Sharp quantitative estimates of Struwe’s decomposition
Bibliographic record
Abstract
In a seminal work, Struwe proved that if 0≤u∈H˙1(Rn) and Γ(u):=‖Δu+u n+2 n−2‖H−1→0, then dist(u,T)→0, where T denotes the manifold of sums of Aubin–Talenti bubbles and dist(u,T) denotes the H˙1(Rn)-distance of u from T. Ciraolo, Figalli, and Maggi obtained the first quantitative version of Struwe’s decomposition with one bubble in all dimensions, namely dist(u,T)≤CΓ(u). For two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely dist(u,T)≤CΓ(u) when 3≤n≤5, while this is false for n≥6. In this article, we show that dist(u,T)≤C Γ(u)|logΓ(u)|1 2ifn=6,|Γ(u)| n+2 2(n−2)ifn≥7. Furthermore, we show that this inequality is sharp.
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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 | 0.003 | 0.013 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.005 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.006 | 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".