{
  "id": "1902.03887",
  "version": "v1",
  "published": "2019-02-07T16:57:07.000Z",
  "updated": "2019-02-07T16:57:07.000Z",
  "title": "Quantization for uniform distributions on hexagonal, semicircular, and elliptical curves",
  "authors": [
    "Gabriela Pena",
    "Hansapani Rodrigo",
    "Mrinal Kanti Roychowdhury",
    "Josef Sifuentes",
    "Erwin Suazo"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1809.08364",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, first we have defined the uniform distribution on the boundary of a regular hexagon, and then investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. We give an exact formula to determine them if $n$ is of the form $n=6k+6$ for some positive integer $k$. We further calculate the quantization dimension, and the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number, which supports the well-known result of Bucklew and Wise (1982), which says that for a Borel probability measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number. Then, we define the uniform distribution on the boundary of a semicircular disc, and obtain a sequence and an algorithm, which help us to determine the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$ with respect to the uniform distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-07T16:57:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60Exx",
      "94A34"
    ],
    "keywords": [
      "uniform distribution",
      "elliptical curve",
      "th quantization errors",
      "positive integer",
      "quantization coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}