The Challenge of the Characteristic height of Bohemian Matrices: experiments with Maple and open problems
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Bibliographic record
Abstract
A family of Bohemian matrices is a set of matrices where the entries are independently sampled from a finite set, usually integers, of bounded height. Such families arise in many applications (e.g. compressed sensing) and the properties of matrices selected "at random" from such families are of practical and mathematical interest. Studying such matrices leads to many unanswered questions. In this paper, we state a new problem arising from the results obtained in a previous work concerning the maximal height of the characteristic polynomials of this family. We present a challenge with a computational flavour. More precisely, we describe, and illustrate by means of some experiments—executed with Maple—the problem of analyzing the behaviour of the set of heights, not only the maximal, of the characteristic polynomials of a family of upper Hessenberg Toeplitz structured Bohemian matrices.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.001 | 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