{"title":"Marie-Hélène Schwartz et les champs radiaux, un parcours mathématique","authors":"J. Brasselet","doi":"10.5802/CRMATH.180","DOIUrl":"https://doi.org/10.5802/CRMATH.180","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"47 1","pages":"329-354"},"PeriodicalIF":0.8,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88870925","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}
By studying the variable denominators introduced by X. Zhou–L. Zhu, we generalize the results of D. Popovici for the L2 extension theorem for jets. As a direct corollary, we also give a generalization of T. Ohsawa–K. Takegoshi’s extension theorem to a jet version.
{"title":"L 2 extension theorem for jets with variable denominators","authors":"S. Rao, Runze Zhang","doi":"10.5802/CRMATH.167","DOIUrl":"https://doi.org/10.5802/CRMATH.167","url":null,"abstract":"By studying the variable denominators introduced by X. Zhou–L. Zhu, we generalize the results of D. Popovici for the L2 extension theorem for jets. As a direct corollary, we also give a generalization of T. Ohsawa–K. Takegoshi’s extension theorem to a jet version.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"16 1","pages":"181-193"},"PeriodicalIF":0.8,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75884995","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 nonlinear Korn inequality on a surface estimates a distance between a surface θ(ω) and another surface φ(ω) in terms of distances between their fundamental forms in the space Lp (ω), 1 < p <∞. We establish a new inequality of this type. The novelty is that the immersion θ belongs to a specific set of mappings of class C 1 from ω into R3 with a unit vector field also of class C 1 over ω. Résumé. Une inégalité de Korn non linéaire sur une surface estime une distance entre une surfaceθ(ω) et une autre surfaceφ(ω) en fonction des distances entre leur formes fondamentales dans l’espace Lp (ω), 1 < p <∞. Nous établissons une nouvelle inégalité de ce type. La nouveauté réside dans l’appartenance de l’immersion θ à un ensemble particulier d’applications de classe C 1 de ω dans R3 avec un champ de vecteurs normaux unitaires aussi de classe C 1 dans ω. Funding. The work of the second author was substantially supported by a grant from City University of Hong Kong (Project No. 7005495). Manuscript received 18th September 2020, accepted 23rd September 2020. ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 106 Maria Malin and Cristinel Mardare 1. Notation and definitions Vector and matrix fields are denoted by boldface letters. Given any open set Ω ⊂ Rn , n > 1, any subset V ⊂ Y of a finite-dimensional vector space Y , and any integer `> 0, the notation C (Ω;V ) designates the set of all fields v = (vi ) :Ω→ Y such that v (x) ∈ V for all x ∈ Ω and vi ∈ C (Ω). Likewise, given any real number p > 1, the notation Lp (Ω;V ), resp. W `, p (Ω;V ), designates the set of all fields v = (vi ) :Ω→ Y such that v (x) ∈ V for almost all x ∈Ω and vi ∈ Lp (Ω), resp. vi ∈W `, p (Ω). The space of all real matrices with k rows and ` columns is denotedMk×`. We also let M :=Mk×k ,S := { A ∈Mk ; A = A } , S> := { A ∈Sk ; A is positive-definite } , and O+ := { A ∈Mk ; A A = I and det A = 1 } . A k × ` matrix whose column vectors are the vectors v 1, . . . , v` ∈ Rk is denoted (v 1| . . . |v`). If A ∈S>, there exists a unique matrix U ∈S> such that U 2 = A; this being the case, we let A1/2 :=U . The Euclidean norm in R3 is denoted | · |. Spaces of matrices are equipped with the Frobenius norm, also denoted | · |. The spaces Lp (Ω), Lp (Ω;Rk ), and Lp (Ω;Mk×`), are respectively equipped with the norms denoted and defined by
表面上的非线性Korn不等式估计距离表面θ(ω),另一个表面φ(ω)的基本形式的空间之间的距离Lp(ω),1 < p 1,有限维向量空间的任何子集V⊂Y Y,和任何整数> 0,符号C(Ω,V)指定所有字段的集合V = (vi):Ω→Y, V (x)∈对于所有x∈Ω和vi∈C(Ω)。同样地，给定任意实数p > 1，符号Lp (Ω;V)， resp。W′，p (Ω;V)表示所有字段V = (vi):Ω→Y的集合，使得对于几乎所有x∈Ω和vi∈Lp (Ω)， V (x)∈V。vi∈W '， p (Ω)。所有有k行和k列的实矩阵的空间记为mkx。令M:=Mk×k,S:= {A∈Mk;A = A}， S>:= {A∈Sk;A为正定}，且0 +:= {A∈Mk;A A = I det A = 1}。一个k × '矩阵，它的列向量是向量v1，…， v′∈Rk记为(v1 |…| v”)。若A∈S>，则存在唯一矩阵U∈S>使得U 2 = A;在这种情况下，令a /2 =U。R3中的欧几里得范数表示为|·|。矩阵空间具有Frobenius范数，也表示为|·|。在空间Lp (Ω)、Lp (Ω;Rk)和Lp (Ω; mkx ')中，分别配备用表示和定义的规范
{"title":"A nonlinear Korn inequality on a surface with an explicit estimate of the constant","authors":"M. Mălin, C. Mardare","doi":"10.5802/CRMATH.122","DOIUrl":"https://doi.org/10.5802/CRMATH.122","url":null,"abstract":"A nonlinear Korn inequality on a surface estimates a distance between a surface θ(ω) and another surface φ(ω) in terms of distances between their fundamental forms in the space Lp (ω), 1 < p <∞. We establish a new inequality of this type. The novelty is that the immersion θ belongs to a specific set of mappings of class C 1 from ω into R3 with a unit vector field also of class C 1 over ω. Résumé. Une inégalité de Korn non linéaire sur une surface estime une distance entre une surfaceθ(ω) et une autre surfaceφ(ω) en fonction des distances entre leur formes fondamentales dans l’espace Lp (ω), 1 < p <∞. Nous établissons une nouvelle inégalité de ce type. La nouveauté réside dans l’appartenance de l’immersion θ à un ensemble particulier d’applications de classe C 1 de ω dans R3 avec un champ de vecteurs normaux unitaires aussi de classe C 1 dans ω. Funding. The work of the second author was substantially supported by a grant from City University of Hong Kong (Project No. 7005495). Manuscript received 18th September 2020, accepted 23rd September 2020. ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 106 Maria Malin and Cristinel Mardare 1. Notation and definitions Vector and matrix fields are denoted by boldface letters. Given any open set Ω ⊂ Rn , n > 1, any subset V ⊂ Y of a finite-dimensional vector space Y , and any integer `> 0, the notation C (Ω;V ) designates the set of all fields v = (vi ) :Ω→ Y such that v (x) ∈ V for all x ∈ Ω and vi ∈ C (Ω). Likewise, given any real number p > 1, the notation Lp (Ω;V ), resp. W `, p (Ω;V ), designates the set of all fields v = (vi ) :Ω→ Y such that v (x) ∈ V for almost all x ∈Ω and vi ∈ Lp (Ω), resp. vi ∈W `, p (Ω). The space of all real matrices with k rows and ` columns is denotedMk×`. We also let M :=Mk×k ,S := { A ∈Mk ; A = A } , S> := { A ∈Sk ; A is positive-definite } , and O+ := { A ∈Mk ; A A = I and det A = 1 } . A k × ` matrix whose column vectors are the vectors v 1, . . . , v` ∈ Rk is denoted (v 1| . . . |v`). If A ∈S>, there exists a unique matrix U ∈S> such that U 2 = A; this being the case, we let A1/2 :=U . The Euclidean norm in R3 is denoted | · |. Spaces of matrices are equipped with the Frobenius norm, also denoted | · |. The spaces Lp (Ω), Lp (Ω;Rk ), and Lp (Ω;Mk×`), are respectively equipped with the norms denoted and defined by","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"78 1","pages":"105-111"},"PeriodicalIF":0.8,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84308818","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 study the existence of ground states of a Schrödinger–Poisson–Slater type equation with pure power nonlinearity. By carrying out the constrained minimization on a special manifold, which is a combination of the Pohozaev manifold and Nehari manifold, we obtain the existence of ground state solutions of this system.
{"title":"On the existence of ground states of an equation of Schrödinger–Poisson–Slater type","authors":"Chun-Yu Lei, Y. Lei","doi":"10.5802/CRMATH.175","DOIUrl":"https://doi.org/10.5802/CRMATH.175","url":null,"abstract":"We study the existence of ground states of a Schrödinger–Poisson–Slater type equation with pure power nonlinearity. By carrying out the constrained minimization on a special manifold, which is a combination of the Pohozaev manifold and Nehari manifold, we obtain the existence of ground state solutions of this system.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"22 1","pages":"219-227"},"PeriodicalIF":0.8,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77632648","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 show that the elliptic problem ∆u + f (u) = 0 in RN , N ≥ 1, with f ∈C 1(R) and f (0) = 0 does not have nontrivial stable solutions that decay to zero at infinity, provided that f is nonincreasing near the origin. As a corollary, we can show that any two nontrivial solutions that decay to zero at infinity must intersect each other, provided that at least one of them is sign-changing. This property was previously known only in the case where both solutions are positive with a different approach. We also discuss implications of our main result on the existence of monotone heteroclinic solutions to the corresponding reaction-diffusion equation. Manuscript received 26th August 2020, revised 13th November 2020, accepted 15th November 2020.
我们证明了在RN中，N≥1，当f∈c1 (R)且f(0) = 0时，椭圆型问题∆u + f (u) = 0不存在在无穷远处衰减为零的非平凡稳定解，只要f在原点附近不增加。作为推论，我们可以证明任意两个在无穷远处衰减为零的非平凡解必须彼此相交，只要其中至少有一个是改变符号的。这个性质以前只在两个解都是正的情况下用不同的方法才知道。我们还讨论了我们的主要结果对相应的反应扩散方程的单调异斜解存在性的影响。2020年8月26日收稿，2020年11月13日改稿，2020年11月15日收稿。
{"title":"Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in ℝ N","authors":"C. Sourdis","doi":"10.5802/CRMATH.150","DOIUrl":"https://doi.org/10.5802/CRMATH.150","url":null,"abstract":"We show that the elliptic problem ∆u + f (u) = 0 in RN , N ≥ 1, with f ∈C 1(R) and f (0) = 0 does not have nontrivial stable solutions that decay to zero at infinity, provided that f is nonincreasing near the origin. As a corollary, we can show that any two nontrivial solutions that decay to zero at infinity must intersect each other, provided that at least one of them is sign-changing. This property was previously known only in the case where both solutions are positive with a different approach. We also discuss implications of our main result on the existence of monotone heteroclinic solutions to the corresponding reaction-diffusion equation. Manuscript received 26th August 2020, revised 13th November 2020, accepted 15th November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"9 1","pages":"131-136"},"PeriodicalIF":0.8,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80498745","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}
Shan E. Farooq, Khurram Shabir, S. Qaisar, Farooq Ahmad, O. A. Almatroud
This article presents some new inequalities of Simpson’s type for differentiable functions by using (α,m)-convexity. Some results for concavity are also obtained. These new estimates improve on the previously known ones. Some applications for special means of real numbers are also provided. 2020 Mathematics Subject Classification. 26A15, 26A51, 26D10. Funding. This work was supported by the Higher Education Commission (Islamabad) thorough the National Research Program for Universities, Grant No. 7359/Punjab/NRPU/R&D/HEC/2017. Manuscript received 19th July 2020, revised 16th August 2020 and 21st October 2020, accepted 17th November 2020.
{"title":"New Inequalities of Simpson’s type for differentiable functions via generalized convex function","authors":"Shan E. Farooq, Khurram Shabir, S. Qaisar, Farooq Ahmad, O. A. Almatroud","doi":"10.5802/CRMATH.152","DOIUrl":"https://doi.org/10.5802/CRMATH.152","url":null,"abstract":"This article presents some new inequalities of Simpson’s type for differentiable functions by using (α,m)-convexity. Some results for concavity are also obtained. These new estimates improve on the previously known ones. Some applications for special means of real numbers are also provided. 2020 Mathematics Subject Classification. 26A15, 26A51, 26D10. Funding. This work was supported by the Higher Education Commission (Islamabad) thorough the National Research Program for Universities, Grant No. 7359/Punjab/NRPU/R&D/HEC/2017. Manuscript received 19th July 2020, revised 16th August 2020 and 21st October 2020, accepted 17th November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"27 1","pages":"137-147"},"PeriodicalIF":0.8,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90779597","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}
For a character χ of a finite group G , the co-degree of χ is χc (1) = [G :kerχ] χ(1) . Let p be a prime and let e be a positive integer. In this paper, we first show that if G is a p-solvable group such that pe+1 χc (1), for every irreducible character χ of G , then the p-length of G is not greater than e. Next, we study the finite groups satisfying the condition that p2 does not divide the co-degrees of their irreducible characters. Mathematical subject classification (2010). 20C15, 20D10, 20D05. Manuscript received 22nd April 2020, revised and accepted 24th November 2020.
{"title":"p-parts of co-degrees of irreducible characters","authors":"Roya Bahramian, N. Ahanjideh","doi":"10.5802/CRMATH.158","DOIUrl":"https://doi.org/10.5802/CRMATH.158","url":null,"abstract":"For a character χ of a finite group G , the co-degree of χ is χc (1) = [G :kerχ] χ(1) . Let p be a prime and let e be a positive integer. In this paper, we first show that if G is a p-solvable group such that pe+1 χc (1), for every irreducible character χ of G , then the p-length of G is not greater than e. Next, we study the finite groups satisfying the condition that p2 does not divide the co-degrees of their irreducible characters. Mathematical subject classification (2010). 20C15, 20D10, 20D05. Manuscript received 22nd April 2020, revised and accepted 24th November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"161 1","pages":"79-83"},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76304151","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}
In this paper, we introduce a new generalization of Bell numbers, the s-associated r -Dowling numbers by combining two investigational directions. Here, r distinguished elements have to be in distinct blocks, some elements are coloured according to a colouring rule, and the cardinality of certain blocks is bounded from below by s. Along with them, we define some relatives, the s-associated r -Dowling factorials and the s-associated r -Dowling–Lah numbers, when the underlying set is decomposed into cycles or ordered blocks. The study of these numbers is based on their combinatorial meaning, and the exponential generating functions of their sequences derived from the so-called r -compositional formula. 2020 Mathematics Subject Classification. 05A15, 05A18, 05A19, 11B73. Funding. Research of the first author Eszter Gyimesi was supported by the ÚNKP-18-3 New National Excellence Program of the Ministry of Human Capacities. Research of the second author Gábor Nyul was supported by Grant 115479 from the Hungarian Scientific Research Fund. Manuscript received 22nd July 2019, revised 17th February 2020, accepted 5th November 2020.
{"title":"Associated r-Dowling numbers and some relatives","authors":"Eszter Gyimesi, Gábor Nyul","doi":"10.5802/CRMATH.145","DOIUrl":"https://doi.org/10.5802/CRMATH.145","url":null,"abstract":"In this paper, we introduce a new generalization of Bell numbers, the s-associated r -Dowling numbers by combining two investigational directions. Here, r distinguished elements have to be in distinct blocks, some elements are coloured according to a colouring rule, and the cardinality of certain blocks is bounded from below by s. Along with them, we define some relatives, the s-associated r -Dowling factorials and the s-associated r -Dowling–Lah numbers, when the underlying set is decomposed into cycles or ordered blocks. The study of these numbers is based on their combinatorial meaning, and the exponential generating functions of their sequences derived from the so-called r -compositional formula. 2020 Mathematics Subject Classification. 05A15, 05A18, 05A19, 11B73. Funding. Research of the first author Eszter Gyimesi was supported by the ÚNKP-18-3 New National Excellence Program of the Ministry of Human Capacities. Research of the second author Gábor Nyul was supported by Grant 115479 from the Hungarian Scientific Research Fund. Manuscript received 22nd July 2019, revised 17th February 2020, accepted 5th November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"8 1","pages":"47-55"},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81456268","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}
In this paper, we use particular polynomials to establish some results on the real rootedness of polynomials. The considered polynomials are Bell polynomials and Hermite polynomials. To cite this article: M. Mihoubi and S. Taharbouchet, C. R. Acad. Sci. Paris, Ser. I 340 (2021). Résumé. Dans ce papier, nous utilisons des polynômes particuliers pour établir quelques résultats sur les polynômes à racines réelles. Les polynômes considérés sont des polynômes de Bell et des polynômes de Hermite. Pour citer cet article : M. Mihoubi and S. Taharbouchet, C. R. Acad. Sci. Paris, Ser. I 340 (2021). Manuscript received 3rd November 2019, accepted 6th November 2020.
= =地理= =根据美国人口普查，这个县的土地面积为。= =地理= =根据美国人口普查，该县的总面积为，其中土地和(3.064平方公里)水。引用本文:M. Mihoubi和S. Taharbouchet, C. R. Acad. Sci。巴黎,爵士。他的父亲是一名医生，母亲是一名医生。摘要。在这篇论文中，我们使用特殊的多项式来建立一些关于实根多项式的结果。考虑的多项式是贝尔多项式和埃尔米特多项式。引用本文:M. Mihoubi和S. Taharbouchet, c.r. Acad. Sci。巴黎,爵士。他的父亲是一名医生，母亲是一名医生。他的父亲是一名律师，母亲是一名律师。
{"title":"Polynomials with real zeros via special polynomials","authors":"M. Mihoubi, Said Taharbouchet","doi":"10.5802/CRMATH.147","DOIUrl":"https://doi.org/10.5802/CRMATH.147","url":null,"abstract":"In this paper, we use particular polynomials to establish some results on the real rootedness of polynomials. The considered polynomials are Bell polynomials and Hermite polynomials. To cite this article: M. Mihoubi and S. Taharbouchet, C. R. Acad. Sci. Paris, Ser. I 340 (2021). Résumé. Dans ce papier, nous utilisons des polynômes particuliers pour établir quelques résultats sur les polynômes à racines réelles. Les polynômes considérés sont des polynômes de Bell et des polynômes de Hermite. Pour citer cet article : M. Mihoubi and S. Taharbouchet, C. R. Acad. Sci. Paris, Ser. I 340 (2021). Manuscript received 3rd November 2019, accepted 6th November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"37 1","pages":"57-64"},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74076710","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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