Generic 2-parameter perturbations of parabolic singular points of vector fields in ℂ
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Bibliographic record
Abstract
We describe the equivalence classes of germs of generic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-parameter families of complex vector fields <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove z With dot equals omega Subscript epsilon Baseline left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot z = \omega _\epsilon (z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> unfolding a singular parabolic point of multiplicity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k plus 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 0 equals z Superscript k plus 1 Baseline plus o left-parenthesis z Superscript k plus 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\omega _0= z^{k+1} +o(z^{k+1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fields <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove z With dot equals z Superscript k plus 1 Baseline plus epsilon 1 z plus epsilon 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot z = z^{k+1}+\epsilon _1z+\epsilon _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C double-struck upper P Superscript 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}\mathbb {P}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) unique normal forms for the equivalence classes of germs.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| 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 it