Recursion relations for the four-electron subsidiary integral<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>;</mml:mo></mml:mrow></mml:mrow></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>
Bibliographic record
Abstract
The subsidiary integral ${W}_{4}(k,l,m,n;\phantom{\rule{0.16em}{0ex}}\ensuremath{\theta},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma})$ plays an essential role in the variational calculation of four-electron atomic systems using Hylleraas coordinates. With respect to the case where the ratio $\ensuremath{\theta}/(\ensuremath{\theta}+\ensuremath{\alpha}+\ensuremath{\beta}+\ensuremath{\gamma})\ensuremath{\sim}1$, an important special situation that may occur in the evaluation of the Bethe logarithm, existing approaches for evaluating the ${W}_{4}$ integral become impractical due to the problem of slow convergence. Based on our recent work for the three-electron subsidiary integral ${W}_{3}(l,m,n;\phantom{\rule{0.16em}{0ex}}\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma})$, we present a computationally efficient and numerically stable method, in which the ${W}_{4}$ integral can be expressed in terms of either a finite series or a finite recursion relation. Numerical experimentation is presented to validate our method.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.003 | 0.002 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.037 | 0.012 |
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".