Geometry of quadratic differential systems in the neighbourhood of the line at infinity. Report no. 2701, Centre de recherches mathematiques et Departement de Mathematiques et de Statistiques, Universite de Montreal
Bibliographic record
Abstract
In this article we consider the behavior in the vicinity of infinity of the class Σ of all planar quadratic differential systems. This family depends on twelve parameters but due to the affine group action, the family actually depends on five parameters. We give simple, integer-valued geometric invariants for this group action which classify this family according to the topology of their phase portraits in the vicinity of infinity. For each one of the classes obtained we give necessary and sufficient conditions in terms of algebraic invariants and comitants so as to be able to easily retrieve for any system, in any chart, the geometric as well as the dynamic characteristics of the systems in the neighborhood of infinity. The program was implemented for computer calculations. Résumé Dans cet article nous considérons le comportement au voisinage de l’infini de la classe Σ de tous les systèmes différentiels quadratiques dans le plan. Cette famille depend de douze paramètres mais a ̀ cause de l’action du groupe affine la famille ne dépend que de cinq paramètres. Nous donnons des invariants simples, a ̀ valeurs dans les entiers, pour cette actions de groupe, classifiant cette famille d’après la topology des portraits de phases au voisinage de l’infini des systèmes. Pour chaque classe ainsi obtenue, nous donnons des conditions necessaires et suffisantes en termes d’invariants algébriques et de comitants afin de pouvoir determiner pour tout système, par rapport a ̀ toute carte, les caractéristiques au voisinage de l’infini, géométriques et dynamiques de ces systèmes. Ces calculs ont éte ́ programmés pour être effectués a ̀ l’ordinateur. 1
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How this classification was reachedexpand
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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)
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 teacher head, 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".