{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,12]],"date-time":"2026-04-12T01:45:17Z","timestamp":1775958317356,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T00:00:00Z","timestamp":1653523200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Shandong Province","award":["ZR2020MA025"],"award-info":[{"award-number":["ZR2020MA025"]}]},{"name":"Natural Science Foundation of Shandong Province","award":["ZR2019PA016"],"award-info":[{"award-number":["ZR2019PA016"]}]},{"name":"Natural Science Foundation of Shandong Province","award":["ZR2021MA090"],"award-info":[{"award-number":["ZR2021MA090"]}]},{"name":"Natural Science Foundation of Shandong Province","award":["11901343"],"award-info":[{"award-number":["11901343"]}]},{"name":"Natural Science Foundation of Shandong Province","award":["2020M682143"],"award-info":[{"award-number":["2020M682143"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2020MA025"],"award-info":[{"award-number":["ZR2020MA025"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2019PA016"],"award-info":[{"award-number":["ZR2019PA016"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2021MA090"],"award-info":[{"award-number":["ZR2021MA090"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11901343"],"award-info":[{"award-number":["11901343"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2020M682143"],"award-info":[{"award-number":["2020M682143"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Postdoctoral Science Foundation of China","award":["ZR2020MA025"],"award-info":[{"award-number":["ZR2020MA025"]}]},{"name":"Postdoctoral Science Foundation of China","award":["ZR2019PA016"],"award-info":[{"award-number":["ZR2019PA016"]}]},{"name":"Postdoctoral Science Foundation of China","award":["ZR2021MA090"],"award-info":[{"award-number":["ZR2021MA090"]}]},{"name":"Postdoctoral Science Foundation of China","award":["11901343"],"award-info":[{"award-number":["11901343"]}]},{"name":"Postdoctoral Science Foundation of China","award":["2020M682143"],"award-info":[{"award-number":["2020M682143"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The C-eigenpair of piezoelectric tensors finds applications in the area of the piezoelectric effect and converse piezoelectric effect. In this paper, we provide some characterizations of C-eigenvectors by exploring the structure of piezoelectric tensors, and establish sharp C-eigenvalue inclusion intervals via Cauchy\u2013Schwartz inequality. Further, we propose the lower and upper bounds of the largest C-eigenvalue and evaluate the efficiency of the best rank-one approximation of piezoelectric tensors. Numerical examples are proposed to verify the efficiency of the obtained results.<\/jats:p>","DOI":"10.3390\/axioms11060250","type":"journal-article","created":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T08:50:22Z","timestamp":1653555022000},"page":"250","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Further Study on C-Eigenvalue Inclusion Intervals for Piezoelectric Tensors"],"prefix":"10.3390","volume":"11","author":[{"given":"Gang","family":"Wang","sequence":"first","affiliation":[{"name":"Institute of Operations Research, Qufu Normal University, Rizhao 276800, China"}]},{"given":"Xiaoxuan","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Management Science, Qufu Normal University, Rizhao 276800, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8559-6442","authenticated-orcid":false,"given":"Wei","family":"Shao","sequence":"additional","affiliation":[{"name":"School of Management Science, Qufu Normal University, Rizhao 276800, China"}]},{"given":"Qiuling","family":"Hou","sequence":"additional","affiliation":[{"name":"School of Management Science, Qufu Normal University, Rizhao 276800, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"025206","DOI":"10.1088\/1751-8121\/aa98a8","article-title":"Octupolar tensors for liquid crystals","volume":"51","author":"Chen","year":"2018","journal-title":"J. Phys. A"},{"key":"ref_2","first-page":"294","article-title":"D\u2019eveloppement, par pression, de l\u2019electricit\u2019e polaire dans les cristaux h\u2019emiedresa faces inclin\u2019ees","volume":"91","author":"Curie","year":"1880","journal-title":"Comptes Rendus"},{"key":"ref_3","unstructured":"Haussuhl, S. (2007). Physical Properties of Crystals: An Introduction, Wiley-VCH Verlag."},{"key":"ref_4","unstructured":"Safari, A., and Akdogan, E.K. (2008). Piezolelectricity and crystal symmetry. Piezoelectric and Acoustic Materials, Springer."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1140\/epje\/i2016-16113-7","article-title":"Octupolar order in three dimensions","volume":"39","author":"Gaeta","year":"2016","journal-title":"Eur. Phys. J. E"},{"key":"ref_6","unstructured":"Qi, L. (2017). Transposes, L-eigenvalues and invariants of third order tensors. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"20120755","DOI":"10.1098\/rspa.2012.0755","article-title":"Symmetric types of the piezotensor and their identification","volume":"469","author":"Zou","year":"2013","journal-title":"Proc. R. Soc. A"},{"key":"ref_8","unstructured":"Chen, Y., Jakli, A., and Qi, L. (2021). The C-eigenvalue of third order tensors and its application in crystals. J. Ind. Manag. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-021-01636-x","article-title":"Shifted eigenvalue decomposition method for computing C-eigenvalues of a piezoelectric-type tensor","volume":"40","author":"Liang","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_10","first-page":"127124","article-title":"Calculating C-eigenpairs of piezoelectric-type tensors via a Z-eigenpair method","volume":"426","author":"Liu","year":"2022","journal-title":"Appl. Math. Comput."},{"key":"ref_11","unstructured":"Zhao, J., and Luo, J. (2021). Properties and calculation for C-eigenvalues of a piezoelectric-type tensor. J. Ind. Manag. Optim."},{"key":"ref_12","first-page":"244","article-title":"C-eigenvalues intervals for piezoelectric-type tensors","volume":"358","author":"Li","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.aml.2018.09.014","article-title":"C-eigenvalue inclusion theorems for piezoelectric-type tensors","volume":"89","author":"Che","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107448","DOI":"10.1016\/j.aml.2021.107448","article-title":"An S-type inclusion set for C-eigenvalues of a piezoelectric-type tensor","volume":"121","author":"He","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3349","DOI":"10.3934\/jimo.2020122","article-title":"C-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices","volume":"17","author":"Liu","year":"2021","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"123","DOI":"10.4208\/eajam.060119.040619","article-title":"A new C-eigenvalue localisation set for piezoelectric-type tensors","volume":"10","author":"Xiong","year":"2020","journal-title":"East Asian J. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1137\/S0895479801387413","article-title":"On the best rank-1 approximation of higher-order supersymmetric tensors","volume":"23","author":"Kofidis","year":"2002","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1324","DOI":"10.1137\/S0895479898346995","article-title":"On the best rank-1 and rank-(R1, R2, \u2026, RN) approximation of higher-order tensors","volume":"21","author":"Lathauwer","year":"2000","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_19","first-page":"187","article-title":"Z-eigenvalue inclusion theorems for tensors. Discrete Contin","volume":"22","author":"Wang","year":"2017","journal-title":"Dyn. Syst. Ser. B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1137\/S0895479899352045","article-title":"Rank-1 approximation of higher-order tensors","volume":"23","author":"Zhang","year":"2001","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"150053","DOI":"10.1038\/sdata.2015.53","article-title":"A database to enable discovery and design of piezoelectric materials","volume":"2","author":"Jong","year":"2015","journal-title":"Sci. Data"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1021\/acsbiomaterials.7b00183","article-title":"Piezoelectric Tensor of Collagen Fibrils Determined at the Nanoscale","volume":"3","author":"Denning","year":"2017","journal-title":"ACS Biomater. Sci. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1137\/100795802","article-title":"The best rank-one approximation ratio of a tensor space","volume":"32","author":"Qi","year":"2011","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_24","first-page":"125","article-title":"Robust extreme learning machine in the presence of outliers by iterative reweighted algorithm","volume":"377","author":"Wang","year":"2020","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/6\/250\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:18:58Z","timestamp":1760138338000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/6\/250"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,26]]},"references-count":24,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["axioms11060250"],"URL":"https:\/\/doi.org\/10.3390\/axioms11060250","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,26]]}}}