{"title":"La vie et l’oeuvre de Jean-Marc Fontaine","authors":"Jean-Pierre Serre","doi":"10.5802/crmath.126","DOIUrl":"https://doi.org/10.5802/crmath.126","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87530677","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}
The eigenvalue problem of stochastic Hamiltonian systems with boundary conditions was studied by Peng cite{peng} in 2000. For one-dimensional case, denoting by ${lambda_n}_{n=1}^{infty}$ all the eigenvalues of such an eigenvalue problem, Peng proved that $lambda_nto +infty$. In this short note, we prove that the growth order of $lambda_n$ is the same as $n^2$ as $nto +infty$. Apart from the interesting of its own, by this result, the statistic period of solutions of FBSDEs can be estimated directly by corresponding coefficients and time duration.
{"title":"A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”","authors":"Guangdong Jing, Penghui Wang","doi":"10.5802/CRMATH.103","DOIUrl":"https://doi.org/10.5802/CRMATH.103","url":null,"abstract":"The eigenvalue problem of stochastic Hamiltonian systems with boundary conditions was studied by Peng cite{peng} in 2000. For one-dimensional case, denoting by ${lambda_n}_{n=1}^{infty}$ all the eigenvalues of such an eigenvalue problem, Peng proved that $lambda_nto +infty$. In this short note, we prove that the growth order of $lambda_n$ is the same as $n^2$ as $nto +infty$. Apart from the interesting of its own, by this result, the statistic period of solutions of FBSDEs can be estimated directly by corresponding coefficients and time duration.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85561974","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}
Le sr iyantra(ou sr icakra) est un diagramme sacre dans les traditions hindoues tantriques. Il afait l’objet de nombreuses etudes dans differentes disciplines. En mathematiques, sa construction pose unprobleme elementaire et non trivial. Dans cette note, on fournit une methode de construction a la regle etau compas. La question est equivalente a celle d’un probleme d’Apollonius qui consiste a trouver un cercletangent a un cercle donne, a une droite donnee et passant par un point donne.
sr iyantra(或sr icakra)是印度教坦陀罗传统中的神圣图表。它在不同的学科中进行了大量的研究。在数学中，它的构造提出了一个基本而非平凡的问题。本说明提供了一种用尺子和罗盘构造的方法。这个问题等价于阿波罗尼厄斯的问题，即在给定的圆上，在给定的直线上，通过给定的点，找到一个圆的切线。
{"title":"Comptes Rendus Mathématique","authors":"Alessandro Chiodo","doi":"10.5802/CRMATH.163>","DOIUrl":"https://doi.org/10.5802/CRMATH.163>","url":null,"abstract":"Le sr iyantra(ou sr icakra) est un diagramme sacre dans les traditions hindoues tantriques. Il afait l’objet de nombreuses etudes dans differentes disciplines. En mathematiques, sa construction pose unprobleme elementaire et non trivial. Dans cette note, on fournit une methode de construction a la regle etau compas. La question est equivalente a celle d’un probleme d’Apollonius qui consiste a trouver un cercletangent a un cercle donne, a une droite donnee et passant par un point donne.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91377806","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 article, we consider the radial Dunkl geometric case k = 1 corresponding to flat Riemannian symmetric spaces in the complex case and we prove exact estimates for the positive valued Dunkl kernel and for the radial heat kernel. Dans cet article, nous considerons le cas geometrique radial de Dunkl k = 1 correspondant aux espaces symetriques riemanniens plats dans le cas complexe et nous prouvons des estimations exactes pour le noyau de Dunkla valeur positive et pour le noyau de chaleur radial.
在这篇文章中，我们考虑了径向Dunkl几何情况k = 1对应于平面黎曼对称空间的复杂情况，并给出了正值Dunkl核和径向热核的精确估计。在本文中，我们考虑了Dunkl k = 1在复情况下对应于黎曼对称平面空间的径向几何情况，并证明了Dunkl核和径向热核的精确估计。
{"title":"Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case A n","authors":"P. Graczyk, P. Sawyer","doi":"10.5802/CRMATH.188","DOIUrl":"https://doi.org/10.5802/CRMATH.188","url":null,"abstract":"In this article, we consider the radial Dunkl geometric case k = 1 corresponding to flat Riemannian symmetric spaces in the complex case and we prove exact estimates for the positive valued Dunkl kernel and for the radial heat kernel. Dans cet article, nous considerons le cas geometrique radial de Dunkl k = 1 correspondant aux espaces symetriques riemanniens plats dans le cas complexe et nous prouvons des estimations exactes pour le noyau de Dunkla valeur positive et pour le noyau de chaleur radial.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74305698","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 article, we correct the classification of unimodal map germs from the plane to the plane of Boardman symbol (2,2) given by Dimca and Gibson. Also, we characterize this classification of unimodal map germs in terms of certain invariants. Moreover, on the basis of this characterization we present an algorithm to compute the type of unimodal map germs of the Boardman symbol (2,2) without computing the normal form and give its implementation in the computer algebra system Singular [8]. 2020 Mathematics Subject Classification. 58Q05, 14H20. Funding. The research of the first author is supported by Higher Education Commission, Pakistan by the Project Number 7495/Punjab/NRPU/R&D/HEC/2017. Manuscript received 21st July 2020, accepted 8th September 2020.
{"title":"Contact unimodal map germs from the plane to the plane","authors":"M. Binyamin, S. Aslam, Khawar Mehmood","doi":"10.5802/crmath.114","DOIUrl":"https://doi.org/10.5802/crmath.114","url":null,"abstract":"In this article, we correct the classification of unimodal map germs from the plane to the plane of Boardman symbol (2,2) given by Dimca and Gibson. Also, we characterize this classification of unimodal map germs in terms of certain invariants. Moreover, on the basis of this characterization we present an algorithm to compute the type of unimodal map germs of the Boardman symbol (2,2) without computing the normal form and give its implementation in the computer algebra system Singular [8]. 2020 Mathematics Subject Classification. 58Q05, 14H20. Funding. The research of the first author is supported by Higher Education Commission, Pakistan by the Project Number 7495/Punjab/NRPU/R&D/HEC/2017. Manuscript received 21st July 2020, accepted 8th September 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87118900","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":"Appendix to the paper “On the Billingsley dimension of Birkhoff average in the countable symbolic space”","authors":"B. Selmi","doi":"10.5802/crmath.116","DOIUrl":"https://doi.org/10.5802/crmath.116","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91277702","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 A and B be two subsets of the nonnegative integers. We call A and B additive complements if all sufficiently large integers n can be written as a +b, where a ∈ A and b ∈ B . Let S = {12,22,32, · · ·} be the set of all square numbers. Ben Green was interested in the additive complement of S. He asked whether there is an additive complement B = {bn }n=1 ⊆Nwhich satisfies bn = π 2 16 n 2+o(n2). Recently, Chen and Fang proved that if B is such an additive complement, then limsup n→∞ π2 16 n 2 −bn n1/2 logn ≥ √ 2 π 1 log4 . They further conjectured that limsup n→∞ π2 16 n 2 −bn n1/2 logn =+∞. In this paper, we confirm this conjecture by giving a much more stronger result, i.e., limsup n→∞ π2 16 n 2 −bn n ≥ π 4 . 2020 Mathematics Subject Classification. 11B13, 11B75. Manuscript received 3rd August 2020, revised 19th August 2020, accepted 20th August 2020.
设A和B是非负整数的两个子集。如果所有足够大的整数n都可以写成A + B，其中A∈A, B∈B，我们称A和B为可加补数。设S ={12,22,32，···}为所有平方数的集合。Ben Green对s的加性补很感兴趣，他问是否存在一个B = {bn}n=1的可加性补，满足bn = π 2 16 n2 +o(n2)。最近，Chen和Fang证明了如果B是这样的可加补，则limsup n→∞π2 16 n 2−bn n /2 logn≥√2 π 1 log4。他们进一步推测limsup n→∞π2 16 n 2 - bn n /2 logn =+∞。在本文中，我们给出了一个更强的结果，即limsup n→∞π2 16 n 2−bn n≥π 4，从而证实了这个猜想。2020数学学科分类。11B13, 11B75。2020年8月3日收稿，2020年8月19日改稿，2020年8月20日收稿。
{"title":"Green’s problem on additive complements of the squares","authors":"Yuchen Ding","doi":"10.5802/crmath.107","DOIUrl":"https://doi.org/10.5802/crmath.107","url":null,"abstract":"Let A and B be two subsets of the nonnegative integers. We call A and B additive complements if all sufficiently large integers n can be written as a +b, where a ∈ A and b ∈ B . Let S = {12,22,32, · · ·} be the set of all square numbers. Ben Green was interested in the additive complement of S. He asked whether there is an additive complement B = {bn }n=1 ⊆Nwhich satisfies bn = π 2 16 n 2+o(n2). Recently, Chen and Fang proved that if B is such an additive complement, then limsup n→∞ π2 16 n 2 −bn n1/2 logn ≥ √ 2 π 1 log4 . They further conjectured that limsup n→∞ π2 16 n 2 −bn n1/2 logn =+∞. In this paper, we confirm this conjecture by giving a much more stronger result, i.e., limsup n→∞ π2 16 n 2 −bn n ≥ π 4 . 2020 Mathematics Subject Classification. 11B13, 11B75. Manuscript received 3rd August 2020, revised 19th August 2020, accepted 20th August 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89605079","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 the following functions fn (x) = 1− ln x + lnGn (x +1) x and gn (x) = x Gn (x +1) x , x ∈ (0,∞), n ∈N, where Gn (z) = (Γn (z))(−1) and Γn is the multiple gamma function of order n. In this work, our aim is to establish that f (2n) 2n (x) and (ln g2n (x)) (2n) are strictly completely monotonic on the positive half line for any positive integer n. In particular, we show that f2(x) and g2(x) are strictly completely monotonic and strictly logarithmically completely monotonic respectively on (0,3]. As application, we obtain new bounds for the Barnes G-function. 2020 Mathematics Subject Classification. 33B15, 26D07. Manuscript received 2nd August 2020, revised and accepted 8th September 2020.
我们考虑以下函数fn (x) = 1−ln x + lnGn (x + 1) x和gn (x) = x gn (x + 1) x, x∈(0,∞),n∈n,在gn (z) =(Γn (z))(−1)和Γn n的多个伽马函数。在这个工作中,我们的目标是建立f (2 n) 2 n (x)和(ln g2n (x)) (2 n)是严格完全单调正半直线上任何正整数n。特别是,我们证明了f2(x)和g2(x)分别在(0,3)上是严格完全单调和严格对数完全单调的。作为应用，我们得到了Barnes g函数的新的界。2020数学学科分类。33B15, 26D07。2020年8月2日收稿，2020年9月8日改稿。
{"title":"A complete monotonicity property of the multiple gamma function","authors":"Sourav Das","doi":"10.5802/crmath.115","DOIUrl":"https://doi.org/10.5802/crmath.115","url":null,"abstract":"We consider the following functions fn (x) = 1− ln x + lnGn (x +1) x and gn (x) = x Gn (x +1) x , x ∈ (0,∞), n ∈N, where Gn (z) = (Γn (z))(−1) and Γn is the multiple gamma function of order n. In this work, our aim is to establish that f (2n) 2n (x) and (ln g2n (x)) (2n) are strictly completely monotonic on the positive half line for any positive integer n. In particular, we show that f2(x) and g2(x) are strictly completely monotonic and strictly logarithmically completely monotonic respectively on (0,3]. As application, we obtain new bounds for the Barnes G-function. 2020 Mathematics Subject Classification. 33B15, 26D07. Manuscript received 2nd August 2020, revised and accepted 8th September 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76046723","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 prove that, in all dimensions, germs of nondegenerate holomorphic vector fields on complex manifolds are univalent in the sense of Palais (semicomplete in the sense of Rebelo), this is, that there exist neighborhoods of their singular points where all their solutions are single-valued. This implies that, in stark contrast with the degenerate case, all germs of nondegenerate holomorphic vector fields give local models for complete holomorphic vector fields on complex manifolds (albeit possibly non-Hausdorff ones). Résumé. On prouve que, en toute dimension, tout germe de champs de vecteurs holomorphe singulier nondégénéré sur une variété est univalent au sens de Palais (semicomplet au sens de Rebelo): en restriction à un voisinage convenable du point singulier, ses solutions n’ont pas de multivaluation. Ceci implique que, à la différence du cas dégénéré, un germe de champ de vecteurs holomorphe non-dégénéré est le modèle local d’un champ de vecteurs holomorphe complet sur une variété complexe (pas nécessairement séparée). 2020 Mathematics Subject Classification. 34M45, 34M35, 57S20, 32M05. Funding. PAPIIT (UNAM, Mexico) grant IN102518. Manuscript received 15th June 2020, revised 22nd July 2020, accepted 23rd July 2020.
{"title":"On the local univalence of nondegenerate holomorphic vector fields","authors":"Adolfo Guillot","doi":"10.5802/crmath.100","DOIUrl":"https://doi.org/10.5802/crmath.100","url":null,"abstract":"We prove that, in all dimensions, germs of nondegenerate holomorphic vector fields on complex manifolds are univalent in the sense of Palais (semicomplete in the sense of Rebelo), this is, that there exist neighborhoods of their singular points where all their solutions are single-valued. This implies that, in stark contrast with the degenerate case, all germs of nondegenerate holomorphic vector fields give local models for complete holomorphic vector fields on complex manifolds (albeit possibly non-Hausdorff ones). Résumé. On prouve que, en toute dimension, tout germe de champs de vecteurs holomorphe singulier nondégénéré sur une variété est univalent au sens de Palais (semicomplet au sens de Rebelo): en restriction à un voisinage convenable du point singulier, ses solutions n’ont pas de multivaluation. Ceci implique que, à la différence du cas dégénéré, un germe de champ de vecteurs holomorphe non-dégénéré est le modèle local d’un champ de vecteurs holomorphe complet sur une variété complexe (pas nécessairement séparée). 2020 Mathematics Subject Classification. 34M45, 34M35, 57S20, 32M05. Funding. PAPIIT (UNAM, Mexico) grant IN102518. Manuscript received 15th June 2020, revised 22nd July 2020, accepted 23rd July 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87852554","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}
The present work aims to approximate the solution of linear time fractional PDE with Caputo Fabrizio derivative. For the said purpose Laplace transform with local radial basis functions is used. The Laplace transform is applied to obtain the corresponding time independent equation in Laplace space and then the local RBFs are employed for spatial discretization. The solution is then represented as a contour integral in the complex space, which is approximated by trapezoidal rule with high accuracy. The application of Laplace transform avoids the time stepping procedure which commonly encounters the time instability issues. The convergence of the method is discussed also we have derived the bounds for the stability constant of the differentiation matrix of our proposed numerical scheme. The efficiency of the method is demonstrated with the help of numerical examples. For our numerical experiments we have selected three different domains, in the first test case the square domain is selected, for the second test the circular domain is considered, while for third case the L-shape domain is selected. Manuscript received 13th August 2019, revised and accepted 20th July 2020.
{"title":"A transform based local RBF method for 2D linear PDE with Caputo–Fabrizio derivative","authors":"Kamran, Amjad Ali, J. F. Gómez‐Aguilar","doi":"10.5802/crmath.98","DOIUrl":"https://doi.org/10.5802/crmath.98","url":null,"abstract":"The present work aims to approximate the solution of linear time fractional PDE with Caputo Fabrizio derivative. For the said purpose Laplace transform with local radial basis functions is used. The Laplace transform is applied to obtain the corresponding time independent equation in Laplace space and then the local RBFs are employed for spatial discretization. The solution is then represented as a contour integral in the complex space, which is approximated by trapezoidal rule with high accuracy. The application of Laplace transform avoids the time stepping procedure which commonly encounters the time instability issues. The convergence of the method is discussed also we have derived the bounds for the stability constant of the differentiation matrix of our proposed numerical scheme. The efficiency of the method is demonstrated with the help of numerical examples. For our numerical experiments we have selected three different domains, in the first test case the square domain is selected, for the second test the circular domain is considered, while for third case the L-shape domain is selected. Manuscript received 13th August 2019, revised and accepted 20th July 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82952541","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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