A Study on Dual-Generalized Complex and Hyperbolic-Generalized Complex Numbers

Abstract

This work is intended to introduce the theories of dual-generalized complex and hyperbolicgeneralized complex numbers. The algebraic properties of these numbers are taken into consideration. Besides, dual-generalized complex and hyperbolic-generalized complex valued functions are defined and different matrix representations of these numbers are examined. Moreover, a remarkable classification are given for special cases and the set of complexgeneralized complex numbers are mentioned.

Keywords

• Generalized complex numbers
• Complex nnumber
• Dual-generalized
• Hyperbolic-generalized
• Complex number

References

1. [1] Yaglom, I. M., Complex Numbers in Geometry, Academic Press, New York, (1968).
2. [2] Sobczyk, G., “The hyperbolic number plane”, The College Mathematics Journal, 26(4): 268-280, (1995).
3. [3] Fjelstad, P., “Extending special relativity via the perplex numbers”, American Journal of Physics, 54(5): 416-422 (1986).
4. [4] Study, E., Geometrie der Dynamen. Leipzig, (1903).
5. [5] Pennestrì, E., Stefanelli, R., “Linear algebra and numerical algorithms using dual numbers”, Multibody System Dynamics, 18(3): 323-344 (2007).
6. [6] Majernik, V., “Multicomponent number systems”, Acta Physica Polonica A, 90: 491-498, (1996).
7. [7] Harkin, A. A., Harkin, J. B., “Geometry of generalized complex numbers”, Mathematics Magazine, 77(2): 118-129, (2004).
8. [8] Cockle, J., “On a new imaginary in algebra”, Philosophical magazine, London-Dublin-Edinburgh, 34(226): 37-47, (1849).

9. [9] Kantor, I., Solodovnikov, A., Hypercomplex Numbers, Springer-Verlag, New York, (1989).
10. [10] Alfsmann, D., “On families of 2N-dimensional hypercomplex algebras suitable for digital signal processing”, 14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, (2006).
11. [11] Fjelstad, P., Sorin, G. Gal, “ n -dimensional hyperbolic complex numbers”, Advances in Applied Clifford Algebras, 8(1): 47-68, (1998).
12. [12] Price G. B., An introduction to multicomplex spaces and functions, New York: M. Dekker, (1991).
13. [13] Toyoshima H., “Computationally efficient bicomplex multipliers for digital signal processing”, IEICE Transactions on Information and Systems, 81(2): 236-238, (1998).
14. [14] Rochon, D., Shapiro, M., “On algebraic properties of bicomplex and hyperbolic numbers”, Analele Universitatii din Oradea. Fascicola Matematica, 11: 71-110, (2004).
15. [15] Cheng, H. H., Thompson, S., “Dual polynomials and complex dual numbers for analysis of spatial mechanisms”, Proc. of ASME 24th Biennial Mechanisms Conference, Irvine, CA, August,19-22, (1996).
16. [16] Cheng, H. H., Thompson, S., “Singularity analysis of spatial mechanisms using dual polynomials and complex dual numbers”, Journal of Mechanical Design, 121(2): 200–205, (1999).
17. [17] Messelmi, F., “Dual-complex numbers and their holomorphic functions”, hal-01114178, (2015).
18. [18] Akar, M., Yüce, S., Şahin, S., “On the dual hyperbolic numbers and the complex hyperbolic numbers”, Journal of Computer Science & Computational Mathematics, 8 (1): 1-6, (2018).
19. [19] Fike, J. A., “Numerically exact derivative calculations using hyper-dual numbers”, 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design, (2009).
20. [20] Fike, J. A., Alonso J. J., “The development of hyper-dual numbers for exact second- derivative calculations”, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, (2011).
21. [21] Fike, J. A., Jongsma, S., Alonso J. J., Weide E. Van Der, “Optimization with gradient and hessian information calculated using hyper-dual numbers”, 29th AIAA Applied Aerodynamics Conference, Honolulu, Hawaii, (2011).
22. [22] Cohen, A., Shoham, M., “Principle of transference-An extension to hyper-dual numbers”, Mechanism and Machine Theory, 125: 101-110, (2018).
23. [23] Fike, J. A., Alonso J. J., “Automatic differentiation through the use of hyper-dual numbers for second derivatives”, Lecture Notes in Computational Science and Engineering book series (LNCSE), 87: 163-173, (2011).
24. [24] Fike, J. A., Alonso J. J., “Automatic differentiation through the use of hyper-dual numbers for second derivatives”, 6th International Conference on Automatic Differentiation, Fort Collins, CO, July 23, (2012).
25. [25] Pogorui, A. A., Rodriguez-Dagnino, R. M., Rodrigue-Said, R .D., “On the set of zeros of bihyperbolic polynomials”, Complex Variables and Elliptic Equations, 53(7): 685–690, (2008).
26. [26] Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P., The mathematics of Minkowski space-time with an introduction to commutative hypercomplex numbers, Birkhauser Verlag, Basel, Boston, Berlin, (2008).
27. [27] Olariu, S., Complex Numbers in n −dimensions, North-Holland Mathematics Studies, Elsevier, Amsterdam, Boston, (2002).
28. [28] Bilgin, M., Ersoy, S., “Algebraic properties of bihyperbolic numbers”, Advances in Applied Clifford Algebras, 30(1): 13, (2020).
29. [29] Apostolova, L. N., Krastev, K. I., Kiradjiev, B., “Hyperbolic double‐complex numbers”, AIP Conference Proceedings, 1184 (1): 193-198, (2009).
30. [30] Apostolova , L. N., Dimiev, S., Stoev, P., “Hyperbolic hypercomplex D-Bar operators, hyperbolic Cr-equations and harmonicity”, arXiv:1012.3420v1, (2010).
31. [31] Özdemir, M., “Introduction to hybrid numbers”, Advances in Applied Clifford Algebras, 28(1): 11, (2018).