ON LAGRANGIAN DISTRIBUTIONS OF THE SECOND KIND
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Bibliographic record
Abstract
Janardan and Rao (SIAM J. Applied Math. 1983, 43, 302–313) have used the second Lagrange expansion, with f(z) and g(z) as two probability generating functions (pgfs) defined on nonnegative integers where g(0) ≠ 0, to define the class of discrete Lagrangian probability distributions of the second kind. They have also studied a number of properties of Lagrangian distributions of the second kind. Different families are generated by various choices of the pgfs f(z) and g(z). In this paper, the class of Lagrange distributions of the second kind is considerably widened to provide many more families. The convolution theorem has been modified and the central moments and cumulants have been obtained.
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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 | 0.002 | 0.004 |
| 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) | 0.001 | 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