The sequential approach to the product of distribution
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
It is well known that the sequential approach is one of the main tools of dealing with product, power, and convolution of distribution (cf. Chen (1981), Colombeau (1985), Jones (1973), and Rosinger (1987)). Antosik, Mikusiński, and Sikorski in 1972 introduced a definition for a product of distributions using a delta sequence. However, δ 2 as a product of δ with itself was shown not to exist (see Antosik, Mikusiński, and Sikorski (1973)). Later, Koh and Li (1992) chose a fixed δ ‐sequence without compact support and used the concept of neutrix limit of van der Corput to define δ k and for some values of k . To extend such an approach from one‐dimensional space to m ‐dimensional, Li and Fisher (1990) constructed a delta sequence, which is infinitely differentiable with respect to x 1 , x 2 , …, x m and r , to deduce a non‐commutative neutrix product of r − k and Δ δ . Li (1999) also provided a modified δ ‐sequence and defined a new distribution ( d k / d r k ) δ ( x ), which is used to compute the more general product of r − k and Δ l δ , where l ≥ 1, by applying the normalization procedure due to Gel′fand and Shilov (1964). We begin this paper by distributionally normalizing Δ r − k with the help of distribution . Then we utilize several nice properties of the δ ‐sequence by Li and Fisher (1990) and an identity of δ distribution to derive the product Δ r − k · δ based on the results obtained by Li (2000), and Li and Fisher (1990).
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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 | 0.004 | 0.006 |
| 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 it