| Title:
|
Investigating generalized quaternions with dual-generalized complex numbers (English) |
| Author:
|
Gürses, Nurten |
| Author:
|
Şentürk, Gülsüm Yeliz |
| Author:
|
Yüce, Salim |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
148 |
| Issue:
|
3 |
| Year:
|
2023 |
| Pages:
|
329-348 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values $\alpha $, $\beta $ and $\mathfrak {p}$. Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given. (English) |
| Keyword:
|
generalized quaternion |
| Keyword:
|
dual-generalized complex number |
| Keyword:
|
matrix representation |
| MSC:
|
11R52 |
| MSC:
|
15B33 |
| idZBL:
|
Zbl 07729580 |
| idMR:
|
MR4628616 |
| DOI:
|
10.21136/MB.2022.0096-21 |
| . |
| Date available:
|
2023-08-11T14:16:26Z |
| Last updated:
|
2023-09-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151763 |
| . |
| Reference:
|
[1] Akar, M., Yüce, S., Şahin, S.: On the dual hyperbolic numbers and the complex hyperbolic numbers.J. Computer Sci. Comput. Math. 8 (2018), 1-6. 10.20967/jcscm.2018.01.001 |
| Reference:
|
[2] Alfsmann, D.: On families of $2^N$-dimensional hypercomplex algebras suitable for digital signal processing.14th European Signal Processing Conference IEEE, Piscataway (2006), 1-4. |
| Reference:
|
[3] Aslan, S., Bekar, M., Yaylı, Y.: Hyper-dual split quaternions and rigid body motion.J. Geom. Phys. 158 (2020), Article ID 103876, 12 pages. Zbl 1444.11022, MR 4145613, 10.1016/j.geomphys.2020.103876 |
| Reference:
|
[4] 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.Frontiers in Mathematics. Birkhäuser, Basel (2008). Zbl 1151.53001, MR 2411620, 10.1007/978-3-7643-8614-6 |
| Reference:
|
[5] Catoni, F., Cannata, R., Catoni, V., Zampetti, P.: Two-dimensional hypercomplex numbers and related trigonometries and geometries.Adv. Appl. Clifford Algebr. 14 (2004), 47-68. Zbl 1118.30300, MR 2236099, 10.1007/s00006-004-0008-2 |
| Reference:
|
[6] Catoni, F., Cannata, R., Catoni, V., Zampetti, P.: $N$-dimensional geometries generated by hypercomplex numbers.Adv. Appl. Clifford Algebr. 15 (2005), 1-25. Zbl 1114.53003, MR 2236622, 10.1007/s00006-005-0001-4 |
| Reference:
|
[7] Cheng, H. H., Thompson, S.: Dual polynomials and complex dual numbers for analysis of spatial mechanisms.Design Engineering Technical Conferences and Computers in Engineering. Conference ASME 1996 ASME, Irvine (1996), 19-22. 10.1115/96-DETC/MECH-1221 |
| Reference:
|
[8] Cheng, H. H., Thompson, S.: Singularity analysis of spatial mechanisms using dual polynomials and complex dual numbers.J. Mech. Des. 121 (1999), 200-205. 10.1115/1.2829444 |
| Reference:
|
[9] Clifford, W. K.: Mathematical Papers.Chelsea Publishing, New York (1968). MR 0238662 |
| Reference:
|
[10] Cockle, J.: On a new imaginary in algebra.Phil. Mag. (3) 34 (1849), 37-47. 10.1080/14786444908646169 |
| Reference:
|
[11] Cockle, J.: On systems of algebra involving more than one imaginary; and on equations of the fifth degree.Phil. Mag. (3) 35 (1849), 434-437. 10.1080/14786444908646384 |
| Reference:
|
[12] Cohen, A., Shoham, M.: Principle of transference: An extension to hyper-dual numbers.Mech. Mach. Theory 125 (2018), 101-110. 10.1016/j.mechmachtheory.2017.12.007 |
| Reference:
|
[13] Dickson, L. E.: On the theory of numbers and generalized quaternions.Am. J. Math. 46 (1924), 1-16 \99999JFM99999 50.0094.02. MR 1506514, 10.2307/2370658 |
| Reference:
|
[14] 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 American Institute of Aeronautics and Astronautics, New York (2011), 1-17. 10.2514/6.2011-886 |
| Reference:
|
[15] Fike, J. A., Alonso, J. J.: Automatic differentiation through the use of hyper-dual numbers for second derivatives.Recent Advances in Algorithmic Differentiation Lecture Notes in Computational Science and Engineering 87. Springer, Berlin (2012), 163-173. Zbl 1251.65026, MR 3241221, 10.1007/978-3-642-30023-3_15 |
| Reference:
|
[16] Fjelstad, P.: Extending special relativity via the perplex numbers.Am. J. Phys. 54 (1986), 416-422. MR 0839491, 10.1119/1.14605 |
| Reference:
|
[17] Fjelstad, P., Gal, G. S.: $n$-dimensional hyperbolic complex numbers.Adv. Appl. Clifford Algebr. 8 (1998), 47-68. Zbl 0937.30029, MR 1648833, 10.1007/BF03041925 |
| Reference:
|
[18] Griffiths, L. W.: Generalized quaternion algebras and the theory of numbers.Am. J. Math. 50 (1928), 303-314 \99999JFM99999 54.0164.01. MR 1506671, 10.2307/2371761 |
| Reference:
|
[19] Gürses, N., Şentürk, G. Y., Yüce, S.: A study on dual-generalized complex and hyperbolic-generalized complex numbers.Gazi Univ. J. Sci. 34 (2021), 180-194. 10.35378/gujs.653906 |
| Reference:
|
[20] Hamilton, W. R.: On quaternions; or on a new system of imaginaries in algebra.Phil. Mag. (3) 25 (1844), 10-13. 10.1080/14786444408644923 |
| Reference:
|
[21] Hamilton, W. R.: Lectures on Quaternions.Hodges and Smith, Dublin (1853). |
| Reference:
|
[22] Hamilton, W. R.: Elements of Quaternions.Chelsea Publishing, New York (1969). MR 0237284, 10.1017/CBO9780511707162 |
| Reference:
|
[23] Harkin, A. A., Harkin, J. B.: Geometry of generalized complex numbers.Math. Mag. 77 (2004), 118-129. Zbl 1176.30070, MR 1573734, 10.1080/0025570X.2004.11953236 |
| Reference:
|
[24] Jafari, M., Yayli, Y.: Generalized quaternions and their algebratic properties.Commun. Fac. Sci. Univ. Ank., Ser. A1 64 (2015), 15-27. MR 3453638, 10.1501/Commua1_0000000724 |
| Reference:
|
[25] Kantor, I. L., Solodovnikov, A. S.: Hypercomplex Numbers: An Elementary Introduction to Algebras.Springer, New York (1989). Zbl 0669.17001, MR 0996029 |
| Reference:
|
[26] Majern{'ık, V.: Multicomponent number systems.Acta Phys. Polon. A 90 (1996), 491-498. MR 1426884, 10.12693/APhysPolA.90.491 |
| Reference:
|
[27] Mamagani, A. B., Jafari, M.: On properties of generalized quaternion algebra.J. Novel Appl. Sci. 2 (2013), 683-689. |
| Reference:
|
[28] Messelmi, F.: Dual-complex numbers and their holomorphic functions.Available at https://hal.archives-ouvertes.fr/hal-01114178 (2015), 11 pages. |
| Reference:
|
[29] Pennestr{\`ı, E., Stefanelli, R.: Linear algebra and numerical algorithms using dual numbers.Multibody Syst. Dyn. 18 (2007), 323-344. Zbl 1128.70002, MR 2355249, 10.1007/s11044-007-9088-9 |
| Reference:
|
[30] Pottman, H., Wallner, J.: Computational Line Geometry.Mathematics and Visualization. Springer, Berlin (2001). Zbl 1006.51015, MR 1849803, 10.1007/978-3-642-04018-4 |
| Reference:
|
[31] Price, G. B.: An Introduction to Multicomplex Spaces and Functions.Pure and Applied Mathematics 140. Marcel Dekker, New York (1991). Zbl 0729.30040, MR 1094818, 10.1201/9781315137278 |
| Reference:
|
[32] Rochon, D., Shapiro, M.: On algebraic properties of bicomplex and hyperbolic numbers.An. Univ. Oradea, Fasc. Mat. 11 (2004), 71-110. Zbl 1114.11033, MR 2127591 |
| Reference:
|
[33] Savin, D., Flaut, C., Ciobanu, C.: Some properties of the symbol algebras.Carpathian J. Math. 25 (2009), 239-245. Zbl 1249.17007, MR 2731200 |
| Reference:
|
[34] Sobczyk, G.: The hyperbolic number plane.Coll. Math. J. 26 (1995), 268-280. 10.1080/07468342.1995.11973712 |
| Reference:
|
[35] Study, E.: Geometrie der Dynamen: Die Zusammensetzung von Kräften und verwandte Gegenstände der Geometrie.B. G. Teubner, Leipzig (1903), German \99999JFM99999 33.0691.01. |
| Reference:
|
[36] Toyoshima, H.: Computationally efficient bicomplex multipliers for digital signal processing.IEICE Trans. Inform. Syst. E81-D (1998), 236-238. |
| Reference:
|
[37] Veldsman, S.: Generalized complex numbers over near-fields.Quaest. Math. 42 (2019), 181-200. Zbl 1437.16044, MR 3918887, 10.2989/16073606.2018.1442884 |
| Reference:
|
[38] Yaglom, I. M.: Complex Numbers in Geometry.Academic Press, New York (1968). Zbl 0147.20201, MR 0220134 |
| Reference:
|
[39] Yaglom, I. M.: A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity.Springer, New York (1979). Zbl 0393.51013, MR 0520230, 10.1007/978-1-4612-6135-3 |
| . |
