On rank-2 Toda systems with arbitrary singularities: local mass and new estimates
Bibliographic record
Abstract
For all rank-2 Toda systems with an arbitrary singular source, we use a unified approach to prove:(1) The pair of local masses . 1 ; 2 / at each blowup point has the expressionwhere N ij 2 ,ޚ i D 1; 2, j D 1; 2; 3.(2) At each vortex point p t if .˛1t ; ˛2 t / are integers and i … 4 ,ގ then all the solutions of Toda systems are uniformly bounded.(3) If the blowup point q is a vortex point p t and ˛1 t ; ˛2 t and 1 are linearly independent over Q, then u k .x/C 2 log jx p t j Ä C:The Harnack-type inequalities of 3 are important for studying the bubbling behavior near each blowup point.where g is the Laplace-Beltrami operator ( g 0), S is a finite set on M, h 1 ; : : : ; h n are positive and smooth functions on M, ˛i t > 1 is the strength of the Dirac mass ı p t and D . 1 ; : : : ; n / is a constant vector with nonnegative components.Here for simplicity we just assume that the total area of M is 1.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| 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".