The notions of skeleton and crack, and singularities of the oriented distance function
Bibliographic record
Abstract
In problems where a geometric object is the variable, the object can be identified with the oriented distance function which can simultaneously deal with the smooth sets of classical Differential Geometry and sets with a lousy boundary.This paper reviews some properties of the distance function d_{A} to a set A , the oriented distance function b_{A}=d_{A}-d_{\complement A} ( \complement A , the complement of A ), and the associated notions of skeletons , b -crack , and crack . It gives the respective partitions of the boundaries \partial A and \partial (\partial A) and the partition of the singularities of \nabla b_{A} . It turns out that the notion of b -crack is possibly too broad since it also includes corners in the core boundary \partial\overline{A}\cap\partial\overline{\complement A} of the set A . On one hand, this analysis leads to the notions of crack-free sets and strongly crack-free sets and, on the other hand, to the notion of cracked sets in Image Segmentation and Mathematical Morphology.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.001 | 0.011 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".