On mapping theorems for numerical range
Bibliographic record
Abstract
Let T T be an operator on a Hilbert space H H with numerical radius w ( T ) ≤ 1 w(T)\le 1 . According to a theorem of Berger and Stampfli, if f f is a function in the disk algebra such that f ( 0 ) = 0 f(0)=0 , then w ( f ( T ) ) ≤ ‖ f ‖ ∞ w(f(T))\le \|f\|_\infty . We give a new and elementary proof of this result using finite Blaschke products. A well-known result relating numerical radius and norm says ‖ T ‖ ≤ 2 w ( T ) \|T\| \leq 2w(T) . We obtain a local improvement of this estimate, namely, if w ( T ) ≤ 1 w(T)\le 1 , then \[ ‖ T x ‖ 2 ≤ 2 + 2 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 | 0.005 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.016 |
| Open science | 0.002 | 0.006 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.018 | 0.004 |
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".