What question did this study set out to answer?

The research aims to extend the Euclidean decomposition framework for multiplicative products under base changes.

April 5, 2026Open Access

Euclidean Transport of Multiplication

Key Points

The research aims to extend the Euclidean decomposition framework for multiplicative products under base changes.
Defines the quotient–remainder array A_d(n,q,r)
Proves a rank-independent transport map for fixed products
Decomposes the array into inherited interior mass and new boundary mass
Corrects the transport formula for arbitrary multiplicative rank
Adds the reverse Euclidean re-expression for a fixed integer
Establishes a transport map that remains effective across different multiplicative ranks
Confirms the congruence relationship R ≡ r−q (mod n+1) is preserved
Enhances the theoretical understanding of Euclidean decompositions in multiplicative contexts

Abstract

A short note on the Euclidean transport of multiplicative products under base change. It defines the quotient–remainder array Ad (n, q, r), proves a rank-independent transport map for fixed products, and decomposes the array at level n+1 into inherited interior mass and new boundary mass. The note extends the Euclidean decomposition framework and generalizes the transport viewpoint previously developed for the multiplication table. Version 2. This version corrects the transport formula in Proposition 1 and Corollary 1 for arbitrary multiplicative rank; the congruence R ≡ r−q (mod n+1) remains unchanged. Version 3. This version completes the transport picture by adding the reverse Euclidean re-expression for a fixed integer.

AI से पूछें

Bookmark

View Full Paper