The Convolution and Fractional Derivative of Distributions
Why this work is in the frame
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Bibliographic record
Abstract
Let {n} be a certain sequence of functions in D converging to 1 in D . The commutative neutrix convolution f * g of two distributions f and g in D is defined to be the neutrix limit of the sequence 1 2 {(f n) * g + f * (gn)} , provided the limit exists. We present relations between this new convolution and other existing distributional convolutions, and demonstrate its strong computational power in evaluating convolutions as well as applications to defining new fractional derivatives and integrals of generalized functions in the new space H which contains D (R + ). The neutrix convolutions x - * x + for , , + = 0, 1, 2, and x - * x s + for = 0, 1, 2, and s = 0, 1, 2, are evaluated, from which other neutrix convolutions are deduced.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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