INTERMEDIATE AND LIMITING BEHAVIOR OF POWERS OF SOME CIRCULANT MATRICES
Bibliographic record
Abstract
Let A be an arbitrary circulant stochastic matrix, and let x 0 be a vector. An canonical form is derived for A k (as k !1 ) as a tensor product of three simple matrices by employing a pseudo-invariant on sections of states for a Markov process with tran- sition matrix A 0 , and by analyzing how A acts on the sections, through its auxiliary polynomial. An element-wise asymptotic characterization of A k is also given, generalizing previous results to cover both periodic and aperiodic cases. For a particular circulant stochastic matrix, identifying the intermediate stage at which fractions first appear in the sequence xk = A k x0 is accomplished by utilizing congruential matrix identities and (0,1)-matrices to determine the minimum 2-adic order of the co- ordinates of xk through their binary expansions. Throughout, results are interpreted in the context of an arbitrary weighted average repeat- edly applied simultaneously to each term of a finite sequence when read cyclically.
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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 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".