Solving higher order linear differential equations having elliptic function coefficients
Bibliographic record
Abstract
We consider the problem of finding closed form solutions of a linear homogeneous ordinary differential equation having coefficients which are elliptic functions. In particular, the input coefficients are assumed to be represented as elements of C(p,p'), where C is the complex number field, and p(x) and p' (x) are the Weierstrass p function and its first derivative, respectively. The specific closed form solutions y(x) which we seek are hyperexponential over C(p,p'), i.e., solutions y(x) such that y' (x)/y(x) is in C(p,p'). Such solutions correspond to first order right-hand factors of the associated linear differential operator, and are analogous to hyperexponential solutions over C(x), in the more well-known case where the coefficients of the ode are in C(x). A previous paper [4] gave an algorithm for equations of second order. The algorithm presented here works for equations of arbitrary order, and will find all such hyperexponential solutions that may exist. It relies on determining the structure of such first order factors to construct an ansatz of a solution, which can then be completely determined by solving a system of multivariate polynomial equations. The algorithm works well for solutions having few singularities and hidden poles, but can slow as the number of such points increases.
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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) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".