Invariant solutions of the equation of motion for a sphere composed of Blatz‐Ko material
Bibliographic record
Abstract
Abstract The nonlinear partial differential equation governing the spherically symmetric dynamic deformations of Blatz‐Ko materials is studied. The Lie groups and three invariant solutions of the governing equation are derived. Two of the invariant solutions are in separable form. The governing equation of the invariant solution in non‐separable form is a singular second order non‐autonomous ordinary differential equation (ODE). We transform this ODE to an autonomous ODE successfully and analyze its trajectories in the phase plane. By doing this we can capture the global behaviors of the solutions for the ODE. Especially, we can figure out what kind of problems will have singularity. The invariant solution in non‐separable form studied in this paper is interesting because it shows that when localization or blowing up of some physical quantities occurs close to the outer boundary of a Blatz‐Ko sphere the sphere may still look normal, that is, the distribution of the radial deformation as well as the boundary data still vary moderately. Therefore this special invariant solution gives us new insights about the behaviors of Blatz‐Ko materials.
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How this classification was reachedexpand
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) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".