Bibliographic record
Abstract
Abstract We show that any nondegenerate vector field u in \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^{\infty}(\Omega, \mathbb{R}^N)\end{align*} \end{document} , where Ω is a bounded domain in \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathbb{R}^N\end{align*} \end{document} , can be written as \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}u(x)= \nabla_1 H(S(x), x)\quad {\text for a.e.\ x \in \Omega}\end{align*} \end{document} }, where S is a measure‐preserving point transformation on Ω such that \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}S^2=I\end{align*} \end{document} a.e. (an involution), and \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}H: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}\end{align*} \end{document} is a globally Lipschitz antisymmetric convex‐concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a self‐dual version of Brenier's polar decomposition for the vector field as \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}u(x)=\nabla \phi (S(x))\end{align*} \end{document} , where ϕ is convex and S is a measure‐preserving transformation. We also describe how our polar decomposition can be reformulated as a (self‐dual) mass transport problem. © 2012 Wiley Periodicals, Inc.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.019 | 0.006 |
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".