An analogue of the Magnus problem for associative algebras

We prove an analogue of Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by $n+k$ generators and $k$ relations and has an $n$-element system of generators, then this algebra is a free algebra of rank $n$.

The 90-degree problem of scattering theory revisited: dynamical turbulence effects versus nonlinear effects

The 90° problem of cosmic-ray transport theory is revisited in this paper. By using standard forms of the wave spectrum in the solar wind, the pitch-angle Fokker–Planck coefficient and the parallel mean free path are computed for different resonance functions. A critical comparison is made of the strength of 90° scattering due to plasmawave effects, dynamical turbulence effects and nonlinear effects. It is demonstrated that, only for low-energy cosmic particles, dynamical effects are usually dominant. The novel results presented here are essential for an effective comparison of heliospheric observations for the parallel mean free path with the theoretical model results.

The conjugacy problem for two-by-two matrices over polynomial rings

We give an effective solution of the conjugacy problem for two-by-two matrices over the polynomial ring in one variable over a finite field.