Projective geometry in robust stabilization problems Part I: Projective lines
Bibliographic record
Abstract
This paper aims to highlight connections between projective geometry and stabilization problems. Within the fractional representation approach, we introduce the definition of the projective line P(A) over a ring A of proper and stable transfer functions and the definition of the projective line P(K) over the quotient field K of A. We show that the groups of homographies of these projective lines correspond to the Möbius transformations defined over A or K. We generalize the definitions of a well-posed system and internal stabilizability to consider plants defined over P(K). The vanishing of the denominator of a plant or controller is no longer considered a pathological case, and the Youla-Kučera parameterization of all stabilizing controllers is always well-defined. Finally, we show that the points of P(A) can be interpreted as transfer functions with coprime factorizations. Concepts of projective geometry, such as distant relation and distant graph on P(A), are introduced, and their system-theoretic interpretations are given.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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 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".