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Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group Heisenberg群上Schrödinger算子的局部色散和Strichartz估计
数学研究通讯 Pub Date : 2020-12-15 DOI: 10.4208/cmr.2021-0101
H. Bahouri, I. Gallagher
{"title":"Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group","authors":"H. Bahouri, I. Gallagher","doi":"10.4208/cmr.2021-0101","DOIUrl":"https://doi.org/10.4208/cmr.2021-0101","url":null,"abstract":"It was proved by H. Bahouri, P. G{'e}rard and C.-J. Xu in [9] that the Schr{\"o}dinger equation on the Heisenberg group $mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $mathbb{H}^d$ for the linear Schr{\"o}dinger equation, by a refined study of the Schr{\"o}dinger kernel $S_t$ on $mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $mathbb{H}^d$ derived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $mathbb{H}^d$.","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42750892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Moving-Water Equilibria Preserving HLL-Type Schemes for the Shallow Water Equations 浅水方程的动水保平衡HLL型格式
数学研究通讯 Pub Date : 2020-06-01 DOI: 10.4208/cmr.2020-0013
C. Klingenberg
{"title":"Moving-Water Equilibria Preserving HLL-Type Schemes for the Shallow Water Equations","authors":"C. Klingenberg","doi":"10.4208/cmr.2020-0013","DOIUrl":"https://doi.org/10.4208/cmr.2020-0013","url":null,"abstract":"We construct new HLL-type moving-water equilibria preserving upwind schemes for the one-dimensional Saint-Venant system of shallow water equations with nonflat bottom topography. The designed firstand secondorder schemes are tested on a number of numerical examples, in which we verify the well-balanced property as well as the ability of the proposed schemes to accurately capture small perturbations of moving-water steady states. AMS subject classifications: 76M12, 65M08, 35L65, 86-08, 86A05","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48539846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
Global Well-Posedness for the 2-D MHD Equations with Magnetic Diffusion 具有磁扩散的二维磁流体动力学方程的全局适定性
数学研究通讯 Pub Date : 2020-06-01 DOI: 10.4208/cmr.2020-0022
Dongyi Zhang
{"title":"Global Well-Posedness for the 2-D MHD Equations with Magnetic Diffusion","authors":"Dongyi Zhang","doi":"10.4208/cmr.2020-0022","DOIUrl":"https://doi.org/10.4208/cmr.2020-0022","url":null,"abstract":"","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43482861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 23
Preface 前言
数学研究通讯 Pub Date : 2020-06-01 DOI: 10.4208/cmr.2020.v36n2.preface
global sci
{"title":"Preface","authors":"global sci","doi":"10.4208/cmr.2020.v36n2.preface","DOIUrl":"https://doi.org/10.4208/cmr.2020.v36n2.preface","url":null,"abstract":"","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49208414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Remark on the Lifespan of Solutions to 3-D Anisotropic Navier Stokes Equations 关于三维各向异性Navier-Stokes方程解的寿命的注记
数学研究通讯 Pub Date : 2020-06-01 DOI: 10.4208/cmr.2020-0002
Siyu Liang
{"title":"Remark on the Lifespan of Solutions to 3-D Anisotropic Navier Stokes Equations","authors":"Siyu Liang","doi":"10.4208/cmr.2020-0002","DOIUrl":"https://doi.org/10.4208/cmr.2020-0002","url":null,"abstract":"The goal of this article is to provide a lower bound for the lifespan of smooth solutions to 3-D anisotropic incompressible Navier-Stokes system, which in particular extends a similar type of result for the classical 3-D incompressible Navier-Stokes system. AMS subject classifications: 35Q30, 76D03","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49408964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Precision Matrix Estimation by Inverse Principal Orthogonal Decomposition 逆主正交分解的矩阵精度估计
数学研究通讯 Pub Date : 2020-06-01 DOI: 10.4208/cmr.2020-0001
C. Tang
{"title":"Precision Matrix Estimation by Inverse Principal Orthogonal Decomposition","authors":"C. Tang","doi":"10.4208/cmr.2020-0001","DOIUrl":"https://doi.org/10.4208/cmr.2020-0001","url":null,"abstract":"We investigate the structure of a large precision matrix in Gaussian graphical models by decomposing it into a low rank component and a remainder part with sparse precision matrix. Based on the decomposition, we propose to estimate the large precision matrix by inverting a principal orthogonal decomposition (IPOD). The IPOD approach has appealing practical interpretations in conditional graphical models given the low rank component, and it connects to Gaussian graphical models with latent variables. Specifically, we show that the low rank component in the decomposition of the large precision matrix can be viewed as the contribution from the latent variables in a Gaussian graphical model. Compared with existing approaches for latent variable graphical models, the IPOD is conveniently feasible in practice where only inverting a low-dimensional matrix is required. To identify the number of latent variables, which is an objective of its own interest, we investigate and justify an approach by examining the ratios of adjacent eigenvalues of the sample covariance matrix. Theoretical properties, numerical examples, and a real data application demonstrate the merits of the IPOD approach in its convenience, performance, and interpretability.","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47965483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Periodic Solutions for a Damped Rayleigh Beam Model with Time Delay 含时滞阻尼瑞利光束模型的周期解
数学研究通讯 Pub Date : 2020-06-01 DOI: 10.4208/cmr.2020-0015
Bochao Li
{"title":"Periodic Solutions for a Damped Rayleigh Beam Model with Time Delay","authors":"Bochao Li","doi":"10.4208/cmr.2020-0015","DOIUrl":"https://doi.org/10.4208/cmr.2020-0015","url":null,"abstract":"Vibrations of a beam can be described as an Euler-Bernoulli beam, or as a Rayleigh beam or as a Timoshenko beam. In this paper, we establish the existence of periodic solutions in time for a damped Rayleigh beam model with time delay, which is treated as a bifurcation parameter. The main proof is based on a Lyapunov-Schmidt reduction together with the classical implicit function theorem. Moreover, we give a sufficient condition for a direction of bifurcation. AMS subject classifications: 35L75, 37K50","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47861790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Bounding Topology via Geometry, $A$-Simple Fundamental Groups 通过几何的有界拓扑，$A$-简单基群
数学研究通讯 Pub Date : 2020-06-01 DOI: 10.4208/cmr.2020-0030
Xiao Yao
{"title":"Bounding Topology via Geometry, $A$-Simple Fundamental Groups","authors":"Xiao Yao","doi":"10.4208/cmr.2020-0030","DOIUrl":"https://doi.org/10.4208/cmr.2020-0030","url":null,"abstract":"","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44566873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Impact of Multilateral Imported Cases of COVID-19 on the Epidemic Control in China 多边输入病例对中国疫情防控的影响
数学研究通讯 Pub Date : 2020-04-06 DOI: 10.4208/cmr.2020-0017
J. Jia, Siyu Liu, Jing Ding, G. Liao, Lihua Zhang, Ran Zhang
{"title":"The Impact of Multilateral Imported Cases of COVID-19 on the Epidemic Control in China","authors":"J. Jia, Siyu Liu, Jing Ding, G. Liao, Lihua Zhang, Ran Zhang","doi":"10.4208/cmr.2020-0017","DOIUrl":"https://doi.org/10.4208/cmr.2020-0017","url":null,"abstract":"Nowadays, the epidemic of COVID-19 in China is under control. However, the epidemic are developing rapidly around the world. Due to the normal migration of population, China is facing high risk from imported cases. The potential specific medicine and vaccine is still in the process of clinical trials. Currently, controlling the impact of imported cases is the key to prevent new outbreak of COVID-19 in China. In this paper, we propose two impulsive systems to describe the impact of multilateral imported cases of COVID-19. Based on the published data, we simulate and discussed the epidemic trends under different control strategies. We compare four different scenarios and show the corresponding medical burden. The results help to design appropriate control strategy for imported cases in practice.","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48023309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
On Two Problems About Isogenies of Elliptic Curves Over Finite Fields 有限域上椭圆曲线的等同源性的两个问题
数学研究通讯 Pub Date : 2020-01-01 DOI: 10.4208/cmr.2020-0071
Lixia Luo, Guanju Xiao, Yingpu Deng
{"title":"On Two Problems About Isogenies of Elliptic Curves Over Finite Fields","authors":"Lixia Luo, Guanju Xiao, Yingpu Deng","doi":"10.4208/cmr.2020-0071","DOIUrl":"https://doi.org/10.4208/cmr.2020-0071","url":null,"abstract":"Isogenies occur throughout the theory of elliptic curves. Recently, the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols. Given two elliptic curves $E_1, E_2$ defined over a finite field $k$ with the same trace, there is a nonconstant isogeny $beta$ from $E_2$ to $E_1$ defined over $k$. This study gives out the index of $rm{Hom}_{it k}(it E_{rm 1},E_{rm 2})beta$ as a left ideal in $rm{End}_{it k}(it E_{rm 2})$ and figures out the correspondence between isogenies and kernel ideals. In addition, some results about the non-trivial minimal degree of isogenies between the two elliptic curves are also provided.","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43663402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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