Abundances of C, N and O in Planetary Nebulae

Invariant solutions for equations of axion electrodynamics

“…Some of these solutions, e.g., (18), (21) with µ = 1 and c 2 1 + c 2 2 > 1, describe the wave propagation with the group velocity higher then the velocity of light. Moreover, these solutions are smooth and bounded functions which correspond to positive definite and bounded energy density [14]. In spite of that these solutions are causal since their energy velocity does not exceed the velocity of light [14].…”

“…It has been proven in [14] that the maximal continuous invariance group of system (2), (3) with arbitrary function F (θ) is the group P (1,3). The corresponding Lie algebra p(1,3) is spanned on the following basis elements:…”

“…Here P µ and J µν are generators given by relations (7), G 1 = J 01 −J 13 , G 2 = J 02 −J 23 , α and β are arbitrary parameters. Using subalgebras (14) we can deduce exact solutions for system (2), (3). Notice that to make an effective reduction using the Lie algorithm, we can use only such subalgebras whose basis elements satisfy conditions (5).…”

“…the 3-dimensional subalgebras of the Poincaré algebra. Basis elements of these subalgebras are given by relations (14).…”

“…We have described the finite dimensional indecomposable vector representations of the homogeneous Galilei group and construct Lagrangians which admit the corresponding symmetries [11]- [13] . The Lagrangian of axion electrodynamics appears to be the relativistic counterpart of some of these Galilei invariant Lagrangians [14].…”