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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.019 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.016 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".