Spectral Residues of Second‐order Differential Equations: A New Method for Summation Identities and Inversion Formulas
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Bibliographic record
Abstract
This article deals with differential equations with spectral parameter from the point of view of formal power series.The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residue. The article focuses on second‐order, self‐adjoint problems. In such a setting, every potential function determines a sequence of spectral residues. This correspondence is invertible and gives rise to a combinatorial inversion formula. Other interesting combinatorial consequences are obtained by considering spectral residues of exactly solvable potentials of one‐dimensional quantum mechanics. It is also shown that the Darboux transformation of one‐dimensional potentials corresponds to a simple negation of the corresponding spectral residues. This fact leads to another combinatorial inversion formula. Finally, there is a brief discussion of applications. The topics considered are enumeration problems and integrable systems.
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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 | 0.000 |
| 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