A theory of anti-selfdual Lagrangians: dynamical case
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Bibliographic record
Abstract
We consider the class of time-dependent anti-selfdual Lagrangians , which – just like the stationary case announced in Ghoussoub [C. R. Acad. Sci. Paris, Ser. I 340 (2005)] – enjoys remarkable permanence properties and provides variational formulations and resolutions for several initial-value parabolic equations including gradient flows and other dissipative systems. Even though these evolutions do not fit in the standard Euler–Lagrange theory, we show that their solutions can be obtained as minima – but also as zeroes – of action functionals of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mtext>,</mml:mtext> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mtext>,</mml:mtext> <mml:mover accent="true"> <mml:mi>u</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="0.2em"/> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> where L is a time-dependent anti-selfdual Lagrangian and where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> is a flow of skew-adjoint operators.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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) | 0.002 | 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