The family of cubic differential systems with two real and two complex distinct infinite singularities and invariant straight lines of the type <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mo stretchy="false">(</a:mo> <a:mn>3</a:mn> <a:mo>,</a:mo> <a:mn>1</a:mn> <a:mo>,</a:mo> <a:mn>1</a:mn> <a:mo>,</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo></a:math>
Bibliographic record
Abstract
In this article we consider the class <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msubsup> <a:mrow class="MJX-TeXAtom-ORD"> <a:mtext mathvariant="bold">CSL</a:mtext> </a:mrow> <a:mn>7</a:mn> <a:mrow class="MJX-TeXAtom-ORD"> <a:mn>2</a:mn> <a:mi>r</a:mi> <a:mn>2</a:mn> <a:mi>c</a:mi> <a:mi mathvariant="normal">∞</a:mi> </a:mrow> </a:msubsup> </a:math> of non-degenerate real planar cubic vector fields, which possess two real and two complex distinct infinite singularities and invariant straight lines of total multiplicity 7, including the line at infinity. The classification according to the configurations of invariant lines of systems possessing invariant straight lines was given in articles published from 2014 up to 2022. We continue our investigation for the family <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:msubsup> <f:mrow class="MJX-TeXAtom-ORD"> <f:mtext mathvariant="bold">CSL</f:mtext> </f:mrow> <f:mn>7</f:mn> <f:mrow class="MJX-TeXAtom-ORD"> <f:mn>2</f:mn> <f:mi>r</f:mi> <f:mn>2</f:mn> <f:mi>c</f:mi> <f:mi mathvariant="normal">∞</f:mi> </f:mrow> </f:msubsup> </f:math> possessing configurations of invariant lines of type <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mo stretchy="false">(</k:mo> <k:mn>3</k:mn> <k:mo>,</k:mo> <k:mn>1</k:mn> <k:mo>,</k:mo> <k:mn>1</k:mn> <k:mo>,</k:mo> <k:mn>1</k:mn> <k:mo stretchy="false">)</k:mo> </k:math> and prove that there are exactly 42 distinct configurations of this type. Moreover we construct all the orbit representatives of the systems in this class with respect to affine group of transformations and a time rescaling.
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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 | 0.010 | 0.005 |
| Meta-epidemiology (narrow) | 0.004 | 0.003 |
| Meta-epidemiology (broad) | 0.007 | 0.003 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.005 |
| 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; both teacher heads agree on what is shown here.
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".