Bibliographic record
Abstract
Our voyage through the preceding eight chapters has shown that we have certainly not yet reached the end of the story on collocation methods for Volterra functional integral and integro-differential equations. Many important questions remain unanswered. It is my belief that we have to find new mathematical approaches and tools (likely from very unexpected areas) if we are to make substantial progress towards finding complete solutions to these open problems. It is the purpose of this brief final chapter to point to some possible, and seemingly very promising, new approaches for the numerical analysis of collocation solutions to Volterra functional equations. Semigroups and abstract resolvent theory The long-time integration of Volterra integral and integro-differential equations by collocation methods, in particular the asymptotic behaviour of collocation solutions, is not yet understood. As a number of papers and books have shown (see, e.g. Ito and Kappel (1989, 1991, 2002), Ito and Turi (1991), Brunner, Kauthen and Ostermann (1995), Bellen and Maset (1999), Maset (1999, 2003), and Bellen and Zennaro (2003, pp. 56–60)) the appropriate reformulation of the given equation as an abstract Cauchy problem and the exploitation of the underlying semigroup or abstract resolvent framework (integrability and asymptotic behaviour of resolvents) will often lead to deep insight into the qualitative properties of approximate solutions.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.303 | 0.138 |
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".