Pseudospectral discretization of delay differential equations in <i>sun-star</i> formulation: Results and conjectures
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Bibliographic record
Abstract
In this paper we study the pseudospectral approximation of delay differential equations formulated as abstract differential equations in the $ \odot* $-space. This formalism also allows us to define rigorously the abstract variation-of-constants formula, where the $ \odot* $-shift operator plays a fundamental role. By applying the pseudospectral discretization technique we derive a system of ordinary differential equations, whose dynamics can be efficiently analyzed by existing bifurcation tools. To better understand to what extent the resulting finite-dimensional system 'mimics' the dynamics of the original infinite-dimensional one, we study the pseudospectral approximations of the $ \odot* $-shift operator and of the $ \odot* $-generator in the supremum norm, which is the natural choice for delay differential equations, when the discretization parameter increases. In this context there are still open questions. We collect the most relevant results from the literature and we present some conjectures, supported by various numerical experiments, to illustrate the behavior w.r.t. the discretization parameter and to indicate the direction of ongoing and future research.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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