American Institute of Mathematical Sciences

Advanced
  • Previous Article
    How's the performance of the optimized portfolios by safety-first rules: Theory with empirical comparisons
  • JIMO Home
  • This Issue
  • Next Article
    Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies
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, 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

[1]

Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153
[2]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[3]

Zhen Wang, Wei Wu. Bounds for the greatest eigenvalue of positive tensors. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1031-1039. doi: 10.3934/jimo.2014.10.1031
[4]

Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial & Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431
[5]

Shenglong Hu, Zheng-Hai Huang, Hong-Yan Ni, Liqun Qi. Positive definiteness of Diffusion Kurtosis Imaging. Inverse Problems & Imaging, 2012, 6 (1) : 57-75. doi: 10.3934/ipi.2012.6.57
[6]

Haibin Chen, Liqun Qi. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1263-1274. doi: 10.3934/jimo.2015.11.1263
[7]

Gang Wang, Yiju Wang, Yuan Zhang. Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019069
[8]

Julio C. Rebelo, Ana L. Silva. On the Burnside problem in Diff(M). Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 423-439. doi: 10.3934/dcds.2007.17.423
[9]

Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013
[10]

Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895
[11]

Zsolt Saffer, Wuyi Yue. M/M/c multiple synchronous vacation model with gated discipline. Journal of Industrial & Management Optimization, 2012, 8 (4) : 939-968. doi: 10.3934/jimo.2012.8.939
[12]

Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1
[13]

Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463
[14]

Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008
[15]

Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026
[16]

Mengmeng Zheng, Ying Zhang, Zheng-Hai Huang. Global error bounds for the tensor complementarity problem with a P-tensor. Journal of Industrial & Management Optimization, 2019, 15 (2) : 933-946. doi: 10.3934/jimo.2018078
[17]

Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1
[18]

Hideaki Takagi. Times until service completion and abandonment in an M/M/$ m$ preemptive-resume LCFS queue with impatient customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1701-1726. doi: 10.3934/jimo.2018028
[19]

Patrick Bonckaert, Timoteo Carletti, Ernest Fontich. On dynamical systems close to a product of $m$ rotations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 349-366. doi: 10.3934/dcds.2009.24.349
[20]

Grzegorz Graff, Michał Misiurewicz, Piotr Nowak-Przygodzki. Periodic points of latitudinal maps of the $m$-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6187-6199. doi: 10.3934/dcds.2016070

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (48)
  • HTML views (204)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences