We have studied the effect of time-dependent ionization and recombination processes on magnetic reconnection in the solar corona. Petschek-type steady reconnection, in which model the magnetic energy is mainly converted at the slow-mode shocks, was a
ssumed. We carried out the time-dependent ionization calculation in the magnetic reconnection structure. We only calculated the transient ionization of iron; the other species were assumed to be in ionization equilibrium. The intensity of line emissions at specific wavelengths were also calculated for comparison with {it Hinode} or other observations in future. What we found is as follows: (1) iron is mostly in non-equilibrium ionization in the reconnection region, (2) the intensity of line emission estimated by the time-dependent ionization calculation is significantly different from that with the ionization equilibrium assumption, (3) the effect of time-dependent ionization is sensitive to the electron density in the case that the electron density is less than $10^{10}$ cm$^{-3}$, (4) the effect of thermal conduction lessens the time-dependent ionization effect, (5) the effect of radiative cooling is negligibly small even if we take into account time-dependent ionization.
It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic.
As an instance of a linear action of a Hopf algebra on a free associative algebra, we consider finite group gradings of a free algebra induced by gradings on the space spanned by the free generators. The homogeneous component corresponding to the ide
ntity of the group is a free subalgebra which is graded by the usual degree. We look into its Hilbert series and prove that it is a rational function by giving an explicit formula. As an application, we show that, under suitable conditions, this subalgebra is finitely generated if and only if the grading on the base vector space is trivial.
