Continuous and Discontinuous Piecewise Linear Solutions of the Linearly Forced Inviscid Burgers Equation
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Bibliographic record
Abstract
We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term.Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints.We derive and solve ODEs governing the soliton dynamics, first for continuous solutions, and then for more general shock wave solutions with discontinuities.We show that triple collisions of solitons cannot take place for continuous solutions, but give an example of a triple collision in the presence of a shock.which makes sense for a much larger class of functions than just u C 1 (R).For example, if u L 2 loc (R) we can interpret (1.2) to hold in the sense of distributions.One could work with distributions acting on test functions (x,t) D(R 2 ), but the following simpler interpretation is
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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 | 0.000 | 0.000 |
| 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)
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