算子的谱分析

In this paper,a systematic direct perturbation method of dark solitons is found.Having analyzed the mistakes in earlier works on perturbation method for dark solitonsand essence of the direct perturbation method for bright solitons,we notice that to in-troduce the adjoint solutions of the squared Jost solutions and to prove the completenessare crucial to the problem.Giving up the unnecessary scheme of introducing the adjointoperator in the bright soliton case,we directly find the adjoint solutions by meetingthe demand for the orthogonality that inner product of the squared Jost solutions andits adjoint should be proportional to a δ function in the case of continuous spectra.The corresponding adjoint operator is thus found.Taking into account the reductiontransformation,we find a correct description for the completeness of the squared Jostsolutions and directly verify its validity with explicit expressions of the squared Jostsolutions.

本论文建立了系统的暗孤子直接微扰方法，在对前人关于暗孤子微扰方法的错误以及亮孤子直接微扰方法的本质作了充分的分析后，认识到引入平方Jost解的伴随解和证明完备性是问题的关键，撇开过去亮孤子情况首先引入伴随算子的非必要作法，直接从平方Jost解与其伴随解的内积在连续谱时正比于δ函数这一正交性要求出发，找出了伴随解，同时得出了应有的伴随算子，在考虑到约化变换性后，得到了暗孤子情况的平方Jost解的完备性的正确表述，并在单个暗孤子的情况利用平方Jost解的显式直接验证了它的正确性。

In order to find a stable approximate solution of linear compact operator equation, the article introduces general theories about ill-posed problems, it bases on spectral theory of self-adjiont compact operators and the singular value decomposition for compact operators, avails singular system to give expression of the solution, and explains ill-posedness of compact operator equation roots in the property that the singular values trends to zero. Thereout, it is provided with theoretic support of building up regularization method by inducting regularization filter to weaken or filtrate the influence that the nature of the singular value being very close to zero has on the solutions stability.

为了得到线性紧算子方程稳定的近似解，介绍了不适定问题正则化的一般理论，以自伴紧算子的谱分析与紧算子奇异值分解为理论基础，利用奇异系给出了解的表达式，说明了紧算子方程不适定性的根源在于紧算子的奇异值趋于零的性质，由此通过引入正则化滤子函数来减弱或滤掉奇异值趋于零的性质对解的稳定性的影响，构造正则算子，从而提供了建立正则化方法的理论依据。

Three methods for the computation of the isotropic tensor function,spectral decomposition method,eigenprojection operator method and representation theorem method,are compared.Different approaches to calculate the eigenvalues and their corresponding eigenvectors are also discussed.As an example,the exponent tensor function is calculated.

对计算各向同性对称张量函数的3种方法(谱分解法、特征投影算子法和表示定理法)进行了分析比较，同时也对如何计算特征根与特征向量的技巧进行了探讨，并利用3种方法对指数张量函数进行了实例计算。

According to singular entropy and Percent of contribution to total energy, the de-noising rank is obtained and the signal is reconstructed. The thesis compares reconstruct wave form of different reconstruct rank and the error of original signal and reconstruct signal is contrasted.According to voltage flicker measurement, voltage flicker detection method based on Mathe Matical filtering and Teager Energy Operator is proposed. The multiple MM filter filters out pulse noise and white noise contained in fliker signal, then the filtered signal is tracked accurately and quikly via TEO.

提出了基于数学形态滤波和TEO能量算子的电压闪变快速测量方法，含噪电压闪变信号经过多结构元素复合数学形态滤波后，采用TEO能量算子计算跟踪电压闪变信号的包络，得到的包络信号用Pisarenco谱分析方法获得闪变调制信号的幅值频率等参数，对不同情况下的电压闪变信号进行了仿真验证。

The main contents include: Some preliminary theory (introduction to Sobolev spaces and variational formulations for differential equations); finite element methods for one-dimensional elliptic problems; the construction methods for general finite elements; error estimates for interpolation operators and inverse inequalities for finite element spaces; a priori and a posteriori error estimates for the finite element method for high-dimensional elliptic problems; some typical spectral methods for partial differential equations; error analysis for the spectral approximation for some linear and nonlinear partial differential equations.

主要内容有：准备知识（Sobolev空间的基本概念和主要结果，微分方程的变分描述）；一维椭圆型方程有限元方法；一般有限元的构造；插值算子误差估计和逆不等式；高维椭圆型方程的先验、后验误差估计；求解偏微分方程的几类谱方法；线性与非线性问题谱逼近的误差分析等。

This course covers linear functional analysis, which include basic operator theoty, Riesz representative theorem and basic spectrum theory.

该课程涉及线性泛函分析的基本知识，基本算子理论，Riesz表示和基本的谱分解理论。

This article gives an analysis,simulation and comparison of the performance of square demodulation ,Hilbert transform,wavelet transform,and nonlinear operator methods in extracting envelope spectrum of ship noise,and comes to the conclusion that their performanes are close.

分析并仿真比较了检波滤波法、希尔伯特变换法、小波变换法与非线性算子法等在提取舰船噪声包络谱中的性能，得出了它们性能接近的结论。

The closed-form expressions are derived so that all required differentiations are obtained in closed form and numerical differentiation can be avoided.

特别是对微带和缝隙混合结构的空域MoM分析，旋度算子的微分在谱域进行，并可以给出它们的解析表达式，其对应的空域格林函数可通过高阶的Sommerfeld恒等式获得，从而避免了数值微分。

In Chapter 4, We consider rational maps of degree d＞2 on the Riemann sphere and prove that they satisfy the large deviation theorem.

我们证明的方法是转移算子的傅里叶变换的谱分析和一些概率论中的经典方法。

In this thesis, we mainly study problems on global geometry and geometricanalysis of Riemannian submanifolds, including vanishing theorem for homologygroups, topological sphere theorem, L~2 harmonic 1-forms, finiteness of end andthe spectrum of the Laplacian. In 1973, by using the Federer-Fleming existence theorem and the techniquesfrom the calculus of variations in the geometric measure theory, H.

本文着重研究黎曼子流形上整体几何与几何分析的若干问题，主要内容包括子流形的同调群消没定理、拓扑球面定理、L~2调和1-形式、端的有限性和Laplace算子谱等问题。

I am accused of being overreligious," she said in her quiet, frank manner,"but that does not prevent me thinking the children very cruel who obstinately commit such suicide.""

客人们在卡罗利娜·埃凯家里，举止就文雅一些，因为卡罗利娜的母亲治家很严厉。

Designed by French fashion house Herm è s, this elegant uniform was manufactured in our home, Hong Kong, and was the first without a hat.

由著名品牌 Herm è s 设计，这件高贵的制服是香港本土制造，是我们第一套不配帽子的制服。