等价方程

Secondly, we define a monotone operator and prove that the orthogonal projection is the solution for the monotone operator equation.

其次定义了一个单调算子，利用单调算子的理论，证明了求解这个正交投影等价于求解一个单调算子方程。

Bifurcation behaviour of nonlinear systems under the resonance condition isstudied when there are parametric and external excitations at the same time.

本文提出的映射迭代法可以有效地用于强非线性系统的定性分析，且对于弱非线性系统，其一次近似的平均方程与其它方法等价。

This paper focuses on four important aspects as follows. The first one is the establishment of variational quations which provides the mathematics theories for the finite element method of HMT; the second one is for the study of scalar/node and vector/edge finite element method; the third one is about how to solve faster for the large-sacled matrix system of scalar/node and vector/edge finite element; and the last one is achieving the forward modeling with high precision of HMT method.

主要包括四部分：建立与高频大地电磁法边值问题相等价的变分问题，为有限元数值计算奠定数理基础；标量/矢量有限元方法的研究；加速标量/矢量有限元矩阵系统方程迭代求解技术的研究；实现高频大地电磁参数精确快速正演模拟的研究。

By theory says, the variational principle and the Schrodinger equation are equal.

从原理上讲，变分原理与薛定谔方程等价。

Because the items be introduced,difference of composite with imperfect interface from one with perfect interface is that stationary value principles of the former is no longer equivalent of variation equation s corresponding the principles.

与完美界面不同的是，由于这个积分项的引入，非完美界面复合材料的驻值原理不再与相应的变分方程等价。

The system with the Bouc-Wen hysteretic force is converted into an equivalent non-hysteretic nonlinear stochastic control system. Then an It stochastic differential equation for total energy is derived by using the stochastic averaging method.

先将该系统变换成等价的非迟滞的非线性随机控制系统，再运用随机平均法导出关於能量的It随机微分方程。

This paper has obtained following three results about the new scheme:(1) The hardness of recovering the private key from the public key is based on the hardness of the shortest vector problems of several lattices;(2) The hardness of forging a signature is equivalent to the hardness of the closest vector problem of some lattice;(3) Each signature will leak information on the private key,but the final shape of the unlimited leakage is just a group of complicated non-linear equations.

文中给出新方案的3个结果：(1)由公钥恢复出私钥的困难性基于若干格上的最小向量问题；(2)由公钥伪造签名的困难性等价于某个格上的最近向量问题；(3)每个签名值仍然会泄露私钥的一些信息，但无限制泄露的最终形式只是关于私钥的一组复杂的非线性方程。

The obtained results indicate that in the case of Riemann-Liouville definition, theequivalence between FDEs and weak formulation does not require any initial conditions.

上述研究结果表明在Riemann-Liouville分数阶导数定义的情况下，分数阶扩散方程与弱形式的等价性证明不需要添加初值条件。

In Chapter Six,we translate the equation of the iterative step of nonsmooth Newton-type method for nonlinear SDP into the conversional linear equations by the tensor technique. By using the equivalency conditions in Chapter Five, we establish the local convergence of this method.

在第六章中，我们通过张量技术把非线性半定规划问题的非光滑牛顿型法迭代步方程转化为一般的线性方程组，并利用第五章中的等价性结论分析了这种方法的局部二次收敛性。

In addition, we improve Nivens method and our new method to solve the equation more quickly.2 Complete Solution Structure: We obtain that the whole solutions of real quaternionic polynomial equation are just composed of some quaternion equivalence classes and some isolated points.

另外，从新方法及四元数带余除法理论出发，进一步改进了新方法和Niven的传统方法。 2解的完整结构：由求解的新方法得到四元数多项式方程全体解的完整结构为四元数等价类与孤立点的集合，以往的结果成为其自然推论。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失