Comment on ‘invariants of differential equations defined by vector fields’
Bibliographic record
Abstract
In Ndogmo ( 2008 J. Phys. A: Math. Theor. 41 025207 ), the author claims to have determined a complete list of functionally independent differential invariants up to order 2 for the equivalence group of differential equations defined by vector fields when there are two and three independent variables. In this comment, we show that this is not the case. Using the equivariant moving frame method, we derive a complete set of functionally independent differential invariants of orders 1 and 2 for an arbitrary number n ⩾ 2 of independent variables. In the particular case n = 2, we obtain six functionally independent invariants, two of which were not found in Ndogmo ( 2008 J. Phys. A: Math. Theor. 41 025207 ). In the case n = 3, we get 21 functionally independent invariants, six of which are new. We also give a complete classification of the differential invariants.
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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 | 0.004 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.008 |
| Scholarly communication | 0.002 | 0.007 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.006 | 0.009 |
| Insufficient payload (model declined to judge) | 0.005 | 0.002 |
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".