Inverse Spectral Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions
Bibliographic record
Abstract
Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(−(py′)′+qy), l ˜ y = ( 1 / r ˜ ) ( - ( p ˜ y ′ ) ′ + q ˜ y ) , Δ = [ a b c d ] and ∑ = [ r s t u ] where det Δ = δ > 0, c ≠ 0, det ∑ > 0, t ≠ 0 and (cs + dr − au − tb)2 < 4(cr − ta)(ds − ub). Denote by (l; α; Δ) the eigenvalue problem ly = λy with boundary conditions y(0)cosα+y′(0)sinα = 0 and (aλ+b)y(1) = (cλ+d)(py′)(1). Define ( l ˜ ; α; Δ) as above but with l replaced by l ˜ . Let wn denote the eigenfunction of (l; α; Δ) having eigenvalue λn and initial conditions wn(0) = sin α and pw′n(0) = −cos α and let γn = −awn(1)+cpw′n(1). Define w ˜ n and γ ˜ n similarly. As sample results, it is proved that if (l; α; Δ) and ( l ˜ ; α; Δ) have the same spectrum, and (l; α; Σ) and ( l ˜ ; α; Σ) have the same spectrum or ∫ | w n | 0 1 r d t + ( | γ n | 2 / δ ) = ∫ | w ˜ n | 0 1 r ˜ d t + ( | γ ˜ n | 2 / δ ) for all n, then q/r = q ˜ / r ˜ .
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".