1,387 research outputs found
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
The Ring of Quasimodular Forms for a Cocompact Group
We describe the additive structure of the graded ring of
quasimodular forms over any discrete and cocompact group \Gamma \subset
\rm{PSL}(2, \RM). We show that this ring is never finitely generated. We
calculate the exact number of new generators in each weight . This number is
constant for sufficiently large and equals \dim_{\CM}(I / I \cap
\widetilde{I}^2), where and are the ideals of modular
forms and quasimodular forms, respectively, of positive weight. We show that
is contained in some finitely generated ring
of meromorphic quasimodular forms with i.e. the same order of growth as Comment: 22 pages, 1 figur
Inverse problems of symbolic dynamics
This paper reviews some results regarding symbolic dynamics, correspondence
between languages of dynamical systems and combinatorics. Sturmian sequences
provide a pattern for investigation of one-dimensional systems, in particular
interval exchange transformation. Rauzy graphs language can express many
important combinatorial and some dynamical properties. In this case
combinatorial properties are considered as being generated by substitutional
system, and dynamical properties are considered as criteria of superword being
generated by interval exchange transformation. As a consequence, one can get a
morphic word appearing in interval exchange transformation such that
frequencies of letters are algebraic numbers of an arbitrary degree.
Concerning multydimensional systems, our main result is the following. Let
P(n) be a polynomial, having an irrational coefficient of the highest degree. A
word (w=(w_n), n\in \nit) consists of a sequence of first binary numbers
of i.e. . Denote the number of different subwords
of of length by .
\medskip {\bf Theorem.} {\it There exists a polynomial , depending only
on the power of the polynomial , such that for sufficiently
great .
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
Siciak-Zahariuta extremal functions, analytic discs and polynomial hulls
We prove two disc formulas for the Siciak-Zahariuta extremal function of an
arbitrary open subset of complex affine space. We use these formulas to
characterize the polynomial hull of an arbitrary compact subset of complex
affine space in terms of analytic discs. Similar results in previous work of
ours required the subsets to be connected
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced
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