In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = （F1, F2） with orders {k1, k2} （k1≥ k2） preserves the invariant subspace Wn1^1× Wn2^2 （n1 ≥ n2）, then n1 - n2 ≤ k2, n1 ≤2（k1 ＋ k2） ＋ 1, where Wnq^q is the space generated by solutions of a linear ordinary differential equation of order nq （q = 1, 2）. Several examples including the （1＋1）-dimensional diffusion system and Ito＇s type, Drinfel＇d-Sokolov-Wilson＇s type and Whitham-Broer-Kaup＇s type equations are presented to illustrate the result. Furthermore, the estimate of dimension for m-component nonlinear systems is also given.
Differential-difference equations of the form un = Fn（t, un-1,Un,Unn＋1,Un-1,un,Un＋1） are clas- sifted according to their intrinsic Lie point symmetries, equivalence group and some low-dimensional Lie algebras including the Abelian symmetry algebras, nilpotent nonAbelian symmetry algebras, solvable symmetry algebras with nonAbelian nilradicals, solvable symmetry algebras with Abelian nilradicals and nonsolvable symmetry al- gebras. Here Fn is a nonlinear function of its arguments and the dot over u denotes differentiation with respect to t.
We prove some sharp Hardy inequality associated with the gradient △γ=（△x,｜x｜γ [x△y） by a direct and simple approach. Moreover, similar method is applied to ob- tain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L^p. We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.
