A weighted and exponential generalization of Wilker's inequality and its applications
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Bibliographic record
Abstract
Abstract In this paper, the authors first prove a weighted and exponential generalization of Wilker's inequality. The main result presented here is then applied with a view to deriving an improved version of the Sándor–Bencze conjectured inequality. Some other closely-related inequalities are also considered. Keywords: Wilker's inequalityHuygens's inequalityWeighted generalizationSándor–Bencze conjectured inequalityWeighted inequalitiesExponential generalization Acknowledgements The present investigation was supported, in part, by the Natural Science Foundation of the Fujian Province of the People's Republic of China under Grant 2006J0197 and, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.000 | 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 it