Generalized vector matrices, hypercomplex numbers, and dual numbers : theory and its aplications to some mathematical problems in mechanics, robotics, and optics

เมทริกซ์เชิงเวกเตอร์ในนัยทั่วไปจำนวนเชิงซ้อนมิติเกินและจำนวนทวิภาค : ทฤษฎีและการประยุกต์เพื่อแก้ปัญหาทางคณิตศาสตร์ในกลศาสตร์ วิทยาการหุ่นยนต์และทัศนศาสตร์

Name: Vimolrat Ngamaramvaranggul

MeSH: Mathematics

MeSH: Matrics

MeSH: Optics

MeSH: Robotics

Abstract: The main purpose of this thesis is to study the algebraic and geometrical properties of the generalized vector matrices (that include elements of both scalars and vectors), hypercomplex numbers, and dual numbers, and to investigate their applications to the solution of some mathematical problems" in physical sciences and engineering. The development of "number systems is first described, from natural numbers, thorough rational, real, complex, and hypercomplex numbers, to general hspernumbers . A brief survey of the p-adic, nonstandard (hyperreal), and surreal (Conway) numbers, is also given. The algebraic characterizations of the hrpercomplex structures of the quaternions, octonions (or Cayley numbers), biquaternions (of Hamilton and of Clifford), dual numbers, etc., lead naturally to the study of the Zorn vector matrices, and their recent generalizations by Anargyros G. Fellouris, Hyo Chul Myung, and Susumu Okubo. The connections of these structures with the familiar Lie and Jordan algebras are indicated. Some related algebraic structures of physics are discussed, including the Poisson algebras and the Heisenberg and important matrix groups in classical and quantum mechanics, the Jones and Mueller matrices of the algebra and calculus of polarization in optics, and the spinors and twistors in relativity and relativistic quantum theory. To provide an adequate basis for the study of the general hypernumbers (of Charles Muses), that may include a wide variety of units, zero-divisors, nilpotents and idempotents, a broader view of algebra is needed. This involves the study of more general nonassociative algebras, and other algebraic structures, with various combinations of such properties as power-associativity, alternativity, flexibility, Lie- or Jordan- or Malcev-admissibility, nilpotence, and grading. Some of these have recently been introduced into mathematical and theoretical physics, for example, the graded Lie structures that are now referred to as superalgebras and supergeroups. As examples of their applications to some interesting mathematical problems in mechanics and robotics, hypercomplex and dual numbers have been used in the dynamics of gyroscopes, and in the kinematics of spatial mechanisms and robot-arm manipulators. In optics, a suggestion is made for the USE of generalized vector matrices in developing a theory of partial polarization of light with fluctuating mode and degree of polarization A pictographical computer method has been introduced to provide an easy "visualization" of the generalized vector matrices.

Mahidol University. Mahidol University Library and Knowledge Center

Address: NAKHONPATHOM

Email: liwww@mahidol.ac.th

Name: Preedeeporn Limcharoen

Role: Thesis Advisors

Name: Wudhibhan Prachyabrued

Role: Thesis Advisors

Created: 1991

Modified: 2561-02-19

Issued: 2018-02-17

วิทยานิพนธ์/Thesis

application/pdf

eng

DegreeName: Master of Science

Level: Master's Degree

Descipline: Applied Mathematics

Grantor: Mahidol University

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