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双语推荐:迹恒等式

应用牛顿恒等式,得到矩阵的特征值的对称多项式与等幂和之间的关系,以此为基础给出行列式的表示,另由克莱姆法则导出的行列式表示.
The relationship between symmetric polynomial and power sum polynomial of matrix eigenvalues is obtained from Newton identities .The trace representation of determinant is therefore followed .On the other hand ,the determinant representation of trace is deduced from Cramer rule .

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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.

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给出一个广义 WKI 谱的负梯队,其中包含了一些已知的无色散可积系统。基于迹恒等式,构造了该负孤子梯队的广义双哈密顿结构。应用达布阵方法和符号计算技巧,建立了该负孤子梯队的达布变换。
A new negative order generalized WKI hierarchy is presented.It contains some well known coupled integrable dispersionless systems as special members.Based on the trace identity approach,bi-Hamiltonian struc-ture of the negative order generalized WKI hierarchy is proposed.With the aid of the Darboux matrix method and symbolic computation,a Darboux transformation for this soliton hierarchy is established.

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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.

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基于一类新的Lie超代数,介绍了构造超孤子族非线性可积耦合的一般方法.由相应圈超代数上的超迹恒等式,可以得到超孤子族非线性可积偶的超哈密顿结构.作为应用,给出了超Kaup-Newell族的非线性可积耦合及其超哈密顿结构,这种方法还可以推广到其他的超孤子族.
10.7498/aps.62.120202

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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.

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在讨论组合恒等式时通常采用组合方法,如取系数法、Riordan阵法等,作为基本研究工具。在本文中我们运用比较少见的一种方法来研究了一些特殊组合序列的恒等式。特别的,我们根据前人得出的一些组合序列的概率表达式,利用数学期望的性质、二项式恒等式以及多项式恒等式等方法,得到了有关两类Stirling数、二项式系数倒数、调和数、Bell数以及错排数的一些新的恒等式
In many cases we can use combinatorial methods, e.g., the coe?cient method and the Riordan arrays etc., to prove combinatorial identities. In this paper, by making use of some unusual techniques, we achieve some specific combinatorial identities. Specifically, we derive some new identities involving two kinds of Stirling numbers, reciprocal of binomial coe?cient, harmonic numbers, Bell numbers and the number of derangements, by utilizing given probability expressions, properties of mathematical expectation, binomial identities and polynomial identities.

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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.

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Szigeti-Tuza-Revesz使用Swan图论定理构造了Mn(F)的欧拉恒等式.该文证明这些恒等式可用简单方法由Amitsur-Levitzki定理得到.特别地,用这一方法还可得到Chang[3]、Giambruno-Sehgal[4]关于Mn(F)的多项式恒等式.
Using Swan''s graph theoretic theorem , Szigeti-Tuza-Revesz constructed Eulerian identities on Mn (F) .We showed it could be derived from the Amitsur-Levitzki theorem using a very simple way ,As a special case of this method .We obtained Chang [2]-Giambruno-Sehgal[3] polynomial identities on Mn (F) .

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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 .

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