夏永辉学术报告

发布时间:2019年04月22日 作者:徐宇峰   消息来源:    阅读次数:[]

报告题目:Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

报告时间:2019513日上午9:00-11:00

报告地点:中南大学新校区数学楼145报告厅

报告摘要: Quaternion-valued differential equations (QDEs) are a new kind of differential equations which have many applications in physics and life sciences. The largest difference between QDEs and ordinary differential equations (ODEs) is the algebraic structure. Due to the noncommutativity of the quaternion algebra, the set of all the solutions to the linear homogenous QDEs is completely different from ODEs. It is actually a right-free module, not a linear vector space.This report shall discuss a systematic frame work for the theory of linear QDEs, which can be applied to quantum mechanics, fluid mechanics, Frenet frame in differential geometry, kinematic modeling, attitude dynamics, Kalman filter design, spatial rigid body dynamics, etc. We prove that the set of all the solutions to the linear homogenous QDEs is actually a right-free module, not a linear vector space. On the noncommutativity of the quaternion algebra, many concepts and properties for the ODEs cannot 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 ODEs. 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. Furthermore, we present two algorithms to evaluate the fundamental matrix. Some concrete examples are given to show the feasibility of the obtained algorithms. Finally, a conclusion and discussion ends the lecture.

报告人简介 夏永辉,浙江师范大学特聘教授、博士生导师,获省部级科技奖励3项,曾入选福建省“闽江学者”特聘教授。近年来主持国家自然科学基金3项(包括面上项目2项),在本学科方向的SCI期刊《Proc. Amer. Math. Soc.》、《J. Differential Equations》、《SIAM J. Appl. Math.》、《Studies. Appl. Math.》、《Proc. Edinburgh Math. Soc.》、《Phys. Rew. E.》、《中国科学》等上发表60余篇学术论文。系统建立了四元数体上微分方程的基本框架;改进了非自治Hartman-Grobman线性化的主要结果。与合作者一起推广了庞加莱和李雅普诺夫关于二维平面系统可积的充要条件的经典理论,将此可积理论推广到了任意有限维;与合作者一起改进了张芷芬教授关于广义Lineard系统的一个经典定理,并将她的结果推广到不连续系统。



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