The stability of the parametric Cauchy problem of initial-value ordinary differential equations revisited
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Bibliographic record
Abstract
In this paper, given a function f : I ×V → R m , where V is an open subset of R m , x 0 ∈ V , and I = [0, T ] is the interval of interest, we consider the Cauchy ordinary differential equation initial-value problem ẋ( f , x 0 ) = f (t, x(t)), x(0) = x 0 .We first present a new quantitative stability result under a partial and/or global variation of the data of the problem by involving exact and/or approximate fixed points for which we apply Lim's Lemma either in its exact format or in its very recent approximate version.Our main result is then applied to parametric linear control systems.Finally, we demonstrate that our treatment is coherent with the management of perturbations generated in the classic one-step numerical method.A numerical example written in Scilab 6.1 illustrates the obtained stability.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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) | 0.000 | 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 it