Bibliographic record
Abstract
The paper is concerned with the asymptotic distribution of Laplace eigenvalues on Liouville tori.Liouville metrics are the largest known class of integrable metrics on two-dimensional tori; they contain flat metrics and metrics of revolution as special cases.Using separation of variables, we reduce the eigenvalue counting problem to the problem of counting lattice points in certain planar domains.This allows us to improve the remainder estimate in Weyl's law on a large class of Liouville tori.For flat metrics, such an estimate has been known for more than a century due to classical results of W. Sierpiński and J.G. van der Corput.Our proof combines the method of Y. Colin de Verdière, who proved an analogous result for metrics of revolution on a sphere, with the techniques developed by P. Bleher, D. Kosygin, A. Minasov and Y. Sinai in their study of the almost periodic properties of the remainder in Weyl's law on Liouville tori.
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".