The Experts below are selected from a list of 24 Experts worldwide ranked by ideXlab platform

Abraham A. Ungar - One of the best experts on this subject based on the ideXlab platform.

  • beyond the Einstein addition law and its gyroscopic thomas precession the theory of gyrogroups and Gyrovector Spaces
    2001
    Co-Authors: Abraham A. Ungar
    Abstract:

    List of Figures. List of Tables. Preface. Acknowledgments. Introduction A.A. Ungar. 1. Thomas Precession: The Missing Link. 2. Gyrogroups: Modeled on Einstein's Addition. 3. The Einstein Gyrovector Space. 4. Hyperbolic Geometry of Gyrovector Spaces. 5. The Ungar Gyrovector Space. 6. The Mobius Gyrovector Space. 7. Gyrogeometry. 8. Gyrooperations -- The SL(2,C) Approach. 9. The Cocycle Form. 10. The Lorentz Group and its Abstraction. 11. The Lorentz Transformation Link. 12. Other Lorentz Groups. 13. References. About the Author. Topic Index. Author Index.

  • The Einstein Gyrovector Space
    Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession, 2001
    Co-Authors: Abraham A. Ungar
    Abstract:

    In this chapter we introduce scalar multiplication ⊗E in the Einstein gyrogroup (Vc,⊕E), obtaining the Einstein Gyrovector Space (Vc, ⊕E, ⊗E). This, in turn, results in the emergence of the hyperbolic analytic geometry of the Einstein Gyrovector Space, which turns out to be the familiar Beltrami ball model of hyperbolic geometry. The ball Vc is equipped with the coordinates it inherits from its real inner product Space V, relative to which Gyrovectors are represented. We close the chapter with the observation that the unique hyperbolic ‘straight line’ called a geodesic, passing through two given points a, b ∈ Vc is the set of all points $$a{{\oplus }_{E}}\left( {{\ominus }_{E}}a{{\oplus }_{E}}b \right){{\otimes }_{E}}t $$ of Vc, t ∈ ℝ, ⊖Ea = −a, which is analogous to its counterpart in Euclidean analytic geometry.

Osamu Hatori - One of the best experts on this subject based on the ideXlab platform.

  • A Gyrogeometric Mean in the Einstein Gyrogroup
    Symmetry, 2020
    Co-Authors: Takuro Honma, Osamu Hatori
    Abstract:

    In this paper, we define a gyrogeometric mean on the Einstein Gyrovector Space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation.

Takuro Honma - One of the best experts on this subject based on the ideXlab platform.

  • A Gyrogeometric Mean in the Einstein Gyrogroup
    Symmetry, 2020
    Co-Authors: Takuro Honma, Osamu Hatori
    Abstract:

    In this paper, we define a gyrogeometric mean on the Einstein Gyrovector Space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation.