726 research outputs found
Convergence of the Number of Period Sets in Strings
Consider words of length n. The set of all periods of a word of length n is a subset of {0, 1, 2, . . ., nâ1}. However, any subset of {0, 1, 2, . . ., nâ1} is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko proposed to encode the set of periods of a word into an n long binary string, called an autocorrelation, where a one at position i denotes the period i. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for strings of length n, denoted Îșn. They conjectured that ln(Îșn) asymptotically converges to a constant times ln2(n). Although improved lower bounds for ln(Îșn)/ln2(n) were proposed in 2001, the question of a tight upper bound has remained open since Guibas and Odlyzkoâs paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this longstanding conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings
Remarks on Two Nonstandard Versions of Periodicity in Words
In this paper, we study some periodicity concepts on words. First, we extend the notion of full tilings which was recently introduced by KarhumÀki, Lifshits, and Rytter to partial tilings. Second, we investigate the notion of quasiperiods and show in particular that the set of quasiperiodic words is a context-sensitive language that is not context-free, answering a conjecture by Dömösi, Horvåth and Ito
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
Clusters of repetition roots: single chains (Algebraic system, Logic, Language and Related Areas in Computer Sciences II)
This work proposes a new approach towards solving an over 20 years old conjecture regarding the maximum number of distinct squares that a word can contain. To this end we look at clusters of repetition roots, that is, the set of positions where the root u of a repetition u^[l] occurs. We lay the foundation of this theory by proving basic properties of these clusters and establishing upper bounds on the number of distinct squares when their roots form a chain with respect to the prefix order
Rhombic Tilings and Primordia Fronts of Phyllotaxis
We introduce and study properties of phyllotactic and rhombic tilings on the
cylin- der. These are discrete sets of points that generalize cylindrical
lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical
system S that models plant pattern formation by stacking disks of equal radius
on the cylinder. This system has the advantage of allowing several disks at the
same level, and thus multi-jugate config- urations. We provide partial results
toward proving that the attractor for S is entirely composed of rhombic tilings
and is a strongly normally attracting branched manifold and conjecture that
this attractor persists topologically in nearby systems. A key tool in
understanding the geometry of tilings and the dynamics of S is the concept of
pri- mordia front, which is a closed ring of tangent disks around the cylinder.
We show how fronts determine the dynamics, including transitions of parastichy
numbers, and might explain the Fibonacci number of petals often encountered in
compositae.Comment: 33 pages, 10 picture
Convergence of the number of period sets in strings
Consider words of length n. The set of all periods of a word of length n is a subset of {0,1,2,âŠ,nâ1}. However, any subset of {0,1,2,âŠ,nâ1} is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko have proposed to encode the set of periods of a word into an n long binary string, called an autocorrelation, where a one at position i denotes a period of i. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for length n, denoted Îș_n. They conjectured that ln(Îș_n) asymptotically converges to a constant times ln^2(n). If improved lower bounds for ln(Îș_n)/ln^2(n) were proposed in 2001, the question of a tight upper bound has remained opened since Guibas and Odlyzko's paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this long standing conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings
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