Quantization for a condensation system
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913811843252224 |
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| author | Dubey, Shivam Roychowdhury, Mrinal Kanti Verma, Saurabh |
| author_facet | Dubey, Shivam Roychowdhury, Mrinal Kanti Verma, Saurabh |
| contents | For a given $r \in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, represents the rate at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. If $D_r(μ)$ does not exist, we define $\underline{D}_r(μ)$ and $\overline{D}_r(μ)$ as the lower and the upper quantization dimensions of $μ$ of order $r$, respectively. In this paper, we investigate the quantization dimension of the condensation measure $μ$ associated with a condensation system $(\{S_j\}_{j=1}^N, (p_j)_{j=0}^N, ν).$ We provide two examples: one where $ν$ is an infinite discrete distribution on $\mathbb{R}$, and one where $ν$ is a uniform distribution on $\mathbb{R}$. For both the discrete and uniform distributions $ν$, we determine the optimal sets of $n$-means, and calculate the quantization dimensions of condensation measures $μ$, and show that the $D_r(μ)$-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantization for a condensation system Dubey, Shivam Roychowdhury, Mrinal Kanti Verma, Saurabh Dynamical Systems Probability 60Exx, 94A34, 28A80 For a given $r \in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, represents the rate at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. If $D_r(μ)$ does not exist, we define $\underline{D}_r(μ)$ and $\overline{D}_r(μ)$ as the lower and the upper quantization dimensions of $μ$ of order $r$, respectively. In this paper, we investigate the quantization dimension of the condensation measure $μ$ associated with a condensation system $(\{S_j\}_{j=1}^N, (p_j)_{j=0}^N, ν).$ We provide two examples: one where $ν$ is an infinite discrete distribution on $\mathbb{R}$, and one where $ν$ is a uniform distribution on $\mathbb{R}$. For both the discrete and uniform distributions $ν$, we determine the optimal sets of $n$-means, and calculate the quantization dimensions of condensation measures $μ$, and show that the $D_r(μ)$-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive. |
| title | Quantization for a condensation system |
| topic | Dynamical Systems Probability 60Exx, 94A34, 28A80 |
| url | https://arxiv.org/abs/2503.14344 |