构造了一个新的等谱问题，利用相容性条件，推导出离散晶格方程的正族和负族。再利用迹恒等式，建立其Hamilton 结构。获得的离散方程族的达布变换、双线性化、对称、守恒率及其精确解也值得进一步研究。

A new discrete spectral problem in the paper is devised, whose compatibility condition gives rise to a new positive hierarchy and a negative hierarchy of discrete integrable equations. Making use of the trace identity, their Hamiltonian structures are worked out respectively. The Darboux translation, bilinear form, symmetries, conservation laws, conserved quantities and their exact solutions of the resulting equation hierarchies are worth investigating in the future.

通过构造一个新的Lie代数,利用它相应的Loop代数设计等谱Lax对,根据其相容性条件,得到了一族Lax可积方程族,其一种约化形式为著名的AKNS族.根据迹恒等式得到该方程族的Hamilton结构.利用该可积方程族可以进一步研究它的达布变换、对称、代数几何解等相关性质.

By constructing a new Lie algebra and its corresponding Loop algebra, an isospectral Lax pair is established whose compatibility condition gives rise to a Lax integrable hierarcy, whose reduced form is the well-known AKNS hierarchy. Its Hamilton structure is obtained by the use of the trace identity. Then, its Darboux transformations, symmetry, algebro-geometric solutions, and so on will be investigated further.

在代数学中，牛顿恒等式是联系多项式根的幂和与其系数关系的一个重要恒等式。用数学归纳法给出牛顿恒等式的一个自然证明。

Newton’s identities are an important identity that links the power of polynomial root and its coefficient in algebra.This paper shows a natural proof of Newton’s identities by using mathematical induction.

推广了Coker用代数方法证明的一个组合恒等式,在此基础上得到一些与Narayana和Catalan数相关的恒等式。

One of the identities of Coker is generalized,which is proved by an algebraic method.Some useful identities related to Narayana number and Catalan number are also presented.

首先给出了环R=Fp＋vFp＋v2Fp上线性码及其对偶码的结构及其Gray象的性质.定义了环R上线性码的各种重量计数器并讨论了它们之间的关系,特别的,确定了该环上线性码与其对偶码之间关于完全重量计数器的MacWilliams恒等式,利用该恒等式,进一步建立了该环上线性码与其对偶码之间的一种对称形式的MacWilliams恒等式.最后,利用该对称形式的MacWilliams恒等式得到了该环上的Hamming重量计数器和Lee重量计数器的MacWilliams恒等式,利用不同的方法推广了文献[7]中的结果.

We first give the structures of linear codes and the proposition of their Gray images .A complete weight enumera-tor of linear codes over ring R = Fp + vFp + v2 Fp is defined ;we define some weight enumerators of linear codes and their dual codes ,and then discuss the relations between them .A MacWilliams identity between linear codes and their dual over R with respect to the complete weight enumerator is given .By using this identity ,a symmetrized form MacWilliams identitiy between linear codes and their dual over the ring is also established ,and MacWilliams identities with respect to Hamming and Lee enumerator can be as results of the symmetrized MacWilliams identitiy ,we generalize the results in[7] using different method .