A necessary condition for path-finding by the homotopy continuation method
Bibliographic record
Abstract
This paper is concerned with a path-finding method proposed by H. J. Sussmann, known as the homotopy continuation method (HCM). Given a control system initialized at some state, the HCM can be used to find an admissible open-loop control that performs a desired state transfer. This is accomplished by lifting curves in the state space of the system to curves in the space of regular admissible controls. The lifted curves are constructed as solutions of an ordinary differential equation called the path-lifting equation (PLE). The obstruction to applying the HCM, in general, is that solutions of the PLE may not be globally defined. We show that if this obstruction does not exist, then the endpoint map of the control system, restricted to the space of regular admissible controls, is necessarily a locally trivial fiber bundle. This result is valid for wide classes of control systems and controls.
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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".