Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus
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Bibliographic record
Abstract
Abstract In a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in H s for s > 1 − α /2 and globally well-posed for s > 10 α − 1/12. In this paper we define an invariant probability measure μ on H s for s < α − 1/2, so that for any ∊ > 0 there is a set Ω ⊂ H s such that μ (Ω c ) < ∊ and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense for in an almost sure sense.
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.006 |
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