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2025
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| Acceso en línea: | https://doi.org/10.5281/zenodo.17848505 |
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| _version_ | 1866902088221458432 |
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| author | Mohammed Farhaan Sanchit Kamat |
| author_facet | Mohammed Farhaan Sanchit Kamat |
| contents | <p>In this paper, we investigate the role of commutative Bennett operations (commutative<br>hyperoperations) in Min-Plus and Max-Plus algebras, and in the dequantization of classical<br>polynomials into tropical polynomials. We reformulate the dequantization process using<br>commutative Bennett operations, yielding a framework that suggests potential alternatives<br>to Maslov dequantization. The formal development of these alternatives is left as an open<br>problem.<br>We apply these commutative Bennett operations to derive generalized forms of the raw<br>and central moments, emphasizing the value of studying these moments collectively rather<br>than in isolation. This motivates the introduction of structural constraints on the family of<br>resulting coefficients, along with interpretations for such structure.<br>The paper further proposes an iterative method for fitting probability density func-<br>tions to empirical data, including the use of a simple kernel K(x)=x. Additional statis-<br>tical constructions-such as generalized inner products, covariance, and Pearson’s correlation<br>coefficient-are reexpressed through commutative Bennett operations.<br>Finally, we introduce two approaches for defining commutative Bennett operations be-<br>tween matrices: one by equipping each matrix with a compatible algebra and transferring<br>the operations to these derived structures, and another derived directly from our generalized<br>inner product.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17848505 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Unified Framework for Generalized Arithmetic: Hyperoperations, Tropical Geometry, and Statistical Moments Mohammed Farhaan Sanchit Kamat <p>In this paper, we investigate the role of commutative Bennett operations (commutative<br>hyperoperations) in Min-Plus and Max-Plus algebras, and in the dequantization of classical<br>polynomials into tropical polynomials. We reformulate the dequantization process using<br>commutative Bennett operations, yielding a framework that suggests potential alternatives<br>to Maslov dequantization. The formal development of these alternatives is left as an open<br>problem.<br>We apply these commutative Bennett operations to derive generalized forms of the raw<br>and central moments, emphasizing the value of studying these moments collectively rather<br>than in isolation. This motivates the introduction of structural constraints on the family of<br>resulting coefficients, along with interpretations for such structure.<br>The paper further proposes an iterative method for fitting probability density func-<br>tions to empirical data, including the use of a simple kernel K(x)=x. Additional statis-<br>tical constructions-such as generalized inner products, covariance, and Pearson’s correlation<br>coefficient-are reexpressed through commutative Bennett operations.<br>Finally, we introduce two approaches for defining commutative Bennett operations be-<br>tween matrices: one by equipping each matrix with a compatible algebra and transferring<br>the operations to these derived structures, and another derived directly from our generalized<br>inner product.</p> |
| title | A Unified Framework for Generalized Arithmetic: Hyperoperations, Tropical Geometry, and Statistical Moments |
| url | https://doi.org/10.5281/zenodo.17848505 |