Geometry of Arithmetic Expressions: I. Basic Concepts and Unsolved Problems
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2025
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| _version_ | 1866902216833499136 |
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| author | Yuan, Mingli |
| author_facet | Yuan, Mingli |
| contents | <p>This paper introduces a novel geometric framework for studying arithmetic expressions, establishing a <span>rigorous connection between algebraic operations and hyperbolic geometry. We formalize arithmetic </span>expressions as syntactic structures and demonstrate how they can be embedded into continuous geometric spaces where addition and multiplication correspond to movements along orthogonal directions. Central to our approach is a flow equation that governs how expression values propagate through this geometric space. We construct the first kind arithmetic expression space <span>E</span><span>1 </span>on the upper <span>half-plane with a hyperbolic metric, where the assignment function satisfies the flow equation and </span><span>serves as an eigenfunction of the Laplacian. This construction reveals that arithmetic torsion—the </span>non-commutativity of addition and multiplication—directly corresponds to geometric area, analogous <span>to how curvature measures deviation from flatness. The paper establishes arithmetic expressions as </span><span>geometric objects with intrinsic invariants, opening new avenues for exploring the interplay between </span><span>computation and geometry.</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16938961 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Geometry of Arithmetic Expressions: I. Basic Concepts and Unsolved Problems Yuan, Mingli arithmetic expressions hyperbolic geometry <p>This paper introduces a novel geometric framework for studying arithmetic expressions, establishing a <span>rigorous connection between algebraic operations and hyperbolic geometry. We formalize arithmetic </span>expressions as syntactic structures and demonstrate how they can be embedded into continuous geometric spaces where addition and multiplication correspond to movements along orthogonal directions. Central to our approach is a flow equation that governs how expression values propagate through this geometric space. We construct the first kind arithmetic expression space <span>E</span><span>1 </span>on the upper <span>half-plane with a hyperbolic metric, where the assignment function satisfies the flow equation and </span><span>serves as an eigenfunction of the Laplacian. This construction reveals that arithmetic torsion—the </span>non-commutativity of addition and multiplication—directly corresponds to geometric area, analogous <span>to how curvature measures deviation from flatness. The paper establishes arithmetic expressions as </span><span>geometric objects with intrinsic invariants, opening new avenues for exploring the interplay between </span><span>computation and geometry.</span></p> |
| title | Geometry of Arithmetic Expressions: I. Basic Concepts and Unsolved Problems |
| topic | arithmetic expressions hyperbolic geometry |
| url | https://doi.org/10.5281/zenodo.16938961 |