Erdélyi-Kober fractional integral operators from a statistical perspective -II
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
In this paper we examine the densities of a product and a ratio of two real positive definite matrix-variate random variables $ X_1 $ and $ X_2 $, which are statistically independently distributed, and we consider the density of the product $ U_1=X_2^{\\frac{1}{2}}X_1X_2^{\\frac{1}{2}} $ as well as the density of the ratio $ U_2=X_2^{\\frac{1}{2}}X_1^{-1}X_2^{\\frac{1}{2}} $. We define matrix-variate Kober fractional integral operators of the first and second kinds from a statistical perspective, making use of the derivation in the predecessor of this paper for the scalar variable case, by deriving the densities of product cand ratios where one variable has a matrix-variate type-1 beta density and the other matrix variable has an arbitrary density, in the sense, any real-valued scalar function f(X) of matrix argument X, such that f(X) is non-negative for all X and the total integral over all X, on the support of f(X), is unity. A number of generalizations are considered, by using pathway models, by appending matrix variate hypergeometric series etc. During this process matrix-variate Saigo operator and other operators are also defined and properties studied.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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