基本解释：二次型的等价
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Integrable couplings of the KdV hierarchy is obtained by use of the new subalgebra of the loop algebra,then the Hamiltonian structure of the above system is given by the quadratic-form identity.利用loop代数的半直和得到KdV族的可积耦合,通过二次型恒等式得到它的哈密顿结构。
Since the trace iden-tity fails to generate Hamiltonian structures of integrable couplings,the quadratic-form identity was proposed, which is an extension ofthe trace identity.由于迹恒等式无法寻求可积耦合的Hamilton结构,所以人们提出了二次型恒等式,它是迹恒等式的推广,是寻求可积耦合的Hamilton结构的有效方法。
Further more,the Hamilton ian structure of its integrable couplings is worked out by using of the quadratic-form identity,which is of Liouville intergrable.首先构造了一个loop代数,根据(2+1)维零曲率方程计算得到(2+1)维KdV族的可积耦合,然后通过二次型恒等式得到它的哈密顿结构。
In this paper,we set the conception of quadratic form of a graph and quartic form of a graph, the necessary sufficient condition on 1-factor of graphs and 2-factor of paar graphs and obtain a counting formula for 1-factor of graphs.本文应用图的二次型与四次型的概念，得到图有１－因子、偶图有２－因子的充要条件，并且得到了图的１－因子的计数公式。
