On the Analytic Properties of the Perturbing Function in the PCR3Body Problem
Bibliographic record
Abstract
We provide a new expansion of the Fourier coefficient of the perturbing function of the PCR3Body problem in terms of Hansen coefficients. This gives us a precise asymptotic formula for the coefficient in the region of application of KAM theory (i. e., small value of eccentricity and semimajor axis. See, e. g., [17]). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $$(m,k)\in\mathbb{Z}^{2}$$ and the presence of common zeros as functions of actions between coefficients relative to modes $$(m,k)$$ , $$(2m,2k)$$ and $$(m,k)$$ , $$(2m,2k)$$ , $$(3m,3k)$$ . Thanks to the previous expansion, this numerical analysis is done up to order $$60$$ in the power of eccentricity and semimajor axis. This is the first step for a possible application of [4, 9] to the PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so-called “non-torus” set from $$O(1-\sqrt{\varepsilon})$$ (implied by standard KAM theory) to $$O(1-\varepsilon|\log\varepsilon|^{c})$$ for some $$c>0$$ .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".