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Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball in ℝ2 ℝ2中单位球上特鲁丁格-莫泽函数极值的紧凑性
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-11-15 DOI: 10.1007/s10114-024-3046-9
Wei Wei Shan, Xiao Meng Li
{"title":"Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball in ℝ2","authors":"Wei Wei Shan,&nbsp;Xiao Meng Li","doi":"10.1007/s10114-024-3046-9","DOIUrl":"10.1007/s10114-024-3046-9","url":null,"abstract":"<div><p>Let <span>(mathbb{B})</span> be a unit ball in ℝ<sup>2</sup>, <span>(W_{0}^{1,2}(mathbb{B}))</span> be the standard Sobolev space. For any <i>ϵ</i> &gt; 0, de Figueiredo, do Ó, dos Santons, Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball. Precisely, </p><div><div><span>$$mathop {sup }limits_{u in W_0^{1,2}left( mathbb{B} right),int_mathbb{B} {|nabla u{|^2}dx le 1} } int_mathbb{B} {{{left| x right|}^{2epsilon }}{rm{e}^{4pi left( {1 + epsilon } right){u^2}}}} dx$$</span></div></div><p> can be attained by some radially symmetric function <span>(u_{epsilon}in W_{0}^{1,2}(mathbb{B}))</span> with <span>(int_{mathbb{B}}vertnabla u_{epsilon}vert^{2}dx=1)</span>. In this note, we concern the compactness of the function family {<i>u</i><sub><i>ϵ</i></sub>}<sub><i>ϵ</i>&gt;0</sub> and prove that up to a subsequence <i>u</i><sub><i>ϵ</i></sub> converges to some function <i>u</i>\u0000<sub>0</sub> in <span>(C^{1}(overline{mathbb{B}}))</span> as <i>ϵ</i> → 0. Furthermore, <i>u</i><sub>0</sub> is an extremal function of the supremum </p><div><div><span>$$mathop {sup }limits_{u in W_0^{1,2}left( mathbb{B} right),int_mathbb{B} {|nabla u{|^2}dx le 1} } int_{mathbb{B}}rm{e}^{4pi u^{2}}dx.$$</span></div></div><p> Let us explain the result in geometry. Denote <span>(omega_{0}=dx_{1}^{2}+dx_{2}^{2})</span> be the standard Euclidean metric. Define a conical metric <span>(omega_{epsilon}=vert xvert^{2epsilon}omega_{0})</span> for <span>(xinmathbb{B})</span>. Then the extremal family {<i>u</i>\u0000<sub><i>ϵ</i></sub>}<sub><i>ϵ</i>&gt;0</sub> of the following Trudinger-Moser functionals </p><div><div><span>$$int_{mathbb{B}}rm{e}^{4pi(1+epsilon)u^{2}}dv_{w_{epsilon}}$$</span></div></div><p> under the constraint <span>(uin W_{0}^{1,2}(mathbb{B}))</span> and <span>(int_{mathbb{B}}vertnabla_{omega_{epsilon}u}vert^{2}dv_{omega_{epsilon}}leq 1)</span> is compact as <i>ϵ</i> → 0.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 11","pages":"2840 - 2854"},"PeriodicalIF":0.8,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142694742","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}
引用次数: 0
On pj-rank of Even K-groups of Rings of Integers 论整数环偶数 K 群的 pj 级
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-10-15 DOI: 10.1007/s10114-024-1312-5
Meng Fai Lim
{"title":"On pj-rank of Even K-groups of Rings of Integers","authors":"Meng Fai Lim","doi":"10.1007/s10114-024-1312-5","DOIUrl":"10.1007/s10114-024-1312-5","url":null,"abstract":"<div><p>Let <i>L/F</i> be a finite Galois extension of number fields of degree <i>n</i> and let <i>p</i> be a prime which does not divide <i>n</i>. We shall study the <i>p</i><sup><i>j</i></sup>-rank of <span>(K_{2i}(mathcal{O}_{L}))</span> via its Galois module structure following the approaches of Iwasawa and Komatsu–Nakano. Along the way, we generalize previous observations of Browkin, Wu and Zhou on <i>K</i><sub>2</sub>-groups to higher even <i>K</i>-groups. We also give examples to illustrate our results. Finally, we apply our discussion to refine a result of Kitajima pertaining to the <i>p</i>-rank of even <i>K</i>-groups in the cyclotomic ℤ<sub><i>l</i></sub>-extension, where <i>l</i> ≠ <i>p</i>.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2481 - 2496"},"PeriodicalIF":0.8,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595981","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}
引用次数: 0
Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers 有根四元组表示范畴中诱导共旋对的完备性
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-18 DOI: 10.1007/s10114-024-3041-1
Zhen Xing Di, Li Ping Li, Li Liang, Fei Xu
{"title":"Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers","authors":"Zhen Xing Di,&nbsp;Li Ping Li,&nbsp;Li Liang,&nbsp;Fei Xu","doi":"10.1007/s10114-024-3041-1","DOIUrl":"10.1007/s10114-024-3041-1","url":null,"abstract":"<div><p>This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs (Φ(X), Φ(X)<sup>⊥</sup>) and (<sup>⊥</sup>Ψ(Y), Ψ(Y)) in Rep(<i>Q</i>, A)—the category of all A-valued representations of a quiver <i>Q</i>—are complete whenever (X, Y) is a complete cotorsion pair in an abelian category A satisfying some mild conditions. We give an affirmative answer if the quiver <i>Q</i> is rooted. As an application, we show under certain mild conditions that if a subcategory L, which is not necessarily closed under direct summands, of A is special precovering (resp., preenveloping), then Φ(L)(resp., Ψ(L)) is special precovering (resp., preenveloping) in Rep(<i>Q</i>, A).</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2436 - 2452"},"PeriodicalIF":0.8,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141824347","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}
引用次数: 0
On Some Sums Involving Small Arithmetic Functions 论涉及小算术函数的一些和
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-18 DOI: 10.1007/s10114-024-2129-y
Wen Guang Zhai
{"title":"On Some Sums Involving Small Arithmetic Functions","authors":"Wen Guang Zhai","doi":"10.1007/s10114-024-2129-y","DOIUrl":"10.1007/s10114-024-2129-y","url":null,"abstract":"<div><p>Let <i>f</i> be any arithmetic function and define <span>({S_{f}}(x):=sumnolimits_{{n le x}}f([x/n]))</span>. If the function <i>f</i> is small, namely, <i>f</i>(<i>n</i>) ≪ <i>n</i><sup><i>ε</i></sup>, then the error term <i>E</i><sub><i>f</i></sub>(<i>x</i>) in the asymptotic formula of <i>S</i><sub><i>f</i></sub>(<i>x</i>) has the form <i>O</i>(<i>x</i><sup>1/2+<i>ε</i></sup>). In this paper, we shall study the mean square of <i>E</i><sub><i>f</i></sub>(<i>x</i>) and establish some new results of <i>E</i><sub><i>f</i></sub>(<i>x</i>) for some special functions.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2497 - 2518"},"PeriodicalIF":0.8,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141825074","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}
引用次数: 0
Supnorm Estimates for ({bar partial}) on Product Domains in ℂn ℂn中积域上$${bar partial}$ 的超矩形估计值
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-18 DOI: 10.1007/s10114-024-2463-0
Martino Fassina, Yi Fei Pan
{"title":"Supnorm Estimates for ({bar partial}) on Product Domains in ℂn","authors":"Martino Fassina,&nbsp;Yi Fei Pan","doi":"10.1007/s10114-024-2463-0","DOIUrl":"10.1007/s10114-024-2463-0","url":null,"abstract":"<div><p>Using methods from complex analysis in one variable, we define an integral operator that solves <span>({bar partial})</span> with supnorm estimates on product domains in ℂ<sup><i>n</i></sup>.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2307 - 2323"},"PeriodicalIF":0.8,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141825922","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}
引用次数: 0
On Variational Principles of Metric Mean Dimension on Subsets in Feldman–Katok Metric 论费尔德曼-卡托克公设子集上公设平均维度的变分原理
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-10 DOI: 10.1007/s10114-024-2517-3
Kun Mei Gao, Rui Feng Zhang
{"title":"On Variational Principles of Metric Mean Dimension on Subsets in Feldman–Katok Metric","authors":"Kun Mei Gao,&nbsp;Rui Feng Zhang","doi":"10.1007/s10114-024-2517-3","DOIUrl":"10.1007/s10114-024-2517-3","url":null,"abstract":"<div><p>In this paper, we studied the metric mean dimension in Feldman–Katok (FK for short) metric. We introduced the notions of FK-Bowen metric mean dimension and FK-Packing metric mean dimension on subsets. And we established two variational principles.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2519 - 2536"},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572163","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}
引用次数: 0
Essentially Commuting Truncated Toeplitz Operators 本质归一的截断托普利兹算子
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-10 DOI: 10.1007/s10114-024-2696-y
Xi Zhao, Tao Yu
{"title":"Essentially Commuting Truncated Toeplitz Operators","authors":"Xi Zhao,&nbsp;Tao Yu","doi":"10.1007/s10114-024-2696-y","DOIUrl":"10.1007/s10114-024-2696-y","url":null,"abstract":"<div><p>A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2453 - 2480"},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572167","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}
引用次数: 0
Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process 连续状态分支过程后向微分方程的唯一性问题
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-10 DOI: 10.1007/s10114-024-3107-0
Pei Sen Li, Zeng Hu Li
{"title":"Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process","authors":"Pei Sen Li,&nbsp;Zeng Hu Li","doi":"10.1007/s10114-024-3107-0","DOIUrl":"10.1007/s10114-024-3107-0","url":null,"abstract":"<div><p>The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (<i>Theory Probab. Appl.</i>, 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 8","pages":"1825 - 1836"},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572166","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}
引用次数: 0
Reducibility of Three Dimensional Skew Symmetric System with High Dimensional Weak Liouvillean Frequencies 具有高维弱刘维尔频率的三维斜对称系统的可还原性
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-10 DOI: 10.1007/s10114-024-2540-4
Jie Liu, Yuan Shan, Jing Wang
{"title":"Reducibility of Three Dimensional Skew Symmetric System with High Dimensional Weak Liouvillean Frequencies","authors":"Jie Liu,&nbsp;Yuan Shan,&nbsp;Jing Wang","doi":"10.1007/s10114-024-2540-4","DOIUrl":"10.1007/s10114-024-2540-4","url":null,"abstract":"<div><p>In this paper, we consider the reducibility of three-dimensional skew symmetric systems. We obtain a reducibility result if the base frequency is high-dimensional weak Liouvillean and the parameter is sufficiently small. The proof is based on a modified KAM theory for 3-dimensional skew symmetric systems.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2388 - 2410"},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572162","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}
引用次数: 0
Boundary Behaviors for a Continuous-state Nonlinear Neveu’s Branching Process 连续态非线性内韦乌分支过程的边界行为
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-10 DOI: 10.1007/s10114-024-2741-x
Lin Yu Bai, Xu Yang
{"title":"Boundary Behaviors for a Continuous-state Nonlinear Neveu’s Branching Process","authors":"Lin Yu Bai,&nbsp;Xu Yang","doi":"10.1007/s10114-024-2741-x","DOIUrl":"10.1007/s10114-024-2741-x","url":null,"abstract":"<div><p>By generalizing a criterion of Mufa Chen for Markov jump processes, we establish the necessary and sufficient conditions for the extinction, explosion and coming down from infinity of a continuous-state nonlinear Neveu’s branching process.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 8","pages":"2005 - 2016"},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141577890","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}
引用次数: 0
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