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Beyond pseudo-rotations in pseudo-euclidean spaces : an introduction to the theory of bi-gyrogroups and bi-gyrovector spaces /
Beyond Pseudo-Rotations in Pseudo-Euclidean Spaces presents for the first time a unified study of the Lorentz transformation group SO(m, n) of signature (m, n), m, n? N, which is fully analogous to the Lorentz group SO(1, 3) of Einstein's special theory of relativity. It is based on a novel par...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
London, United Kingdom :
Academic Press, an imprint of Elsevier,
[2018]
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| Colección: | Mathematical analysis and its applications.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Front Cover; Beyond Pseudo-rotations in Pseudo-Euclidean Spaces: An Introduction to the Theory of Bi-gyrogroups and Bi-gyrovector Spaces Introduction to the Theory of Bi-gyrogroups and Bi-gyrovector Spaces; Copyright; Dedication; Contents; Acknowledgments; Preface; About the Author; CHAPTER 1: Introduction; 1.1. Introduction; 1.2. Quantum Entanglement and Geometric Entanglement; 1.3. From Galilei to Lorentz Transformations; 1.4. Galilei and Lorentz Transformations of Particle Systems; 1.5. Chapters of the Book; CHAPTER 2: Einstein Gyrogroups; 2.1. Introduction; 2.2. Einstein Velocity Addition.
- 2.3. Einstein Addition with Respect to Cartesian Coordinates2.4. Einstein Addition vs. Vector Addition; 2.5. Gyrations; 2.6. From Einstein Velocity Addition to Gyrogroups; 2.7. Gyrogroup Cooperation (Coaddition); 2.8. First Gyrogroup Properties; 2.9. Elements of Gyrogroup Theory; 2.10. The Two Basic Gyrogroup Equations; 2.11. The Basic Gyrogroup Cancellation Laws; 2.12. Automorphisms and Gyroautomorphisms; 2.13. Gyrosemidirect Product; 2.14. Basic Gyration Properties; 2.15. An Advanced Gyrogroup Equation; 2.16. Gyrocommutative Gyrogroups; CHAPTER 3: Einstein Gyrovector Spaces.
- 3.1. The Abstract Gyrovector Space3.2. Einstein Special Relativistic Scalar Multiplication; 3.3. Einstein Gyrovector Spaces; 3.4. Einstein Addition and Differential Geometry; 3.5. Euclidean Lines; 3.6. Gyrolines �a#x80;#x93; The Hyperbolic Lines; 3.7. Gyroangles �a#x80;#x93; The Hyperbolic Angles; 3.8. The Parallelogram Law; 3.9. Einstein Gyroparallelograms; 3.10. The Gyroparallelogram Law; 3.11. Euclidean Isometries; 3.12. The Group of Euclidean Motions; 3.13. Gyroisometries �a#x80;#x93; The Hyperbolic Isometries; 3.14. Gyromotions �a#x80;#x93; The Motions of Hyperbolic Geometry.
- CHAPTER 4: Bi-gyrogroups and Bi-gyrovector Spaces �a#x80;#x93; P4.1. Introduction; 4.2. Pseudo-Euclidean Spaces and Pseudo-Rotations; 4.3. Matrix Representation of SO(m, n); 4.4. Parametric Realization of SO(m, n); 4.5. Bi-boosts; 4.6. Lorentz Transformation Decomposition; 4.7. Inverse Lorentz Transformation; 4.8. Bi-boost Parameter Composition; 4.9. On the Block Entries of the Bi-boost Product; 4.10. Bi-gyration Exclusion Property; 4.11. Automorphisms of the Parameter Bi-gyrogroupoid; 4.12. Squared Bi-boosts; 4.13. Commuting Relations Between Bi-gyrations and Bi-rotations.
- 4.14. Product of Lorentz Transformations4.15. The Bi-gyrocommutative Law in Bi-gyrogroupoids; 4.16. The Bi-gyroassociative Law in Bi-gyrogroupoids; 4.17. Bi-gyration Reduction Properties in Bi-gyrogroupoids; 4.18. Bi-gyrogroups �a#x80;#x93; P; 4.19. Bi-gyration Decomposition and Polar Decomposition; 4.20. The Bi-gyroassociative Law in Bi-gyrogroups; 4.21. The Bi-gyrocommutative Law in Bi-gyrogroups; 4.22. Bi-gyrogroup Gyrations; 4.23. Bi-gyrogroups are Gyrocommutative Gyrogroups; 4.24. Bi-gyrovector Spaces; 4.25. On the Pseudo-inverse of a Matrix.