Moving frames, Bäcklund’s theorem, and its affine extension
Bibliographic record
Abstract
The sine-Gordon equation (SGE) is a partial differential equation which has received renewed interest in recent decades due to its connections to the physical phenomenon of solitons.Of particular interest to geometers is a well-established correspondence between solutions to the SGE and pseudospherical surfaces.Elaborating on the work of Chern and Terng [CT80], this thesis rederives an appropriate Bäcklund transformation, allowing for the generation of new Euclidean pseudospherical surfaces from known ones.In doing so, we make use of the geometric structure of pseudospherical line congruences.Taking advantage of the aforementioned correspondence, we then demonstrate how known solutions to the SGE enable the discovery of novel ones.The thesis proceeds to consider an analogous procedure for surfaces in affine space, replacing the notion of a pseudospherical line congruence with the more abstract notion of a Weingarten congruence.This consideration then allows for the generalization of Bäcklund's theorem and the Euclidean integrability theorem to affine space.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".