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Studying Word Equations by a Method of Weighted Frequencies

Saarela A.

Fundamenta Informaticae

2018

Vol. 162, nr 2/3

223--235

We briefly survey some results and open problems on word equations, especially on those equations where the right-hand side is a power of a variable. We discuss a method that was recently used to prove one of the results, and we prove improved versions of some lemmas that are related to the method and can be used as tools when studying word equations. We use the method and the tools to give new, simple proofs for several old results.

Herz-Schur multipliers and non-uniformly bounded representations of locally compact groups

Steenstrup T.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

213--223

Let G be a second countable, locally compact group and let φ be a continuous Herz-Schur multiplier on G. Our main result gives the existence of a (not necessarily uniformly bounded) strongly continuous representation π of G on a Hilbert space ℋ , together with vectors ξ, η ϵ ℋ, such that φ(y−1x) = ⟨π(x)ξ, π(y−1) * η⟩ for x, y ϵ G and supx ϵ G ∥π(x)ξ supy ϵ G ∥π(y−1) * η∥ = ∥φ∥ M0A(G). Moreover, we obtain control over the growth of the representation in the sense that ∥π(g)∥≤ exp ( c/2 d(g, e)) for g ϵ G, where e ϵ G is the identity element, c is a constant, and d is a metric on G.

Context-Free and E0L Derivations over Free Groups

Meduna A., Bildo R., Blatny' P.

Schedae Informaticae

2007

Vol. 16

9-20

In the context-free and E0L grammars discussed in this pa- per, the derivations are introduced over free groups rather than free monoids. It is proved that both grammars with derivations introduced in this way characterize the family of recursively enumerable languages in a very succinct way. Specifically, this characterization is based on the eight-nonterminal context-free grammars and six-nonterminal E0L grammars over free groups.