Critical sets of inertias for matrix patterns
Bibliographic record
Abstract
An n by n zero-nonzero (resp. sign) pattern is a matrix with entries in {*, 0} (resp. {+, −, 0}), where * denotes a nonzero real number. If allows all (n + 1)(n + 2)/2 possible inertias, then is inertially arbitrary. A nonempty, proper subset S of the set of all possible inertias is a critical set of inertias of order n provided an n by n pattern is inertially arbitrary whenever allows all inertias in S, and S is a minimal critical set if no proper subset of S is a critical set. This new concept of critical sets of inertias facilitates the identification of inertially arbitrary patterns. Minimal critical sets of inertias for irreducible zero-nonzero patterns of orders n = 2, 3, 4 are identified. Using these critical sets of inertias, it is shown that a direct sum of two non-inertially-arbitrary irreducible zero-nonzero patterns of orders ≤ 4 is not inertially arbitrary. However, it is shown that there exists a reducible inertially arbitrary zero-nonzero pattern that is a direct sum of non-inertially-arbitrary irreducible zero-nonzero patterns of order 5. Minimal critical sets of inertias for irreducible sign patterns of orders n = 2, 3 are also identified. It is shown that in general critical sets of inertias for irreducible sign patterns are different from those of irreducible zero-nonzero patterns.
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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 | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".