Exponential asymptotics with a small exponent
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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 <italic>numerical</italic> solution is available and <italic>without</italic> recourse to the general term of the outer or inner expansions.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.008 | 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.
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