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Some inequalities for the minimum M-eigenvalue of elasticity M-tensors
School of Mathematics, Zunyi Normal College, Zunyi, Guizhou 563006, China |
In this paper, we derive some lower bounds for the minimum M-eigenvalue of elasticity M-tensors, these bounds only depend on the elements of the elasticity M-tensors and they are easy to be verified. Comparison theorems for elasticity M-tensors are also given.
References:
[1] |
K. C. Chang, L. Q. Qi and G. L. Zhou,
Singular values of a real rectangular tensor, J. Math. Anal. Appl., 370 (2010), 284-294.
doi: 10.1016/j.jmaa.2010.04.037. |
[2] |
J. Cui, G. Peng, Q. Lu and Z. Huang,
Several new estimates of the minimum H -eigenvalue for nonsingular M-tensors, Bull. of the Malaysian Math. Sciences Soc., 42 (2019), 1213-1236.
doi: 10.1007/s40840-017-0544-2. |
[3] |
W. Ding, J. Liu, L. Q. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911v2. Google Scholar |
[4] |
W. Ding, L. Q. Qi and Y. Wei,
M-tensors and nonsingular M-tensors, Linear Algebra and its Appl., 439 (2013), 3264-3278.
doi: 10.1016/j.laa.2013.08.038. |
[5] |
D. Han, H. Dai and L. Q. Qi,
Conditions for strong ellipticity of anisotropic elastic materials, J. of Elasticity, 97 (2009), 1-13.
doi: 10.1007/s10659-009-9205-5. |
[6] |
Z. Huang and L. Q. Qi,
Positive definiteness of paired symmetric tensors and elasticity tensors, J. of Computational and Appl. Math., 388 (2018), 22-43.
doi: 10.1016/j.cam.2018.01.025. |
[7] |
Z. Huang, L. Wang, Z. Xu and J. Cui,
Some new inequalities for the minimum H-eigenvalue of nonsingular M-tensors, Linear Algebra and its Appl., 558 (2018), 146-173.
doi: 10.1016/j.laa.2018.08.023. |
[8] |
C. Q. Li and Y. T. Li,
An eigenvalue localization set for tensors with applications to determine the positive (semi-) deffiniteness of tensors, Linear and Multilinear Algebra, 64 (2016), 587-601.
doi: 10.1080/03081087.2015.1049582. |
[9] |
C. Q. Li, Y. T. Li and X. Kong,
New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.
doi: 10.1002/nla.1858. |
[10] |
L. Q. Qi, H. Dai and D. Han,
Conditions for strong ellipticity and M-eigenvalues, Frontiers of Math. in China, 4 (2009), 349-364.
doi: 10.1007/s11464-009-0016-6. |
[11] |
Y. J. Wang, L. Q. Qi and X. Z. Zhang,
A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.
doi: 10.1002/nla.633. |
[12] |
Y. N. Yang and Q. Z. Yang,
Further results for Perron-Frobenius Theorem for nonnegative tensors II, SIAM. J. Matrix Anal. Appl., 32 (2011), 1236-1250.
doi: 10.1137/100813671. |
[13] |
L. Zhang, L. Qi and G. Zhou,
M-tensors and some applications, SIAM. J. Matrix Anal. Appl., 32 (2014), 437-452.
doi: 10.1137/130915339. |
[14] |
J. X. Zhao and C. Q. Li,
Singular value inclusion sets for rectangular tensors, Linear Multilinear Algebra, 66 (2018), 1333-1350.
doi: 10.1080/03081087.2017.1351518. |
[15] |
L. M. Zubov and A. N. Rudev,
On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials, ZAMM - J. of Appl. Math. and Mechanics, 96 (2016), 1096-1102.
doi: 10.1002/zamm.201500167. |
show all references
References:
[1] |
K. C. Chang, L. Q. Qi and G. L. Zhou,
Singular values of a real rectangular tensor, J. Math. Anal. Appl., 370 (2010), 284-294.
doi: 10.1016/j.jmaa.2010.04.037. |
[2] |
J. Cui, G. Peng, Q. Lu and Z. Huang,
Several new estimates of the minimum H -eigenvalue for nonsingular M-tensors, Bull. of the Malaysian Math. Sciences Soc., 42 (2019), 1213-1236.
doi: 10.1007/s40840-017-0544-2. |
[3] |
W. Ding, J. Liu, L. Q. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911v2. Google Scholar |
[4] |
W. Ding, L. Q. Qi and Y. Wei,
M-tensors and nonsingular M-tensors, Linear Algebra and its Appl., 439 (2013), 3264-3278.
doi: 10.1016/j.laa.2013.08.038. |
[5] |
D. Han, H. Dai and L. Q. Qi,
Conditions for strong ellipticity of anisotropic elastic materials, J. of Elasticity, 97 (2009), 1-13.
doi: 10.1007/s10659-009-9205-5. |
[6] |
Z. Huang and L. Q. Qi,
Positive definiteness of paired symmetric tensors and elasticity tensors, J. of Computational and Appl. Math., 388 (2018), 22-43.
doi: 10.1016/j.cam.2018.01.025. |
[7] |
Z. Huang, L. Wang, Z. Xu and J. Cui,
Some new inequalities for the minimum H-eigenvalue of nonsingular M-tensors, Linear Algebra and its Appl., 558 (2018), 146-173.
doi: 10.1016/j.laa.2018.08.023. |
[8] |
C. Q. Li and Y. T. Li,
An eigenvalue localization set for tensors with applications to determine the positive (semi-) deffiniteness of tensors, Linear and Multilinear Algebra, 64 (2016), 587-601.
doi: 10.1080/03081087.2015.1049582. |
[9] |
C. Q. Li, Y. T. Li and X. Kong,
New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.
doi: 10.1002/nla.1858. |
[10] |
L. Q. Qi, H. Dai and D. Han,
Conditions for strong ellipticity and M-eigenvalues, Frontiers of Math. in China, 4 (2009), 349-364.
doi: 10.1007/s11464-009-0016-6. |
[11] |
Y. J. Wang, L. Q. Qi and X. Z. Zhang,
A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.
doi: 10.1002/nla.633. |
[12] |
Y. N. Yang and Q. Z. Yang,
Further results for Perron-Frobenius Theorem for nonnegative tensors II, SIAM. J. Matrix Anal. Appl., 32 (2011), 1236-1250.
doi: 10.1137/100813671. |
[13] |
L. Zhang, L. Qi and G. Zhou,
M-tensors and some applications, SIAM. J. Matrix Anal. Appl., 32 (2014), 437-452.
doi: 10.1137/130915339. |
[14] |
J. X. Zhao and C. Q. Li,
Singular value inclusion sets for rectangular tensors, Linear Multilinear Algebra, 66 (2018), 1333-1350.
doi: 10.1080/03081087.2017.1351518. |
[15] |
L. M. Zubov and A. N. Rudev,
On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials, ZAMM - J. of Appl. Math. and Mechanics, 96 (2016), 1096-1102.
doi: 10.1002/zamm.201500167. |
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