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Copositivity of Three-Dimensional Symmetric Tensors

Author

Listed:
  • Liqun Qi

    (School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, P. R. China)

  • Yisheng Song

    (School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P. R. China)

  • Xinzhen Zhang

    (School of Mathematics, Tianjin University, Tianjin 300354, P. R. China)

Abstract

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 ℤ3 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.

Suggested Citation

  • Liqun Qi & Yisheng Song & Xinzhen Zhang, 2023. "Copositivity of Three-Dimensional Symmetric Tensors," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 40(03), pages 1-16, June.
    • Handle: RePEc:wsi:apjorx:v:40:y:2023:i:03:n:s0217595922500324
    DOI: 10.1142/S0217595922500324
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