On some refinements of the embedding of critical Sobolev spaces into BMO, and a study of stability for parabolic equations with time delay
Bibliographic record
Abstract
Van Schaftinen showed that the inequalities of Bourgain and Brezis give rise to new function spaces that refine the classical embedding of critical Sobolev spaces into BMO. It was suggested by Van Schaftingen that similar results should hold in the setting of bounded domains. \n \nThe first part of this thesis contains the proofs of this conjecture as well as the development of a non-homogeneous theory of Van Schaftingen spaces. Based on the results in the non-homogeneous setting, we are able to show that the refined embeddings can also be established for bmo spaces on Riemannian manifolds with bounded geometry, introduced by Taylor. \n \nThe stability of parabolic equations with time delay plays an important role in the study of non-linear reaction-diffusion equations with time delay. While the stability regions for such equations without convection on bounded time intervals were described by Travis and Webb, the problem remained unaddressed for the equations with convection. The need to determine exact regions of stability for such equations appeared in the context of the work of Mei and Wang on the Nicholson equation with delay. \n \nIn the second part of this thesis, we study the parabolic equations with and without convection. It has been shown that the presence of convection terms can change the regions of stability. The implications for the stability problems for non-linear equations are also discussed.
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How this classification was reachedexpand
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".