Bibliographic record
Abstract
This doctoral dissertation comprises two peer-reviewed papers and one paper that has been submitted for publication. The first paper has appeared in a peer-reviewed journal, while the second paper is published in an open access journal. The third paper has just been submitted. These three papers comprise chapters 2-4 herein. These articles address issues stemming from the investigation of fractional Sturm-Liouville problems under ordinary and fractional boundary conditions. In the first paper (Chapter 2), we present the general solution of three such two-term fractional equations of mixed Caputo/Riemann-Liouville type and then, in the corresponding eigenvalue problem, we find two-sided estimates as to their location as well as their asymptotic behavior as a function of the fractional order. In the second paper (Chapter 3), we investigate a non-self-adjoint fractional Sturm-Liouville problem with left Caputo and right Riemann-Liouville fractional integrals with Dirichlet boundary conditions. Our study reveals the existence of a finite set of real eigenvalues for certain $\alpha$ where $0 < \alpha < 1$, and no real eigenvalues for $\alpha$ near 1/2. As $\alpha$ approaches 1 (i.e., the classical Sturm-Liouville case), the number of real eigenvalues approaches infinity, and so the spectrum resembles that of a standard Dirichlet problem. In the third paper (Chapter 4), we convert a fractional Sturm-Liouville equation, with fractional boundary conditions, to a hybrid fractional equation with classical boundary conditions and then prove that the eigenvalues of this problem are real and their eigenfunctions have an orthogonality property. We then prove a result analogous to Sturm's comparison theorem and state a number of results and conjectures regarding the zeros of solutions of such problems. We end by showing that, depending on the definition of the fractional operators, transformations of dependent variables can greatly simplify the presentation and resolution of the problems and conjectures we stated.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".