A characterization result for equationally de nable classes of certain coalgebras (including basic hidden algebra) shows that a class of coalgebras is de nable by equations if and only if it is closed under coproducts, quotients, sources of morphisms and representative inclusions. The notions of equation and satisfaction are axiomatized in order to include the di erent approaches in the literat...

We present an explicit description of the 'fine group gradings' (i.e. group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras sl(4, C), sp(4, C), and o(4, C). All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed. The inclusions sl(4, C) ⊃ sp(4, C) ⊃ o(4, C) are an important tool in our presentation...

and Applied Analysis 3 Proof. As argued in [23], the solution of cDp(cDq + λ)x(t) = h(t) can be written as x (t) = ∫ t