学术讲座-Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

讲座题目

Linear     Quaternion Differential Equations: Basic Theory and Fundamental Results

讲 座 人

夏永辉

讲座人

职称、职务

闽江学者特聘教授

主持人

龙建辉

讲座类型

R自然科学

讲 座对象

全校师生

举办时间

11月21日

14:20-16:20

□社会科学

举办地点

数理学院C4-203

讲

座

人

简

介

2017年获浙江省自然科学奖三等奖  (排名第一，共3人)，2017年获“ 福建青年科技奖 ”。入选泉州市“ 海纳百川 ” 高端人才 聚集计划, 入选泉州市引进 高层次创业创新人才 。2012年入选浙江省“新世纪151人才工程”第二层次； 2013年获 “浙江省优秀科技工作者” 荣誉称号(全省共100名)；2011年度浙江省科学技术奖一等奖1项（排名第三，共7人）;2009年度福建省科学技术奖三等奖1项（排名第一）。连续三年入选“中国高被引学者名单”; 2016年获“泉州市青年科技奖”。近年来主持国家自然科学基金3项（面上项目2项，青年项目各1项），主持浙江省自然科学基金2项，获得欧盟研究基金项目资助(MSCA-IF-2014-EF, ILDS - DLV-655209：Marie Curie Individual Fellowship居里夫人奖学金) 1项。2012年7月-2013年7月在斯洛文尼亚Maribor大学做研究员一年。2015.7.1-2016.8.31为澳门大学兼职研究人员。多次作为国家基金和其他省市基金的通讯评委，教育部部分重要人才项目的通讯评委。部分省市科技进步奖评审组专家。一直从事微分方程和动力系统的研究工作，研究兴趣包括微分方程的线性化理论、微分方程的周期解和稳定性、概周期微分方程等方面。在本学科方向的重要SCI期刊《 J. Differential Equations 》、《 SIAM J. Appl.     Math. 》、《 Proc. Edinburgh Math. Soc. 》等上发表50余篇论文。

讲座

主要内容

This paper establishes a systematic frame     work for the theory of linear quaternion-valued differential equations (QDEs),     which can be applied to quantum mechanics, Frenet frame in differential     geometry, kinematic modelling, attitude dynamics, Kalman filter design,     spatial rigid body dynamics and fluid mechanics, etc. On the     non-commutativity of the quaternion algebra, the algebraic structure of the     solutions to the QDEs is not a linear vector space. It is actually a right-free     module. Moreover, many concepts and properties for the ordinary differential     equations (ODEs) can not be used. They should be redefined accordingly. A     definition of Wronskian   is     introduced under the framework of quaternions which is different from     standard one in the ordinary differential equations. Liouville formula for     QDEs is given. Also, it is necessary to treat the eigenvalue problems with     left- and right-sides, accordingly. Upon these, we studied the solutions to     the linear QDEs.

An algorithm to evaluate the fundamental matrix by employing the     eigenvalues and eigenvectors was presented. The fundamental matrix can be     constructed differently providing that the eigenvalues are simple and     multiple eigenvalues. If the linear system has multiple eigenvalues, how to     construct the fundamental matrix? In particular, if the number of independent     eigenvectors might be less than the dimension of the system. That is, the     numbers of the eigenvectors is not enough to construct a fundamental matrix.     How to find the ``missing solutions?

        Moreover, we presented an algorithm for finding a solution of the     linear nonhomogeneous quaternionic-valued differential equations. The     variation of constants formula was established for the nonhomogeneous     quaternionic-valued differential equations.