报告题目： Linear Quaternion Differential Equations: Basic Theory and Fundamental Results
报告时间：2016.11.26 星期六 15:30-16:30
报告地点：7JC306
Abstract: 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 ofWronskianis 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.
夏永辉个人简介
夏永辉、博士、闽江学者特聘教授，现为华侨大学特聘教授。2012年入选浙江省“新世纪151人才工程”第二层次；2015年入选“中国高被引学者名单”；3篇论文曾进入ESI高被引名单；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；2009年度福建省科学技术奖三等奖1项。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，主持欧盟研究基金项目(MSCA-IF-2014-EF：Marie Curie Individual Fellowship) 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（包括省杰青）的通讯评委，2016年浙江省科技进步奖评审组专家。2015年被邀请在中国数学会第十二次全国代表大会暨80周年纪念学术会议上做报告。
一直从事微分方程和动力系统的研究工作，研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、差分方程理论等究方面。这些结果发表在本学科方向SCI的重要期刊上《J. DifferentialEquations》、《SIAM J. Appl. Math.》、《Proc. Edinburgh Math. Soc.》、《Nonl. Anal. RWA》、《J. Math. Anal. Appl.》、《Int. J. Bifurcat. Chaos》。
