On signs of certain Toeplitz-Hessenberg determinants whose elements involve Bernoulli numbers
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Bibliographic record
Abstract
In the paper, by virtue of Wronski's formula and Kaluza's theorem related to a power series and its reciprocal, by means of Cahill and Narayan's recursive relation, and with the aid of the logarithmic convexity of the sequence of the Bernoulli numbers, the author presents the signs of certain Toeplitz-Hessenberg determinants whose elements involve the Bernoulli numbers and combinatorial numbers. Moreover, with the help of a derivative formula for the ratio of two differentiable functions, the author provides an alternative proof of Wronski's formula.
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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 | 0.004 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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