Several results on the commutative neutrix product of distributions
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Bibliographic record
Abstract
Abstract Let H(x) denote Heaviside's function. The goal of this paper is to evaluate the commutative neutrix products f + (x) ⋄ δ(r) (x) and f − (x) ⋄ δ(r) (x) for r=0, 1, 2, …, where f(x) is only the r-th differentiable on an open interval containing the origin and f + (x)=H(x)f(x) and f − (x)=H(−x)f(x). We also obtain the products (ψ (x)/x) ⋄ δ (x), including a few examples as well as x + −r−(1/2) ⋄ x − −r−(1/2). Keywords: Productδ-sequenceNeutrix and distribution Acknowledgements The author is grateful to Dr. Brian Fisher who made several productive suggestions, which improved the quality of this paper. This research is supported by NSERC and BURC. Additional informationNotes on contributorsC. K. Li Email: lic@brandonu.ca
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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 | 0.001 | 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 | 0.000 | 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