. G R ] 1 7 Ju n 20 06 THE WORD PROBLEM DISTINGUISHES COUNTER LANGUAGES

The Word Problem Distinguishes Counter Languages

Counter automata are more powerful versions of finitestate automata where addition and subtraction operations are permitted on a set of n integer registers, called counters. We show that the word problem of Z is accepted by a nondeterministic m-counter automaton if and only if m ≥ n.

ar X iv : m at h / 06 06 67 1 v 1 [ m at h . A C ] 2 7 Ju n 20 06 PRÜFER ⋆ – MULTIPLICATION DOMAINS AND ⋆ – COHERENCE

G-automata, Counter Languages and the Chomsky Hierarchy

We consider how the languages of G-automata compare with other formal language classes. We prove that if the word problem of G is accepted by a machine in the class M then the language of any G-automaton is in the class M. It follows that the so called counter languages (languages of Zn-automata) are context-sensitive, and further that counter languages are indexed if and only if the word probl...

ar X iv : m at h / 06 06 67 7 v 1 [ m at h . D S ] 2 7 Ju n 20 06 COMPLEX MAPS WITHOUT INVARIANT DENSITIES

We consider complex polynomials f (z) = z ℓ + c1 for ℓ ∈ 2N and c1 ∈ R, and find some combinatorial types and values of ℓ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when ℓ sufficiently large and also for a class of 'long–branched' maps of any critical order.

ar X iv : m at h / 06 06 33 9 v 1 [ m at h . SP ] 1 4 Ju n 20 06 Eigenfunction expansions associated with 1 d periodic differential operators of order 2 n

We prove an explicit formula for the spectral expansions in L(R) generated by selfadjoint differential operators (−1) d dx2n + n−1