恒等式

In addition, by utilizing Jacobi two square numbers theorem and Lagrange four square numbers theorem and some theta function identities, we also prove the known results of number theory: two triangular number theorem, four triangular number theorem, and the number of representations of a positive integer by various quadratic forms in terms of divisor functions

包括Jacobi二平方数定理，Lagrange四平方数定理等，然后利用这些结果结合几个theta函数恒等式，我们获得了把任意一个正整数表示成两个三角数或四个三角数的和以及其他的二次形式的方法数，这些方法数都是用因子函数来表示的。

In chapter 1, we review the background and the historical facts of variational identity.

主要有以下几方面的内容：在第一章中回顾了变分恒等式的研究背景及发展状况。

In view of the identity, the sufficient conditions of nonexistence of weak solution for two classes of quasilinear probles are obtained.

利用该恒等式，得到了两类拟线性问题弱解不存在的充分性条件。

The structural form of the model consists of 66 simultaneous equations, including 28 stochastic equations and 38 definitional equations and identities.

模型的结构型由66个方程联立组成，其中随机方程28个，恒等式和定义式38个。

In 1993, Yen in her doctoral dissertation [45] gave the first a priori estimate of the number for hypergeometric identities, it is extremely large.

1993年，Yen在她的博士论文中第一次对超几何恒等式给出了这个项数的一个先验估计，但是非常大。

In this thesis, we focus on this question for q-hypergeometric identities.In 1993, Yen in her doctoral dissertation [45] gave the first a priori estimate of the number for hypergeometric identities, it is extremely large.

1993年，Yen在她的博士论文(来源：Ab6BCd0论2d文网www.abclunwen.com)中第一次对超几何恒等式给出了这个项数的一个先验估计，但是非常大。

When Zeilberger studied Sister Celine's Method in the late 1970s, he realized that one can prove a hypergeometric identity by checking a finite number of special cases. In the book "A=B" [37, p. 70], it is pointed that it is an interesting research question to ask how small we can make this a priori estimate of the number.

当Zeilberger在70年代后期研究Sister Celine方法的时候，他意识到一个超几何恒等式可以通过验证有限项来证明，在书《A=B》中进一步指出，研究这个项数的先验估计到底能达到多么小是一个有趣的问题。

When Zeilberger studied Sister Celine\'s Method in the late 1970s, he realized that one can prove a hypergeometric identity by checking a finite number of special cases. In the book "A=B" 37, p.

当Zeilberger在70年代后期研究Sister Celine方法的时候，他意识到一个超几何恒等式可以通过验证有限项来证明，在书《A=B》中进一步指出，研究这个项数的先验估计到底能达到多么小是一个有趣的问题。

Thirdly, a multi-component form of equation(8)in reference[40]is worked out, then its Hamiltonian structure is presented byuse of the extended trace identity. Nextly, the expanding integrable model ofmulti-component integrable equation hierarchy is obtianed.

最后，将文献[40]中Liouville可积的方程族(8)式扩为多分量的形式，然后利用扩展的迹恒等式得到了该多分量方程族的HamiIton结构，接下来求出了多分量方程族的一类扩展可积模型。

In this thesis, we study some aspects in analytic combinatorics. The main contents can be summarized as follows. In Chapter 2, by classical umbral method, we obtain some properties and identities of invariant sequences. More generally, we study the self-inverse sequences related to sequences of polynomials of binomial type and the self-inverse sequences related to Sheffer sets, and give some interesting results of these sequences.

本文研究组合分析学中的若干问题，具体内容如下：第二章用经典哑运算方法研究了不变序列的性质及满足的恒等式，随后又推广到更一般的情形，即关于二项式型多项式序列的自反序列和关于Sheffer序列的自反序列。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。