94 research outputs found
On growth and fluctuation of k-abelian complexity
An extension of abelian complexity, so called k-abelian complexity, has been considered recently in a number of articles. This paper considers two particular aspects of this extension: First, how much the complexity can increase when moving from a level k to the next one. Second, how much the complexity of a given word can fluctuate. For both questions we give optimal solutions. (C) 2017 Elsevier Ltd. All rights reserved
Nivat's conjecture holds for sums of two periodic configurations
Nivat's conjecture is a long-standing open combinatorial problem. It concerns
two-dimensional configurations, that is, maps where is a finite set of symbols. Such configurations are often
understood as colorings of a two-dimensional square grid. Let denote
the number of distinct block patterns occurring in a configuration
. Configurations satisfying for some
are said to have low rectangular complexity. Nivat conjectured that such
configurations are necessarily periodic.
Recently, Kari and the author showed that low complexity configurations can
be decomposed into a sum of periodic configurations. In this paper we show that
if there are at most two components, Nivat's conjecture holds. As a corollary
we obtain an alternative proof of a result of Cyr and Kra: If there exist such that , then is periodic. The
technique used in this paper combines the algebraic approach of Kari and the
author with balanced sets of Cyr and Kra.Comment: Accepted for SOFSEM 2018. This version includes an appendix with
proofs. 12 pages + references + appendi
Specular sets
We introduce the notion of specular sets which are subsets of groups called
here specular and which form a natural generalization of free groups. These
sets are an abstract generalization of the natural codings of linear
involutions. We prove several results concerning the subgroups generated by
return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352
Words with the Maximum Number of Abelian Squares
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain distinct factors that
are abelian squares. We study infinite words such that the number of abelian
square factors of length grows quadratically with .Comment: To appear in the proceedings of WORDS 201
Garside and quadratic normalisation: a survey
Starting from the seminal example of the greedy normal norm in braid monoids,
we analyse the mechanism of the normal form in a Garside monoid and explain how
it extends to the more general framework of Garside families. Extending the
viewpoint even more, we then consider general quadratic normalisation
procedures and characterise Garside normalisation among them.Comment: 30 page
Homotopy bases and finite derivation type for Schutzenberger groups of monoids
Given a finitely presented monoid and a homotopy base for the monoid, and
given an arbitrary Schutzenberger group of the monoid, the main result of this
paper gives a homotopy base, and presentation, for the Schutzenberger group. In
the case that the R-class R' of the Schutzenberger group G(H) has only finitely
many H-classes, and there is an element s of the multiplicative right pointwise
stabilizer of H, such that under the left action of the monoid on its R-classes
the intersection of the orbit of the R-class of s with the inverse orbit of R'
is finite, then finiteness of the presentation and of the homotopy base is
preserved.Comment: 24 page
Scalar Ambiguity and Freeness in Matrix Semigroups over Bounded Languages
There has been much research into freeness properties of
finitely generated matrix semigroups under various constraints, mainly
related to the dimensions of the generator matrices and the semiring over
which the matrices are defined. A recent paper has also investigated freeness
properties of matrices within a bounded language of matrices, which
are of the form M1M2 · · · Mk ⊆ F
n×n
for some semiring F [9]. Most freeness
problems have been shown to be undecidable starting from dimension
three, even for upper-triangular matrices over the natural numbers.
There are many open problems still remaining in dimension two.
We introduce a notion of freeness and ambiguity for scalar reachability
problems in matrix semigroups and bounded languages of matrices.
Scalar reachability concerns the set {ρ
TMτ |M ∈ S}, where ρ, τ ∈ F
n
are vectors and S is a finitely generated matrix semigroup. Ambiguity
and freeness problems are defined in terms of uniqueness of factorizations
leading to each scalar. We show various undecidability results
Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems
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