Algebraic method of group classification for semi-normalized classes of differential equations
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Bibliographic record
Abstract
We generalize the notion of semi-normalized classes of systems of differential equations, study properties of such classes and extend the algebraic method of group classification to them. In particular, we prove the important theorems on factoring out symmetry groups and invariance algebras of systems from semi-normalized classes and on splitting such groups and algebras within disjointedly semi-normalized classes. Nontrivial particular examples of classes that arise in real-world applications and showcase the relevance of the developed theory are provided. To convincingly illustrate the efficiency of the proposed method, we apply it to the group classification problem for the class of linear Schrödinger equations with complex-valued potentials and the general value of the space dimension. We compute the equivalence groupoid of the class by the direct method and thus show that this class is uniformly semi-normalized with respect to the linear superposition of solutions. This is why the group classification problem reduces to the classification of specific low-dimensional subalgebras of the associated equivalence algebra, which is completely realized for the case of space dimension two. Splitting into different classification cases is based on three integer parameters that are invariant with respect to equivalence transformations. We also single out those of the obtained results that are relevant to linear Schrödinger equations with real-valued potentials.
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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 | 0.000 |
| 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 | 0.001 | 0.001 |
| 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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