Theoretical Article
Year 2024,
Volume: 7 Issue: 1, 48 - 55, 31.01.2024
Abstract
The study of bicomplex numbers, specifically commutative-quaternions, offers a fascinating exploration into the properties of complexified quaternions with commutative multiplication. Understanding the gradient and partial derivatives within this mathematical framework is crucial for analyzing the behavior of bicomplex functions. Real quaternions are not commutative but bicomplex numbers are commutative by multiplication. Bicomplex numbers are the special case of real quaternions. In this study, gradient and partial derivatives are obtained for bicomplex number valued functions.
References
- B. Akyar, Dual quaternions in spatial kinematics in an algebraic sense, Turkish Journal of Mathematics, Vol.32, pp.373-391 (2008).
- D. P. Mandic, C. C. Took, A Quaternion Gradient Operator and Its Aplications, IEEE Signal Processing Letters, Vol.18, No.1., pp.47-49 (2011).
- J. F. Weisz, Comments on mathematical analysis over quaternions, Int. J. Math. Educ. Sci. Technol., Vol.22, No.4, pp.499-506 (1991).
- N. Masrouri, Y. Yaylı and M. H. Faroughi M. Mirshafizadeh, Comments On Differentiable Over Function of Split Quaternions, Revista Notas de Matematica, Vol.7(2), No.312, pp.128-134 (2011).
- G. B. Price, An Introduction to Multi-complex Spaces and Functions, Marcel Dekker Inc., New York, (1991).
- W. R. Hamilton, On quaternions. The London, Edinburgh, and Dublin Phil. Mag. J. Sci. Vol.25(169), pp.489-495 (1844).
- W. R. Hamilton, Elements of Quaternions, Chelsea, New York, (1866).
- M. Jiang, Y. Li and W. Liu, Properties of a general quaternion-valued gradient operator and its applications to signal processing, Frontiers Inf Technol Electronic Eng., Vol.17, pp.83-95 (2016).
- T. A. Ell and S. J. Sangwine, Quaternion involutions and anti-involutions, Computers Mathematics with Applications, Vol. 53, N.1, pp. 137-143 (2007).
- D. Rochon, M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat., Vol.11, pp. 71-110 (2004).
- F. Catoni, R. Cannata and P. Zampetti, An Introduction to Commutative Quaternions, Advances in Applied Clifford Algebras, Vol.16, pp.1-28 (2006).
Year 2024,
Volume: 7 Issue: 1, 48 - 55, 31.01.2024
Abstract
References
- B. Akyar, Dual quaternions in spatial kinematics in an algebraic sense, Turkish Journal of Mathematics, Vol.32, pp.373-391 (2008).
- D. P. Mandic, C. C. Took, A Quaternion Gradient Operator and Its Aplications, IEEE Signal Processing Letters, Vol.18, No.1., pp.47-49 (2011).
- J. F. Weisz, Comments on mathematical analysis over quaternions, Int. J. Math. Educ. Sci. Technol., Vol.22, No.4, pp.499-506 (1991).
- N. Masrouri, Y. Yaylı and M. H. Faroughi M. Mirshafizadeh, Comments On Differentiable Over Function of Split Quaternions, Revista Notas de Matematica, Vol.7(2), No.312, pp.128-134 (2011).
- G. B. Price, An Introduction to Multi-complex Spaces and Functions, Marcel Dekker Inc., New York, (1991).
- W. R. Hamilton, On quaternions. The London, Edinburgh, and Dublin Phil. Mag. J. Sci. Vol.25(169), pp.489-495 (1844).
- W. R. Hamilton, Elements of Quaternions, Chelsea, New York, (1866).
- M. Jiang, Y. Li and W. Liu, Properties of a general quaternion-valued gradient operator and its applications to signal processing, Frontiers Inf Technol Electronic Eng., Vol.17, pp.83-95 (2016).
- T. A. Ell and S. J. Sangwine, Quaternion involutions and anti-involutions, Computers Mathematics with Applications, Vol. 53, N.1, pp. 137-143 (2007).
- D. Rochon, M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat., Vol.11, pp. 71-110 (2004).
- F. Catoni, R. Cannata and P. Zampetti, An Introduction to Commutative Quaternions, Advances in Applied Clifford Algebras, Vol.16, pp.1-28 (2006).
There are 11 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Research Article |
| Authors | |
| Publication Date | January 31, 2024 |
| Submission Date | July 29, 2023 |
| Acceptance Date | January 30, 2024 |
| Published in Issue | Year 2024 Volume: 7 Issue: 1 |
