The embedding problem with kernel PSL (2, p 2)

The embedding problem of number fields is considered. It is proved that the problem is solvable if and only if all associated local problems corresponding to infinite points are solvable. It is also proved that the solvability of the adjoined problem with Sylow 2-group implies the solvability of the original problem. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 135–145.     

A max version of the generalized spectral radius theorem

Let Ψ be a bounded set of n × n nonnegative matrices in max algebra. In this paper we propose the notions of the max algebra version of the generalized spectral radius μ(Ψ) of Ψ, and the max algebra version of the joint spectral radius η(Ψ) of Ψ. The max algebra version of the generalized spectral radius theorem μ(Ψ) = η(Ψ) is established. We propose the relationship between the generalized spectral radius ρ(Ψ) of Ψ (in the sense of Daubechies and Lagarias) and its max algebra version μ(Ψ). Moreover, a generalization of Elsner and van den Driessche’s lemma is presented as well.