2,559 research outputs found
Entropy sensitivity of languages defined by infinite automata, via Markov chains with forbidden transitions
A language L over a finite alphabet is growth-sensitive (or entropy
sensitive) if forbidding any set of subwords F yields a sub-language L^F whose
exponential growth rate (entropy) is smaller than that of L. Let (X, E, l) be
an infinite, oriented, labelled graph. Considering the graph as an (infinite)
automaton, we associate with any pair of vertices x,y in X the language
consisting of all words that can be read as the labels along some path from x
to y. Under suitable, general assumptions we prove that these languages are
growth-sensitive. This is based on using Markov chains with forbidden
transitions.Comment: to appear in Theoretical Computer Science, 201
The prime spectrum of algebras of quadratic growth
We study prime algebras of quadratic growth. Our first result is that if
is a prime monomial algebra of quadratic growth then has finitely many
prime ideals such that has GK dimension one. This shows that prime
monomial algebras of quadratic growth have bounded matrix images. We next show
that a prime graded algebra of quadratic growth has the property that the
intersection of the nonzero prime ideals such that has GK dimension 2
is non-empty, provided there is at least one such ideal. From this we conclude
that a prime monomial algebra of quadratic growth is either primitive or has
nonzero locally nilpotent Jacobson radical. Finally, we show that there exists
a prime monomial algebra of GK dimension two with unbounded matrix images
and thus the quadratic growth hypothesis is necessary to conclude that there
are only finitely many prime ideals such that has GK dimension 1.Comment: 23 page
The word problem for some uncountable groups given by countable words
We investigate the fundamental group of Griffiths' space, and the first
singular homology group of this space and of the Hawaiian Earring by using
(countable) reduced tame words. We prove that two such words represent the same
element in the corresponding group if and only if they can be carried to the
same tame word by a finite number of word transformations from a given list.
This enables us to construct elements with special properties in these groups.
By applying this method we prove that the two homology groups contain
uncountably many different elements that can be represented by infinite
concatenations of countably many commutators of loops. As another application
we give a short proof that these homology groups contain the direct sum of
2^{\aleph_0} copies of \mathbb{Q}. Finally, we show that the fundamental group
of Griffith's space contains \mathbb{Q}.Comment: 24 pages, 7 figure
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