Shape and topological derivatives as Hadamard semidifferentials
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Bibliographic record
Abstract
Abstract The object of this paper is to further investigate the notion of shape and topological derivatives in the light of the general notion of Hadamard semidifferential for a function defined on a subset of a topological vector space. The use of semitrajectories and the characterization of the adjacent tangent cone provide simple tools for defining Hadamard semi-differentials and differentials without a priori introduction of geometric structures such as, for instance, a differential manifold. Such a simple notion retains all the operations of the classical differential calculus, including the chain rule, for a large class of nondifferentiable functions, in particular, the norms and the convex functions. It also provides a direct access to functions defined on a lousy set or a manifold with boundary. This direct approach is first illustrated in the context of the classical matrix subgroups of the general linear group GL( n ) of invertible n × n matrices, which are the prototypes of Lie groups . For the shape derivative we have groups of diffeomorphisms of the Euclidean space ℝ n with the composition operation, and the adjacent tangent cone is a linear space; for the topological derivative we have the group of characteristic functions with the symmetric difference operation and the adjacent tangent cone is only a cone at some points.
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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 | 0.000 |
| 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