An elementary algorithm for the automatic derivation and proof of tensor product identities via computer algebra
Bibliographic record
Abstract
Tensor product identities in two variables are quite common in mathematics: Exponential, logarithmic, trigonometric, and hyperbolic functions all satisfy tensor product identities, and the binomial theorem is a familiar example of a tensor product identity for polynomial functions. This article presents a new elementary technique which can derive and prove all of these identities---automatically! This unified approach is based on the author's recent research on uniqueness theory for dual asymptotic expansions and remainder theory for Taylor interpolation on two lines. These results provide a simple iterative algorithm which derives a tensor product from a closed-form expression using the author's asymptotic splitting operator, and a simple hyperbolic eigenfunction criterion which proves that the two forms are identically equal. The author has implemented these methods as a complete derivation and proof system in the Maple 8 computer algebra system. The Maple code, which is surprisingly brief, is included in its entirety. The article also includes numerous examples which illustrate a variety of novel techniques for deriving and proving tensor product identities using this simple but effective system. (Maple is a registered trademark of Waterloo Maple Inc.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.025 | 0.016 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".