A Fractal Function Related to the John–Nirenberg Inequality for<i>Q</i><sub>α</sub>(ℝ<sup>n</sup>)
Bibliographic record
Abstract
Abstract A borderline case function f for Q α (ℝ n ) spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter β > 0. On its support I 0 = [0, 1] n , f ( x ) can be expressed by the binary expansions of the coordinates of x . In particular, f = f β ∈ Q α (ℝ n ) if and only if α < β < , while for β = α, it was shown by Yue and Dafni that f satisfies a John–Nirenberg inequality for Q α (ℝ n ). When β ≠ 1, f is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside I 0 . In addition, it is not monotone along any coordinate direction in any small cube. When the parameter β ∈ (0, 1), f is onto from I 0 to , and the graph of f has a non-integer fractal dimension n + 1 − β .
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
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".