Helmholtz solutions for the fractional Laplacian and other related operators
Why this work is in the frame
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Bibliographic record
Abstract
We show that the bounded solutions to the fractional Helmholtz equation, [Formula: see text] for [Formula: see text] in [Formula: see text], are given by the bounded solutions to the classical Helmholtz equation [Formula: see text] in [Formula: see text] for [Formula: see text] when [Formula: see text] is additionally assumed to be vanishing at [Formula: see text]. When [Formula: see text], we show that the bounded fractional Helmholtz solutions are again given by the classical solutions [Formula: see text]. We show that this classification of fractional Helmholtz solutions extends for [Formula: see text] and [Formula: see text] when [Formula: see text]. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation [Formula: see text] in [Formula: see text], when [Formula: see text] is complete Bernstein and certain regularity conditions are imposed on the associated weight [Formula: see text].
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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