Bibliographic record
Abstract
Abstract In this chapter, we shall discuss how fast a polynomial f can possibly grow in the complex plane. Assuming that f is of degree n, we need to know something about f(z), such as its modulus, its real part, or its imaginary part on a setε, containing at least n + 1 points. Except for this requirement, the set ε, may be almost any bounded subset of ℂ. We wish to find out how large its modulus or its real part (or its imaginary part) can be at a given point outside £,, or on any other set. For example, we may suppose | f (z) | to be bounded above by M on the unit circle and look for its sharp upper bound on the concentric circle of radius R > 1. We start with the Bernstein Walsh lemma, which provides us with a method for handling very general situations. We shall present two other methods, namely, 1The Convolution Method’ and ‘The Method of Functionals’. The latter two yield more precise results, but they are not as general as the first method.
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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 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.041 | 0.011 |
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".