On the index of sequences over cyclic groups
Bibliographic record
Abstract
Let G be a finite cyclic group of order n 2. Every sequence S over G can be written in the form S = (n1g) · ... · (nlg) where g 2 G and n1,...,nl 2 (1,ord(g)), and the index ind(S) of S is defined as the minimum of (n1 + ... + nl)/ord(g) over all g 2 G with ord(g) = n. In this paper we prove that a sequence S over G of length |S| = n having an element with multiplicity at least n has a subsequence T with ind(T) = 1, and if the group order n is a prime, then the assumption on the multiplicity can be relaxed to n 2 10 . On the other hand, if n = 4k + 2 with k 5, we provide an example of a sequence S having length |S| > n and an element with multiplicity n 1 which has no subsequence T with ind(T) = 1. This disproves a conjecture given twenty years ago by Lemke and Kleitman.
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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 | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".