Several Asymptotic Products of Particular Distributions
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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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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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 it