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Algebraic Construction for Dual Quaternions with GCN

Year 2022, Volume: 11 Issue: 2, 586 - 593, 30.06.2022

Gülsüm Yeliz Şentürk Nurten Gürses Salim Yüce

https://doi.org/10.17798/bitlisfen.1063550

Abstract

In this paper, we explain how dual quaternion theory can extend to dual quaternions with generalized complex number (GCN) components. More specifically, we algebraically examine this new type dual quaternion and give several matrix representations both as a dual quaternion and as a GCN.

Keywords

Generalized complex number Dual quaternion Matrix form

Supporting Institution

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Project Number

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Thanks

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References

  • Hamilton W.R. 1969. Elements of Quaternions. Chelsea Pub. Com. New York, 1–242.
  • Hamilton W.R. 1853. Lectures on Quaternions. Hodges and Smith. Dublin, 1–736.
  • Hamilton W.R. 1844–1850. On Quaternions; or on a New System of Imaginaries in Algebra. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series). xxv-xxxvi.
  • Clifford W.K. 1873. Preliminary Sketch of Bi-quaternions. Proceedings of the London Mathematical Society. s1–4(1):381–395.
  • Jr. Edmonds J.D. 1997. Relativistic Reality: A Modern View. World Scientific. Singapore, 1–352.
  • Ercan Z., Yüce S. 2011. On Properties of the Dual Quaternions. European Journal of Pure and Applied Mathematics. 4(2):142–146.
  • Majernik V. 2006. Quaternion Formulation of the Galilean Space-time Transformation. Acta Physica Slovaca. 56:9–14.
  • Majernik V., Nagy M. 1976. Quaternionic Form of Maxwell’s Equations with Sources. Lettere al Nuovo Cimento. 16:165–169.
  • Majernik V. 1995. Galilean Transformation Expressed by the Dual Four-Component Numbers. Acta Physica Polonica. 87(6):919–923.
  • Yaylı Y., Tutuncu E. E. 2009. Generalized Galilean Transformations and Dual Quaternions. Scientia Magna. 5(1):94–100.
  • Kantor I., Solodovnikov A. (1989). Hypercomplex Numbers. Springer-Verlag. New York, 1–169.
  • Catoni F., Boccaletti D., Cannata R., Catoni V., Nichelatti E., Zampetti P. 2008. The Mathematics of Minkowski Space-time and an Introduction to Commutative Hypercomplex Numbers. Birkhauser Verlag. Basel, 1–255.
  • Catoni F., Cannata R., Catoni V., Zampetti P. 2004. Two-dimensional Hypercomplex Numbers and Related Trigonometries and Geometries. Advances in Applied Clifford Algebras. 14:47–68.
  • Catoni F., Cannata R., Catoni V., Zampetti P. 2005. N-dimensional Geometries Generated by Hypercomplex Numbers. Advances in Applied Clifford Algebras. 15(1):1–25.
  • Harkin A.A., Harkin J.B. 2004. Geometry of Generalized Complex Numbers. Mathematics Magazine. 77(2):118–129.
  • Veldsman S. 2019. Generalized Complex Numbers over Near-Fields. Quaestiones Mathematicae 42(2):181–200.
  • Clifford W.K. 1968. Mathematical Papers. (ed. R. Tucker). Chelsea Pub. Co., Bronx, New York.
  • Fjelstad P. 1986. Extending Special Eelativity via the Perplex Numbers. American Journal of Physics. 54(5):416–422.
  • Sobczyk G. 1995. The Hyperbolic Number Plane. The College Mathematics Journal. 26(4):268–280.
  • Yaglom I.M. 1979. A Simple Non-Euclidean Geometry and its Physical Basis. Springer-Verlag. NewYork, 1–307.
  • Yaglom I.M. 1968. Complex Numbers in Geometry. Academic Press. New York, 1–243.
  • Pennestrì E., Stefanelli R. 2007. Linear Algebra and Numerical Algorithms using Dual Numbers. Multibody System Dynamics. 18(3):323–344.
  • Study E. 1903. Geometrie der Dynamen. Mathematiker Deutschland Publisher. Leibzig, 1–603.
  • Zhang F. 1997. Quaternions and Matrices of Quaternions. Linear Algebra and its Applications. 251:21–57.
  • Messelmi F. 2015. Generalized Numbers and Their Holomorphic Functions. International Journal of Open Problems in Complex Analysis. 7(1):35–47.
  • Hamilton W. R. 1853. On the Geometrical Interpretation of Some Results Obtained by Calculation with Biquaternions, In Proceedings of the Royal Irish Academy. 5:388–390.
  • Tian Y. 2013. Biquaternions and Their Complex Matrix Representations. Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry. 54(2):575–592.
  • Karaca E., Yılmaz F., Çalışkan M. 2020. A Unified Approach: Split Quaternions with Quaternion Coefficients and Quaternions with Dual Coefficients. Mathematics. 8(12):2149.
  • Kotelnikov A.P. 1895. Screw Calculus and Some Applications to Geometry and Mechanics. Annal. Imp. Univ. Kazan.
  • McAulay A. 1898. Octonions: a development of Clifford's Biquaternions. University Press, 1–253.
  • Jafari M. Introduction to Dual Quasi-quaternions: Algebra and Geometry. https://www.researchgate.net/publication/281068240_On_the_properties_of_quasi-quaternion_Algebra. (Date of Access: 26.01.2022).

GCN Katsayılı Dual Kuaterniyonlar için Cebirsel Yapı

Year 2022, Volume: 11 Issue: 2, 586 - 593, 30.06.2022

Gülsüm Yeliz Şentürk Nurten Gürses Salim Yüce

https://doi.org/10.17798/bitlisfen.1063550

Abstract

Bu çalışmada, dual kuaterniyon teorisinin genelleştirilmiş kompleks sayı (GCN) katsayılı dual kuaterniyonlara nasıl genişletilebileceğini açıklıyoruz. Daha özel olarak, bu yeni tip dual kuaterniyonun cebirsel olarak incelenmesini ve çeşitli matris temsillerini hem dual kuaterniyon hem de GCN olarak veriyoruz.

Keywords

Genelleştirilmiş kompleks sayı Dual kuaterniyon Matris form

Project Number

-

References

  • Hamilton W.R. 1969. Elements of Quaternions. Chelsea Pub. Com. New York, 1–242.
  • Hamilton W.R. 1853. Lectures on Quaternions. Hodges and Smith. Dublin, 1–736.
  • Hamilton W.R. 1844–1850. On Quaternions; or on a New System of Imaginaries in Algebra. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series). xxv-xxxvi.
  • Clifford W.K. 1873. Preliminary Sketch of Bi-quaternions. Proceedings of the London Mathematical Society. s1–4(1):381–395.
  • Jr. Edmonds J.D. 1997. Relativistic Reality: A Modern View. World Scientific. Singapore, 1–352.
  • Ercan Z., Yüce S. 2011. On Properties of the Dual Quaternions. European Journal of Pure and Applied Mathematics. 4(2):142–146.
  • Majernik V. 2006. Quaternion Formulation of the Galilean Space-time Transformation. Acta Physica Slovaca. 56:9–14.
  • Majernik V., Nagy M. 1976. Quaternionic Form of Maxwell’s Equations with Sources. Lettere al Nuovo Cimento. 16:165–169.
  • Majernik V. 1995. Galilean Transformation Expressed by the Dual Four-Component Numbers. Acta Physica Polonica. 87(6):919–923.
  • Yaylı Y., Tutuncu E. E. 2009. Generalized Galilean Transformations and Dual Quaternions. Scientia Magna. 5(1):94–100.
  • Kantor I., Solodovnikov A. (1989). Hypercomplex Numbers. Springer-Verlag. New York, 1–169.
  • Catoni F., Boccaletti D., Cannata R., Catoni V., Nichelatti E., Zampetti P. 2008. The Mathematics of Minkowski Space-time and an Introduction to Commutative Hypercomplex Numbers. Birkhauser Verlag. Basel, 1–255.
  • Catoni F., Cannata R., Catoni V., Zampetti P. 2004. Two-dimensional Hypercomplex Numbers and Related Trigonometries and Geometries. Advances in Applied Clifford Algebras. 14:47–68.
  • Catoni F., Cannata R., Catoni V., Zampetti P. 2005. N-dimensional Geometries Generated by Hypercomplex Numbers. Advances in Applied Clifford Algebras. 15(1):1–25.
  • Harkin A.A., Harkin J.B. 2004. Geometry of Generalized Complex Numbers. Mathematics Magazine. 77(2):118–129.
  • Veldsman S. 2019. Generalized Complex Numbers over Near-Fields. Quaestiones Mathematicae 42(2):181–200.
  • Clifford W.K. 1968. Mathematical Papers. (ed. R. Tucker). Chelsea Pub. Co., Bronx, New York.
  • Fjelstad P. 1986. Extending Special Eelativity via the Perplex Numbers. American Journal of Physics. 54(5):416–422.
  • Sobczyk G. 1995. The Hyperbolic Number Plane. The College Mathematics Journal. 26(4):268–280.
  • Yaglom I.M. 1979. A Simple Non-Euclidean Geometry and its Physical Basis. Springer-Verlag. NewYork, 1–307.
  • Yaglom I.M. 1968. Complex Numbers in Geometry. Academic Press. New York, 1–243.
  • Pennestrì E., Stefanelli R. 2007. Linear Algebra and Numerical Algorithms using Dual Numbers. Multibody System Dynamics. 18(3):323–344.
  • Study E. 1903. Geometrie der Dynamen. Mathematiker Deutschland Publisher. Leibzig, 1–603.
  • Zhang F. 1997. Quaternions and Matrices of Quaternions. Linear Algebra and its Applications. 251:21–57.
  • Messelmi F. 2015. Generalized Numbers and Their Holomorphic Functions. International Journal of Open Problems in Complex Analysis. 7(1):35–47.
  • Hamilton W. R. 1853. On the Geometrical Interpretation of Some Results Obtained by Calculation with Biquaternions, In Proceedings of the Royal Irish Academy. 5:388–390.
  • Tian Y. 2013. Biquaternions and Their Complex Matrix Representations. Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry. 54(2):575–592.
  • Karaca E., Yılmaz F., Çalışkan M. 2020. A Unified Approach: Split Quaternions with Quaternion Coefficients and Quaternions with Dual Coefficients. Mathematics. 8(12):2149.
  • Kotelnikov A.P. 1895. Screw Calculus and Some Applications to Geometry and Mechanics. Annal. Imp. Univ. Kazan.
  • McAulay A. 1898. Octonions: a development of Clifford's Biquaternions. University Press, 1–253.
  • Jafari M. Introduction to Dual Quasi-quaternions: Algebra and Geometry. https://www.researchgate.net/publication/281068240_On_the_properties_of_quasi-quaternion_Algebra. (Date of Access: 26.01.2022).
There are 31 citations in total.

Details

Primary Language English
Journal Section Araştırma Makalesi
Authors

Gülsüm Yeliz Şentürk 0000-0002-8647-1801

Nurten Gürses 0000-0001-8407-854X

Salim Yüce 0000-0002-8296-6495

Project Number -
Publication Date June 30, 2022
Submission Date January 26, 2022
Acceptance Date June 26, 2022
Published in Issue Year 2022 Volume: 11 Issue: 2

Cite

IEEE G. Y. Şentürk, N. Gürses, and S. Yüce, “Algebraic Construction for Dual Quaternions with GCN”, Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol. 11, no. 2, pp. 586–593, 2022, doi: 10.17798/bitlisfen.1063550.
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