Completely degenerate responsive tori in Hamiltonian systems <sup>*</sup>
Bibliographic record
Abstract
Abstract We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian H ( θ , I , x , y , ϵ ) = ⟨ ω , I ⟩ + λ x n n + y m m + ϵ P ( θ , x , y , ϵ ) , ( θ , I , x , y ) ∈ T d × R d + 2 , which is associated with the standard symplectic structure, where λ = ±1 and n > 2, n ⩾ m ⩾ 2 are integers. With P satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and ϵ sufficiently small, responsive tori exist for each ω satisfying a weak non-resonant condition; (2) For λ = 1 and ϵ * sufficiently small, there exists a Cantor set E ⊂ ( 0 , ϵ * ) with almost full Lebesgue measure such that responsive tori exist for each ϵ ∈ E if ω satisfies a Diophantine condition. Non-existence of responsive tori are also discussed when P fails to satisfy the non-degenerate condition. Our results are directly applicable to the existence problem of quasi-periodic responsive solutions of degenerate harmonic oscillators.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.012 | 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".