scholarly journals Sharp Bounds on the Minimum M-Eigenvalue of Elasticity M-Tensors

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 250
Author(s):  
Ying Zhang ◽  
Linxuan Sun ◽  
Gang Wang
Keyword(s):  
Lower Bounds ◽  
Elastic Material ◽  
Upper Bound ◽  
Nonlinear Elastic Material ◽  
Sharp Bounds ◽  
Numerical Examples ◽  
Material Analysis ◽  
Nonlinear Elastic

The M-eigenvalue of elasticity M-tensors play important roles in nonlinear elastic material analysis. In this paper, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors without irreducible conditions, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.

scholarly journals Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuyan Yao ◽  
Gang Wang
Keyword(s):  
Fourth Order ◽  
Elastic Material ◽  
Quantum Physics ◽  
Upper Bounds ◽  
Nonlinear Elastic Material ◽  
Numerical Examples ◽  
Nonnegative Tensors ◽  
Material Analysis ◽  
Nonlinear Elastic ◽  
Partially Symmetric

<p style='text-indent:20px;'><inline-formula><tex-math id="M1">\begin{document}$ M $\end{document}</tex-math></inline-formula>-eigenvalues of partially symmetric nonnegative tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we establish two upper bounds for the maximum <inline-formula><tex-math id="M2">\begin{document}$ M $\end{document}</tex-math></inline-formula>-eigenvalue of partially symmetric nonnegative tensors, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.</p>

scholarly journals M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Gang Wang ◽  
Linxuan Sun ◽  
Lixia Liu
Keyword(s):  
Spectral Radius ◽  
Fourth Order ◽  
Quantum Physics ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Symmetric Tensors ◽  
Nonlinear Elastic Material ◽  
Numerical Examples ◽  
Nonlinear Elastic ◽  
Partially Symmetric

M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.

scholarly journals Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge

2020 ◽  
Vol 117 (28) ◽  
pp. 16181-16186
Author(s):  
Rocco Martinazzo ◽  
Eli Pollak
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Spectral Features ◽  
Numerical Examples ◽  
Tunneling Splittings ◽  
Order Of Magnitude ◽  
Bound Theorem ◽  
Math Phys ◽  
Better Than

The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple,Proc. R. Soc. A Math. Phys. Eng. Sci.119, 276–293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple’s lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple’s result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz’s upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry.

Mixing of two co-directional Rayleigh surface waves in a nonlinear elastic material

2015 ◽  
Vol 137 (1) ◽  
pp. 281-292 ◽  
Author(s):  
Merlin B. Morlock ◽  
Jin-Yeon Kim ◽  
Laurence J. Jacobs ◽  
Jianmin Qu
Keyword(s):  
Surface Waves ◽  
Elastic Material ◽  
Nonlinear Elastic Material ◽  
Rayleigh Surface Waves ◽  
Nonlinear Elastic
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