Bibliographic record
Abstract
Abstract Glasser, in 2011, introduced a remarkable integral identity of physical interest and suggested that the evaluation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:mi>k</mml:mi> <mml:msup> <mml:mi>K</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:mfrac> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msup> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msup> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mo>⋯</mml:mo> </mml:math> denotes Catalan’s constant. We show how a case of Clausen’s product identity related to Ramanujan’s series for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>π</mml:mi> </mml:mfrac> </mml:math> 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 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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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; both teacher heads agree on what is shown here.
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".