线性算子

In this thesis, we express one dimension wave equation by arithmetic theory and prove D'Alembert Solution in the light of method on one stage linear partial differential equations.

本文从另一角度即算子的方法，将弦振动方程写成算子的形式，再根据一阶线性偏微分方程的求解方法，最终推导出D'Alembert公式。

In addition,we also discuss the boundedness results for a class of multilinear commutators generated by fractional integral operators and BMOfunctions on MK_~.

在MK_~上的有界性，同时还讨论了一类由分数次积分算子与BMO函数所生成的多线性交换子在MK_~上的有界性结果。

This paper is the continuance of that of Lewalle (2001). Lewalle has made use of Gaussian wavelet to obtain the N-S equations with flexion as fundamental quantity in the infinite domain. In that work, Lapalace operator has been degraded to one-order operator. Moreover the resulted N-S equations take the form of diffusion equations with local bilinear mutual interaction kernel.

本文是Lewalle工作的继续，Lewalle利用高斯子波，得到了以弯曲度为基本量的无穷域中N-S方程，此时Lapalace算子降低为一阶导数，并且N-S方程变形为具有局部化双线性相互作用的扩散方程的形式。

Using this method,we discussed for generalized Burger equation with variable coefficients and linear danping,obtained its symmetry group,simliar reduced equations,infinitesimal generator and Lie algebras;In chapter 6,using the Exp-function method,the generalized solutions of combined KdV-mKdV equation with variable coefficients are discussed.

首先介绍了Lie群分析法的基本思想，其次用Lie群分析法得到了带线性阻尼项的变系数广义Burgers方程的无穷小变换、无穷小算子的李代数结构，并具体求出了带线性阻尼项的变系数广义Burger方程的群不变解和约化方程。

The circular convolution operator is compared with the linear convolution operator in the calculation of local statistics.

比较了循环卷积算予和线性卷积算子在计算局部统计量上的异同点。

To solve a heavily constrained optimization problem includes linear constraint conditions and non-linear constraint conditions, in this paper, a new comprehensive solution mechanism, based on specialized genetic operators, exact penalty function and initialization with multiple linear feasible solutions, is proposed.

针对一类同时具有线性约束条件和非线性约束条件的重约束优化问题，本文成功地综合采用了专门化遗传算子、精确惩罚函数及多个线性可行解初始化的求解方法，取得了很好的效果。

Secondly,we present internal stability of feedback systems.The fundamental idea is to represent a linear system as the range of a 2×1 operator matrix with causal entries.

其次介绍了线性时不变系统中的内部稳定性理论，基本思想是将一个线性系统表示成一个2×1算子矩阵的值域，其中的矩阵元素是关联的。

The abstract result contains several concrete results in the literature and can also be used to deal with some new cases for resonant differential equations.In the introduction, we briefly introduce the development process of the variational methods. In Chapter 2, we list some basic knowledges refering to the variational methods, including the Sobobev space,—△ operator, the weak solution and the minimizing sequence methods and some minimax theorems. In Chapter 3, we introduce the research process of Hamiltonian system of second order and the semilinear elliptic problems, using the methods introduced previously. In Chapter 4, we prove the main theorem of the thesis, and apply it to the problems in the previous Chapter, and can also be applied to some new resonant cases.

在前言中，简要介绍了变分法的产生、发展过程，在第二章中我们介绍了有关变分法的一些基本知识，包括Sobolev空间，—△算子，弱解，极小化序列方法和一些极小极大定理，在第三章中我们介绍了非线性项有界或满足次线性条件，以及它满足推广的Ahmad-Lazer-Paul条件时，二阶Hamiltonian系统和半线性椭圆问题的研究历程，最后在第四章中我们证明了本论文的主要定理，并把它应用到第三章的问题中，使得前面的几种共振的情形都可以统一到这个抽象的结果中。

Finite element methods are analyzed, where the stress is approximated by the discontinuous Raviart-Thomas element of lowestorder,and the velocity by the piecewise constant element,and the Lagranian multiplier by the trace space of the piecewise continuous linear element.

然后发展其有限元方法：对应四变量分别用间断型Raviart-Thomas最低阶元，分片常数元，连续线性元和连续线性元的迹空间，我们获得了稳定性和最优误差界，通过后处理办法，我们得到一个适合于计算的速度-压力格式，该格式可视为"mini"元方法的一个变形（本文格式中引用了局部投影算子）。

According to the number of unknown variables in ideal element fitting, variable substitution is adopted to convert the overdetermined nonlinear equations into overdetermined linear equations. Cumulative sum operator is used to solve the overdetermined linear equations.

根据理想要素拟合过程中待求未知量的个数，运用变量代换，将非线性超定方程组转化为线性超定方程组；对变换后的超定线性方程组施加累积算子，进行未知数求解；最后用仿真实例进行了分析，并与CIRP方法进行比较。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。