This paper presents a standardized natural gradient ICA(Independent Component Analysis) algorithm through improving the natural gradient ICA algorithm based on Lie Group Invariance.

Based on the assumptions of semi logarithmic relationship between coefficient of permeability and void ratio as well as the relationship between effective stress and void ratio of soil, the method of Lie Group Transformation is applied to solve the non linear partial differential equation of large strain consolidation of homogeneous saturated clay in semi infinite domain.

In the article we advance the concept of prolongation group of Lie transformation group in a visual and pithy way, and resolve the coefficient problem of prolongation operator.

It is discovered that based on the prolongation group concept of Lie transformation group in a visual and pithy way,the resolved coefficient problem of the prolongation operator is used as a lemma by the fiber bundle method.

In this paper,the symmetries and Lie algebra of the Kdv-Burgers equation were discussed,and the symmetry reductions were applied to get some group-invariant solutions of the KdV-Burgers equation.

The Symmetry and Group-invariant solutions are discussed for the following KdV equation with distributed delay ut=uxxx+6(f*u)ux,where f is a delay kernel function.

This paper considers the symmetries and Lie algebra ot the Coupled KdVequations,and uses symmetry reductions to get some group-invariant solutions of the Couples KdV equations.

This paper considers the symmetries and Lie algebra of Collapse equation, uses symmetries to obtain one-parameter invariant groups, and utilizes symmetry reductions to give some group invariant solutions of the Collapse equation.