The Second Lie-Group $SO_o(n,1)$ Used to Solve Ordinary Differential Equations
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Bibliographic record
Abstract
Liu (2001) derived the first augmented Lie-group $SO_o(n,1)$ symmetry for the nonlinear ordinary differential equations (ODEs): $\dot{\bf x}={\bf f}({\bf x},t)$, and developed the corresponding group-preserving scheme (GPS). However, the earlier formulation did not consider the rotational effect of nonlinear ODEs. In this paper, we derive the second augmented Lie-group $SO_o(n,1)$ symmetry by taking the rotational effect into account. The numerical algorithm exhibits two solutions of the Lie-group ${\bf G} \in SO_o(n,1)$, depending on the sign of $\|{\bf f}\|^2 \|{\bf x}\|^2-2({\bf f}\cdot{\bf x})^2$, which means that the algorithm may be switched between two states, depending on ${\bf x}$. We give numerical examples to assess the new algorithm GPS2, which upon comparing with the GPS can raise the accuracy about three orders. It is interesting that for the chaotic system the signum function sign$(\|{\bf f}\|^2 \|{\bf x}\|^2-2({\bf f}\cdot{\bf x})^2)$ is frequently switched between $+1$ and $-1$ in time.
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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 | 0.009 | 0.018 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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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