Configurations of quadratic systems possessing three distinct infinite singularities and one or more invariant parabolas
Bibliographic record
Abstract
Denote by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QS</mml:mtext> </mml:mrow> </mml:math> the class of all non-degenerate planar quadratic differential systems and by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> </mml:mrow> </mml:math> the subclass of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QS</mml:mtext> </mml:mrow> </mml:math> of all systems possessing at least one invariant parabola. In this paper we consider the subfamily of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> </mml:mrow> </mml:math> defined by the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:math> , which means the presence of three distinct infinite singularities real or complex. We denote this subfamily by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We investigate all possible configurations of invariant parabolas and invariant straight lines which systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> could possess and their geometric properties encoded in such configurations. The classification presented here is taken modulo the action of the group of real affine transformations and time rescaling and it is given in terms of affine invariant polynomials. It yields a total of 146 distinct configurations. The obtained classification is an algorithm which makes it possible for any given real quadratic differential system in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> to specify its configuration of invariant parabolas and straight lines. This work will prove helpful in studying the integrability of the systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .
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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 | 0.004 |
| 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 | 0.001 |
| 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".