Bibliographic record
Abstract
This is an overview paper. This paper is an attempt to show that fractional calculus can be reached through statistical distribution theory. This paper brings together results on fractional integrals and fractional derivatives of the first and second kinds in the real and complex domains in the scalar, vector, and matrix-variate cases, and shows that all these results can be reached through statistical distribution theory. It is shown that the whole area of fractional integrals can be reached through distributions of products and ratios in the scalar variable case and distributions of symmetric products and symmetric ratios in the matrix-variate cases. While summarizing the materials, the real domain results are also listed side by side with the complex domain results so that a comparative study is possible. Fractional integrals and derivatives in the real domain mean that the parameters involved could be real or complex with appropriate conditions, the arbitrary function is real-valued, and the variables involved are all real. These in the complex domain mean that the parameters could be real or complex and the arbitrary function is still real-valued but the variables involved are in the complex domain. Fully complex domain means the variables as well as the arbitrary function are in the complex domain. Most of the materials on fractional integrals and fractional derivatives involving a single matrix or a number of matrices in the real or complex domain are of this author. Slight modifications of the results, compared with the published works in various papers, are there in various sections. In the paragraph on notations, the lemmas that are taken from this author’s own book on Jacobians are common with published works and hence the similarity index with this author’s works will be high. Section Matrix-Variate Joint Distributions and Fractional Integrals in Many Matrix-Variate Cases material on a statistical approach to Kiryakova’s multi-index fractional integral and its extension to the real scalar case of second kind integrals as well as extensions of first and second kind integrals to real and complex matrix-variate cases are believed to be new. Matrix differential operators are introduced in Section Fractional Derivatives and, with the help of these operators, fractional derivatives are constructed from the corresponding fractional integrals. These operators are applicable in a large variety of functions. Applicability is shown through identities created from scale transformed gamma random variables. Some concluding remarks are given and some open problems are pointed out in Section Concluding Remarks.
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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 | 0.003 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 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".