The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial words that are sequences that may have a number of “do not know ” symbols also called “holes”. This paper is devoted to a fundamental result on periods of words, the Critical Factorization Theorem, which states that the period of a word is always locally detectable in at least one position of the word resulting in a corresponding critical factorization. Here, we describe precisely the class of partial words w with one hole for which the weak period is locally detectable in at least one position of w. Our proof provides an algorithm which computes a critical factorization when one exists. A World Wide Web server interface at

The border correlation function β: A ∗ → A ∗ , for A = {a, b}, specifies which conjugates (cyclic shifts) of a given word w of length n are bordered, in other words, β(w) = c0c1... cn−1, where ci = a or b according to whether the i-th cyclic shift σ i (w) of w is unbordered or bordered. Except for some special cases, no binary word w has two consecutive unbordered conjugates (σ i (w) and σ i+1 (w)). We show that this is optimal: in every cyclically overlap-free word every other conjugate is unbordered. We also study the relationship between unbordered conjugates and critical points, as well as, the dynamic system given by iterating the function β. We prove that, for each word w of length n, the sequence w, β(w), β 2 (w),... terminates either in b n or in the cycle of conjugates of the word ab k ab k+1 for n = 2k + 3.

We give a short proof of a result by C. M. Weinbaum [Proc. AMS, 109(3):615– 619, 1990] stating that each a primitive word of length at least 2 has a conjugate w ′ = uv such that both u and v have a unique position in the cyclic word of w.