Singular Perturbation of Nonlinear Systems with Regular Singularity
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Bibliographic record
Abstract
We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>ε</mml:mi><mml:mi>z</mml:mi><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">′</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>F</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>ε</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi></mml:mrow></mml:mfenced></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:math> a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>-valued function, holomorphic in a polydisc <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mo>-</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mo>-</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mo>-</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math>. We show that its unique formal solution in power series of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>ε</mml:mi></mml:mrow></mml:math>, whose coefficients are holomorphic functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:math>, is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:math>-summable under a Siegel-type condition on the eigenvalues of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">0,0</mml:mn><mml:mo>,</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>. The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| 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