We present a novel approach for solving systems of polynomialequations via graph partitioning. The concept of a variable-sharing graph of asystem of polynomial equations is defined. If such graph is disconnected, thenthe system of equations is actually two separate systems that can be solvedindividually. This can provide a significant speed-up in computing the solutionto the...

Polynomial-in-time dependent symmetries are analysed for polynomial-in-time dependent evolution equations. Graded Lie algebras, especially Virasoro algebras, are used to construct nonlinear variable-coefficient evolution equations, both in 1+1 dimensions and in 2+1 dimensions, which possess higher-degree polynomial-in-time dependent symmetries. The theory also provides a kind of new realisation...

In this paper, methods of mechanical deriving of formulas from a set of differential polynomial equations and a set of differential polynomial inequations are presented. The methods have been used successfully to many problems in space curve theory and mechanics. In particular, a mechanical derivation of Newton’s gravitational law from Kepler’s laws has been given without knowing Newton’s laws ...

We have discussed general methods for solving arbitrary equations, and looked at the special class of polynomial equations. A subclass of the latter comprises all the systems of linear equations to which the area of linear algebra is devoted. In fact, many a problem in numerical analysis can be reduced to one of solving a system of linear equations. We already witnessed this in the use of Newto...