Quantization for a Condensation System
For a given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mo&g...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/9/1424 |
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| Summary: | For a given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, the quantization dimension of order <i>r</i>, if it exists, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>r</mi></msub><mrow><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, represents the rate at which the <i>n</i>th quantization error of order <i>r</i> approaches zero as the number of elements <i>n</i> in an optimal set of <i>n</i>-means for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> tends to infinity. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>r</mi></msub><mrow><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> does not exist, we define <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><munder><mi>D</mi><mo>̲</mo></munder><mi>r</mi></msub><mrow><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover><mi>D</mi><mo>¯</mo></mover><mi>r</mi></msub><mrow><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> as the lower and the upper quantization dimensions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> of order <i>r</i>, respectively. In this paper, we investigate the quantization dimension of the condensation measure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> associated with a condensation system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mrow><mo>{</mo><msub><mi>S</mi><mi>j</mi></msub><mo>}</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mo>,</mo><mo> </mo><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>p</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></msubsup><mo>,</mo><mi>ν</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></semantics></math></inline-formula> We provide two examples: one where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula> is an infinite discrete distribution on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>, and one where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula> is a uniform distribution on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>. For both the discrete and uniform distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>, we determine the optimal sets of <i>n</i>-means, calculate the quantization dimensions of condensation measures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>, and show that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>r</mi></msub><mrow><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive. |
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| ISSN: | 2227-7390 |