The product of <i>r</i><sup>−<i>k</i></sup> and ∇<i>δ</i> on <i>ℝ</i><sup><i>m</i></sup>
Bibliographic record
Abstract
In the theory of distributions, there is a general lack of definitions for products and powers of distributions. In physics (Gasiorowicz (1967), page 141), one finds the need to evaluate δ2 when calculating the transition rates of certain particle interactions and using some products such as (1/x) · δ. In 1990, Li and Fisher introduced a “computable” delta sequence in an m‐dimensional space to obtain a noncommutative neutrix product of r−k and Δδ (Δ denotes the Laplacian) for any positive integer k between 1 and m − 1 inclusive. Cheng and Li (1991) utilized a net δϵ(x) (similar to the δn(x)) and the normalization procedure of to deduce a commutative neutrix product of r−k and δ for any positive real number k. The object of this paper is to apply Pizetti′s formula and the normalization procedure to derive the product of r−k and ∇δ (∇ is the gradient operator) on ℝm. The nice properties of the δ‐sequence are fully shown and used in the proof of our theorem.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".