Comptes Rendus Mathematique最新文献

{"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}

{"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}

{"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}

{"title":"A Berry–Esseen bound of order $protect frac{1}{protect sqrt{n}} $ for martingales","authors":"Songqi Wu, Xiaohui Ma, Hailin Sang, Xiequan Fan","doi":"10.5802/CRMATH.81","DOIUrl":"https://doi.org/10.5802/CRMATH.81","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"25 1","pages":"701-712"},"PeriodicalIF":0.8,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82230652","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 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}

{"title":"Convex maps on $protect mathbb{R}^n$ and positive definite matrices","authors":"J. Bourin, J. Shao","doi":"10.5802/CRMATH.25","DOIUrl":"https://doi.org/10.5802/CRMATH.25","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"178 1","pages":"645-649"},"PeriodicalIF":0.8,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81630916","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":"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}

{"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}

{"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}