The problem whether the set of all equations that are satisfiable in some free semigroup or, equivalently, in an algebra of words with concatenation is recursive (usually called the satisfiability problem for semigroup equations) was first formulated by A.A. Markov in early sixties (see [3]). Special cases of the problem were solved affirmatively by A.A. Markov (see [3]), Yu.I. Khmelevskiı̆ [8],...

We present a method, based on series expansions and symmetric polynomials, by which a mean of two variables can be extended to several variables. We apply it mainly to the logarithmic mean.

Small world models are networks consisting of many local links and fewer long range ‘shortcuts’. In this paper, we consider some particular instances, and rigorously investigate the distribution of their inter–point network distances. Our results are framed in terms of approximations, whose accuracy increases with the size of the network. We also give some insight into how the reduction in typi...