This work is concerned with developing asymptotically sharp geometric rigidity estimates in thin domains. A thin domainΩ in space is roughly speaking a shell with non-constant thickness around a regular enough two dimensional compact surface. We prove a sharp geometric rigidity interpolation inequality that permits one to bound the Lp distance of the gradient of a u ∈W 1,p field from any constant proper rotation R , in terms of the average Lp distance (nonlinear strain) of the gradient from the rotation group, and the average Lp distance of the field itself from the set of rigid motions corresponding to the rotation R . The constants in the estimate are sharp in terms of the domain thickness scaling. If the domain mid-surface has a constant sign Gaussian curvature then the inequality reduces the problem of estimating the gradient ∇u in terms of the nonlinear strain ∫ Ωdist p (∇u(x),SO(3))dx to the easier problem of estimating only the vector field u in terms of the nonlinear strain with no asymptotic loss in the constants. This being said, the new interpolation inequality reduces the problem of proving “any” geometric one well rigidity problem in thin domains to estimating the vector field itself instead of the gradient, thus reducing the complexity of the problem. Funding. This material is based upon work partially supported by the National Science Foundation under Grants No. DMS-1814361, and partially supported by the Regents’ Junior Faculty Fellowship 2018 by UCSB. Manuscript received 8th February 2019, revised 6th June 2020 and 19th June 2020, accepted 18th June 2020.
{"title":"On the Geometric Rigidity interpolation estimate in thin bi-Lipschitz domains","authors":"D. Harutyunyan","doi":"10.5802/crmath.87","DOIUrl":"https://doi.org/10.5802/crmath.87","url":null,"abstract":"This work is concerned with developing asymptotically sharp geometric rigidity estimates in thin domains. A thin domainΩ in space is roughly speaking a shell with non-constant thickness around a regular enough two dimensional compact surface. We prove a sharp geometric rigidity interpolation inequality that permits one to bound the Lp distance of the gradient of a u ∈W 1,p field from any constant proper rotation R , in terms of the average Lp distance (nonlinear strain) of the gradient from the rotation group, and the average Lp distance of the field itself from the set of rigid motions corresponding to the rotation R . The constants in the estimate are sharp in terms of the domain thickness scaling. If the domain mid-surface has a constant sign Gaussian curvature then the inequality reduces the problem of estimating the gradient ∇u in terms of the nonlinear strain ∫ Ωdist p (∇u(x),SO(3))dx to the easier problem of estimating only the vector field u in terms of the nonlinear strain with no asymptotic loss in the constants. This being said, the new interpolation inequality reduces the problem of proving “any” geometric one well rigidity problem in thin domains to estimating the vector field itself instead of the gradient, thus reducing the complexity of the problem. Funding. This material is based upon work partially supported by the National Science Foundation under Grants No. DMS-1814361, and partially supported by the Regents’ Junior Faculty Fellowship 2018 by UCSB. Manuscript received 8th February 2019, revised 6th June 2020 and 19th June 2020, accepted 18th June 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"41 1","pages":"811-816"},"PeriodicalIF":0.8,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87432390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider star-shaped tubular domains consisting of a number of non intersecting semi-infinite strips of small thickness that are connected by a central region of diameter proportional to the thickness of the strips. At the thin-domain limit, the region reduces to a network of half-lines with the same end point (junction). We show that the solutions of uniformly elliptic partial differential equations set on the domain with Neumann boundary conditions converge, in the thin-domain limit, to the unique solution of a second-order partial differential equation on the network satisfying an effective Kirchhoff-type transmission condition at the junction. The latter is found by solving an “ergodic”-type problem at infinity obtained after a first-order blow up at the junction. 2020 Mathematics Subject Classification. 35J15, 35J99, 35B40, 35B25, 49L25, 47H25. Funding. The first author was partially supported by the Air Force Office for Scientific Research grant FA955018-I-0494. The second author was partially supported by the Air Force Office for Scientific Research grant FA9550-18-1-0494, the Office for Naval Research grant N000141712095 and the National Science Foundation grants DMS-1600129 and DMS-1900599. Manuscript received 23rd April 2020, accepted 5th June 2020.
{"title":"Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains","authors":"P. Lions, P. Souganidis","doi":"10.5802/crmath.83","DOIUrl":"https://doi.org/10.5802/crmath.83","url":null,"abstract":"We consider star-shaped tubular domains consisting of a number of non intersecting semi-infinite strips of small thickness that are connected by a central region of diameter proportional to the thickness of the strips. At the thin-domain limit, the region reduces to a network of half-lines with the same end point (junction). We show that the solutions of uniformly elliptic partial differential equations set on the domain with Neumann boundary conditions converge, in the thin-domain limit, to the unique solution of a second-order partial differential equation on the network satisfying an effective Kirchhoff-type transmission condition at the junction. The latter is found by solving an “ergodic”-type problem at infinity obtained after a first-order blow up at the junction. 2020 Mathematics Subject Classification. 35J15, 35J99, 35B40, 35B25, 49L25, 47H25. Funding. The first author was partially supported by the Air Force Office for Scientific Research grant FA955018-I-0494. The second author was partially supported by the Air Force Office for Scientific Research grant FA9550-18-1-0494, the Office for Naval Research grant N000141712095 and the National Science Foundation grants DMS-1600129 and DMS-1900599. Manuscript received 23rd April 2020, accepted 5th June 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"24 1","pages":"797-809"},"PeriodicalIF":0.8,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89238961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A family of infinite subsets of a countable set X is called positive iff it is closed under supersets and finite changes and contains a co-infinite set. We show that a countable ultrahomogeneous relational structure X has the strong amalgamation property iff the setP(X)={A⊂X :A∼=X} contains a positive family. In that case the family of large copies of X (i.e. copies having infinite intersection with each orbit) is the largest positive family in P(X), and for each R-embeddable Boolean linear order Lwhose minimum is non-isolated there is a maximal chain isomorphic to L {minL} in 〈P(X),⊂〉. 2020 Mathematics Subject Classification. 03C15, 03C50, 20M20, 06A06, 06A05. Funding. The authors acknowledge financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2020-14/200125). Manuscript received 6th April 2019, revised 27th May 2020, accepted 2nd June 2020.
{"title":"Positive families and Boolean chains of copies of ultrahomogeneous structures","authors":"Miloš S. Kurilić, Boriša Kuzeljević","doi":"10.5802/crmath.82","DOIUrl":"https://doi.org/10.5802/crmath.82","url":null,"abstract":"A family of infinite subsets of a countable set X is called positive iff it is closed under supersets and finite changes and contains a co-infinite set. We show that a countable ultrahomogeneous relational structure X has the strong amalgamation property iff the setP(X)={A⊂X :A∼=X} contains a positive family. In that case the family of large copies of X (i.e. copies having infinite intersection with each orbit) is the largest positive family in P(X), and for each R-embeddable Boolean linear order Lwhose minimum is non-isolated there is a maximal chain isomorphic to L {minL} in 〈P(X),⊂〉. 2020 Mathematics Subject Classification. 03C15, 03C50, 20M20, 06A06, 06A05. Funding. The authors acknowledge financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2020-14/200125). Manuscript received 6th April 2019, revised 27th May 2020, accepted 2nd June 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"23 1","pages":"791-796"},"PeriodicalIF":0.8,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89233027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ŵ be an unbiased estimate of an unknown w ∈ R. Given a function t (w), we show how to choose a function fn (w) such that for w∗ = ŵ + fn (w), E t ( w∗ )= t (w). We illustrate this with t (w) = w a for a given constant a. For a = 2 and ŵ normal, this leads to the convolution equation cr = cr ⊗ cr . Manuscript received 10th January 2019, accepted 9th April 2020.
设´´是未知w∈r的无偏估计。给定一个函数t (w)，我们展示如何选择一个函数fn (w)，使得对于w∗=´´+ fn (w)， E t (w∗)= t (w)。对于给定的常数a，我们用t (w) = w a来说明这一点。对于a = 2和´正态，这导致卷积方程cr = cr⊗cr。收稿日期2019年1月10日，收稿日期2020年4月9日。
{"title":"A note on bias reduction","authors":"C. Withers, S. Nadarajah","doi":"10.5802/CRMATH.49","DOIUrl":"https://doi.org/10.5802/CRMATH.49","url":null,"abstract":"Let ŵ be an unbiased estimate of an unknown w ∈ R. Given a function t (w), we show how to choose a function fn (w) such that for w∗ = ŵ + fn (w), E t ( w∗ )= t (w). We illustrate this with t (w) = w a for a given constant a. For a = 2 and ŵ normal, this leads to the convolution equation cr = cr ⊗ cr . Manuscript received 10th January 2019, accepted 9th April 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"64 1","pages":"641-644"},"PeriodicalIF":0.8,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91109635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper provides a conceptual analysis of forcing and generic extensions. Our goal is to give general axioms for the concept of standard forcing-generic extension and to show that the usual (poset) constructions are unified and explained as realizations of this concept. According to our approach, the basic idea behind forcing and generic extensions is that the latter are uniform adjunctions which are groundcontrolled by forcing, and forcing is nothing more than that ground-control. As a result of our axiomatization of this idea, the usual definitions of forcing and genericity are derived. Résumé. Cet article présente une analyse conceptuelle du forcing et des extensions génériques. Notre objectif est de donner des axiomes généraux pour le concept d’extension forcing-générique standard, et de montrer que les constructions habituelles sont unifiées et expliquées comme étant des réalisations de ce concept. Selon notre approche, l’idée-clé sous-tendant le forcing et les extensions génériques est que ces dernières sont des adjonctions uniformes qui sont contrôlées par le forcing, ainsi le forcing n’est rien de plus que ce contrôle. Comme conséquence de notre axiomatisation de cette idée, on dérive les définitions habituelles du forcing et de la généricité. Funding. This research was partially supported by fapesp, proccess 2016/25891-3. Manuscript received 10th April 2020, revised and accepted 17th July 2020. 1. Preliminary Remarks Forcing and generic extensions are usually not given as realizations of a concept, rather they are presented as specific constructions serving a specific purpose. Indeed, there are many different constructions with the same effect and differing on technical minutiae which obfuscate its essential components. If we want to make explicit what is this specific purpose, we must first capture the general idea avoiding inessential variations. In order to accomplish that, we turn towards an axiomatic approach. The situation is analogous to that of the real number system up to isomorphism: There are many different constructions of this system, but the axiomatic ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 758 Rodrigo A. Freire approach gives us a concept behind those constructions. We wish to capture a conceptual basis for forcing and generic extensions. Our aim is to characterize forcing and generic extensions through properties (axioms) that are common to all explicit constructions of forcing predicates and generic extensions. For example, textbook definitions of forcing relation in the ground model (which is customarily denoted by ∗), generic filter, P-name and evaluation of a P-name may vary widely, but there are common properties shared by the whole variety of constructions of forcing and generic extensions. The truth lemma and the definability lemma, for instance, hold in all constructions, independently of one’s choice of basic definitions. It is important to keep in mind the analo
本文提供了强制和一般扩展的概念分析。我们的目标是给出标准强迫-一般扩展概念的一般公理，并表明通常的(偏序集)结构是统一的，并解释为这个概念的实现。根据我们的方法，强迫和一般扩展背后的基本思想是，后者是统一的附加物，由强迫控制，强迫只不过是地面控制。由于我们对这种观念的公理化，通常的强迫和一般性的定义就被推导出来了。的简历。这篇文章分析了一些关于强迫和扩展的概念。客观客观地说明了公理的性质，例如，在扩展强迫的条件下，在标准的条件下，在结构习惯的条件下，在统一的条件下，在明确的条件下，在概念的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下，在确定的条件下。根据诺approche l 'idee-cle sous-tendant le迫使et les扩展generiques是ces上最后的是des adjonctions制服是controlees par le迫使依照ainsi le迫使n是不加这个controle。这样后果德诺公理化de这个想法,推导habituelles du迫使et de la genericite les定义。资金。本研究得到fapesp的部分支持，编号2016/ 25893 -3。2020年4月10日收稿，2020年7月17日修订并接受。1. 强制扩展和一般扩展通常不是作为一个概念的实现，而是作为服务于特定目的的特定结构来呈现。事实上，有许多不同的结构具有相同的效果，但在技术细节上有所不同，这混淆了其基本组成部分。如果我们想明确这个特定的目的是什么，我们必须首先抓住一般的想法，避免不必要的变化。为了做到这一点，我们转向一种公理化的方法。这种情况类似于实数系统的同构:这个系统有许多不同的结构，但是公义的ISSN(电子):1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 758 Rodrigo a . Freire方法为我们提供了这些结构背后的概念。我们希望获得强制扩展和通用扩展的概念基础。我们的目标是通过强制谓词和一般扩展的所有显式结构所共有的属性(公理)来表征强制和一般扩展。例如，教科书中关于地面模型中强迫关系(通常用*表示)、一般过滤器、P-name和P-name的求值的定义可能有很大的不同，但是强迫的各种构造和一般扩展都有共同的性质。例如，真性引理和可定义性引理在所有结构中都成立，而不依赖于基本定义的选择。重要的是要记住实数系统的公理化的类似情况。实数的传统构造及其对有理数的运算可能有很大的不同，但所有给定的构造都满足完全有序域的表征公理。我们希望通过强制和泛型扩展的公理化实现同样的目的。然而，公理不是随机选择的。我们需要一个指导思想，大致可以解释如下。首先，我们的策略是将强制和泛型理解为一个概念的主要组成部分，即强制-泛型扩展的概念。然后，理解由一般滤波器G给出的传递模型M的强迫-一般扩展是G与M的一致附加，该附加由强迫从地面控制。被强迫控制的基础概念是由基本的二象性来精确定义的:M |= p φ =⇒∀g3p;M [G] |=φ， M [G] |=φ < =⇒∃p∈G;M |= p φ。以非正式的方式思考上述二元性可能会有所帮助，考虑到“一般扩展是那些通过强制从基础上控制的，而强制是一般扩展的基础控制”的口号。根据我们的抽象描述，强迫和泛型的联系不是偶然的，这个主题的概念核心是这个未解离的强迫-泛型化合物。这里提出的强迫和一般扩展的公理化方法的发展与约瑟夫·舒恩菲尔德在经典论文《非分支强迫》中对这一主题的阐述相似。然而，我们的方法非常不同。实际上，在某种意义上，我们已经逆转了传统的方法:公理构成了我们的出发点，而基础模型中通用过滤器和强制谓词的传统定义是从它们派生出来的。 我们的大多数公理都可以在传统方法的所有变体中找到相关的定理，如真性引理、可定义性引理、一般存在定理等，我们将简要回顾一下它们是如何在这种方法中被证明的。此外，第9节提供了一个标准强制泛型扩展的构造，随后验证了该构造中公理的成立，这相当于对所有相关定理的阐述。然而，我们的准公理并没有在我们的方法中得到证明。我们工作的教训是，如果我们想要基本对偶性(公理(7)和(8))，G对M的一致附加(公理(5)和(6))，一般存在性(公理(4))，以及我们的控制装置的基本性质(公理(1)，(2)和(3))，那么强制和一般性必须以通常的方式定义。如果我们也采用p隶属性的普适性(公理(9))，那么我们就得到了范畴性的结果。随后，作为我们开发的自然结果，完成了由地面模型和通用滤波器唯一决定的标准强迫-通用扩展的构造。我们应该证明这一切，但首先我们必须提供一个可以陈述公理的框架。所有的公理对于传统方法的所有变体都是共同的，并且可以用简单的术语来解释，从而表明整个C. R. mathmatique, 2020,358, n6,757 -775 Rodrigo a . Freire 759主题依赖于一个非常一般的想法，而不是一组特定的，特别的技术结构。2. 假设我们有以下基本数据:一个传递模型M，其中的元素称为个体，用A、b、c和d表示，以及一个绝对偏阶P，最大元素为1。P的定义域是M的一个个体，它的元素称为条件，用P, q, r, s和t表示。绝对序关系用≤表示。如果p≤q，我们说p是一个比q强的条件。个体可以用作公式中的参数。备注1。我们以Azriel Levy在b[3]中所做的方式处理关于传递模型的通常警告，这意味着我们在Z FC的保守扩展中工作，该扩展由一个附加常数M给出，一个说明M是传递的公理和一个说明它反映原始语言的每个句子的公理图式。集合模型M的作用是允许通用过滤器，但这并不是严格必要的。由于一般的延伸必须从地面控制，我们可以呆在地面上，把延伸作为一个虚构。因此，我们可以对V进行强制，其中(i)通过满足某些公理的一元谓词符号给出正确条件的选择(参见[4,p. 282])和(ii)从V强制的陈述解释关于虚拟一般扩展的陈述(参见[4,p. 285])。除了我们的基本数据外，我们还需要控制装置。我们需要从基层开始控制M [G]的成员。为了实现这一点，M [G]必须是M中一个二元关系的可传递坍缩，这样我们就可以将M [G]中的隶属度拉回到M中的一个关系中。这种关系由下一段解释的附加数据给出，用∃p∈G表示;M |= a∈p b，表示坍缩的a是符合正确条件的坍缩b的一个元素。因此，我们需要一个三元关系R在M中，我们要求它对于传递模型是可定义的和绝对的。这种关系称为p隶属关系，它由三元组(p, a,b)满足，表示为M |= a∈p b。粗略地说，这是一个“隶属于p”的关系，这是迈向控制装置的第一步。然而，由于相应的崩溃不是内注入的，我们将通过强制谓词调整成员控制。正如我们刚才提到的，控制装置需要改进和完善，这就产生了强制谓词。它们被设计成最终的控制装置，如下所示。对于每个有n个自由变量的公式φ，在M中有一个可定义的n+1元谓词φ，称为与φ对应的强制谓词。用M |= (p φ)
{"title":"An axiomatic approach to forcing and generic extensions","authors":"R. A. Freire","doi":"10.5802/CRMATH.97","DOIUrl":"https://doi.org/10.5802/CRMATH.97","url":null,"abstract":"This paper provides a conceptual analysis of forcing and generic extensions. Our goal is to give general axioms for the concept of standard forcing-generic extension and to show that the usual (poset) constructions are unified and explained as realizations of this concept. According to our approach, the basic idea behind forcing and generic extensions is that the latter are uniform adjunctions which are groundcontrolled by forcing, and forcing is nothing more than that ground-control. As a result of our axiomatization of this idea, the usual definitions of forcing and genericity are derived. Résumé. Cet article présente une analyse conceptuelle du forcing et des extensions génériques. Notre objectif est de donner des axiomes généraux pour le concept d’extension forcing-générique standard, et de montrer que les constructions habituelles sont unifiées et expliquées comme étant des réalisations de ce concept. Selon notre approche, l’idée-clé sous-tendant le forcing et les extensions génériques est que ces dernières sont des adjonctions uniformes qui sont contrôlées par le forcing, ainsi le forcing n’est rien de plus que ce contrôle. Comme conséquence de notre axiomatisation de cette idée, on dérive les définitions habituelles du forcing et de la généricité. Funding. This research was partially supported by fapesp, proccess 2016/25891-3. Manuscript received 10th April 2020, revised and accepted 17th July 2020. 1. Preliminary Remarks Forcing and generic extensions are usually not given as realizations of a concept, rather they are presented as specific constructions serving a specific purpose. Indeed, there are many different constructions with the same effect and differing on technical minutiae which obfuscate its essential components. If we want to make explicit what is this specific purpose, we must first capture the general idea avoiding inessential variations. In order to accomplish that, we turn towards an axiomatic approach. The situation is analogous to that of the real number system up to isomorphism: There are many different constructions of this system, but the axiomatic ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 758 Rodrigo A. Freire approach gives us a concept behind those constructions. We wish to capture a conceptual basis for forcing and generic extensions. Our aim is to characterize forcing and generic extensions through properties (axioms) that are common to all explicit constructions of forcing predicates and generic extensions. For example, textbook definitions of forcing relation in the ground model (which is customarily denoted by ∗), generic filter, P-name and evaluation of a P-name may vary widely, but there are common properties shared by the whole variety of constructions of forcing and generic extensions. The truth lemma and the definability lemma, for instance, hold in all constructions, independently of one’s choice of basic definitions. It is important to keep in mind the analo","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"14 1","pages":"757-775"},"PeriodicalIF":0.8,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73193211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let f be a transcendental meromorphic function on C, and P (z),Q(z) be two polynomials with degP (z) > degQ(z). In this paper, we prove that: if f (z) = 0 ⇒ f ′(z) = a(a nonzero constant), except possibly finitely many, then f ′(z)−P (z)/Q(z) has infinitely many zeros. Our result extends or improves some previous related results due to Bergweiler–Pang, Pang–Nevo–Zalcman, Wang–Fang, and the author, et. al. 2020 Mathematics Subject Classification. 30D35, 30D45. Funding. This work was supported by NSFC(Grant No.11471163). Manuscript received 7th January 2020, accepted 4th July 2020.
{"title":"Picard-Hayman behavior of derivatives of meromorphic functions","authors":"Yan Xu, Shirong Chen, P. Niu","doi":"10.5802/CRMATH.96","DOIUrl":"https://doi.org/10.5802/CRMATH.96","url":null,"abstract":"Let f be a transcendental meromorphic function on C, and P (z),Q(z) be two polynomials with degP (z) > degQ(z). In this paper, we prove that: if f (z) = 0 ⇒ f ′(z) = a(a nonzero constant), except possibly finitely many, then f ′(z)−P (z)/Q(z) has infinitely many zeros. Our result extends or improves some previous related results due to Bergweiler–Pang, Pang–Nevo–Zalcman, Wang–Fang, and the author, et. al. 2020 Mathematics Subject Classification. 30D35, 30D45. Funding. This work was supported by NSFC(Grant No.11471163). Manuscript received 7th January 2020, accepted 4th July 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"31 1","pages":"753-756"},"PeriodicalIF":0.8,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72824154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Mozolyako, G. Psaromiligkos, A. Volberg, Pavel Zorin Kranich
We prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces of holomorphic functions in bi-disc and tri-disc this proves the embedding theorem of those spaces on biand tri-disc. We completely describe the Carleson measures for such embeddings. Funding. We acknowledge the support of the following grants: NSF grant DMS-1900286, Theorem 2 was obtained in the frameworks of the Russian Science Foundation grant 17-11-01064-P, the third author was supported also by Alexander von Humboldt foundation. Manuscript received 10th February 2020, revised 25th June 2020, accepted 26th June 2020. Version française abrégée Un n -arbre T n , n ≥ 1, est un produit cartésien de n arbres dyadiques identiques avec un ordre partiel induit par la structure du produit. Etant donné un point β ∈ T n , nous définissons son successeur en posant S (β) = {α ∈ T n : α≤ β}. Soient w,μ deux fonctions positives sur T n , nous définissons la constante de boîte comme le plus petit nombre [w,μ]Box tel que ES (β)[μ] := ∑ α≤β w(α)(I∗μ(α))2 ≤ [w,μ]Boxμ(S (β)), ∀β ∈ T n . (1) La constante de plongement de Carleson est la plus petite constante [w,μ]C E telle que l’inégalité suivante ait lieu: E [ψμ] ≤ [w,μ]C E ∑ ω∈T n |ψ(ω)|2μ(ω) (4) Le résultat principal de cet article est le théorème suivant: ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 722 Pavel Mozolyako, Georgios Psaromiligkos, Alexander Volberg and Pavel Zorin Kranich Theorem 1. Soit μ : T n → R+, n = 1,2,3 et soit w : T n → [0,∞) un poids d’une forme tensorielle. Alors l’inégalité suivante a lieu [w,μ]C E . [w,μ]Box . 1. Hardy inequality on the n-tree and energy of measures A (finite) tree T is a finite partially ordered set such that for every ω ∈ T the set {α ∈ T : α ≥ ω} is totally ordered (here we identify the tree with its vertex set). In what follows we consider rooted dyadic trees, i.e. there is a unique maximal element in T , and every element (except for the minimal ones) has exactly two children. An n-tree T n , n ≥ 1 is a Cartesian product of n identical dyadic trees with order induced by the product structure. In what follows no estimate will depend on the depth of the tree. A subset U (resp. D) of a partially ordered set T n is called an up-set (resp. down-set) if, for every α ∈U and β ∈ T with α≤β (resp. β≤α), we also have β ∈U (resp. β ∈D). Given a point β ∈ T n we define its successor set S (β) = {α ∈ T n : α≤β}, clearly it is a down-set. From now on we assume that the weight w : T n → R+ is fixed. The Hardy operator associated with w is defined by Iwφ(γ) := ∑ γ′≥γ w(γ′)φ(γ′) and I∗ψ(γ) = ∑ γ′≤γ ψ(γ′). For a measure (non-negative function) μ on T n we define the (w-) potential to be Vμw (α) := (Iw I∗μ)(α), α ∈ T n , again we usually drop the index w . Let E ⊂ T n andμ be a measure on T n . The E-truncated energy of μ is EE [μ] := ∑ α∈E (I∗μ)2(α)w(α). If E = T n , we wri
证明了多树上Hardy算子的多参数并矢嵌入定理。对于双盘和三盘上的一类全纯函数的Dirichlet空间，证明了这些空间在双盘和三盘上的嵌入定理。我们完整地描述了这种嵌入的Carleson测度。资金。我们感谢以下基金的支持:NSF基金DMS-1900286，定理2是在俄罗斯科学基金会基金17-11-01064-P的框架下获得的，第三作者也得到了Alexander von Humboldt基金会的支持。稿收到2020年2月10日，修改2020年6月25日，接受2020年6月26日。法语版本abregee联合国n -arbre T n, n≥1,是联合国产品cartesien de n arbres dyadiques identiques用范围partiel代购契约par杜拉结构产品。当点β∈t_n时，n (β) = {α∈t_n: α≤β}。∑α≤β w(α)(I∗μ(α))2≤[w，μ]∑β μ(S (β))，∑β∈tn，∀β∈tn。(1) La constant de plongement de Carleson est La + petite constant [w，μ]C E telle que l ' in: E [ψμ]≤[w，μ]C E∑ω∈T n |ψ(ω)|2μ(ω) (4) Le rsultat principal de cet article est Le th:∗通讯作者。Pavel Mozolyako, Georgios Psaromiligkos, Alexander Volberg和Pavel Zorin Kranich定理。Soit μ: t_n→R+， n = 1,2,3; Soit μ: t_n→[0，∞)unpoids d 'une forme tensororielle。[j]李晓明。基于遗传算法的生物信息学研究[j]。(w,μ)盒子。1. n树上的Hardy不等式和A(有限)树T的测度能是一个有限偏序集合，使得对于每一个ω∈T，集合{α∈T: α≥ω}是全序的(这里我们用它的顶点集来识别树)。在接下来的内容中，我们考虑有根并矢树，即在T中有一个唯一的最大元素，并且每个元素(除了最小元素)都有两个子元素。n树tn, n≥1是n棵相同并矢树的笛卡尔积，并由积结构引起序。在接下来的内容中，任何估计都不取决于树的深度。子集U (p。偏序集合T n的D)称为逆集(逆集)。对于α∈U， β∈T，且α≤β (resp;β≤α)，我们也有β∈U (resp。β∈D)。给定一个点β∈tn，我们定义它的后继集S (β) = {α∈tn: α≤β}，显然它是一个下集。从现在开始，我们假设权w: tn→R+是固定的。定义的哈代运营商与w是信息战φ(γ):=∑γ’≥γw(γ)φ(γ)我∗ψ(γ)=∑γ’≤γψ(γ)。对于T n上的测度(非负函数)μ，我们定义(w-)势为Vμw (α):= (Iw I∗μ)(α)， α∈T n，我们通常省略指标w。设E∧t_n， μ是t_n上的测度。μ的E-truncated能量EE(μ):=∑α∈E(我∗μ)2(α)w(α)。如果T E = n,我们写E(μ)相反,所以E(μ)=∫T n Vdμ:=∑α∈T n V(α)μ(α)。如果E是V在δ> 0时的δ水平集，即E = {α: V≤δ}，则我们写Eδ[μ]:= EE [μ]。我们定义框常数最小的数量(w,μ)箱,ES(β)(μ):=∑α≤βw(α)(我∗μ(α))2≤(w,μ)盒子μ(S(β)),∀β∈T n。(1) Carleson常数是使ED [μ]≤[w，μ]Cμ(D)，∀D∧n的最小数[w，μ]C。(2)遗传的卡里森常数(或限制能条件常数，或REC常数)是最小的常数[w，μ]HC，使得E [μ 1e]≤[w，μ]HCμ(E)，∀E∧n。(3)最后Carleson嵌入常数是最小的常数[w，μ]C E，使得伴随嵌入E [ψμ]≤[w，μ]C E∑ω∈T n |ψ(ω)|2μ(ω)(4)对所有函数ψ在T n上成立。当[w，μ]C E < +∞时，我们称(w，μ)为T n上加权Hardy不等式的迹对。对于正数A,B，我们写A。如果A≤C, B具有绝对常数C，则特别不依赖于(w，μ)对。不等式[w，μ]Box≤[w，μ]C≤[w，μ]HC≤[w，μ]C E是明显的。[6]证明了1-树的逆不等式。我们的主要结果是2树和3树的扩展。读者可以在预印本[3]、[4]和[1]中看到2-tree的部分细节。张建军，张建军，张建军，等。723定理[j] .数学学报，2016,35(1):349 - 349。设μ: T n→R+， n = 1,2,3。设w: T n→[0，∞)为张量积形式。则上述不等式的反转也成立:[w，μ]C E。(w,μ)HC。(w,μ)C。(w,μ)盒子。证明所有这些不等式的关键要素是所谓的替代极大原理。为了详细说明，让我们考虑单参数的情况。那么对于任意α∈T, V μ δ (α)≤δ，因此，特别地，对于任意度量ρ在T上，我们有平凡的∫T V μ δ ρ≤δ|ρ|，其中|ρ| = ρ(T)是ρ的总质量。在更高的维度，情况是不同的，即。
{"title":"Combinatorial property of all positive measures in dimensions $2$ and $3$","authors":"P. Mozolyako, G. Psaromiligkos, A. Volberg, Pavel Zorin Kranich","doi":"10.5802/CRMATH.90","DOIUrl":"https://doi.org/10.5802/CRMATH.90","url":null,"abstract":"We prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces of holomorphic functions in bi-disc and tri-disc this proves the embedding theorem of those spaces on biand tri-disc. We completely describe the Carleson measures for such embeddings. Funding. We acknowledge the support of the following grants: NSF grant DMS-1900286, Theorem 2 was obtained in the frameworks of the Russian Science Foundation grant 17-11-01064-P, the third author was supported also by Alexander von Humboldt foundation. Manuscript received 10th February 2020, revised 25th June 2020, accepted 26th June 2020. Version française abrégée Un n -arbre T n , n ≥ 1, est un produit cartésien de n arbres dyadiques identiques avec un ordre partiel induit par la structure du produit. Etant donné un point β ∈ T n , nous définissons son successeur en posant S (β) = {α ∈ T n : α≤ β}. Soient w,μ deux fonctions positives sur T n , nous définissons la constante de boîte comme le plus petit nombre [w,μ]Box tel que ES (β)[μ] := ∑ α≤β w(α)(I∗μ(α))2 ≤ [w,μ]Boxμ(S (β)), ∀β ∈ T n . (1) La constante de plongement de Carleson est la plus petite constante [w,μ]C E telle que l’inégalité suivante ait lieu: E [ψμ] ≤ [w,μ]C E ∑ ω∈T n |ψ(ω)|2μ(ω) (4) Le résultat principal de cet article est le théorème suivant: ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 722 Pavel Mozolyako, Georgios Psaromiligkos, Alexander Volberg and Pavel Zorin Kranich Theorem 1. Soit μ : T n → R+, n = 1,2,3 et soit w : T n → [0,∞) un poids d’une forme tensorielle. Alors l’inégalité suivante a lieu [w,μ]C E . [w,μ]Box . 1. Hardy inequality on the n-tree and energy of measures A (finite) tree T is a finite partially ordered set such that for every ω ∈ T the set {α ∈ T : α ≥ ω} is totally ordered (here we identify the tree with its vertex set). In what follows we consider rooted dyadic trees, i.e. there is a unique maximal element in T , and every element (except for the minimal ones) has exactly two children. An n-tree T n , n ≥ 1 is a Cartesian product of n identical dyadic trees with order induced by the product structure. In what follows no estimate will depend on the depth of the tree. A subset U (resp. D) of a partially ordered set T n is called an up-set (resp. down-set) if, for every α ∈U and β ∈ T with α≤β (resp. β≤α), we also have β ∈U (resp. β ∈D). Given a point β ∈ T n we define its successor set S (β) = {α ∈ T n : α≤β}, clearly it is a down-set. From now on we assume that the weight w : T n → R+ is fixed. The Hardy operator associated with w is defined by Iwφ(γ) := ∑ γ′≥γ w(γ′)φ(γ′) and I∗ψ(γ) = ∑ γ′≤γ ψ(γ′). For a measure (non-negative function) μ on T n we define the (w-) potential to be Vμw (α) := (Iw I∗μ)(α), α ∈ T n , again we usually drop the index w . Let E ⊂ T n andμ be a measure on T n . The E-truncated energy of μ is EE [μ] := ∑ α∈E (I∗μ)2(α)w(α). If E = T n , we wri","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"11 1","pages":"721-725"},"PeriodicalIF":0.8,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79239725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A nonlinear Korn inequality in $protect mathbb{R}^n$ with an explicitly bounded constant","authors":"M. Mălin, C. Mardare","doi":"10.5802/CRMATH.84","DOIUrl":"https://doi.org/10.5802/CRMATH.84","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"98 1","pages":"621-626"},"PeriodicalIF":0.8,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73618184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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