The integer part of <i>qα + β</i>
Bibliographic record
Abstract
We turn to the study of a class of generating functions that, somewhat like the folds and ripples of the previous chapter, lead to remarkable continued fractions and rational approximations. They rely on an inhomogeneous continuous function algorithm discussed below. Inhomogeneous Diophantine approximation As we have already learnt, the convergents p n /q n of a continued fraction [ a 0 ; a 1 , a 2 , …] representing the real irrational number α minimise the quantity | q α− p |. The related homogeneous Diophantine problem initiated by Dirichlet's theorem (Theorem 1.36) and discussed in Sections 1.4 and 1.5 was in fact our motivation to develop the theory of continued fractions. It is not therefore completely unreasonable to believe that a slightly more general minimisation problem for the quantity | q α + β − p |, where β is another (not necessarily irrational) real number already considered in Chebyshev's theorem (Exercise 1.39), gives rise to a natural extension of continued fractions. More important is not so much the algorithmic solution of this inhomogeneous Diophantine approximation problem but its many consequences. Because the replacement of α and β by their fractional parts does not affect the Diophantine problem, we will assume that they lie between 0 and 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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".