30,929 research outputs found
On Pansiot Words Avoiding 3-Repetitions
The recently confirmed Dejean's conjecture about the threshold between
avoidable and unavoidable powers of words gave rise to interesting and
challenging problems on the structure and growth of threshold words. Over any
finite alphabet with k >= 5 letters, Pansiot words avoiding 3-repetitions form
a regular language, which is a rather small superset of the set of all
threshold words. Using cylindric and 2-dimensional words, we prove that, as k
approaches infinity, the growth rates of complexity for these regular languages
tend to the growth rate of complexity of some ternary 2-dimensional language.
The numerical estimate of this growth rate is about 1.2421.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
Binary Patterns in Binary Cube-Free Words: Avoidability and Growth
The avoidability of binary patterns by binary cube-free words is investigated
and the exact bound between unavoidable and avoidable patterns is found. All
avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the
growth rates of the avoiding languages are studied. All such languages, except
for the overlap-free language, are proved to have exponential growth. The exact
growth rates of languages avoiding minimal avoidable patterns are approximated
through computer-assisted upper bounds. Finally, a new example of a
pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons
Days 2012
A family of formulas with reversal of high avoidability index
We present an infinite family of formulas with reversal whose avoidability index is bounded between 4 and 5, and we show that several members of the family have avoidability index 5. This family is particularly interesting due to its size and the simple structure of its members. For each k ∈ {4,5}, there are several previously known avoidable formulas (without reversal) of avoidability index k, but they are small in number and they all have rather complex structure.http://dx.doi.org/10.1142/S021819671750024
Avoidability of formulas with two variables
In combinatorics on words, a word over an alphabet is said to
avoid a pattern over an alphabet of variables if there is no
factor of such that where is a
non-erasing morphism. A pattern is said to be -avoidable if there exists
an infinite word over a -letter alphabet that avoids . We consider the
patterns such that at most two variables appear at least twice, or
equivalently, the formulas with at most two variables. For each such formula,
we determine whether it is -avoidable, and if it is -avoidable, we
determine whether it is avoided by exponentially many binary words
Why did NEP fail?
Why did NEP fail? I should like to distinguish three ways in which this question
has been answered, indicating why the third appears to me to be the most
satisfactory. In the first view, NEP was abandoned because it was inconsistent with
any further industrial development of a socialist kind, and its abandonment was
therefore a rational economic decision. In the second view, strongly reacting against
the first, NEP is seen as consistent with a wide variety of development patterns,
including the industrial development actually achieved in the inter-war Five Year
Plans. Therefore the abandonment of NEP had no strictly economic rationale, but
was an outcome of brute political struggles and the formation of the Stalinist
political system. In the third view, NEP is seen as inconsistent with the degree and
rate of industrialization actually undertaken from 1928 onwards, but contained the
possibility of alternative development patterns involving a lesser commitment to
industrial growth. In this case, the abandonment of NEP was neither simply rational
(according to the first view) nor irrational (according to the second), but was the
outcome of a political conflict over the course of Soviet economic development
Growth rate of binary words avoiding
Consider the set of those binary words with no non-empty factors of the form
. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words
grows polynomially or exponentially with length. In this paper, we demonstrate
the existence of upper and lower bounds on the number of such words of length
, where each of these bounds is asymptotically equivalent to a (different)
function of the form , where , are constants
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