We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.
{"title":"On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations","authors":"Clément Cancès, Juliette Venel","doi":"10.5802/crmath.421","DOIUrl":"https://doi.org/10.5802/crmath.421","url":null,"abstract":"We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135659035","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 Brezzi–Douglas–Marini interpolation error on anisotropic elements has been analyzed in two recent publications, the first focusing on simplices with estimates in L 2 , the other considering parallelotopes with estimates in terms of L p -norms. This contribution provides generalized estimates for anisotropic simplices for the L p case, 1≤p≤∞, and shows new estimates for anisotropic prisms with triangular base.
最近的两篇论文分析了各向异性元素上的Brezzi-Douglas-Marini插值误差，第一篇论文关注的是具有l2估计的简单体，另一篇论文考虑了具有L p -范数估计的平行四边形。这一贡献提供了L p(1≤p≤∞)情况下各向异性简单体的广义估计，并给出了三角形基底的各向异性棱镜的新估计。
{"title":"Brezzi–Douglas–Marini interpolation on anisotropic simplices and prisms","authors":"Volker Kempf","doi":"10.5802/crmath.424","DOIUrl":"https://doi.org/10.5802/crmath.424","url":null,"abstract":"The Brezzi–Douglas–Marini interpolation error on anisotropic elements has been analyzed in two recent publications, the first focusing on simplices with estimates in L 2 , the other considering parallelotopes with estimates in terms of L p -norms. This contribution provides generalized estimates for anisotropic simplices for the L p case, 1≤p≤∞, and shows new estimates for anisotropic prisms with triangular base.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135995595","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}
Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee, Seong-Deog Yang
We construct a two discrete parameter family of compact minimal surfaces embedded in the Berger sphere which may be considered as the analogue of the helicoidal Karcher-Scherk surfaces.
{"title":"Addendum to the paper: Compact embedded minimal surfaces in the Berger sphere","authors":"Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee, Seong-Deog Yang","doi":"10.5802/crmath.403","DOIUrl":"https://doi.org/10.5802/crmath.403","url":null,"abstract":"We construct a two discrete parameter family of compact minimal surfaces embedded in the Berger sphere which may be considered as the analogue of the helicoidal Karcher-Scherk surfaces.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135997151","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 a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.
{"title":"On uniform controllability of 1D transport equations in the vanishing viscosity limit","authors":"Camille Laurent, Matthieu Léautaud","doi":"10.5802/crmath.405","DOIUrl":"https://doi.org/10.5802/crmath.405","url":null,"abstract":"We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135948610","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}
Five binomial sums are extended by a free parameter m, that are shown, through the generating function method, to have the same value. These sums generalize the ones by Ruehr (1980), who discovered them in the study of two unexpected equalities between definite integrals. 2020 Mathematics Subject Classification. 11B65, 05A10. Manuscript received 22nd December 2020, revised and accepted 2nd February 2021. In 1980, Kimura [14] proposed a monthly problem about two curious identities of definite integrals. If f is continuous on [− 1 2 , 3 2 ], then for δ= 0, 1, prove that ∫ 3 2 − 2 x f ( 3x −2x3)dx = 2∫ 1 0 x f ( 3x −2x3)dx. In his (trigonometric) proof, Ruehr [14] observed by linearity that to prove these identities, it is enough to verify them for monomials f (x) = xn . This led him to discover the following interesting identities An =Cn and Bn = Dn , where for a natural number n, the four binomial sums are defined by An = n ∑ j=0 3 j ( 3n − j 2n ) , Bn = n ∑ j=0 2 j ( 3n +1 2n + j +1 ) , Cn = 2n ∑ j=0 (−3) j ( 3n − j n ) , Dn = 2n ∑ j=0 (−4) j ( 3n +1 n + j +1 ) . ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 422 Mei Bai and Wenchang Chu Allouche [1, 2] examined the related integrals and reviewed these identities in a more elegant manner. These identities were also reconfirmed by Meehan et al [16] who found, through the WZ-algorithm, that these four sequences satisfy also the common recurrence relation: X0 = 1 and Xn+1 = 27 4 Xn − 3 (3n+1 n ) 4(n +1) . By introducing a variable, Alzer and Prodinger [3] recently considered the polynomial analogues, that were also examined by Kilic–Arikan [13] through bijections. By applying the generating function approach to the binomial convolutions Ωn = n ∑ k=0 ( 3k k )( 3n −3k n −k ) the authors [4] not only confirmed the identities Ωn = An = Bn =Cn = Dn ; (1) but also found the two additional ones Ωn = En = Fn , (2)
五个二项式和由一个自由参数m展开，通过生成函数法显示出它们具有相同的值。这些和推广了Ruehr(1980)的和，Ruehr在研究定积分之间的两个意想不到的等式时发现了它们。2020数学学科分类[j] . 11B65, 05A10。稿件于2020年12月22日收稿，2021年2月2日修订并接受。1980年，Kimura[14]提出了关于两个奇异的定积分恒等式的月问题。如果f在[- 1,2,3,2]上连续，那么对于δ= 0,1，证明∫32 - 2x f (3x - 2x3)dx = 2∫10 x f (3x - 2x3)dx。在他的(三角)证明中，Ruehr[14]通过线性观察到，为了证明这些恒等式，只要对单项式f (x) = xn进行验证就足够了。这使他发现以下有趣的身份= Cn和Bn = Dn,为自然数n,定义的四个二项金额= n∑j = 0 3 (3 n−j 2 n) Bn n =∑j = 0 2 (3 n + 1 2 n + j + 1), Cn = 2 n∑j = 0(−3)(3 n−j n), Dn = 2 n∑j = 0(−4)(3 n + 1 n + j + 1)。∗通讯作者。422白梅、楚文昌[1,2]考察了相关的积分，更优雅地回顾了这些恒等式。Meehan等人[16]也再次证实了这些等式，他们通过wz算法发现这四个序列也满足共同递归关系:X0 = 1和Xn+1 = 274 Xn−3 (3n+1 n) 4(n +1)。通过引入一个变量，Alzer和Prodinger[3]最近考虑了多项式类似物，Kilic-Arikan[13]也通过双射检验了多项式类似物。将生成函数方法应用于二项式卷积Ωn = n∑k=0 (3k k)(3n−3k n−k)，作者[4]不仅证实了等式Ωn = An = Bn =Cn = Dn;(1)还发现了另外两个Ωn = En = Fn， (2)
{"title":"Further Equivalent Binomial Sums","authors":"M. Bai, W. Chu","doi":"10.5802/CRMATH.184","DOIUrl":"https://doi.org/10.5802/CRMATH.184","url":null,"abstract":"Five binomial sums are extended by a free parameter m, that are shown, through the generating function method, to have the same value. These sums generalize the ones by Ruehr (1980), who discovered them in the study of two unexpected equalities between definite integrals. 2020 Mathematics Subject Classification. 11B65, 05A10. Manuscript received 22nd December 2020, revised and accepted 2nd February 2021. In 1980, Kimura [14] proposed a monthly problem about two curious identities of definite integrals. If f is continuous on [− 1 2 , 3 2 ], then for δ= 0, 1, prove that ∫ 3 2 − 2 x f ( 3x −2x3)dx = 2∫ 1 0 x f ( 3x −2x3)dx. In his (trigonometric) proof, Ruehr [14] observed by linearity that to prove these identities, it is enough to verify them for monomials f (x) = xn . This led him to discover the following interesting identities An =Cn and Bn = Dn , where for a natural number n, the four binomial sums are defined by An = n ∑ j=0 3 j ( 3n − j 2n ) , Bn = n ∑ j=0 2 j ( 3n +1 2n + j +1 ) , Cn = 2n ∑ j=0 (−3) j ( 3n − j n ) , Dn = 2n ∑ j=0 (−4) j ( 3n +1 n + j +1 ) . ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 422 Mei Bai and Wenchang Chu Allouche [1, 2] examined the related integrals and reviewed these identities in a more elegant manner. These identities were also reconfirmed by Meehan et al [16] who found, through the WZ-algorithm, that these four sequences satisfy also the common recurrence relation: X0 = 1 and Xn+1 = 27 4 Xn − 3 (3n+1 n ) 4(n +1) . By introducing a variable, Alzer and Prodinger [3] recently considered the polynomial analogues, that were also examined by Kilic–Arikan [13] through bijections. By applying the generating function approach to the binomial convolutions Ωn = n ∑ k=0 ( 3k k )( 3n −3k n −k ) the authors [4] not only confirmed the identities Ωn = An = Bn =Cn = Dn ; (1) but also found the two additional ones Ωn = En = Fn , (2)","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90924742","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 construction of the Śrī Yantra","authors":"A. Chiodo","doi":"10.5802/CRMATH.163","DOIUrl":"https://doi.org/10.5802/CRMATH.163","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86845100","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 note we shall show that a lattice Zω1 +Zω2 in C has Q-linearly dependent quasi-periods if and only if ω2/ω1 is equivalent to a zero of the Eisenstein series E2 under the action of SL2(Z) on the upper half plane of C. 2020 Mathematics Subject Classification. 11J72, 11J89. Manuscript received 23rd March 2020, revised 7th August 2020, accepted 17th December 2020.
{"title":"Linear dependence of quasi-periods over the rationals","authors":"K. S. Kumar","doi":"10.5802/CRMATH.171","DOIUrl":"https://doi.org/10.5802/CRMATH.171","url":null,"abstract":"In this note we shall show that a lattice Zω1 +Zω2 in C has Q-linearly dependent quasi-periods if and only if ω2/ω1 is equivalent to a zero of the Eisenstein series E2 under the action of SL2(Z) on the upper half plane of C. 2020 Mathematics Subject Classification. 11J72, 11J89. Manuscript received 23rd March 2020, revised 7th August 2020, accepted 17th December 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86599515","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 analysis of Cartesian Perfectly Matched Layers (PMLs) in the context of time-domain electromagnetic wave propagation in a 3D unbounded anisotropic homogeneous medium modelled by a diagonal dielectric tensor is presented. Contrary to the 3D scalar wave equation or 2D Maxwell's equations some diagonal anisotropies lead to the existence of backward waves giving rise to instabilities of the PMLs. Numerical experiments confirm the presented result.
{"title":"On a surprising instability result of Perfectly Matched Layers for Maxwell’s equations in 3D media with diagonal anisotropy","authors":"É. Bécache, S. Fliss, M. Kachanovska, M. Kazakova","doi":"10.5802/CRMATH.165","DOIUrl":"https://doi.org/10.5802/CRMATH.165","url":null,"abstract":"The analysis of Cartesian Perfectly Matched Layers (PMLs) in the context of time-domain electromagnetic wave propagation in a 3D unbounded anisotropic homogeneous medium modelled by a diagonal dielectric tensor is presented. Contrary to the 3D scalar wave equation or 2D Maxwell's equations some diagonal anisotropies lead to the existence of backward waves giving rise to instabilities of the PMLs. Numerical experiments confirm the presented result.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85403141","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 have proved two main theorems under more weaker conditions dealing with absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series. We have also obtained certain new results on the different absolute summability methods. 2020 Mathematics Subject Classification. 26D15, 40D15, 42A24, 46A45. Manuscript received 14th November 2020, accepted 13th January 2021.
{"title":"A new note on factored infinite series and trigonometric Fourier series","authors":"H. Bor","doi":"10.5802/CRMATH.179","DOIUrl":"https://doi.org/10.5802/CRMATH.179","url":null,"abstract":"In this paper, we have proved two main theorems under more weaker conditions dealing with absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series. We have also obtained certain new results on the different absolute summability methods. 2020 Mathematics Subject Classification. 26D15, 40D15, 42A24, 46A45. Manuscript received 14th November 2020, accepted 13th January 2021.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86897353","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 discuss the local properties of quasihyperbolic mappings in metric spaces, which are related to an open problem raised by Huang et al in 2016. Our result is a partial solution to this problem, which is also a generalization of the corresponding result obtained by Huang et al in 2016. 2020 Mathematics Subject Classification. 30L10, 53C23, 30L99, 30F10. Funding. The first author was supported by NNSF of China (No. 11901090), and by Department of Education of Guangdong Province, China (No. 2018KQNCX285). The second author was partly supported by NNSF of China (No. 11601529,11971124), and by Scientifific Research Fund of Hunan Provincial Education Department (No. 20B118). Manuscript received 9th June 2020, revised 4th October 2020, accepted 22nd November 2020.
{"title":"Quasihyperbolic mappings in length metric spaces","authors":"Qingshan Zhou, Yaxiang Li, Yuehui He","doi":"10.5802/CRMATH.154","DOIUrl":"https://doi.org/10.5802/CRMATH.154","url":null,"abstract":"In this paper, we discuss the local properties of quasihyperbolic mappings in metric spaces, which are related to an open problem raised by Huang et al in 2016. Our result is a partial solution to this problem, which is also a generalization of the corresponding result obtained by Huang et al in 2016. 2020 Mathematics Subject Classification. 30L10, 53C23, 30L99, 30F10. Funding. The first author was supported by NNSF of China (No. 11901090), and by Department of Education of Guangdong Province, China (No. 2018KQNCX285). The second author was partly supported by NNSF of China (No. 11601529,11971124), and by Scientifific Research Fund of Hunan Provincial Education Department (No. 20B118). Manuscript received 9th June 2020, revised 4th October 2020, accepted 22nd November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76035154","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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