Comptes Rendus Mathematique最新文献

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IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-25 DOI: 10.5802/CRMATH.137

D. C. Sertbas

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.

We show that many hyperharmonic痕迹exist无限integers, and this refutes of Mező猜想了。特别地，对于r = 64·(2α−1)+32，超谐波数h) 33是α (mod 748 440)的153个不同值的整数，其中最小的r等于64·(22659−1)+32。2019数学学科分类。11B83, 05A10, 11B75。手稿于2020年2月12日收到，2020年7月20日和2020年10月22日修订，2020年10月23日接受。在[4]中，Conway和Guy引入了超谐波数，这是普通谐波数的推广。[8]先是conjecturéMezőhyperharmoniques数应不整的。文献中的几篇文章[1 - 3,5]支持这一猜想;然而，他们都没有证明这一点。在该说明中,我们证明存在无穷多个信封hyperharmoniques Mező,并驳斥了这样猜测。特别地，我们证明了当r = 64·(2α−1)+32时，超调和数h) 33是α(mod748440) 153个不同值的整数，其中最小的r是64·(22659−1)+32。

引用次数: 2

On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis 在Fekete-Szegö上为虚轴方向上的凸函数键入函数

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-25 DOI: 10.5802/CRMATH.144

P. Zaprawa

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.

本文考虑了两个Fekete-Szegö型泛函:Φ f (μ) = a2a4−μ a3和Θ f (μ) = a4−μa2a3，适用于解析函数f (z) = z +a2z +a3z +…，z∈∆，(∆= {z∈C: |z| < 1})和实数μ。对于虚轴方向上的一元凸函数f，我们得到了函数Φ f (μ)和Θ f (μ)的明确界限。可以将结果转移到KR(i)类的虚轴方向凸函数与实系数，也可以转移到T类的典型实数函数。作为推论，我们得到了在KR(i)和T中的第二汉克尔行列式的界。2020数学学科分类。30C50。资金。该项目/研究由波兰科学和高等教育部资助的卢布林科技大学区域卓越倡议项目(合同编号030/RID/2018/19)框架内资助。收稿于2019年8月14日，改稿于2020年6月24日，收稿于2020年11月3日。

引用次数: 3

Asymptotic behavior of solutions of fully nonlinear equations over exterior domains 外域上全非线性方程解的渐近性质

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-25 DOI: 10.5802/CRMATH.138

Xiaobiao Jia

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.

本文考虑了一类完全非线性椭圆方程F (D2u) = F (x)在外域上解在无穷远处的渐近性，其中Hessian矩阵(D2u)在无穷远处趋向于某个对称正定矩阵，F (x)在无穷远处= O(|x|−t)，且尖锐条件为t |。此外，当(D2u)仅在无穷远处非常接近某个对称正定矩阵时，我们也得到了同样的结果。2020数学学科分类。35J60, 35B40。2020年9月4日收稿，2020年10月9日改稿，2020年10月25日收稿。

引用次数: 3

La vie et l’oeuvre de John Tate 约翰·泰特的生平和作品

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-25 DOI: 10.5802/CRMATH.125

Jean-Pierre Serre

引用次数: 0

Influence of the number of Sylow subgroups on solvability of finite groups Sylow子群数目对有限群可解性的影响

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-25 DOI: 10.5802/CRMATH.146

C. Anabanti, A. Moretó, M. Zarrin

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.

设G是一个有限群。证明了如果G的Sylow 3-子群的个数不大于7,G的Sylow 5-子群的个数不大于1455，则G是可解的。这是Robati最近猜想的一种强形式。2020数学学科分类。20D10, 20D20, 20F16, 20F19。资金。第一作者得到格拉茨工业大学(R-1501000001)和奥地利科学基金(FWF)的部分资助:P30934-N35, F05503, F05510。他还在尼日利亚恩苏卡大学(UNN)工作。第二作者的研究得到了Ministerio de Ciencia e Innovación PID−2019−103854GB−100、Generalitat Valenciana AICO/2020/298和federer基金的支持。2020年10月4日收稿，2020年11月5日修改并验收。

引用次数: 0

A question of Malinowska on sizes of finite nonabelian simple groups in relation to involution sizes 有限非abel单群的大小与对合大小的关系的Malinowska问题

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-25 DOI: 10.5802/CRMATH.130

C. Anabanti

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.

令In (G)表示有限群G中n阶元素的个数。Malinowska最近问了一个问题:“当存在两个非阿贝尔有限简单群S和G，且|G|和|S|的素数因子p1，···，pk满足2 = p1 <···< pk且对于所有i∈{1，···，k}，我们有|G| = |S|，那么最小的正整数k是什么?”本文解决了马林诺夫斯卡的问题。2020数学学科分类。20D60,20D06。资金。作者得到格拉茨工业大学和奥地利科学基金(FWF)的部分资助:P30934-N35, F05503, F05510。他还在尼日利亚恩苏卡大学工作。收稿2020年5月25日，改稿2020年10月6日，收稿2020年10月7日。

引用次数: 2

A nonlocal Dirichlet problem with impulsive action: estimates of the growth for the solutions 一类具有脉冲作用的非局部狄利克雷问题:解的增长估计

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-25 DOI: 10.5802/CRMATH.109

J. C. Ferreira, M. Pereira

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.

本文研究了一类具有脉冲作用和狄利克雷条件的非局部扩散问题解在t→+∞时的渐近性质。在脉冲函数和非线性反应的适当假设下，建立了解的衰减率。

引用次数: 2

Comportement extrémal des copules diagonales et de Bertino 对角线和Bertino copula的极端行为

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-25 DOI: 10.5802/CRMATH.135

Christian Genest, M. Sabbagh

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

在适当的正则性条件下，确定了二元对角和Bertino联结的最大吸引子。给出了这些事实的一些结果，即具有给定对角线截面的对称联结的最大吸引子的界，以及上尾相关系数已知的可交换极值联结的Spearman的rho和Kendall的tau的界。然后将其中的一些结果推广到任意二元copula和多元copula的情况。分类mathassimatique(2020)。62年60 g70 g32。Financement。加拿大研究委员会主席、加拿大自然科学研究委员会主席、加拿大公共政治与科学研究所主席、加拿大公共政治与科学研究所主席。2020年6月9日复稿，2020年10月15日复稿，2020年10月21日收稿。A copula C是随机向量(U1，…)的分布函数。，Uk)，在单位间隔上具有均匀的边距。其对角线截面∆(C)为max(U1，…)的分布。英国)。几位作者考虑了当∆(C)已知时，对C能说些什么。在k = 2维，可交换一致随机变量随机对(U,V)的联合分布C的点向下界和上界由fr切特-霍夫丁公式给出。后者对应于V = 1−U或V =U几乎为共单调依赖的情况，ISSN(电子版):1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 1158 Christian Genest et Magid Sabbagh肯定。Nelsen等人[26]表明，当∆(C) = δ已知且C是对称的时，可以收紧这些界限。具体来说，有Bδ(u, v)≤C (u, v)≤Kδ(u, v)，对于所有(u, v)∈[0,1]，其中Bδ(u, v) = (u∧v)−inf{t−δ(t): t∈[u∧v,u∨v]}，定义了具有对角截面δ的Bertino copula [2]， Kδ(u, v) = u∧v∧{δ(u)+δ(v)}/2是另一个具有对角截面δ的copula, Fredricks和Nelsen[8]称之为“对角copula”。对于任意实数a和b, a∧b = min(a,b)和a∨b = max(a,b)。本文在δ上适当的正则性假设下，确定了copulbδ和Kδ的极值性。在第2节中首先证明，如果δ在1处有左导数δ '，即d = δ '(1−)，则Kδ属于参数θ = d/2∈[1/2,1]的联结函数的最大吸引域，对于所有(u, v)∈[0,1]2，由Dθ(u, v) = u∧v∧(uv)定义。进一步假设存在一个实数2∈(0,1)，使得映射δ δ:[0,1]→[0,1]在每个t∈[0,1]处由δ δ(t) = t - δ(t)定义在区间(2,1)上递减。在此附加条件下，在第3节中证明了Bδ属于参数θ = 2−d∈[0,1]的cuadras - aug copula的最大吸引域，对于所有(u, v)∈[0,1]2，定义为Qθ(u, v) = (uv)1−θ(u∧v)。这些结果的各种后果将在第4节中提到。首先，如果C是一个对角截面δ满足上述要求的对称二元copula，且C∗表示它的最大吸引子，且假设存在，则对于所有(u, v)∈[0,1]2,Q2−d (u, v)≤C∗(u, v)≤Dd/2(u, v)。这串不等式立即推导出与C相关的上尾相关系数为Λ(C) =Λ(C∗)=Λ(Q2−d) =Λ(Dd/2) = 2 - d。此外，如果C∗对称且具有上尾依赖性coefficientΛ(C∗)=λ，且ρ(C∗)和τ(C∗)分别表示与C∗相关的Spearman 's rho和Kendall 's tau的值，则3λ/(4−λ)≤ρ(C∗)≤3λ(8−5λ)/(4−λ)2和λ/(2−λ)≤τ(C∗)≤λ。这些边界解决了[22]中提出的一个问题，Jaworski[18]最近以不同的方式解决了这个问题。然后，第5节中的命题4展示了如何放宽C上的对称性假设。最后，第6节简要说明了对任意维度k > 2的可能扩展。文中指出，二元Bertino copula的k变量扩展通常不是一个分布，除非对角线部分是k/(k−1)次的Lipschitz增量，如Arias-García等人[1]所报道的那样，这阻碍了下界的搜索。相反，由Jaworski[17]引入的k变量对角copula的扩展在与二元情况相同的假设下确实具有最大吸引子。详见提案5。

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引用次数: 0

Effective André–Oort for non-compact curves in Hilbert modular varieties Hilbert模变中非紧曲线的有效andr<s:1> - oort

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-16 DOI: 10.5802/CRMATH.177

Gal Binyamini, D. Masser

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.

在大多数andr - oort猜想的证明中，有两个不同的步骤，其有效性尚不清楚:使用Brauer-Siegel的概括和使用Pila-Wilkie的概括。目前只有C曲线的情况是有效的(通过其他方法)。我们给出了非紧曲线在每一个Hilbert模曲面和每一个奇属Hilbert模变体上的andr - oort的有效证明(在一个次要的一般简单性条件下)。特别地，我们证明了在这些情况下，第一步可以用w stholz和第二作者的自同态估计以及andr通过g函数的专门化方法来代替，第二步可以用Novikov、Yakovenko和第一作者的q函数来实现。

引用次数: 7

Finite groups with Quaternion Sylow subgroup 四元数Sylow子群有限群

IF 0.8 4区数学 Q2 MATHEMATICS

Comptes Rendus Mathematique

Pub Date : 2021-01-05 DOI: 10.5802/crmath.131

Hamid Mousavi

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.

本文证明了具有四元数Sylow 2-子群的有限群G是2幂零的，如果3 |G|或G是可解的，并且它的Sylow 2-子群的阶严格大于16。数学学科分类(2010)。20 d99 20 e45。稿于2020年10月7日收到，2020年10月11日和2020年10月13日修订，2020年10月13日接受。

引用次数: 2

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