Comptes Rendus Mathematique最新文献

{"title":"Hyperharmonic integers exist","authors":"D. C. Sertbas","doi":"10.5802/CRMATH.137","DOIUrl":"https://doi.org/10.5802/CRMATH.137","url":null,"abstract":"We show that there exist infinitely many hyperharmonic integers, and this refutes a conjecture of Mező. In particular, for r = 64 ·(2α−1)+32, the hyperharmonic number h ) 33 is integer for 153 different values of α (mod 748 440), where the smallest r is equal to 64 · (22659 −1)+32. 2020 Mathematics Subject Classification. 11B83, 05A10, 11B75. Manuscript received 12th February 2020, revised 20th July 2020 and 22nd October 2020, accepted 23rd October 2020. Version française abrégée Dans [4], Conway et Guy ont introduit des nombres hyperharmoniques qui sont une généralisation des nombres harmoniques ordinaires. Mező [8] a d’abord conjecturé que les nombres hyperharmoniques n’étaient pas des entiers. Plusieurs articles [1–3, 5] dans la littérature soutiennent cette conjecture ; cependant, aucun d’entre eux ne la prouve. Dans cette note, nous prouvons qu’il existe une infinité d’entiers hyperharmoniques, et cela réfute la conjecture de Mező. En particulier, nous montrons que pour r = 64 · (2α− 1)+ 32, le nombre hyperharmonique h ) 33 est un entier pour 153 valeurs différentes de α(mod748440), où le plus petit r est 64 · (22659 −1)+32.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"20 1","pages":"1179-1185"},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84392040","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":"On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis","authors":"P. Zaprawa","doi":"10.5802/CRMATH.144","DOIUrl":"https://doi.org/10.5802/CRMATH.144","url":null,"abstract":"In this paper we consider two functionals of the Fekete–Szegö type: Φ f (μ) = a2a4 − μa3 and Θ f (μ) = a4 −μa2a3 for analytic functions f (z) = z +a2z +a3z + . . ., z ∈∆, (∆= {z ∈C : |z| < 1}) and for real numbers μ. For f which is univalent and convex in the direction of the imaginary axis, we find sharp bounds of the functionals Φ f (μ) and Θ f (μ). It is possible to transfer the results onto the class KR(i ) of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class T of typically real functions. As corollaries, we obtain bounds of the second Hankel determinant in KR(i ) and T . 2020 Mathematics Subject Classification. 30C50. Funding. The project/research was financed in the framework of the project Lublin University of Technology Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract no.030/RID/2018/19). Manuscript received 14th August 2019, revised 24th June 2020, accepted 3rd November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"1 1","pages":"1213-1226"},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90148219","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":"Asymptotic behavior of solutions of fully nonlinear equations over exterior domains","authors":"Xiaobiao Jia","doi":"10.5802/CRMATH.138","DOIUrl":"https://doi.org/10.5802/CRMATH.138","url":null,"abstract":"In this paper, we consider the asymptotic behavior at infinity of solutions of a class of fully nonlinear elliptic equations F (D2u) = f (x) over exterior domains, where the Hessian matrix (D2u) tends to some symmetric positive definite matrix at infinity and f (x) = O(|x|−t ) at infinity with sharp condition t > 2. Moreover, we also obtain the same result if (D2u) is only very close to some symmetric positive definite matrix at infinity. 2020 Mathematics Subject Classification. 35J60, 35B40. Manuscript received 4th September 2020, revised 9th October 2020, accepted 25th October 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"397 1","pages":"1187-1197"},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77461065","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":"La vie et l’oeuvre de John Tate","authors":"Jean-Pierre Serre","doi":"10.5802/CRMATH.125","DOIUrl":"https://doi.org/10.5802/CRMATH.125","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"39 1","pages":"1129-1133"},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76936324","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":"Influence of the number of Sylow subgroups on solvability of finite groups","authors":"C. Anabanti, A. Moretó, M. Zarrin","doi":"10.5802/CRMATH.146","DOIUrl":"https://doi.org/10.5802/CRMATH.146","url":null,"abstract":"Let G be a finite group. We prove that if the number of Sylow 3-subgroups of G is at most 7 and the number of Sylow 5-subgroups of G is at most 1455, then G is solvable. This is a strong form of a recent conjecture of Robati. 2020 Mathematics Subject Classification. 20D10, 20D20, 20F16, 20F19. Funding. The first author is supported by both TU Graz (R-1501000001) and partial funding from the Austrian Science Fund (FWF): P30934–N35, F05503, F05510. He is also at the University of Nigeria, Nsukka (UNN). The research of the second author is supported by Ministerio de Ciencia e Innovación PID−2019−103854GB−100, Generalitat Valenciana AICO/2020/298 and FEDER funds. Manuscript received 4th October 2020, revised and accepted 5th November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"84 1","pages":"1227-1230"},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90417806","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 question of Malinowska on sizes of finite nonabelian simple groups in relation to involution sizes","authors":"C. Anabanti","doi":"10.5802/CRMATH.130","DOIUrl":"https://doi.org/10.5802/CRMATH.130","url":null,"abstract":"Let In (G) denote the number of elements of order n in a finite group G . Malinowska recently asked “what is the smallest positive integer k such that whenever there exist two nonabelian finite simple groups S and G with prime divisors p1, · · · , pk of |G| and |S| satisfying 2 = p1 < ·· · < pk and Ipi (G) = Ipi (S) for all i ∈ {1, · · · , k}, we have that |G| = |S|?”. This paper resolves Malinowska’s question. 2020 Mathematics Subject Classification. 20D60,20D06. Funding. The author is supported by both TU Graz and partial funding from the Austrian Science Fund (FWF): P30934-N35, F05503, F05510. He is also at the University of Nigeria, Nsukka. Manuscript received 25th May 2020, revised 6th October 2020, accepted 7th October 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"18 1","pages":"1135-1138"},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81811721","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 nonlocal Dirichlet problem with impulsive action: estimates of the growth for the solutions","authors":"J. C. Ferreira, M. Pereira","doi":"10.5802/CRMATH.109","DOIUrl":"https://doi.org/10.5802/CRMATH.109","url":null,"abstract":"Through this paper we deal with the asymptotic behaviour as t→ +∞ of the solutions for the nonlocal diffusion problem with impulsive actions and Dirichlet condition. We establish a decay rate for the solutions assuming appropriate hypotheses on the impulsive functions and the nonlinear reaction.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"79 1","pages":"1119-1128"},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75846421","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":"Comportement extrémal des copules diagonales et de Bertino","authors":"Christian Genest, M. Sabbagh","doi":"10.5802/CRMATH.135","DOIUrl":"https://doi.org/10.5802/CRMATH.135","url":null,"abstract":"The maximal attractors of bivariate diagonal and Bertino copulas are determined under suitable regularity conditions. Some consequences of these facts are drawn, namely bounds on the maximal attractor of a symmetric copula with a given diagonal section, and bounds on Spearman’s rho and Kendall’s tau for an exchangeable extreme-value copula whose upper-tail dependence coefficient is known. Some of these results are then extended to the case of arbitrary bivariate copulas and to multivariate copulas. Classification Mathématique (2020). 60G70, 62G32. Financement. Ce travail a bénéficié de l’appui financier du Secrétariat des Chaires de recherche du Canada, du Conseil de recherches en sciences naturelles et en génie du Canada, ainsi que de l’Institut Trottier pour la science et la politique publique. Manuscrit reçu le 9 juin 2020, révisé le 15 octobre 2020, accepté le 21 octobre 2020. Abridged English version A copula C is the distribution function of a random vector (U1, . . . ,Uk ) with uniform margins on the unit interval. Its diagonal section ∆(C ) is the distribution of max(U1, . . . ,Uk ). Several authors have considered the question of what can be said about C when ∆(C ) is known. In dimension k = 2, point-wise lower and upper bounds on the joint distribution C of a random pair (U ,V ) of exchangeable uniform random variables are given by the Fréchet–Hoeffding copulas. The latter correspond to the cases of comonotonic dependence in which either V = 1−U or V =U almost ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 1158 Christian Genest et Magid Sabbagh surely. Nelsen et al. [26] showed that when∆(C ) = δ is known and C is symmetric, it is possible to tighten these bounds. Specifically, one has Bδ(u, v) ≤C (u, v) ≤ Kδ(u, v), for all (u, v) ∈ [0,1], where Bδ(u, v) = (u ∧ v)− inf{t −δ(t ) : t ∈ [u ∧ v,u ∨ v]}, defines the Bertino copula [2] with diagonal section δ and Kδ(u, v) = u ∧ v ∧ {δ(u)+δ(v)}/2 is another copula with diagonal section δ which Fredricks and Nelsen [8] called a “diagonal copula.” Here and below, a ∧b = min(a,b) and a ∨b = max(a,b) for any reals a and b. In this paper, the extremal behavior of the copulas Bδ and Kδ is determined under suitable regularity assumptions on δ. It is first shown in Section 2 that if δ admits a left-sided derivative δ′ at 1, say d = δ′(1−), then Kδ belongs to the max domain of attraction of the copula with parameter θ = d/2 ∈ [1/2,1] defined, for all (u, v) ∈ [0,1]2, by Dθ(u, v) = u ∧ v ∧ (uv) . Further assume that there exits a real 2 ∈ (0,1) such that the map δ̂ : [0,1] → [0,1] defined at each t ∈ [0,1] by δ̂(t ) = t−δ(t ) is decreasing on the interval (2,1). Under this additional condition, it is shown in Section 3 that Bδ belongs to the max domain of attraction of the Cuadras–Augé copula with parameter θ = 2−d ∈ [0,1] defined, for all (u, v) ∈ [0,1]2, by Qθ(u, v) = (uv)1−θ(u ∧ v) . Various consequences of these results are mentioned in Section 4. First and f","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"146 1","pages":"1157-1167"},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80546457","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 André–Oort for non-compact curves in Hilbert modular varieties","authors":"Gal Binyamini, D. Masser","doi":"10.5802/CRMATH.177","DOIUrl":"https://doi.org/10.5802/CRMATH.177","url":null,"abstract":"In the proofs of most cases of the André-Oort conjecture, there are two different steps whose effectivity is unclear: the use of generalizations of Brauer-Siegel and the use of Pila-Wilkie. Only the case of curves in C is currently known effectively (by other methods). We give an effective proof of André-Oort for non-compact curves in every Hilbert modular surface and every Hilbert modular variety of odd genus (under a minor generic simplicity condition). In particular we show that in these cases the first step may be replaced by the endomorphism estimates of Wüstholz and the second author together with the specialization method of André via G-functions, and the second step may be effectivized using the Q-functions of Novikov, Yakovenko and the first author.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"6 1","pages":"313-321"},"PeriodicalIF":0.8,"publicationDate":"2021-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88281706","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":"Finite groups with Quaternion Sylow subgroup","authors":"Hamid Mousavi","doi":"10.5802/crmath.131","DOIUrl":"https://doi.org/10.5802/crmath.131","url":null,"abstract":"In this paper we show that a finite group G with Quaternion Sylow 2-subgroup is 2-nilpotent if, either 3 |G| or G is solvable and the order of its Sylow 2-subgroup is strictly greater than 16. Mathematical subject classification (2010). 20D99, 20E45. Manuscript received 7th October 2020, revised 11th October 2020 and 13th October 2020, accepted 13th October 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"45 1","pages":"1097-1099"},"PeriodicalIF":0.8,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73811343","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}