The average-distance problem with an Euler elastica penalization
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Bibliographic record
Abstract
We consider the minimization of an average-distance functional defined on a two-dimensional domain \Omega with an Euler elastica penalization associated with \partial\Omega , the boundary of \Omega . The average distance is given by \int_{\Omega}\operatorname{dist}^p(x,\partial\Omega)\operatorname{d}x, where p\geq 1 is a given parameter and \operatorname{dist}(x,\partial\Omega) is the Hausdorff distance between \{x\} and \partial\Omega . The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve {\partial\Omega} , which is proportional to the integrated squared curvature defined on \partial\Omega , as given by \lambda\int_{\partial\Omega} \kappa_{\partial\Omega}^2 \operatorname{d}\mathcal{H}_{\llcorner\partial\Omega}^1, where \kappa_{\partial\Omega} denotes the (signed) curvature of \partial\Omega and \lambda>0 denotes a penalty constant. The domain \Omega is allowed to vary among compact, convex sets of \mathbb{R}^2 with Hausdorff dimension equal to two. Under no a priori assumptions on the regularity of the boundary \partial\Omega , we prove the existence of minimizers of E_{p,\lambda} . Moreover, we establish the C^{1,1} -regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.003 | 0.001 |
| Scholarly communication | 0.001 | 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