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On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy
$ Z $-eigenvalue exclusion theorems for tensors
School of Management Science, Qufu Normal University, Rizhao, Shandong, China |
To locate all $ Z$-eigenvalues of a tensor more precisely, we establish three $Z$-eigenvalue exclusion sets such that all $ Z$-eigenvalues do not belong to them and get three tighter $Z$-eigenvalue inclusion sets of tensor by using these $Z$-eigenvalue exclusion sets. Furthermore, we show that the new inclusion sets are tighter than the existing results via two running examples.
References:
| [1] |
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, in Medical Image Computing and Computer-Assisted Intervention, Springer, (2008), 1–8.
doi: 10.1007/978-3-540-85988-8_1. |
| [2] |
K. Chang, K. Pearson and T. Zhang,
Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra and its Applications, 438 (2013), 4166-4182.
doi: 10.1016/j.laa.2013.02.013. |
| [3] |
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. |
| [4] |
H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite 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, 16pp.
doi: 10.1002/nla.2125. |
| [5] |
L. De Lathauwer, B. De Moor and J. Vandewalle,
A multilinear singular value decomposition, SIAM Journal on Matrix Analysis and Applications, 21 (2000), 1253-1278.
doi: 10.1137/S0895479896305696. |
| [6] |
S. Friedland, S. Gaubert and L. Han,
Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, 438 (2013), 738-749.
doi: 10.1016/j.laa.2011.02.042. |
| [7] |
L. Gao, D. Wang and G. Wang,
Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects, Applied Mathematics and Computation, 268 (2015), 186-200.
doi: 10.1016/j.amc.2015.06.023. |
| [8] |
L. Gao, D. Wang and G. Zong.,
Exponential stability for generalized stochastic impulsive functional differential equations with delayed impulses and Markovian switching, Nonlinear Analysis: Hybrid Systems, 30 (2018), 199-212.
doi: 10.1016/j.nahs.2018.05.009. |
| [9] |
J. He and T. Huang,
Upper bound for the largest $Z$-eigenvalue of positive tensors, Applied Mathematics Letters, 38 (2014), 110-114.
doi: 10.1016/j.aml.2014.07.012. |
| [10] |
E. Kofidis and R. Regalia,
On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM Journal on Matrix Analysis and Applications, 23 (2002), 863-884.
doi: 10.1137/S0895479801387413. |
| [11] |
T. Kolda and J. Mayo,
Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis and Applications, 34 (2011), 1095-1124.
doi: 10.1137/100801482. |
| [12] |
W. Li and M. Ng,
On the Limiting probability distribution of a transition probability tensor, Linear Multilinear Algebra, 62 (2014), 362-385.
doi: 10.1080/03081087.2013.777436. |
| [13] |
W. Li, D. Liu and S. W. Vong,
$Z$-eigenpair bound for an irreducible nonnegative tensor, Linear Algebra and its Applications, 483 (2015), 182-199.
doi: 10.1016/j.laa.2015.05.033. |
| [14] |
C. Li, F. Wang, J. Zhao, Y. Zhu and Y. Li,
Criterions for the positive definiteness of real supersymmetric tensors, Journal of Computational and Applied Mathematics, 255 (2014), 1-14.
doi: 10.1016/j.cam.2013.04.022. |
| [15] |
C. Li, S. Li, Q. Liu and Y. Li, Exclusion sets in eigenvalue inclusion sets for tensors, arXiv: 1706.00944. |
| [16] |
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach., Proceedings of the IEEE International Workshop on Computational Advances, in Multi-Sensor Adaptive Processing, Puerto Vallarta, (2005), 129–132. |
| [17] |
Q. Liu and Y. Li,
Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Mathematics, 14 (2016), 181-194.
doi: 10.1515/math-2016-0017. |
| [18] |
M. Ng, L. Qi and G. Zhou,
Finding the largest eigenvalue of a nonnegative tensor, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
| [19] |
Q. Ni, L. Qi and F. Wang,
An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107.
doi: 10.1109/TAC.2008.923679. |
| [20] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [21] |
L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, 2017.
doi: 10.1137/1.9781611974751.ch1. |
| [22] |
C. Sang,
A new Brauer-type $ Z$-eigenvalue inclusion set for tensors, Numerical Algorithms, 80 (2019), 781-794.
doi: 10.1007/s11075-018-0506-2. |
| [23] |
Y. Song and L. Qi,
Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 1581-1595.
doi: 10.1137/130909135. |
| [24] |
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. |
| [25] |
G. Wang, G. Zhou and L. Caccetta,
Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pacific journal of Optimization, 14 (2018), 227-244.
|
| [26] |
G. Wang, Y. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear and Multilinear Algebra, (2019).
doi: 10.1080/03081087.2018.1561823. |
| [27] |
G. Wang, Y. Wang and L. Liu, Bound estimations on the eigenvalues for Fan product of $M $-tensors, Taiwanese Journal of Mathematics, (2019), 16pp.
doi: 10.11650/tjm/180905. |
| [28] |
Y. Wang and G. Wang,
Two S-type $ Z$-eigenvalue inclusion sets for tensors, Journal of Inequalities and Application, 2017 (2017), 1-12.
doi: 10.1186/s13660-017-1428-6. |
| [29] |
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. |
| [30] |
Y. Wang, G. Zhou and L. Caccetta,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra with applications, 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
| [31] |
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. |
| [32] |
Y. Wang, G. Zhou and L. Caccetta,
Nonsingular $ H$-tensor and its criteria, Journal of Industrial and Management Optimization, 12 (2016), 1173-1186.
doi: 10.3934/jimo.2016.12.1173. |
| [33] |
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. |
| [34] |
J. Zhao and C. Sang,
A new $ Z$-eigenvalue localization set for tensors, Journal of Inequalities and Application, 2017 (2017), 1-9.
doi: 10.1186/s13660-017-1363-6. |
| [35] |
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, 10pp.
doi: 10.1002/nla.2134. |
show all references
References:
| [1] |
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, in Medical Image Computing and Computer-Assisted Intervention, Springer, (2008), 1–8.
doi: 10.1007/978-3-540-85988-8_1. |
| [2] |
K. Chang, K. Pearson and T. Zhang,
Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra and its Applications, 438 (2013), 4166-4182.
doi: 10.1016/j.laa.2013.02.013. |
| [3] |
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. |
| [4] |
H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite 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, 16pp.
doi: 10.1002/nla.2125. |
| [5] |
L. De Lathauwer, B. De Moor and J. Vandewalle,
A multilinear singular value decomposition, SIAM Journal on Matrix Analysis and Applications, 21 (2000), 1253-1278.
doi: 10.1137/S0895479896305696. |
| [6] |
S. Friedland, S. Gaubert and L. Han,
Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, 438 (2013), 738-749.
doi: 10.1016/j.laa.2011.02.042. |
| [7] |
L. Gao, D. Wang and G. Wang,
Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects, Applied Mathematics and Computation, 268 (2015), 186-200.
doi: 10.1016/j.amc.2015.06.023. |
| [8] |
L. Gao, D. Wang and G. Zong.,
Exponential stability for generalized stochastic impulsive functional differential equations with delayed impulses and Markovian switching, Nonlinear Analysis: Hybrid Systems, 30 (2018), 199-212.
doi: 10.1016/j.nahs.2018.05.009. |
| [9] |
J. He and T. Huang,
Upper bound for the largest $Z$-eigenvalue of positive tensors, Applied Mathematics Letters, 38 (2014), 110-114.
doi: 10.1016/j.aml.2014.07.012. |
| [10] |
E. Kofidis and R. Regalia,
On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM Journal on Matrix Analysis and Applications, 23 (2002), 863-884.
doi: 10.1137/S0895479801387413. |
| [11] |
T. Kolda and J. Mayo,
Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis and Applications, 34 (2011), 1095-1124.
doi: 10.1137/100801482. |
| [12] |
W. Li and M. Ng,
On the Limiting probability distribution of a transition probability tensor, Linear Multilinear Algebra, 62 (2014), 362-385.
doi: 10.1080/03081087.2013.777436. |
| [13] |
W. Li, D. Liu and S. W. Vong,
$Z$-eigenpair bound for an irreducible nonnegative tensor, Linear Algebra and its Applications, 483 (2015), 182-199.
doi: 10.1016/j.laa.2015.05.033. |
| [14] |
C. Li, F. Wang, J. Zhao, Y. Zhu and Y. Li,
Criterions for the positive definiteness of real supersymmetric tensors, Journal of Computational and Applied Mathematics, 255 (2014), 1-14.
doi: 10.1016/j.cam.2013.04.022. |
| [15] |
C. Li, S. Li, Q. Liu and Y. Li, Exclusion sets in eigenvalue inclusion sets for tensors, arXiv: 1706.00944. |
| [16] |
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach., Proceedings of the IEEE International Workshop on Computational Advances, in Multi-Sensor Adaptive Processing, Puerto Vallarta, (2005), 129–132. |
| [17] |
Q. Liu and Y. Li,
Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Mathematics, 14 (2016), 181-194.
doi: 10.1515/math-2016-0017. |
| [18] |
M. Ng, L. Qi and G. Zhou,
Finding the largest eigenvalue of a nonnegative tensor, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
| [19] |
Q. Ni, L. Qi and F. Wang,
An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107.
doi: 10.1109/TAC.2008.923679. |
| [20] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [21] |
L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, 2017.
doi: 10.1137/1.9781611974751.ch1. |
| [22] |
C. Sang,
A new Brauer-type $ Z$-eigenvalue inclusion set for tensors, Numerical Algorithms, 80 (2019), 781-794.
doi: 10.1007/s11075-018-0506-2. |
| [23] |
Y. Song and L. Qi,
Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 1581-1595.
doi: 10.1137/130909135. |
| [24] |
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. |
| [25] |
G. Wang, G. Zhou and L. Caccetta,
Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pacific journal of Optimization, 14 (2018), 227-244.
|
| [26] |
G. Wang, Y. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear and Multilinear Algebra, (2019).
doi: 10.1080/03081087.2018.1561823. |
| [27] |
G. Wang, Y. Wang and L. Liu, Bound estimations on the eigenvalues for Fan product of $M $-tensors, Taiwanese Journal of Mathematics, (2019), 16pp.
doi: 10.11650/tjm/180905. |
| [28] |
Y. Wang and G. Wang,
Two S-type $ Z$-eigenvalue inclusion sets for tensors, Journal of Inequalities and Application, 2017 (2017), 1-12.
doi: 10.1186/s13660-017-1428-6. |
| [29] |
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. |
| [30] |
Y. Wang, G. Zhou and L. Caccetta,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra with applications, 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
| [31] |
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. |
| [32] |
Y. Wang, G. Zhou and L. Caccetta,
Nonsingular $ H$-tensor and its criteria, Journal of Industrial and Management Optimization, 12 (2016), 1173-1186.
doi: 10.3934/jimo.2016.12.1173. |
| [33] |
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. |
| [34] |
J. Zhao and C. Sang,
A new $ Z$-eigenvalue localization set for tensors, Journal of Inequalities and Application, 2017 (2017), 1-9.
doi: 10.1186/s13660-017-1363-6. |
| [35] |
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, 10pp.
doi: 10.1002/nla.2134. |
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