BESSEL FUNCTIONS IN A QUANTUM-BILLIARD CONFIGURATION PROBLEM
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Bibliographic record
Abstract
In studying various quantum-billiard configurations, R. L. Liboff (J. Math. Phys.35 (1994) 2218), was led to investigate the vanishing of f(ν)=j 2ν,1 - j ν2 , where j μk is the kth positive zero of the Bessel function J μ (x). Here we show that the even more general function f α (ν)=c αν,k - c ν,k+l is increasing and vanishes once (and only once) in 0<ν<∞, provided α≥π/2 and [Formula: see text], k, l=1,2,3,…. As usual, c μn is the nth positive zero of the cylinder function C μ (x)=J μ (x) cos θ - Y μ (x) sin θ. Specialized to Liboff's case, f(ν), this yields not only the existence of a zero of f(ν) but also its uniqueness.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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) | 0.000 | 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