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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| author | Shivam Dubey Mrinal Kanti Roychowdhury Saurabh Verma |
| author_facet | Shivam Dubey Mrinal Kanti Roychowdhury Saurabh Verma |
| author_sort | Shivam Dubey |
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| description | 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. |
| format | Article |
| id | doaj-art-96b2ef7def324d10aeb4c7aeb2cc7c1f |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-96b2ef7def324d10aeb4c7aeb2cc7c1f2025-08-20T01:49:11ZengMDPI AGMathematics2227-73902025-04-01139142410.3390/math13091424Quantization for a Condensation SystemShivam Dubey0Mrinal Kanti Roychowdhury1Saurabh Verma2Department of Applied Sciences, Indian Institute of Information Technology Allahabad, Prayagraj 211015, UP, IndiaSchool of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USADepartment of Applied Sciences, Indian Institute of Information Technology Allahabad, Prayagraj 211015, UP, IndiaFor 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.https://www.mdpi.com/2227-7390/13/9/1424condensation measureoptimal quantizersquantization errorquantization dimensionquantization coefficientdiscrete distribution |
| spellingShingle | Shivam Dubey Mrinal Kanti Roychowdhury Saurabh Verma Quantization for a Condensation System Mathematics condensation measure optimal quantizers quantization error quantization dimension quantization coefficient discrete distribution |
| title | Quantization for a Condensation System |
| title_full | Quantization for a Condensation System |
| title_fullStr | Quantization for a Condensation System |
| title_full_unstemmed | Quantization for a Condensation System |
| title_short | Quantization for a Condensation System |
| title_sort | quantization for a condensation system |
| topic | condensation measure optimal quantizers quantization error quantization dimension quantization coefficient discrete distribution |
| url | https://www.mdpi.com/2227-7390/13/9/1424 |
| work_keys_str_mv | AT shivamdubey quantizationforacondensationsystem AT mrinalkantiroychowdhury quantizationforacondensationsystem AT saurabhverma quantizationforacondensationsystem |