We prove an ΩOmega-result for the quadratic Dirichlet LL-function |L(1/2,χP)||L(1/2, chi _P)| over irreducible polynomials PP associated with the hyperelliptic curve of genus gg over a fixed finite field Fqmathbb {F}_q in the large genus limit. In particular, we showed that for any
我们证明了二次 Dirichlet L L -函数 | L ( 1 / 2 , χ P ) | L(1/2, chi _P)| 上不可还原多项式 P P 的 Ω Omega 结果。 |L(1/2, chi _P)| 与大属极限中固定有限域 F q 上属 g g 的超椭圆曲线 P P 相关的不可约多项式。特别是，我们证明了对于任何 ∈ ( 0 , 1 / 2 ) epsilon in (0, 1/2) , [ max P ∈ P 2 g + 1 | L ( 1 / 2 , χ P ) ≫ exp ( ( 1 / 2 - ϵ ) ln q + o ( 1 ) ) g ln 2 g ln g ) , max _{substack {Pin mathcal {P}_{2g+1}}}||L(1/2, chi _P)|gg exp left (left (sqrt {left (1/2-epsilon right )ln q}+o(1)right )sqrt {frac {g ln _2 g}{ln g}}right )、 其中 P 2 g + 1 {P}_{2g+1} 是所有度数为 2 g + 1 2g+1 的一元不可约多项式的集合。这与邦达连科-塞普边界的数量级相吻合。
{"title":"Large values of quadratic Dirichlet 𝐿-functions over monic irreducible polynomial in 𝔽_{𝕢}[𝕥]","authors":"Pranendu Darbar, Gopal Maiti","doi":"10.1090/proc/16828","DOIUrl":"https://doi.org/10.1090/proc/16828","url":null,"abstract":"<p>We prove an <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Ω</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Omega</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-result for the quadratic Dirichlet <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\u0000 <mml:semantics>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-function <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue upper L left-parenthesis 1 slash 2 comma chi Subscript upper P Baseline right-parenthesis EndAbsoluteValue\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>χ</mml:mi>\u0000 <mml:mi>P</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">|L(1/2, chi _P)|</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over irreducible polynomials <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P\">\u0000 <mml:semantics>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">P</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> associated with the hyperelliptic curve of genus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g\">\u0000 <mml:semantics>\u0000 <mml:mi>g</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">g</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over a fixed finite field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper F Subscript q\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>q</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {F}_q</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the large genus limit. In particular, we showed that for any <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon element-of left-parenthesis 0 comma 1 slash 2 right-parenthesis\">\u0000 ","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141339598","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 compute the Hausdorff dimension of the set of ψpsi-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than 22 and for approximating functions ψpsi with order at infinity less than or equal to −2-2. Our method relies on the analogous result in dimension 11, proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.
{"title":"A remark on the set of exactly approximable vectors in the simultaneous case","authors":"Reynold Fregoli","doi":"10.1090/proc/16790","DOIUrl":"https://doi.org/10.1090/proc/16790","url":null,"abstract":"<p>We compute the Hausdorff dimension of the set of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"psi\">\u0000 <mml:semantics>\u0000 <mml:mi>ψ</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">psi</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and for approximating functions <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"psi\">\u0000 <mml:semantics>\u0000 <mml:mi>ψ</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">psi</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with order at infinity less than or equal to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"negative 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>−</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">-2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Our method relies on the analogous result in dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\u0000 <mml:semantics>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141339471","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, given a prescribed measure on S1mathbb {S}^1 whose density is bounded and positive, we establish a uniform diameter estimate for solutions to the planar LpL_p dual Minkowski problem when 0>p>10>p>1 and q≥2qge 2. We also prove the uniqueness and positivity of solutions to the LpL_p Minkowski problem when the density of the measure is sufficiently close to a constant in CαC^alpha.
在本文中，给定 S 1 mathbb {S}^1 上密度有界且为正的规定度量，当 0 > p > 1 0>p>1 且 q≥ 2 qge 2 时，我们建立了平面 L p L_p 对偶闵科夫斯基问题解的均匀直径估计。我们还证明了当度量密度足够接近 C α C^alpha 中的一个常数时，L p L_p Minkowski 问题解的唯一性和实在性。
{"title":"Diameter estimate for planar 𝐿_{𝑝} dual Minkowski problem","authors":"Minhyun Kim, Taehun Lee","doi":"10.1090/proc/16464","DOIUrl":"https://doi.org/10.1090/proc/16464","url":null,"abstract":"<p>In this paper, given a prescribed measure on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper S Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {S}^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> whose density is bounded and positive, we establish a uniform diameter estimate for solutions to the planar <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">L_p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> dual Minkowski problem when <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 greater-than p greater-than 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">0>p>1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q greater-than-or-equal-to 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mo>≥</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">qge 2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We also prove the uniqueness and positivity of solutions to the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">L_p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> Minkowski problem when the density of the measure is sufficiently close to a constant in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript alpha\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mi>α</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141109399","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 the context of ZF+DCmathsf {ZF}+mathsf {DC}, we force DCκmathsf {DC}_kappa for relations on P(κ)mathcal {P}(kappa ) for arbitrarily large κ>ℵωkappa >aleph _omega over the Chang model L(O
在 Z F + D C （ZF}+DC} 的背景下，我们强制 D C κ （DC}_kappa）为 P ( κ ) 上的mathcal{P}（kappa）关系。 对于任意大 κ > ℵ ω kappa >aleph _omega 在 Chang 模型 L ( O r d ω ) 上的关系，我们强制 D C κmathsf {DC}_kappa 对由 Þ 0 = ω Þ_0=omega 定义的刺序列做一些假设、 Þ α + 1 Þ{alpha +1} 作为不是 Þ α ω Þ_alpha ^omega 的射影的最小序数，并且 Þ γ = sup α > γ Þ α Þ_gamma =sup _{alpha >gamma }Þ_alpha 对于极限 γ gamma 。这些假设的动机来自于确定性背景下关于 Θ Theta 的结果，也可能是思考 Chang 模型的合理方法。明确地说，我们假定荆棘序列上的λ lambda 的后继点是强正则的--意思是正则的，并且只要 κ > λ kappa > lambda 在荆棘序列上，函数 f : κ > κ → λ f:kappa ^{>kappa }rightarrow lambda 都是有界的--意思是 P ( κ ω ) ∩ L ( O r d ω ) 是有理的。 ⊆ L λ ( λ ω )
{"title":"Forcing more 𝖣𝖢 over the Chang model using the Thorn sequence","authors":"James Holland, Grigor Sargsyan","doi":"10.1090/proc/16700","DOIUrl":"https://doi.org/10.1090/proc/16700","url":null,"abstract":"<p>In the context of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper Z sans-serif upper F plus sans-serif upper D sans-serif upper C\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"sans-serif\">Z</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"sans-serif\">D</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">C</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathsf {ZF}+mathsf {DC}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we force <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper D sans-serif upper C Subscript kappa\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"sans-serif\">D</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>κ</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathsf {DC}_kappa</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for relations on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P left-parenthesis kappa right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>κ</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {P}(kappa )</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for arbitrarily large <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"kappa greater-than normal alef Subscript omega\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>κ</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi mathvariant=\"normal\">ℵ</mml:mi>\u0000 <mml:mi>ω</mml:mi>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">kappa >aleph _omega</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over the Chang model <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper L left-parenthesis normal upper O normal r normal d Superscript omega Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">O</mml:mi>\u0000 <mml:mi mathvarian","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141109352","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}
J. Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy
In this paper we provide a detailed topological and measure-theoretic study of Lebesgue measure-preserving continuous circle maps that are composed with independent rotations on each of the sides. In particular, we analyze the stability of the locally eventually onto and measure-theoretic mixing properties.
{"title":"Are generic dynamical properties stable under composition with rotations?","authors":"J. Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy","doi":"10.1090/proc/16800","DOIUrl":"https://doi.org/10.1090/proc/16800","url":null,"abstract":"In this paper we provide a detailed topological and measure-theoretic study of Lebesgue measure-preserving continuous circle maps that are composed with independent rotations on each of the sides. In particular, we analyze the stability of the locally eventually onto and measure-theoretic mixing properties.","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141115764","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 only pseudo-spherical surface whose caustic becomes a minimal surface is Dini surface family. Moreover, we give the Weierstrass data for corresponding minimal surface to the caustic.
{"title":"On the minimality condition for caustics of pseudo-spherical surfaces","authors":"Yoshiki Jikumaru, Keisuke Teramoto","doi":"10.1090/proc/16780","DOIUrl":"https://doi.org/10.1090/proc/16780","url":null,"abstract":"We show that only pseudo-spherical surface whose caustic becomes a minimal surface is Dini surface family. Moreover, we give the Weierstrass data for corresponding minimal surface to the caustic.","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141118260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the 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":null,"pages":null},"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}
We consider the existence of invariant manifolds to evolution equations u′(t)=Au(t)u’(t)=Au(t), A:D(A)⊂X→XA:D(A)subset mathbb {X}to mathbb {X} near its equilibrium A(0)=0A(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)
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":null,"pages":null},"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}
C. Bellavita, N. Chalmoukis, V. Daskalogiannis, G. Stylogiannis
<p>For a finite, positive Borel measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding="application/x-tex">mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we consider an infinite matrix <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript mu"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Gamma _mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, related to the classical Hausdorff matrix defined by the same measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding="application/x-tex">mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding="application/x-tex">mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Lebesgue measure, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript mu"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Gamma _mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reduces to the classical Hilbert matrix. We prove that the matrices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript mu"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Gamma _mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are not Hankel, unless <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding="application/x-tex">mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a constant multiple of the Lebesgue measure, we give necessary and sufficien
对于 ( 0 , 1 ) (0 , 1) 上的有限正伯尔量μ mu，我们考虑一个无穷矩阵Γ μ Gamma _mu ，它与由相同量μ mu 定义的经典豪斯多夫矩阵相关，其代数方式与希尔伯特矩阵与塞萨罗矩阵相关的代数方式相同。当 μ mu 是 Lebesgue 度量时，Γ μ Gamma _mu 等同于经典的希尔伯特矩阵。我们证明了矩阵 Γ μ Gamma _mu 不是汉克尔矩阵，除非 μ mu 是 Lebesgue 量的常数倍，我们给出了它们在哈代空间 H p , 1 ≤ p > ∞ H^p, , 1 leq p > infty 的尺度上有界的必要条件和充分条件，并研究了它们的紧凑性和完全连续性。在 2 ≤ p > ∞ 2leq p>infty 的情况下，我们能够计算出算子的精确规范值。
{"title":"Generalized Hilbert operators arising from Hausdorff matrices","authors":"C. Bellavita, N. Chalmoukis, V. Daskalogiannis, G. Stylogiannis","doi":"10.1090/proc/16917","DOIUrl":"https://doi.org/10.1090/proc/16917","url":null,"abstract":"<p>For a finite, positive Borel measure <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding=\"application/x-tex\">mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 0 comma 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we consider an infinite matrix <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma Subscript mu\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">Gamma _mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, related to the classical Hausdorff matrix defined by the same measure <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding=\"application/x-tex\">mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding=\"application/x-tex\">mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Lebesgue measure, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma Subscript mu\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">Gamma _mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reduces to the classical Hilbert matrix. We prove that the matrices <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma Subscript mu\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">Gamma _mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are not Hankel, unless <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding=\"application/x-tex\">mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a constant multiple of the Lebesgue measure, we give necessary and sufficien","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197099","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":null,"pages":null},"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}
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