講座名稱：Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

講座人：夏永輝 教授

講座時間：12月9日19:00-20:00

地點：騰訊會議 880-306-588

講座人介紹：

夏永輝，佛山大學教授。曾獲省部級科技進步獎3項，其中浙江省科學技術進步一等獎1項(前三完成人),獲福建青年科技獎。入選閩江學者特聘教授；2012年入選浙江省“151人才工程”第二層次。2021年，2023年科技部重點研發計劃答辯會評專家組成員。多次擔任科技部、教育部以及各省市基金、人才項目和科技獎勵的通訊評議或者會評專家。主持國家自然科學基金3項(其中面上2項)，參與國家重點1項，主持浙江省基金重點項目1項。曾任浙江師范大學“杰出學者”特聘教授、博士生導師，與合作者一起推廣了著名學者龐加萊和李雅普諾夫關于二維平面系統可積的充要條件的經典理論,將此可積理論推廣到了任意有限維；建立了四元數體上微分方程理論與應用的基本框架(已經形成專著在中國科學出版社出版)；改進了經典的全局Hartman-Grobman線性化定理。

講座內容：

本報告介紹四元數體上方程的基礎理論和基本框架。系統性指出四元數體上微分方程與常微分方程的區別。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 paper establishes 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.

主辦單位：數學與統計學院