Bibliographic record
Abstract
The problem of defining products of distributions is a difficult and not completely understood problem, studied from several points of views since Schwartz established the theory of distributions around 1950. Many fields, such as wave propagation or quantum mechanics, require such multiplications. The product of an infinitely differentiable function φ(x) and distribution 4δ(x) in R is well defined by (φ(x)4δ(x), ψ) = (δ(x), 4(φψ)), since 4(φψ) ∈ D(R). Using an induction, we derive an interesting formula for 4(φ(x)ψ(x)) and hence we are able to write out an explicit expression of the product φ(x)4δ(x). In particular, we imply the product X4δ(x) with a few applications in further simplifying existing distributional products. Furthermore, we obtain an asymptotic expression for δ(r−a) in terms of4δ(x), which is equivalent to the well-known Pizzetti’s formula. Several asymptotic products including φ(x) δ(r−1), X δ(r−1) as well as the more generalized φ(x) δ(r−1) are calculated and presented as infinitely series.
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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 | 0.002 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 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".