Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces
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Bibliographic record
Abstract
Abstract We establish sufficient conditions on the shape of a set A included in the space ( X , Y ) of the n -linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a C n -smooth mapping f : X → Y , with bounded support, and such that f ( n ) ( X ) = A , provided that X admits a C n -smooth bump with bounded n -th derivative and dens X = dens ℒ n ( X , Y ). For instance, when X is infinite-dimensional, every bounded connected and open set U containing the origin is the range of the n -th derivative of such amapping. The same holds true for the closure of U , provided that every point in the boundary of U is the end point of a path within U . In the finite-dimensional case, more restrictive conditions are required. We also study the Fréchet smooth case for mappings from ℝ n to a separable infinite-dimensional Banach space and the Gâteaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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