Application of numerical algebraic geometry and numerical linear algebra to PDE
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Bibliographic record
Abstract
The computational difficulty of completing nonlinear pde to involutive form by differential elimination algorithms is a significant obstacle in applications. We apply numerical methods to this problem which, unlike existing symbolic methods for exact systems, can be applied to approximate systems arising in applications.We use Numerical Algebraic Geometry to process the lower order leading nonlinear parts of such pde systems. The irreducible components of such systems are represented by certain generic points lying on each component and are computed by numerically following paths from exactly given points on components of a related system. To check the conditions for involutivity Numerical Linear Algebra techniques are applied to constant matrices which are the leading linear parts of such systems evaluated at the generic points. Representations for the constraints result from applying a method based on Polynomial Matrix Theory.Examples to illustrate the new approach are given. The scope of the method, which applies to complexified problems, is discussed. Approximate ideal and differential ideal membership testing are also discussed.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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 it