There exist exactly four homomorphisms ϕ from the pseudo-orthogonal group of index one G = O(n, 1, R) into the group of real numbers R0. Thus we have four G-spaces of ϕ-scalars (R, G, hϕ) in the geometry of the group G. The group G operates also on the sphere S n−2 forming a G-space of isotropic directions (S n−2 , G, ∗). In this note, we have solved the functional equation F(A∗q1, A∗q2, . . . , A∗qm) = ϕ(A)·F(q1, q2, . . . , qm) for given independent points q1, q2, . . . , qm ∈ S n−2 with 1 ≤ m ≤ n and an arbitrary matrix A ∈ G considering each of all four homomorphisms. Thereby we have determined all equivariant mappings F : (S n−2 ) m → R.
The paper deals with the design of stator channels of the aerator using CFD code ANSYS Fluent. The main problem is to design proper inclination of the channels corresponding with the direction of flow at the impeller outlet. The direction of flow is variable along the channel and represented by the absolute velocity angle. Therefore, this angle is computed first, and according to it, the inclination of stator channels is designed. Numerical simulations are made as single-phase flow for two different shapes of channels and for two different channel incllnations - for already computed ones and for ones used in older type of aerator which this work develops. Stator channels inclined by computed angle that corresponded with the direction of flow had the best results. On the other hand, the channels inclined by the same angle as the channels of older aerator had the worst efficiency. The decrease of aerator efficiency was caused by the large vortexes in the stator channels. and Obsahuje seznam literatury a názvosloví