Convex decompositions and the valence of some functions
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Bibliographic record
Abstract
The aim of this paper is twofold. On one hand, we provide examples of R n -valued functions on some open subsets D of R n whose restrictions to the convex subsets of D are all injective. Such applications are shortly called CIP functions. On the other hand, we provide alternative descriptions of the maximal convex subsets of the convex open sets with compact convex subsets removed. The maximal convex subsets of R n with convex sets removed were characterized before by Martnez-Legaz and Singer [Compatible Preorders and Linear Operators on R n , Linear Algebra Appl. 153 (1991), 53-66] as being the convex subsets of R n , shortly called hemispaces, whose complements are convex too. The two topics merge together as the smallest number k of convex subsets, of the considered open set, needed to cover it, is an upper bound for the valence of every CIP function on that open set.
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Full frame distilled prediction
Teacher imitationNot 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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it