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

Yıl 2024, Cilt: 7 Sayı: 1, 48 - 55, 31.01.2024

Ali Atasoy

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

Öz

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.

Anahtar Kelimeler

Bicomplex number Quaternion Partial derivate Gradient

Kaynakça

  • 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).

Yıl 2024, Cilt: 7 Sayı: 1, 48 - 55, 31.01.2024

Ali Atasoy

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

Öz

Kaynakça

  • 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).
Toplam 11 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Temel Matematik (Diğer)
Bölüm Araştırma Makalesi
Yazarlar

Ali Atasoy 0000-0002-1894-7695

Yayımlanma Tarihi 31 Ocak 2024
Gönderilme Tarihi 29 Temmuz 2023
Kabul Tarihi 30 Ocak 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 7 Sayı: 1

Kaynak Göster

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. Ocak 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, sy. 1 (Ocak 2024): 48-55. https://doi.org/10.33773/jum.1334588.
EndNote Atasoy A (01 Ocak 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, c. 7, sy. 1, ss. 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 (Ocak 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, c. 7, sy. 1, 2024, ss. 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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