Bibliographic record
Abstract
Most nonlinear systems cannot be solved exactly so we must resort to a variety of approaches in order to obtain an approximate solution. Where applicable, the phase plane portrait can serve as a valuable tool for qualitatively determining the types of possible solutions before resorting to numerical or (usually approximate) analytical methods for specific initial conditions. In this chapter, the concept of phase plane analysis will be examined in some depth, not for a specific problem, but for a wide class of physical problems described by the following system of first order equations: 4.1 $$ \frac{{dx}}{{dt}} = P(x,y),\quad \frac{{dy}}{{dt}} = Q(x,y), $$ where P, Q are, in general, nonlinear functions of x and y and the independent variable has been taken here to be time t. In the laser competition equations (2.32), t would, of course, be replaced with z,the spatial coordinate. The mathematician would refer to this set of equations as being autonomous, meaning that P and Q do not explicitly depend on t. Why it is desirable to restrict the discussion for the moment to autonomous equations will become readily apparent1.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".