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(xué)教授。曾獲省部級科技進(jìn)步獎3項，其中浙江省科學(xué)技術(shù)進(jìn)步一等獎1項(前三完成人),獲福建青年科技獎。入選閩江學(xué)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(fā)計劃答辯會評專家組成員。多次擔(dān)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(xué)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(xué)“杰出學(xué)者”特聘教授、博士生導(dǎo)師，與合作者一起推廣了著名學(xué)者龐加萊和李雅普諾夫關(guān)于二維平面系統(tǒng)可積的充要條件的經(jīng)典理論,將此可積理論推廣到了任意有限維；建立了四元數(shù)體上微分方程理論與應(yīng)用的基本框架(已經(jīng)形成專著在中國科學(xué)出版社出版)；改進(jìn)了經(jīng)典的全局Hartman-Grobman線性化定理。

講座內(nèi)容：

本報告介紹四元數(shù)體上方程的基礎(chǔ)理論和基本框架。系統(tǒng)性指出四元數(shù)體上微分方程與常微分方程的區(qū)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(shù)學(xué)與統(tǒng)計學(xué)院