Families of vector fields with many numerical invariants
Bibliographic record
Abstract
We study bifurcations in finite-parameter families of vector fields on \begin{document}$S^2$\end{document} . Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable \begin{document}$3$\end{document} -parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of \begin{document}$(2D+1)$\end{document} -parameter families such that the topological classification of these families has at least \begin{document}$D$\end{document} numerical invariants and used those examples to construct families with functional invariants of topological classification. In this paper, we construct locally generic \begin{document}$4$\end{document} -parameter families with any prescribed number of numerical invariants and use them to construct \begin{document}$5$\end{document} -parameter families with functional invariants. We also describe a locally generic class of \begin{document}$3$\end{document} -parameter families with a tail of an infinite number sequence as an invariant of topological classification.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
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".