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November  2020, 16(6): 3035-3045. doi: 10.3934/jimo.2019092

Some inequalities for the minimum M-eigenvalue of elasticity M-tensors

Jun He , , Guangjun Xu and  Yanmin Liu

School of Mathematics, Zunyi Normal College, Zunyi, Guizhou 563006, China

* Corresponding author: Jun He

Received  October 2018 Revised  February 2019 Published  July 2019

Fund Project: This work is supported by National Natural Science Foundations of China (11661084); Science and Technology Foundation of Guizhou province (Qian Ke He Ji Chu [2016]1161, [2017]1201); Innovative talent team in Guizhou Province(Qian Ke He Pingtai Rencai[2016]5619); High-level innovative talents of Guizhou Province(Zun Ke He Ren Cai[2017]8)

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.

Keywords: Elasticity tensor, M-positive definiteness, M-eigenvalue, bounds.
Mathematics Subject Classification: Primary: 15A18, 15A69; Secondary: 65F15, 65F10.
Citation: Jun He, Guangjun Xu, Yanmin Liu. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. Journal of Industrial & Management Optimization, 2020, 16 (6) : 3035-3045. doi: 10.3934/jimo.2019092
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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