# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020139

## New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors

 1 School of Mathematics and Information Science, Weifang University, Weifang 261061, Shandong, China 2 School of Management Science, Qufu Normal University, Rizhao 276800, Shandong, China 3 Department of Mathematics and Statistics, Curtin University, Perth 6102, Western Australia, Australia

* Corresponding author: Haibin Chen

Received  January 2020 Revised  June 2020 Published  September 2020

Fund Project: This paper is supported by NSF grant(11401438, 11671228, 11601261, 11571120), Shandong Provincial Natural Science Foundation (ZR2019MA022), Project of Shandong Province Higher Educational Science and Technology Program(Grant No. J14LI52), and China Postdoctoral Science Foundation (Grant No. 2017M622163)

M-eigenvalues of fourth-order partially symmetric tensors play an important role in nonlinear elasticity and materials. In this paper, we present some M-eigenvalue intervals to locate all M-eigenvalues of fourth-order partially symmetric tensors. It is proved that the new interval is tighter than the one proposed by He, Li and Wei [16]. Furthermore, we obtain some new checkable sufficient conditions for the strong ellipticity of fourth-order partially symmetric tensors. Three numerical examples arising from anisotropic materials are presented to verify the efficiency of the proposed results.

Citation: Haitao Che, Haibin Chen, Guanglu Zhou. New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020139
##### References:
 [1] H. Che, H. Chen and Y. Wang, On the M-eigenvalue estimation of fourth-order partially symmetric tensors, Journal of Industrial and Management Optimization, 16 (2020), 309-324.  doi: 10.3934/jimo.2018153.  Google Scholar [2] H. Che, H. Chen and Y. Wang, M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors, Journal of Inequalities and Applications, (2019), Paper No. 32, 18 pp. doi: 10.1186/s13660-019-1986-x.  Google Scholar [3] H. Che, H. Chen and Y. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Applied Mathematics Letters, 89 (2019), 41-49.  doi: 10.1016/j.aml.2018.09.014.  Google Scholar [4] H. Chen, Y. Wang and G. Zhou, High-order sum-of-squares structured tensors: Theory and applications, Frontiers of Mathematics in China, 15 (2020), 255-284.  doi: 10.1007/s11464-020-0833-1.  Google Scholar [5] H. Chen, Z. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, Journal of Optimization Theory and Applications, 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.  Google Scholar [6] H. Chen, Y. Chen, G. Li and L. Qi, A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numerical Linear Algebra with Applications, 25 (2018), e2125, 16 pp. doi: 10.1002/nla.2125.  Google Scholar [7] H. Chen, Z. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Computational Optimization and Applications, 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.  Google Scholar [8] H. Chen, L. Qi, Y. Wang and G. Zhou, Further results on sum-of-squares tensors, Optimization Methods and Software, (2020). doi: 10.1080/10556788.2020.1768389.  Google Scholar [9] H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Frontiers of Mathematics in China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.  Google Scholar [10] H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Frontiers of Mathematics in China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.  Google Scholar [11] H. Chen, L. Qi, L. Caccetta and G. Zhou, Birkhoff-von Neumann theorem and decomposition for doubly stochastic tensors, Linear Algebra and Its Applications, 583 (2019), 119-133.  doi: 10.1016/j.laa.2019.08.027.  Google Scholar [12] S. Chirit$\check{a}$, A. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, Journal of Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.  Google Scholar [13] B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete and Continuous Dynamical Systems, Series B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.  Google Scholar [14] M. Dong and H. Chen, Geometry of the Copositive Tensor Cone and Its Dual, Asia-Pacific Journal of Operational Research, 2020. Google Scholar [15] D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, Journal of Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.  Google Scholar [16] J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters, 102 (2020), 106137. doi: 10.1016/j.aml.2019.106137.  Google Scholar [17] J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, Journal of Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.  Google Scholar [18] J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Archive for Rational Mechanics and Analysis, 63 (1976), 321-336.  doi: 10.1007/BF00279991.  Google Scholar [19] S. Li, C. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 356 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013.  Google Scholar [20] C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.  Google Scholar [21] L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Frontiers of Mathematics in China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.  Google Scholar [22] L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar [23] J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, International Journal of Nonlinear Mechanics, 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.  Google Scholar [24] W. Wang, H. Chen and Y. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Applied Mathematics Letters, 100 (2020), 106035. doi: 10.1016/j.aml.2019.106035.  Google Scholar [25] C. Wang, H. Chen, Y. Wang and G. Zhou, On copositiveness identification of partially symmetric rectangular tensors, Journal of Computational and Applied Mathematics, 372 (2020), 112678. doi: 10.1016/j.cam.2019.112678.  Google Scholar [26] X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Frontiers of Mathematics in China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.  Google Scholar [27] G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar [28] Y. Wang, L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra and Applications, 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.  Google Scholar [29] Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Frontiers of Mathematics in China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.  Google Scholar [30] Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.  doi: 10.1002/nla.633.  Google Scholar [31] K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, Journal of Computational and Applied Mathematics, 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.  Google Scholar [32] G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numerical Linear Algebra with Applications, 25 (2018), e2134. doi: 10.1002/nla.2134.  Google Scholar

show all references

##### References:
 [1] H. Che, H. Chen and Y. Wang, On the M-eigenvalue estimation of fourth-order partially symmetric tensors, Journal of Industrial and Management Optimization, 16 (2020), 309-324.  doi: 10.3934/jimo.2018153.  Google Scholar [2] H. Che, H. Chen and Y. Wang, M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors, Journal of Inequalities and Applications, (2019), Paper No. 32, 18 pp. doi: 10.1186/s13660-019-1986-x.  Google Scholar [3] H. Che, H. Chen and Y. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Applied Mathematics Letters, 89 (2019), 41-49.  doi: 10.1016/j.aml.2018.09.014.  Google Scholar [4] H. Chen, Y. Wang and G. Zhou, High-order sum-of-squares structured tensors: Theory and applications, Frontiers of Mathematics in China, 15 (2020), 255-284.  doi: 10.1007/s11464-020-0833-1.  Google Scholar [5] H. Chen, Z. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, Journal of Optimization Theory and Applications, 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.  Google Scholar [6] H. Chen, Y. Chen, G. Li and L. Qi, A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numerical Linear Algebra with Applications, 25 (2018), e2125, 16 pp. doi: 10.1002/nla.2125.  Google Scholar [7] H. Chen, Z. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Computational Optimization and Applications, 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.  Google Scholar [8] H. Chen, L. Qi, Y. Wang and G. Zhou, Further results on sum-of-squares tensors, Optimization Methods and Software, (2020). doi: 10.1080/10556788.2020.1768389.  Google Scholar [9] H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Frontiers of Mathematics in China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.  Google Scholar [10] H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Frontiers of Mathematics in China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.  Google Scholar [11] H. Chen, L. Qi, L. Caccetta and G. Zhou, Birkhoff-von Neumann theorem and decomposition for doubly stochastic tensors, Linear Algebra and Its Applications, 583 (2019), 119-133.  doi: 10.1016/j.laa.2019.08.027.  Google Scholar [12] S. Chirit$\check{a}$, A. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, Journal of Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.  Google Scholar [13] B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete and Continuous Dynamical Systems, Series B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.  Google Scholar [14] M. Dong and H. Chen, Geometry of the Copositive Tensor Cone and Its Dual, Asia-Pacific Journal of Operational Research, 2020. Google Scholar [15] D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, Journal of Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.  Google Scholar [16] J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters, 102 (2020), 106137. doi: 10.1016/j.aml.2019.106137.  Google Scholar [17] J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, Journal of Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.  Google Scholar [18] J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Archive for Rational Mechanics and Analysis, 63 (1976), 321-336.  doi: 10.1007/BF00279991.  Google Scholar [19] S. Li, C. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 356 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013.  Google Scholar [20] C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.  Google Scholar [21] L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Frontiers of Mathematics in China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.  Google Scholar [22] L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar [23] J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, International Journal of Nonlinear Mechanics, 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.  Google Scholar [24] W. Wang, H. Chen and Y. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Applied Mathematics Letters, 100 (2020), 106035. doi: 10.1016/j.aml.2019.106035.  Google Scholar [25] C. Wang, H. Chen, Y. Wang and G. Zhou, On copositiveness identification of partially symmetric rectangular tensors, Journal of Computational and Applied Mathematics, 372 (2020), 112678. doi: 10.1016/j.cam.2019.112678.  Google Scholar [26] X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Frontiers of Mathematics in China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.  Google Scholar [27] G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar [28] Y. Wang, L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra and Applications, 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.  Google Scholar [29] Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Frontiers of Mathematics in China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.  Google Scholar [30] Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.  doi: 10.1002/nla.633.  Google Scholar [31] K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, Journal of Computational and Applied Mathematics, 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.  Google Scholar [32] G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numerical Linear Algebra with Applications, 25 (2018), e2134. doi: 10.1002/nla.2134.  Google Scholar
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