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Growth estimates

2002· book-chapter· en· W4388320534 on OpenAlex
Q I Rahman, Gerhard Schmeißer

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

Venuenot available
Typebook-chapter
Languageen
FieldMathematics
TopicMeromorphic and Entire Functions
Canadian institutionsUniversité de Montréal
Fundersnot available
KeywordsMathematicsComplex planeBounded functionUnit (ring theory)Unit circleDegree (music)Plane (geometry)PolynomialSet (abstract data type)CombinatoricsLemma (botany)Discrete mathematicsPure mathematicsMathematical analysisGeometryPhysicsComputer science

Abstract

fetched live from OpenAlex

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.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.000
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesInsufficient payload (model declined to judge)
Consensus categoriesInsufficient payload (model declined to judge)
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Other · Consensus signal: Other
Teacher disagreement score0.147
Threshold uncertainty score0.997

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0000.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0690.004

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.

Opus teacher head0.108
GPT teacher head0.267
Teacher spread0.160 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it

Quick stats

Citations0
Published2002
Admission routes1
Has abstractyes

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