Bibliographic record
Abstract
Abstract Glasser, in 2011, introduced a remarkable integral identity of physical interest and suggested that the evaluation ∫ 0 1 / 2 k K 2 ( k ) d k = π G 4 provides the unique analytically tractable moment of K 2 on a sub-unit interval, where K denotes the complete elliptic integral of the first kind, and where G = 1 1 2 − 1 3 2 + 1 5 2 − ⋯ denotes Catalan’s constant. We show how a case of Clausen’s product identity related to Ramanujan’s series for 1 π may be applied, via an integration argument derived from our past work in fractional analysis and Fourier–Legendre theory, to show how higher moments of K 2 on the same sub-unit interval may be evaluated analytically in terms of the Γ-function. This and Glasser’s moment formula are motivated by how closely related moment formulas for powers of K arise in the study of Feynman diagrams.
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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 | 0.004 | 0.024 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
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".