Orthonormal Bases on \(\mathit{L}^2\)(\(\mathbb{R}\)\(^+\))
Bibliographic record
Abstract
In addition to orthogonal polynomials, orthogonal functions also play an important role. Their applications are, among others, in the fields of signal and data analysis, dynamic modeling. They are related to the solution of differential equations. In this paper we derive the explicit form of one parameter family of orthonormal bases on space \(\mathit{L}^2\)(\(\mathbb{R}\)\(^+\)). The bases are formed by eigenvectors of the self-adjoint extension \(\mathit{H}_\xi\), parametrized by \(\xi\) \(\in\) \(\small\langle\)0,\(\pi\)), of differential expression \(\mathit{H} = -{ \mathit{d}^2 \over \mathit{d}^{x2}} +{ \mathit{x}^2 \over 4} \) together with the spectrum \(\sigma\)(\(\mathit{H}_\xi\)) on the space \(\mathit{L}^2\)(\(\mathbb{R}^+\)). For each \(\xi\) the set of eigenvectors form an orthonormal basis of \(\mathit{L}^2\)(\(\mathbb{R}\)\(^+\)). From the physical point of view, it is a solution of the Schrodinger equation of a harmonic oscillator on a semi-straight line. To correlate platelet count, splenic index (SI), platelet count/spleen diameter ratio and portal-systemic venous collaterals with the presence of esophageal varices in advanced liver disease to validate other screening parameters.
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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 | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".