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 δ 2 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 Δ δ ( Δ denotes the Laplacian) for any positive integer k between 1 and m − 1 inclusive. Cheng and Li (1991) utilized a net δ ϵ ( x ) (similar to the δ n ( 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 ℝ m . 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 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 | 0.000 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| 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".