Generalized vector matrices, hypercomplex numbers, and dual numbers : theory and its aplications to some mathematical problems in mechanics, robotics, and optics
5
Issued Date
2024
Copyright Date
1991
Resource Type
Language
eng
File Type
application/pdf
No. of Pages/File Size
xv, 251 leaves : ill.
Access Rights
open access
Rights
ผลงานนี้เป็นลิขสิทธิ์ของมหาวิทยาลัยมหิดล ขอสงวนไว้สำหรับเพื่อการศึกษาเท่านั้น ต้องอ้างอิงแหล่งที่มา ห้ามดัดแปลงเนื้อหา และห้ามนำไปใช้เพื่อการค้า
Rights Holder(s)
Mahidol University
Bibliographic Citation
Thesis (M.Sc. (Applied Mathematics))--Mahidol University, 1991
Suggested Citation
Vimolrat Ngamaramvaranggul Generalized vector matrices, hypercomplex numbers, and dual numbers : theory and its aplications to some mathematical problems in mechanics, robotics, and optics. Thesis (M.Sc. (Applied Mathematics))--Mahidol University, 1991. Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/99914
Title
Generalized vector matrices, hypercomplex numbers, and dual numbers : theory and its aplications to some mathematical problems in mechanics, robotics, and optics
Alternative Title(s)
เมทริกซ์เชิงเวกเตอร์ในนัยทั่วไปจำนวนเชิงซ้อนมิติเกินและจำนวนทวิภาค : ทฤษฎีและการประยุกต์เพื่อแก้ปัญหาทางคณิตศาสตร์ในกลศาสตร์ วิทยาการหุ่นยนต์และทัศนศาสตร์
Author(s)
Advisor(s)
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-divlsors, 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.
Description
Applied Mathematics (Mahidol University 1991)
Degree Name
Master of Science
Degree Level
Master's degree
Degree Department
Faculty of Science
Degree Discipline
Applied Mathematics
Degree Grantor(s)
Mahidol University