Faber and Grunsky operators corresponding to bordered Riemann surfaces
Bibliographic record
Abstract
Let R \mathfrak {R} be a compact Riemann surface of finite genus g>0 \mathfrak {g}>0 and let Σ \Sigma be the subsurface obtained by removing n≥1 n\geq 1 simply connected regions Ω1+,…,Ωn+ \Omega _1^+, \dots , \Omega _n^+ from R \mathfrak {R} with non-overlapping closures. Fix a biholomorphism fk f_k from the unit disc onto Ωk+ \Omega _k^+ for each k k and let f=(f1,…,fn) \mathbf {f}=(f_1, \dots , f_n) . We assign a Faber and a Grunsky operator to R \mathfrak {R} and f \mathbf {f} when all the boundary curves of Σ \Sigma are quasicircles in R \mathfrak {R} . We show that the Faber operator is a bounded isomorphism and the norm of the Grunsky operator is strictly less than one for this choice of boundary curves. A characterization of the pull-back of the holomorphic Dirichlet space of Σ \Sigma in terms of the graph of the Grunsky operator is provided.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.017 | 0.003 |
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".