173 research outputs found
On the maximal number of cubic subwords in a string
We investigate the problem of the maximum number of cubic subwords (of the
form ) in a given word. We also consider square subwords (of the form
). The problem of the maximum number of squares in a word is not well
understood. Several new results related to this problem are produced in the
paper. We consider two simple problems related to the maximum number of
subwords which are squares or which are highly repetitive; then we provide a
nontrivial estimation for the number of cubes. We show that the maximum number
of squares such that is not a primitive word (nonprimitive squares) in
a word of length is exactly , and the
maximum number of subwords of the form , for , is exactly .
In particular, the maximum number of cubes in a word is not greater than
either. Using very technical properties of occurrences of cubes, we improve
this bound significantly. We show that the maximum number of cubes in a word of
length is between and . (In particular, we improve the
lower bound from the conference version of the paper.)Comment: 14 page
More Kolakoski Sequences
Our goal in this article is to review the known properties of the mysterious
Kolakoski sequence and at the same time look at generalizations of it over
arbitrary two letter alphabets. Our primary focus will here be the case where
one of the letters is odd while the other is even, since in the other cases the
sequences in question can be rewritten as (well-known) primitive substitution
sequences. We will look at word and letter frequencies, squares, palindromes
and complexity.Comment: 17 pages, 3 tables, 1 figur
Quadratic automaton algebras and intermediate growth
We present an example of a quadratic algebra given by three generators and
three relations, which is automaton (the set of normal words forms a regular
language) and such that its ideal of relations does not possess a finite
Gr\"obner basis with respect to any choice of generators and any choice of a
well-ordering of monomials compatible with multiplication. This answers a
question of Ufnarovski.
Another result is a simple example (4 generators and 7 relations) of a
quadratic algebra of intermediate growth.Comment: To appear in Journal of Cobinatorial Algebr
Hyperbolic surface subgroups of one-ended doubles of free groups
Gromov asked whether every one-ended word-hyperbolic group contains a
hyperbolic surface group. We prove that every one-ended double of a free group
has a hyperbolic surface subgroup if (1) the free group has rank two, or (2)
every generator is used the same number of times in the amalgamating words. To
prove this, we formulate a stronger statement on Whitehead graphs and prove its
specialization by combinatorial induction for (1) and the characterization of
perfect matching polytopes by Edmonds for (2).Comment: 30 pages, 8 figures. This version has been accepted for publication
by the Journal of Topolog
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