{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T10:20:50Z","timestamp":1770891650630,"version":"3.50.1"},"reference-count":18,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["12171064"],"award-info":[{"award-number":["12171064"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["11871369"],"award-info":[{"award-number":["11871369"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Asia Pac. J. Oper. Res."],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:p> In this paper, we seek analytically checkable necessary and sufficient condition for copositivity of a three-dimensional symmetric tensor. We first show that for a general third-order three-dimensional symmetric tensor, checking copositivity is equivalent to solving a quartic equation and some quadratic equations. All of them can be solved analytically. Thus, we present an analytical way to check copositivity of a third-order three-dimensional symmetric tensor. Then, we consider a model of vacuum stability for [Formula: see text] scalar dark matter. This is a special fourth-order three-dimensional symmetric tensor. We show that an analytically expressed necessary and sufficient condition for this model bounded from below can be given, by using a result given by Ulrich and Watson in 1994. <\/jats:p>","DOI":"10.1142\/s0217595922500324","type":"journal-article","created":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T07:22:52Z","timestamp":1661498572000},"source":"Crossref","is-referenced-by-count":3,"title":["Copositivity of Three-Dimensional Symmetric Tensors"],"prefix":"10.1142","volume":"40","author":[{"given":"Liqun","family":"Qi","sequence":"first","affiliation":[{"name":"School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yisheng","family":"Song","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinzhen","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Tianjin University, Tianjin 300354, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2022,11,28]]},"reference":[{"key":"S0217595922500324BIB001","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1007\/s10957-017-1131-2","volume":"174","author":"Chen H","year":"2017","journal-title":"Journal of Optimization Theory and Applications"},{"key":"S0217595922500324BIB002","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s10589-017-9938-1","volume":"69","author":"Chen H","year":"2018","journal-title":"Computational Optimization and Applications"},{"key":"S0217595922500324BIB003","first-page":"1863","volume":"8","author":"Chen H","year":"2018","journal-title":"Journal of Applied Analysis and Computation"},{"key":"S0217595922500324BIB004","volume-title":"Book III: On the Construction of Solid and Supersolid Problems","author":"Descartes R","year":"1954"},{"key":"S0217595922500324BIB005","doi-asserted-by":"crossref","first-page":"035038","DOI":"10.1103\/PhysRevD.100.035038","volume":"100","author":"Faro FS","year":"2019","journal-title":"Physics Review D"},{"key":"S0217595922500324BIB006","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-8176-4771-1","volume-title":"Discriminants, Resultants and Multidimensional Determinants","author":"Gelfand IM","year":"1994"},{"key":"S0217595922500324BIB007","doi-asserted-by":"crossref","first-page":"035038","DOI":"10.1103\/PhysRevD.100.035038","volume":"100","author":"Ivanov IP","year":"2019","journal-title":"Physics Review D"},{"key":"S0217595922500324BIB008","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1140\/epjc\/s10052-016-4160-3","volume":"76","author":"Kannike K","year":"2016","journal-title":"The Europian Physical Journal C"},{"key":"S0217595922500324BIB009","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1140\/epjc\/s10052-018-5837-6","volume":"78","author":"Kannike K","year":"2018","journal-title":"The Europian Physical Journal C"},{"key":"S0217595922500324BIB010","doi-asserted-by":"crossref","first-page":"881","DOI":"10.3934\/jimo.2018075","volume":"15","author":"Li L","year":"2019","journal-title":"Journal of Industrial and Management Optimization"},{"key":"S0217595922500324BIB011","first-page":"135","volume":"45","author":"Liu J","year":"2022","journal-title":"Bulletin of the Malaysian Mathematical Sciences Society"},{"key":"S0217595922500324BIB012","doi-asserted-by":"crossref","first-page":"2902","DOI":"10.1137\/17M115308X","volume":"28","author":"Nie J","year":"2018","journal-title":"SIAM Journal of Optimisation"},{"key":"S0217595922500324BIB013","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.laa.2013.03.015","volume":"439","author":"Qi L","year":"2013","journal-title":"Linear Algebra Applications"},{"key":"S0217595922500324BIB014","doi-asserted-by":"crossref","DOI":"10.1007\/978-981-10-8058-6","volume-title":"Tensor Eigenvalues and Their Applications","author":"Qi L","year":"2018"},{"key":"S0217595922500324BIB015","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611974751","volume-title":"Tensor Analysis: Spectral Theory and Special Tensors","author":"Qi L","year":"2017"},{"key":"S0217595922500324BIB016","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1080\/03081087.2013.851198","volume":"63","author":"Song Y","year":"2015","journal-title":"Linear Multilinear Algebra"},{"key":"S0217595922500324BIB017","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530520500049"},{"key":"S0217595922500324BIB018","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1137\/0915035","volume":"15","author":"Ulrich G","year":"1994","journal-title":"SIAM Journal on Scientific Computing"}],"container-title":["Asia-Pacific Journal of Operational Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0217595922500324","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,15]],"date-time":"2023-06-15T07:25:39Z","timestamp":1686813939000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0217595922500324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,28]]},"references-count":18,"journal-issue":{"issue":"03","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["10.1142\/S0217595922500324"],"URL":"https:\/\/doi.org\/10.1142\/s0217595922500324","relation":{},"ISSN":["0217-5959","1793-7019"],"issn-type":[{"value":"0217-5959","type":"print"},{"value":"1793-7019","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,28]]},"article-number":"2250032"}}