In the recent past, linearly independent isotropic tensors of rank up to 6, under the compact rotation groups SO(2), SO(3) and SO(4) have been studied in some detail. The present paper extends these studies to the case of linearly independent isotropic tensors under the non-compact rotation groups SO(1, 1), SO(1, 2), SO(1, 3) and SO(2, 2). This is done by using the direct method of explicitly constructing these tensors, proving their linear independence and counting their numbers. Interestingly, it is found that these numbers are identical with the corresponding numbers for the case of the compact groups SO(2), SO(3) and SO(4).
| Published in | American Journal of Applied Mathematics (Volume 5, Issue 2) |
| DOI | 10.11648/j.ajam.20170502.11 |
| Page(s) | 39-47 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2017. Published by Science Publishing Group |
Isotropic Tensors, Linear Independence, Non-compact Rotation Groups
| [1] | Faiz Ahmad and Riaz Ahmad Khan, Eigenvectors of a rotation matrix; Q. Jl. Mech. Appl. Maths., 62, (2009), 297 - 310. |
| [2] | Hirth, J. P. and Lothe, Theory of Dislocations, McGraw-Hill, New York, 1968.. |
| [3] | Juretschke, H. J., Crystal Physics, Benjamin, Menlo Park, 1974. |
| [4] | Hearmon, R. F. S., An Introduction to Applied Anisotropic Elasticity; OUP, New York, 1963. |
| [5] | Norris, A. N., Quadratic invariants of elastic moduli; Q. Jl. Mech. Appl. Math., 60, (2007), 367 - 389. |
| [6] | Zhou, S., Li, A. and Wang, B., A reformulation of constitutive relations in the strain gradient elasticity theory for isotropic models; Int. J. Solids Struct. 80, 28e37, 2016. |
| [7] | Lazar, M., Irreducible Decomposition of Strain Gradient Tensor in Isotropic Strain Gradient Elasticity. ArXiv preprint, ArXiv: 1604. 07254. 2016. |
| [8] | Itin, Y. and Hehl, F. W., Irreducible decomposition of elasticity tensor under linear and orthogonal groups and their physical consequences, Journal of Physics: Conference Series 597012046, 2015. |
| [9] | Gusev, A. A. and Lurie, S. A.,. Symmetry conditions in strain elasticity, Math. Mech. Solid, 2015, 1e9. doi.org/10.1177/10812865 15606960. |
| [10] | Ahmad, F. and Rashid, M. A., Linear Invariants of Cartesian Tensors. Q. Jl. Mech. Appl. Maths., 62, 31 - 38 (2009). |
| [11] | Amir, Naila, Linear Invariants of a Cartisian Tensor under SO(4): M. Phil Dissertation, Centre for Advanced Mathematics and Physics, National University of Science and Technology, Islamabad, Pakistan (2010). |
| [12] | Ahmad, F. and Rashid, M. A., Counting Invariants of a Cartisian Tensor. Paper presented in IMACS World Congress on Computational and Applied Mathematics and Applications in Science and Engineering, held in University of Georgia, Athens, USA, 3 - 5 August, 2009. |
APA Style
Ansaruddin Syed. (2017). Isotropic Tensors Under Non-compact Rotation Groups. American Journal of Applied Mathematics, 5(2), 39-47. https://doi.org/10.11648/j.ajam.20170502.11
ACS Style
Ansaruddin Syed. Isotropic Tensors Under Non-compact Rotation Groups. Am. J. Appl. Math. 2017, 5(2), 39-47. doi: 10.11648/j.ajam.20170502.11
AMA Style
Ansaruddin Syed. Isotropic Tensors Under Non-compact Rotation Groups. Am J Appl Math. 2017;5(2):39-47. doi: 10.11648/j.ajam.20170502.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajam.20170502.11,
author = {Ansaruddin Syed},
title = {Isotropic Tensors Under Non-compact Rotation Groups},
journal = {American Journal of Applied Mathematics},
volume = {5},
number = {2},
pages = {39-47},
doi = {10.11648/j.ajam.20170502.11},
url = {https://doi.org/10.11648/j.ajam.20170502.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20170502.11},
abstract = {In the recent past, linearly independent isotropic tensors of rank up to 6, under the compact rotation groups SO(2), SO(3) and SO(4) have been studied in some detail. The present paper extends these studies to the case of linearly independent isotropic tensors under the non-compact rotation groups SO(1, 1), SO(1, 2), SO(1, 3) and SO(2, 2). This is done by using the direct method of explicitly constructing these tensors, proving their linear independence and counting their numbers. Interestingly, it is found that these numbers are identical with the corresponding numbers for the case of the compact groups SO(2), SO(3) and SO(4).},
year = {2017}
}
TY - JOUR T1 - Isotropic Tensors Under Non-compact Rotation Groups AU - Ansaruddin Syed Y1 - 2017/05/27 PY - 2017 N1 - https://doi.org/10.11648/j.ajam.20170502.11 DO - 10.11648/j.ajam.20170502.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 39 EP - 47 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20170502.11 AB - In the recent past, linearly independent isotropic tensors of rank up to 6, under the compact rotation groups SO(2), SO(3) and SO(4) have been studied in some detail. The present paper extends these studies to the case of linearly independent isotropic tensors under the non-compact rotation groups SO(1, 1), SO(1, 2), SO(1, 3) and SO(2, 2). This is done by using the direct method of explicitly constructing these tensors, proving their linear independence and counting their numbers. Interestingly, it is found that these numbers are identical with the corresponding numbers for the case of the compact groups SO(2), SO(3) and SO(4). VL - 5 IS - 2 ER -
Email: