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doi: 10.3934/jimo.2021075

Optimal $ Z $-eigenvalue inclusion intervals of tensors and their applications

1. 

School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China

2. 

College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China

* Corresponding author: Zhen Chen

Received  October 2020 Revised  January 2021 Early access  April 2021

Firstly, a weakness of Theorem 3.2 in [Journal of Industrial and Management Optimization, 17(2) (2021) 687-693] is pointed out. Secondly, a new Geršgorin-type $ Z $-eigenvalue inclusion interval for tensors is given. Subsequently, another Geršgorin-type $ Z $-eigenvalue inclusion interval with parameters for even order tensors is presented. Thirdly, by selecting appropriate parameters some optimal intervals are provided and proved to be tighter than some existing results. Finally, as an application, some sufficient conditions for the positive definiteness of homogeneous polynomial forms as well as the asymptotically stability of time-invariant polynomial systems are obtained. As another application, bounds of $ Z $-spectral radius of weakly symmetric nonnegative tensors are presented, which are used to estimate the convergence rate of the greedy rank-one update algorithm and derive bounds of the geometric measure of entanglement of symmetric pure state with nonnegative amplitudes.

Citation: Caili Sang, Zhen Chen. Optimal $ Z $-eigenvalue inclusion intervals of tensors and their applications. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021075
References:
[1]

A. AmmarF. Chinesta and A. Falcó, On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Eng., 17 (2010), 473-486.  doi: 10.1007/s11831-010-9048-z.  Google Scholar

[2]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Med. Image Comput. Comput. Assist. Interv., 5241 (2008), 1–8. Available from: https://www.ncbi.nlm.nih.gov/pubmed/18979725. doi: 10.1007/978-3-540-85988-8_1.  Google Scholar

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N. Bose and P. Kamt, Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process, 22 (1974), 307-314.  doi: 10.1109/TASSP.1974.1162592.  Google Scholar

[4]

N. K. Bose and R. W. Newcomb, Tellegon's theorem and multivariate realizability theory, Int. J. Electron, 36 (1974), 417-425.  doi: 10.1080/00207217408900421.  Google Scholar

[5]

K. C. ChangK. J. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.  Google Scholar

[6]

C. DengH. Li and C. Bu, Brauer-type eigenvalue inclusion sets of stochastic/irreducible tensors and positive definiteness of tensors, Linear Algebra Appl., 556 (2018), 55-69.  doi: 10.1016/j.laa.2018.06.032.  Google Scholar

[7]

R. A. Devore and V. N. Temlyakov, Some remarks on greedy algorithms, Adv. Comput. Math., 5 (1996), 173-187.  doi: 10.1007/BF02124742.  Google Scholar

[8]

P. V. D. Driessche, Reproduction numbers of infectious disease models, Infectious Disease Model., 2 (2017), 288-303.  doi: 10.1016/j.idm.2017.06.002.  Google Scholar

[9]

A. Falco and A. Nouy, A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach, J. Math. Anal. Appl., 376 (2011), 469-480.  doi: 10.1016/j.jmaa.2010.12.003.  Google Scholar

[10]

J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.   Google Scholar

[11]

J. He, Y.-M. Liu, H. Ke, J.-K. Tian and X. Li, Bounds for the $Z$-spectral radius of nonnegative tensors, Springerplus, 5 (2016), 1727. doi: 10.1186/s40064-016-3338-3.  Google Scholar

[12]

J. He and T.-Z. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[13]

J. C. Hsu and A. U. Meyer, Modern Control Principles and Applications, The McGraw-Hill Series in Advanced Chemistry McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. Google Scholar

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E. I. JuryN. K. Bose and B. D. Anderson, Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control, AC20 (1975), 53-66.  doi: 10.1109/tac.1975.1100846.  Google Scholar

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E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[16]

T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.  doi: 10.1137/100801482.  Google Scholar

[17]

J. C. Kuang, Applied Inequalities (4th ed.), Shandong Science and Technology Press, Jinan, 2010. Google Scholar

[18]

L. D. LathauwerB. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_1, R_2, \ldots, R_N$) approximation of higer-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.  doi: 10.1137/S0895479898346995.  Google Scholar

[19]

C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-)definitenss of tensors, Linear Multilinear Algebra, 64 (2016), 587-601.  doi: 10.1080/03081087.2015.1049582.  Google Scholar

[20]

C. LiY. 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

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C. LiZ. Chen and Y. Li, A new eigenvalue inclusion set for tensors and its applications, Linear Algebra Appl., 481 (2015), 36-53.  doi: 10.1016/j.laa.2015.04.023.  Google Scholar

[22]

C. LiJ. Zhou and Y. Li, A new Brauer-type eigenvalue localization set for tensors, Linear Multiliear Algebra, 64 (2016), 727-736.  doi: 10.1080/03081087.2015.1119779.  Google Scholar

[23]

C. LiA. Jiao and Y. Li, An $S$-type eigenvalue location set for tensors, Linear Algebra Appl., 493 (2016), 469-483.  doi: 10.1016/j.laa.2015.12.018.  Google Scholar

[24]

C. LiY. Liu and Y. Li, Note on $Z$-eigenvalue inclusion theorems for tensors, J. Ind. Manag. Optim., 17 (2021), 687-693.  doi: 10.3934/jimo.2019129.  Google Scholar

[25]

W. LiD. Liu and S.-W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199.  doi: 10.1016/j.laa.2015.05.033.  Google Scholar

[26]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, 2005,129–132. Google Scholar

[27]

Q. Liu and Y. Li, Bounds for the $Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194.  doi: 10.1515/math-2016-0017.  Google Scholar

[28]

Q. NiL. Qi and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Control, 53 (2008), 1096-1107.  doi: 10.1109/TAC.2008.923679.  Google Scholar

[29]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[30]

L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.  doi: 10.1016/j.jsc.2006.02.011.  Google Scholar

[31]

L. QiG. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sciences, 3 (2010), 416-433.  doi: 10.1137/090755138.  Google Scholar

[32]

L. Qi, The best rank-one approximation ratio of a tensor space, SIAM J. Matrix Anal. Appl., 32 (2011), 430-442.  doi: 10.1137/100795802.  Google Scholar

[33]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[34]

L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.  Google Scholar

[35]

C. Sang, A new Brauer-type $Z$-eigenvalue inclusion set for tensors, Numer. Algor., 80 (2019), 781-794.  doi: 10.1007/s11075-018-0506-2.  Google Scholar

[36]

C. Sang and J. Zhao, $E$-eigenvalue inclusion theorems for tensors, Filomat, 33 (2019), 3883-3891.  doi: 10.2298/FIL1912883S.  Google Scholar

[37]

C. Sang and Z. Chen, $E$-eigenvalue localization sets for tensors, J. Ind. Manag. Optim., 16 (2020), 2045-2063.  doi: 10.3934/jimo.2019042.  Google Scholar

[38]

C. Sang and Z. Chen, $Z$-eigenvalue localization sets for even order tensors and their applications, Acta Appl. Math., 169 (2020), 323-339.  doi: 10.1007/s10440-019-00300-1.  Google Scholar

[39]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[40]

L. SunG. Wang and L. Liu, Further Study on $Z$-eigenvalue localization set and positive definiteness of fourth-order tensors, Bull. Malays. Math. Sci. Soc., 44 (2021), 105-129.  doi: 10.1007/s40840-020-00939-2.  Google Scholar

[41]

G. WangG. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst., Ser. B., 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[42]

Y. Wang and L. Qi, On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors, Numer. Linear Algebra Appl., 14 (2007), 503-519.  doi: 10.1002/nla.537.  Google Scholar

[43]

Y. Wang and G. Wang, Two $S$-type $Z$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 152, 12 pp. doi: 10.1186/s13660-017-1428-6.  Google Scholar

[44]

L. Xiong and J. Liu, $Z$-eigenvalue inclusion theorem of tensors and the geometric measure of entanglement of multipartite pure states, Comput. Appl. Math., 39 (2020), Paper No. 135, 11 pp. doi: 10.1007/s40314-020-01166-y.  Google Scholar

[45]

T. Zhang and G. H. Golub, Rank-one approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.  Google Scholar

[46]

J. Zhao, A new $Z$-eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 85, 9 pp. doi: 10.1186/s13660-017-1363-6.  Google Scholar

[47]

J. Zhao and C. Sang, Two new eigenvalue localization sets for tensors and theirs applications, Open Math., 15 (2017), 1267-1276.  doi: 10.1515/math-2017-0106.  Google Scholar

[48]

J. Zhao, $E$-eigenvalue localization sets for fourth-order tensors, Bull. Malays. Math. Sci. Soc., 43 (2020), 1685-1707.  doi: 10.1007/s40840-019-00768-y.  Google Scholar

show all references

References:
[1]

A. AmmarF. Chinesta and A. Falcó, On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Eng., 17 (2010), 473-486.  doi: 10.1007/s11831-010-9048-z.  Google Scholar

[2]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Med. Image Comput. Comput. Assist. Interv., 5241 (2008), 1–8. Available from: https://www.ncbi.nlm.nih.gov/pubmed/18979725. doi: 10.1007/978-3-540-85988-8_1.  Google Scholar

[3]

N. Bose and P. Kamt, Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process, 22 (1974), 307-314.  doi: 10.1109/TASSP.1974.1162592.  Google Scholar

[4]

N. K. Bose and R. W. Newcomb, Tellegon's theorem and multivariate realizability theory, Int. J. Electron, 36 (1974), 417-425.  doi: 10.1080/00207217408900421.  Google Scholar

[5]

K. C. ChangK. J. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.  Google Scholar

[6]

C. DengH. Li and C. Bu, Brauer-type eigenvalue inclusion sets of stochastic/irreducible tensors and positive definiteness of tensors, Linear Algebra Appl., 556 (2018), 55-69.  doi: 10.1016/j.laa.2018.06.032.  Google Scholar

[7]

R. A. Devore and V. N. Temlyakov, Some remarks on greedy algorithms, Adv. Comput. Math., 5 (1996), 173-187.  doi: 10.1007/BF02124742.  Google Scholar

[8]

P. V. D. Driessche, Reproduction numbers of infectious disease models, Infectious Disease Model., 2 (2017), 288-303.  doi: 10.1016/j.idm.2017.06.002.  Google Scholar

[9]

A. Falco and A. Nouy, A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach, J. Math. Anal. Appl., 376 (2011), 469-480.  doi: 10.1016/j.jmaa.2010.12.003.  Google Scholar

[10]

J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.   Google Scholar

[11]

J. He, Y.-M. Liu, H. Ke, J.-K. Tian and X. Li, Bounds for the $Z$-spectral radius of nonnegative tensors, Springerplus, 5 (2016), 1727. doi: 10.1186/s40064-016-3338-3.  Google Scholar

[12]

J. He and T.-Z. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[13]

J. C. Hsu and A. U. Meyer, Modern Control Principles and Applications, The McGraw-Hill Series in Advanced Chemistry McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. Google Scholar

[14]

E. I. JuryN. K. Bose and B. D. Anderson, Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control, AC20 (1975), 53-66.  doi: 10.1109/tac.1975.1100846.  Google Scholar

[15]

E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[16]

T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.  doi: 10.1137/100801482.  Google Scholar

[17]

J. C. Kuang, Applied Inequalities (4th ed.), Shandong Science and Technology Press, Jinan, 2010. Google Scholar

[18]

L. D. LathauwerB. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_1, R_2, \ldots, R_N$) approximation of higer-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.  doi: 10.1137/S0895479898346995.  Google Scholar

[19]

C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-)definitenss of tensors, Linear Multilinear Algebra, 64 (2016), 587-601.  doi: 10.1080/03081087.2015.1049582.  Google Scholar

[20]

C. LiY. 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

[21]

C. LiZ. Chen and Y. Li, A new eigenvalue inclusion set for tensors and its applications, Linear Algebra Appl., 481 (2015), 36-53.  doi: 10.1016/j.laa.2015.04.023.  Google Scholar

[22]

C. LiJ. Zhou and Y. Li, A new Brauer-type eigenvalue localization set for tensors, Linear Multiliear Algebra, 64 (2016), 727-736.  doi: 10.1080/03081087.2015.1119779.  Google Scholar

[23]

C. LiA. Jiao and Y. Li, An $S$-type eigenvalue location set for tensors, Linear Algebra Appl., 493 (2016), 469-483.  doi: 10.1016/j.laa.2015.12.018.  Google Scholar

[24]

C. LiY. Liu and Y. Li, Note on $Z$-eigenvalue inclusion theorems for tensors, J. Ind. Manag. Optim., 17 (2021), 687-693.  doi: 10.3934/jimo.2019129.  Google Scholar

[25]

W. LiD. Liu and S.-W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199.  doi: 10.1016/j.laa.2015.05.033.  Google Scholar

[26]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, 2005,129–132. Google Scholar

[27]

Q. Liu and Y. Li, Bounds for the $Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194.  doi: 10.1515/math-2016-0017.  Google Scholar

[28]

Q. NiL. Qi and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Control, 53 (2008), 1096-1107.  doi: 10.1109/TAC.2008.923679.  Google Scholar

[29]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[30]

L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.  doi: 10.1016/j.jsc.2006.02.011.  Google Scholar

[31]

L. QiG. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sciences, 3 (2010), 416-433.  doi: 10.1137/090755138.  Google Scholar

[32]

L. Qi, The best rank-one approximation ratio of a tensor space, SIAM J. Matrix Anal. Appl., 32 (2011), 430-442.  doi: 10.1137/100795802.  Google Scholar

[33]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[34]

L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.  Google Scholar

[35]

C. Sang, A new Brauer-type $Z$-eigenvalue inclusion set for tensors, Numer. Algor., 80 (2019), 781-794.  doi: 10.1007/s11075-018-0506-2.  Google Scholar

[36]

C. Sang and J. Zhao, $E$-eigenvalue inclusion theorems for tensors, Filomat, 33 (2019), 3883-3891.  doi: 10.2298/FIL1912883S.  Google Scholar

[37]

C. Sang and Z. Chen, $E$-eigenvalue localization sets for tensors, J. Ind. Manag. Optim., 16 (2020), 2045-2063.  doi: 10.3934/jimo.2019042.  Google Scholar

[38]

C. Sang and Z. Chen, $Z$-eigenvalue localization sets for even order tensors and their applications, Acta Appl. Math., 169 (2020), 323-339.  doi: 10.1007/s10440-019-00300-1.  Google Scholar

[39]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[40]

L. SunG. Wang and L. Liu, Further Study on $Z$-eigenvalue localization set and positive definiteness of fourth-order tensors, Bull. Malays. Math. Sci. Soc., 44 (2021), 105-129.  doi: 10.1007/s40840-020-00939-2.  Google Scholar

[41]

G. WangG. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst., Ser. B., 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[42]

Y. Wang and L. Qi, On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors, Numer. Linear Algebra Appl., 14 (2007), 503-519.  doi: 10.1002/nla.537.  Google Scholar

[43]

Y. Wang and G. Wang, Two $S$-type $Z$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 152, 12 pp. doi: 10.1186/s13660-017-1428-6.  Google Scholar

[44]

L. Xiong and J. Liu, $Z$-eigenvalue inclusion theorem of tensors and the geometric measure of entanglement of multipartite pure states, Comput. Appl. Math., 39 (2020), Paper No. 135, 11 pp. doi: 10.1007/s40314-020-01166-y.  Google Scholar

[45]

T. Zhang and G. H. Golub, Rank-one approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.  Google Scholar

[46]

J. Zhao, A new $Z$-eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 85, 9 pp. doi: 10.1186/s13660-017-1363-6.  Google Scholar

[47]

J. Zhao and C. Sang, Two new eigenvalue localization sets for tensors and theirs applications, Open Math., 15 (2017), 1267-1276.  doi: 10.1515/math-2017-0106.  Google Scholar

[48]

J. Zhao, $E$-eigenvalue localization sets for fourth-order tensors, Bull. Malays. Math. Sci. Soc., 43 (2020), 1685-1707.  doi: 10.1007/s40840-019-00768-y.  Google Scholar

Table 1.  Upper bounds of $ \varrho(\mathcal{A}) $
Method $ \varrho(\mathcal{A})\leq $
Theorem 5.5, i.e., Corollary 4.5 of [39] 26.0000
Theorem 3.3 of [25] 25.7771
Theorem 3.4 of [47], where $ S=\{1\},\bar{S}=\{2\} $ 25.7382
Theorem 4.5 of [41] 25.7382
Theorem 3.5 of [10] 25.6437
Theorem 6 of [11] 25.6437
Theorem 4 of [43], where $ S=\{1\},\bar{S}=\{2\} $ 25.6437
Theorem 7 of [35] 25.4807
Theorem 7 of [44] 25.4807
Theorem 2.9 of [27] 23.8617
Theorem 5 of [46] 22.5426
Theorem 3.1 of [36] 21.8172
Theorem 5.6 16.0000
Corollary 9 14.5000
Method $ \varrho(\mathcal{A})\leq $
Theorem 5.5, i.e., Corollary 4.5 of [39] 26.0000
Theorem 3.3 of [25] 25.7771
Theorem 3.4 of [47], where $ S=\{1\},\bar{S}=\{2\} $ 25.7382
Theorem 4.5 of [41] 25.7382
Theorem 3.5 of [10] 25.6437
Theorem 6 of [11] 25.6437
Theorem 4 of [43], where $ S=\{1\},\bar{S}=\{2\} $ 25.6437
Theorem 7 of [35] 25.4807
Theorem 7 of [44] 25.4807
Theorem 2.9 of [27] 23.8617
Theorem 5 of [46] 22.5426
Theorem 3.1 of [36] 21.8172
Theorem 5.6 16.0000
Corollary 9 14.5000
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