An Improved Bound for an Extension of Fine and Wilf’s Theorem and Its Optimality
Bibliographic record
Abstract
Considering two DNA molecules which are Watson-Crick (WK) complementary to each other “equivalent” with respect to the information they encode enables us to extend the classical notions of repetition, period, and power. WK-complementarity has been modelled mathematically by an antimorphic involution θ, i.e., a function θ such that θ(xy) = θ(y)θ(x) for any x, y ∞ Σ*, and θ 2 is the identity. The WK-complementarity being thus modelled, any word which is a repetition of u and θ(u) such as uu, uθ(u)u, and uθ(u)θ(u)θ(u) can be regarded repetitive in this sense, and hence, called a θ-power of u. Taking the notion of θ-power into account, the Fine and Wilf’s theorem was extended as “given an antimorphic involution θ and words u, v, if a θ-power of u and a θ-power of v have a common prefix of length at least b(|u|, |v|) = 2|u| + |v| – gcd(|u|, |v|), then u and v are θ-powers of a same word.” In this paper, we obtain an improved bound b′(|u|, |v|) = b(|u|, |v|) – [gcd(|u|, |v|)/2]. Then we show all the cases when this bound is optimal by providing all the pairs of words (u, v) such that they are not θ-powers of a same word, but one can construct a θ-power of u and a θ-power of v whose maximal common prefix is of length equal to b′(|u|, |v|) − 1. Furthermore, we characterize such words in terms of Sturmian words.
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How this classification was reachedexpand
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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".