Hadamard Semidifferential, Oriented Distance Function, and some Applications
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Bibliographic record
Abstract
<p style='text-indent:20px;'>The <i>Hadamard semidifferential calculus</i> preserves all the operations of the classical differential calculus including the chain rule for a large family of non-differentiable functions including the continuous convex functions. It naturally extends from the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional Euclidean space <inline-formula><tex-math id="M2">\begin{document}$ \operatorname{\mathbb R}^n $\end{document}</tex-math></inline-formula> to subsets of topological vector spaces. This includes most function spaces used in <i>Optimization</i> and the <i>Calculus of Variations</i>, the metric groups used in <i>Shape and Topological Optimization</i>, and functions defined on submanifolds.</p><p style='text-indent:20px;'>Certain set-parametrized functions such as the <i>characteristic function</i> <inline-formula><tex-math id="M3">\begin{document}$ \chi_A $\end{document}</tex-math></inline-formula>of a set <inline-formula><tex-math id="M4">\begin{document}$ A $\end{document}</tex-math></inline-formula>, the <i>distance function</i> <inline-formula><tex-math id="M5">\begin{document}$ d_A $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M6">\begin{document}$ A $\end{document}</tex-math></inline-formula>, and the <i>oriented (signed) distance function</i> <inline-formula><tex-math id="M7">\begin{document}$ b_A = d_A-d_{ \operatorname{\mathbb R}^n\backslash A} $\end{document}</tex-math></inline-formula> can be used to identify a space of subsets of <inline-formula><tex-math id="M8">\begin{document}$ \operatorname{\mathbb R}^n $\end{document}</tex-math></inline-formula> with a metric space of set-parametrized functions. Many geometrical properties of domains (convexity, outward unit normal, curvatures, tangent space, smoothness of boundaries) can be expressed in terms of the analytical properties of <inline-formula><tex-math id="M9">\begin{document}$ b_A $\end{document}</tex-math></inline-formula> and a simple intrinsic differential calculus is available for functions defined on hypersurfaces without appealing to local bases or Christoffel symbols.</p><p style='text-indent:20px;'>The object of this paper is to extend the use of the Hadamard semidifferential and of the oriented distance function from finite to infinite dimensional spaces with some selected illustrative applications from shapes and geometries, plasma physics, and optimization.</p>
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.005 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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