u prime left-parenthesis t right-parenthesis equals upper A u left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>u</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u’(t)=Au(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A colon upper D left-parenthesis upper A right-parenthesis subset-of double-struck upper X right-arrow double-struck upper X"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⊂</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A:D(A)subset mathbb {X}to mathbb {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> near its equilibrium <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis 0 right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">A(0)=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the assumption that its proto-derivative <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper A left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">partial A(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exists and is continuous in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper D left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">xin D(A)</mml:annota
我们考虑演化方程 u ′ ( t ) = A u ( t ) u'(t)=Au(t) , A : D ( A ) ⊂ X → X A 的不变流形的存在性：D(A)subset mathbb {X}to mathbb {X} near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) partial A(x) exists and is continuous in x∈ D ( A ) xin D(A) in the sense of Yosida distance.巴拿赫空间 X 中两个（无界）线性算子 U U 和 V V 之间的约西达距离定义为 d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ - V μ ‖ d_Y(U,V)≔limsup _{mu to +infty }。| U_mu -V_mu | ，其中 U μ U_mu 和 V μ V_mu 分别是 U U 和 V V 的约西达近似值。我们证明，如果 ∂ A partial A 的原支数在 Yosida 距离意义上是连续的，则上述方程在指数二分均衡附近具有局部稳定和不稳定的不变流形。约西达距离方法使我们能够推广众所周知的结果，并可能应用于更大类的偏微分方程和函数微分方程。所获得的结果似乎是新的。
{"title":"Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems","authors":"Xuan-Quang Bui, Nguyen Van Minh","doi":"10.1090/proc/16912","DOIUrl":"https://doi.org/10.1090/proc/16912","url":null,"abstract":"<p>We consider the existence of invariant manifolds to evolution equations <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u prime left-parenthesis t right-parenthesis equals upper A u left-parenthesis t right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>u</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">u’(t)=Au(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A colon upper D left-parenthesis upper A right-parenthesis subset-of double-struck upper X right-arrow double-struck upper X\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>⊂</mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">X</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">→</mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">X</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A:D(A)subset mathbb {X}to mathbb {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> near its equilibrium <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A left-parenthesis 0 right-parenthesis equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A(0)=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the assumption that its proto-derivative <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential upper A left-parenthesis x right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">partial A(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exists and is continuous in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x element-of upper D left-parenthesis upper A right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">xin D(A)</mml:annota","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"42 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945110","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 Diff∂(Dn)operatorname {Diff}_{partial }(D^{n}) be the topological group of diffeomorphisms of DnD^{n} which agree with the identity near the boundary. In this short note, we compute the fundamental group π1Diff∂(D4k)pi _1 operatorname {Diff}_{partial }(D^{4k}) for k≥3kgeq 3.
让 Diff ∂ ( D n ) （operatorname {Diff}_{partial }(D^{n}）是 D n D^{n} 的差分变形的拓扑群，它与边界附近的同一性一致。在这篇短文中，我们计算了 k ≥ 3 kgeq 3 时的基群 π 1 Diff ∂ ( D 4 k ) pi _1 operatorname {Diff}_{partial }(D^{4k}) 。
{"title":"A short note on 𝜋₁(𝐷𝑖𝑓𝑓_{∂}𝐷^{4𝑘}) for 𝑘≥3","authors":"Wei Wang","doi":"10.1090/proc/16908","DOIUrl":"https://doi.org/10.1090/proc/16908","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D i f f Subscript partial-differential Baseline left-parenthesis upper D Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Diff</mml:mi> <mml:mrow> <mml:mi mathvariant=\"normal\">∂</mml:mi> </mml:mrow> </mml:msub> <mml:mo></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">operatorname {Diff}_{partial }(D^{n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the topological group of diffeomorphisms of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D Superscript n\"> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">D^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which agree with the identity near the boundary. In this short note, we compute the fundamental group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi 1 upper D i f f Subscript partial-differential Baseline left-parenthesis upper D Superscript 4 k Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>Diff</mml:mi> <mml:mrow> <mml:mi mathvariant=\"normal\">∂</mml:mi> </mml:mrow> </mml:msub> <mml:mo></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">pi _1 operatorname {Diff}_{partial }(D^{4k})</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=\"k greater-than-or-equal-to 3\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">kgeq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"41 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141880494","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 obtain a Gaussian concentration inequality for a class of non-Lipschitz functions. In the one-dimensional case, our results supplement those established by Paouris and Valettas [Ann. Probab. 46 (2018), pp. 1441–1454].
{"title":"A variance-sensitive Gaussian concentration inequality","authors":"Nguyen Dung","doi":"10.1090/proc/16905","DOIUrl":"https://doi.org/10.1090/proc/16905","url":null,"abstract":"<p>In this note, we obtain a Gaussian concentration inequality for a class of non-Lipschitz functions. In the one-dimensional case, our results supplement those established by Paouris and Valettas [Ann. Probab. 46 (2018), pp. 1441–1454].</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"64 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744658","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>The Petrie conjecture asserts that if a homotopy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C double-struck upper P Superscript n"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {CP}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a non-trivial circle action, its Pontryagin class agrees with that of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C double-struck upper P Superscript n"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {CP}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Petrie proved this conjecture in the case where the manifold admits a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript n"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">T^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-action. An almost complex torus manifold is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional compact connected almost complex manifold equipped with an effective <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript n"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">T^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional almost complex torus manifold <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>
Petrie 猜想断言，如果一个同调 C P n mathbb {CP}^n 承认一个非三维圆作用，那么它的 Pontryagin 类与 C P n mathbb {CP}^n 的 Pontryagin 类一致。Petrie 在流形接受 T n T^n 作用的情况下证明了这一猜想。几乎复环流形是一个 2 n 2n 维紧凑连通的几乎复环流形，它配备了一个有效的 T n T^n 作用，该作用具有定点。对于几乎复杂的环流形，存在一种图，可以编码定点处的权重信息。我们证明，如果一个 2 n 2 n 维的近乎复环流形 M M 只与复投影空间 C P n mathbb {CP}^n 共享欧拉数，则 M M 的图与 C P n mathbb {CP}^n 上的线性 T n T^n 作用的图一致。因此，M M 与 C P n mathbb {CP}^n 具有相同的定点权重、切尔数、共线性类、Hirzebruch χ y chi _y -属、Todd 属和签名，并赋予标准线性作用。此外，如果 M M 是等变形式的，那么 M M 和 C P n mathbb {CP}^n 的等变同调与切尔恩类也是一致的。
{"title":"Almost complex torus manifolds - a problem of Petrie type","authors":"Donghoon Jang","doi":"10.1090/proc/16768","DOIUrl":"https://doi.org/10.1090/proc/16768","url":null,"abstract":"<p>The Petrie conjecture asserts that if a homotopy <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C double-struck upper P Superscript n\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">C</mml:mi> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {CP}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a non-trivial circle action, its Pontryagin class agrees with that of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C double-struck upper P Superscript n\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">C</mml:mi> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {CP}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Petrie proved this conjecture in the case where the manifold admits a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Superscript n\"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">T^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-action. An almost complex torus manifold is a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 n\"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional compact connected almost complex manifold equipped with an effective <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Superscript n\"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">T^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 n\"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional almost complex torus manifold <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>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"121 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059893","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 general elliptic equations with singular/degenerate matrix weights and measurable nonlinearities on nonsmooth bounded domains to obtain a global Calderón-Zygmund type estimate under possibly minimal assumptions that the logarithm of the matrix weight has a small bounded mean oscillation (BMO) norm, the nonlinearity is allowed to be merely measurable in one variable but has a small BMO norm in the other variables and that the boundary of the domain is sufficiently flat in Reifenberg sense.
{"title":"Elliptic equations with matrix weights and measurable nonlinearities on nonsmooth domains","authors":"Sun-Sig Byun, Yumi Cho, Ho-Sik Lee","doi":"10.1090/proc/16770","DOIUrl":"https://doi.org/10.1090/proc/16770","url":null,"abstract":"<p>We study general elliptic equations with singular/degenerate matrix weights and measurable nonlinearities on nonsmooth bounded domains to obtain a global Calderón-Zygmund type estimate under possibly minimal assumptions that the logarithm of the matrix weight has a small bounded mean oscillation (BMO) norm, the nonlinearity is allowed to be merely measurable in one variable but has a small BMO norm in the other variables and that the boundary of the domain is sufficiently flat in Reifenberg sense.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"23 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151832","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}
Consider a unimodular random graph, or just a finitely generated Cayley graph. When does its cycle space have an invariant random generating set of cycles such that every edge is contained in finitely many of the cycles? Generating the free Loop O(1)O(1) model as a factor of iid is closely connected to having such a generating set for FK-Ising cluster. We show that geodesic cycles do not always form such a generating set, by showing for a parameter regime of the FK-Ising model on the lamplighter group the origin is contained in infinitely many geodesic cycles. This answers a question by Angel, Ray and Spinka. Then we take a look at how the property of having an invariant locally finite generating set for the cycle space is preserved by Bernoulli percolation, and apply it to the problem of factor of iid presentations of the free Loop O(1)O(1) model.
{"title":"On invariant generating sets for the cycle space","authors":"Ádám Timár","doi":"10.1090/proc/16910","DOIUrl":"https://doi.org/10.1090/proc/16910","url":null,"abstract":"<p>Consider a unimodular random graph, or just a finitely generated Cayley graph. When does its cycle space have an invariant random generating set of cycles such that every edge is contained in finitely many of the cycles? Generating the free Loop <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> model as a factor of iid is closely connected to having such a generating set for FK-Ising cluster. We show that geodesic cycles do not always form such a generating set, by showing for a parameter regime of the FK-Ising model on the lamplighter group the origin is contained in infinitely many geodesic cycles. This answers a question by Angel, Ray and Spinka. Then we take a look at how the property of having an invariant locally finite generating set for the cycle space is preserved by Bernoulli percolation, and apply it to the problem of factor of iid presentations of the free Loop <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> model.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"79 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197103","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 work we examine the existence of positive classical solutions of <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column minus normal upper Delta u plus u equals StartAbsoluteValue x EndAbsoluteValue Superscript alpha Baseline u Superscript p minus 1 Baseline 2nd Column a m p semicolon in upper B 1 comma 2nd Row 1st Column u greater-than 0 2nd Column a m p semicolon in upper B 1 comma 3rd Row 1st Column partial-differential Subscript nu Baseline u equals 0 2nd Column a m p semicolon on partial-differential upper B 1 comma EndLayout"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <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:mi>α</mml:mi> </mml:msup> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> in </mml:mtext> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> in </mml:mtext> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> on </mml:mtext> <mml:mi mathvariant="normal">∂</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence="true" stretchy="true" symmetric="true"/> </mml:mrow> <mml:annotation encoding="application/x-tex">begin{equation*} begin {cases} -Delta u +u = |x|^alpha u^{p-1} & text { in } B_1, u>0 & text { in } B_1, partial _nu u= 0 & text { on } partial B_1, end{cases} end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xm
在这项工作中，我们考察了 { - Δ u + u = | x |α u p - 1 a m p ; in B 1 , u > 0 a m p ; in B 1 , ∂ ν u = 0 a m p ; on ∂ B 1 , begin{equation*} 的正经典解的存在性。begin {cases} -Delta u +u = |x|^alpha u^{p-1} & text { in }B_1, u>0 & text { in }B_1, partial _nu u= 0 & text { on }B_1, end{cases}end{equation*} 其中 p > 1 p>1 , α > 0 alpha >0 和 B 1 B_1 是 R N {mathbb {R}}^N 中的单位球，其中 N ≥ 4 N ge 4 并且是偶数。我们尤其关注非径向位置经典解的存在。我们证明，在 p , α p,alpha 和 N N 的适当条件下，存在正的经典非径向解。我们的方法是在合适的凸锥上利用变分法。
{"title":"Nonradial solutions of a Neumann Hénon equation on a ball","authors":"Craig Cowan","doi":"10.1090/proc/16897","DOIUrl":"https://doi.org/10.1090/proc/16897","url":null,"abstract":"<p>In this work we examine the existence of positive classical solutions of <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartLayout Enlarged left-brace 1st Row 1st Column minus normal upper Delta u plus u equals StartAbsoluteValue x EndAbsoluteValue Superscript alpha Baseline u Superscript p minus 1 Baseline 2nd Column a m p semicolon in upper B 1 comma 2nd Row 1st Column u greater-than 0 2nd Column a m p semicolon in upper B 1 comma 3rd Row 1st Column partial-differential Subscript nu Baseline u equals 0 2nd Column a m p semicolon on partial-differential upper B 1 comma EndLayout\"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\"> <mml:mtr> <mml:mtd> <mml:mo>−</mml:mo> <mml:mi mathvariant=\"normal\">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <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:mi>α</mml:mi> </mml:msup> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> in </mml:mtext> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> in </mml:mtext> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> on </mml:mtext> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\"/> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">begin{equation*} begin {cases} -Delta u +u = |x|^alpha u^{p-1} & text { in } B_1, u>0 & text { in } B_1, partial _nu u= 0 & text { on } partial B_1, end{cases} end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xm","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"363 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872176","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 minimizers of the LdL^{d}-norm of the Hessian in the unit ball of Rdmathbb {R}^d are locally of class C1,αC^{1,alpha }. Our findings extend previous results on Hessian-dependent functionals to the borderline case and resonate with the Hölder regularity theory available for elliptic equations in double-divergence form.
我们证明，在 R d mathbb {R}^d 的单位球中，Hessian 的 L d L^{d} 准则的最小值局部属于 C 1 类，α C^{1,alpha }。 .我们的发现将之前关于依赖于 Hessian 的函数的结果扩展到了边界情况，并与双发散形式椭圆方程的赫尔德正则性理论产生了共鸣。
{"title":"Improved regularity for a Hessian-dependent functional","authors":"Vincenzo Bianca, Edgard Pimentel, José Urbano","doi":"10.1090/proc/16894","DOIUrl":"https://doi.org/10.1090/proc/16894","url":null,"abstract":"<p>We prove that minimizers of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript d\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^{d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm of the Hessian in the unit ball of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript d\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are locally of class <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript 1 comma alpha\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">C^{1,alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our findings extend previous results on Hessian-dependent functionals to the borderline case and resonate with the Hölder regularity theory available for elliptic equations in double-divergence form.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"64 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945111","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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