{
  "id": "1712.04253",
  "version": "v1",
  "published": "2017-12-12T11:57:15.000Z",
  "updated": "2017-12-12T11:57:15.000Z",
  "title": "Upper bounds for Z$_1$-eigenvalues of generalized Hilbert tensors",
  "authors": [
    "Juan Meng",
    "Yisheng Song"
  ],
  "comment": "9 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "In this paper, we introduce the concept of Z$_1$-eigenvalue to infinite dimensional generalized Hilbert tensors (hypermatrix) $\\mathcal{H}_\\lambda^{\\infty}=(\\mathcal{H}_{i_{1}i_{2}\\cdots i_{m}})$, $$ \\mathcal{H}_{i_{1}i_{2}\\cdots i_{m}}=\\frac{1}{i_{1}+i_{2}+\\cdots i_{m}+\\lambda},\\ \\lambda\\in \\mathbb{R}\\setminus\\mathbb{Z}^-;\\ i_{1},i_{2},\\cdots,i_{m}=0,1,2,\\cdots,n,\\cdots, $$ and proved that its $Z_1$-spectral radius is not larger than $\\pi$ for $\\lambda>\\frac{1}{2}$, and is at most $\\frac{\\pi}{\\sin{\\lambda\\pi}}$ for $\\frac{1}{2}\\geq \\lambda>0$. Besides, the upper bound of $Z_1$-spectral radius of an $m$th-order $n$-dimensional generalized Hilbert tensor $\\mathcal{H}_\\lambda^n$ is obtained also, and such a bound only depends on $n$ and $\\lambda$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-12T11:57:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upper bound",
      "eigenvalue",
      "infinite dimensional generalized hilbert tensors",
      "spectral radius",
      "hypermatrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}