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THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH

Year 2024, Volume: 7 Issue: 1, 48 - 55, 31.01.2024

Ali Atasoy

https://doi.org/10.33773/jum.1334588

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.

Keywords

Bicomplex number Quaternion Partial derivate Gradient

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

Ali Atasoy

https://doi.org/10.33773/jum.1334588

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

Ali Atasoy 0000-0002-1894-7695

Publication Date January 31, 2024
Submission Date July 29, 2023
Acceptance Date January 30, 2024
Published in Issue Year 2024 Volume: 7 Issue: 1

Cite

APA Atasoy, A. (2024). THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH. Journal of Universal Mathematics, 7(1), 48-55. https://doi.org/10.33773/jum.1334588
AMA Atasoy A. THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH. JUM. January 2024;7(1):48-55. doi:10.33773/jum.1334588
Chicago Atasoy, Ali. “THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH”. Journal of Universal Mathematics 7, no. 1 (January 2024): 48-55. https://doi.org/10.33773/jum.1334588.
EndNote Atasoy A (January 1, 2024) THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH. Journal of Universal Mathematics 7 1 48–55.
IEEE A. Atasoy, “THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH”, JUM, vol. 7, no. 1, pp. 48–55, 2024, doi: 10.33773/jum.1334588.
ISNAD Atasoy, Ali. “THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH”. Journal of Universal Mathematics 7/1 (January 2024), 48-55. https://doi.org/10.33773/jum.1334588.
JAMA Atasoy A. THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH. JUM. 2024;7:48–55.
MLA Atasoy, Ali. “THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH”. Journal of Universal Mathematics, vol. 7, no. 1, 2024, pp. 48-55, doi:10.33773/jum.1334588.
Vancouver Atasoy A. THE GRADIENT AND PARTIAL DERIVATIVES OF BICOMPLEX NUMBERS: A COMMUTATIVE-QUATERNION APPROACH. JUM. 2024;7(1):48-55.
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