Uniform Cusp Property, Boundary Integral, and Compactness for Shape Optimization
Bibliographic record
Abstract
In this paper we consider the family of sets verifying the uniform cusp property introduced in [2] and extended in [4] to cusp functions only continuous at the origin. In the latter case we show that to any extended cusp function, we can associate a continuous, non-negative, and monotone strictly increasing cusp function of the type introduced in [2]. We construct an example of a bounded set in R N with a cusp function of the form c|θ|α, 0 < α < 1, for which its boundary integral is infinite and the Hausdorff dimension of its boundary is exactly N − α. We then give compactness theorems for the family of subsets of a bounded open holdall verifying a uniform cusp property with a uniform bound on either the De Georgi [6] or the γ-density perimeter of Bucur and Zolésio [1]. We also give their uniform local C 0-graph versions following [4]. This class forms a much larger family than the one of subsets verifying a uniform cone property.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".