Combinatorial considerations for the number of distinct eigenvalues of a matrix
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Bibliographic record
Abstract
We address the inverse eigenvalue problem of determining the potential number of distinct eigenvalues of a real matrix based on the zero-nonzero structure of the matrix. In particular, a nonzero pattern $\mathcal{A}$ is a matrix with entries in $\{*,0\}$. The allow sequence of distinct eigenvalues for an $n\times n$ pattern $\mathcal{A}$ is a binary vector of length $n$ with the $k$th entry equal to $1$ if and only if there exists a real matrix with pattern $\mathcal{A}$ having exactly $k$ distinct eigenvalues. We develop digraph techniques for identifying properties of the allow sequence and give some general results for cycle patterns. We obtain a classification for all the star patterns according to their allow sequence. We also determine the allow sequence for each $n\times n$ irreducible pattern with $n\leq 4$.
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it