Eigenvalues for stochastic matrices with a prescribed stationary distribution
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Bibliographic record
Abstract
Given a vector $0<w \in \mathbb{R}^n$ whose entries sum to $1$, the region $\sigma_\mathcal{S}(w)$ in the complex plane consisting of all eigenvalues of all stochastic matrices having $w^\top$ as a left Perron vector is considered. Some general observations about this region are made, it is proven that $\bigcap_{w \in \mathbb{R}^n, w>0, w^\top \mathbf{1} =1} \sigma_\mathcal{S}(w) =[0,1],$ and a characterization is given of the vectors $w$ such that $\sigma_\mathcal{S}(w)$ contains an element $\lambda \ne 1$ with $|\lambda|=1.$ The corresponding problem for reversible stochastic matrices with given left Perron vector is also considered, as is the corresponding region $\sigma_\mathcal{R}(w),$ which is a subset of $[-1,1].$ Under a mild hypothesis on $w, $ it is proven that the smallest element of $\sigma_\mathcal{R}(w)$ corresponds to a reversible stochastic matrix whose graph is a tree with a loop at one vertex. A general lower bound on the eigenvalues of reversible stochastic matrices with given left Perron vector is also given, as is a complete description of $\sigma_\mathcal{R}(w)$ when $w$ has two or three entries.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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 it