Extension of functions and metrics with variable domains
Bibliographic record
Abstract
Let ( X , d ) be a complete, bounded, metric space. For a nonempty, closed subset A of X denote by C ⁎ ( A × A ) the set of all continuous, bounded, real-valued functions on A × A . Denote by C † = ⋃ { C ⁎ ( A × A ) | A is a nonempty closed subset of X } the set of all partial, continuous and bounded functions. We prove that there exists a linear, regular extension operator from C † endowed with the topology of convergence in the Hausdorff distance of graphs of partial functions to the space C ⁎ ( X × X ) with the topology of uniform convergence on compact sets. The constructed extension operator preserves constant functions, pseudometrics, metrics and admissible metrics. For a fixed, nonempty, closed subset A of X the restricted extension operator from C ⁎ ( A × A ) to C ⁎ ( X × X ) is continuous with respect to the topologies of pointwise convergence, uniform convergence on compact sets and uniform convergence considered on both C ⁎ ( A × A ) and C ⁎ ( X × X ) .
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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 | 0.003 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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".