Bibliographic record
Abstract
The main theme of this thesis is further scrutinizing classic Hardy inequalities and expanding the study on "Optimal Hardy Inequality for General Elliptic Operators with Improvement". We rediscovered explicit integral form of Hardy inequality with the main focus on its functional aspects, including density of Sobolev space. We discuss our motivation for exploring the main Hardy inequality, with a focus on the quadratic case. The fundamental solution of Laplacian is applied for illustrating the proof of classical Hardy inequality. Furthermore, we conclude the classic Hardy inequalities for a bounded domain and Hardy's boundary inequality for smooth boundary of domain. Then, the inequalities are considered for operators more general than Laplacian and various Hardy inequalities are explored in terms of different boundary weight and interior weight of positive function E. Also The generalization of Cafferelli-Kohn-Nirenberg inequality are examined to find the optimal and not attained constant. Furthermore, the possibility of more general weighted inequalities is investigated. Finally, we consider one common type of improvement for the above-mentioned inequalities.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.009 | 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".