Exponential asymptotics with a small exponent
Bibliographic record
Abstract
Analytic and numerical solutions are considered to a simple model problem which contains a surprisingly complicated solution structure. Asymptotic solutions are sought when a parameter that appears as an exponent in the independent variable is small, the solution then exhibiting a sudden change in slope over a region that is exponentially thin. A straightforward approach using matched asymptotic expansions immediately reveals inadequacies of this method due to the requirement of an outer solution that needs to be evaluated beyond all orders in order to match to a suitable inner solution. This behaviour is elucidated by studying first the asymptotic structure of the solution using an exact integral, which explicitly reveals the need for the inclusion of exponentially small terms in the expansions. It is then shown how a direct asymptotic solution of the differential equation can be obtained by using Borel summation to evaluate the outer solution to exponential accuracy. Further, as a practical alternative, it is shown how these exponentially improved approximations can be made when an exact numerical solution is available and without recourse to the general term of the outer or inner expansions.
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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 | 0.017 |
| 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 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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 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".