谱定理

The main subjects of the volume include:- spectral analysis of periodic differential operators and delay equations, stabilizing controllers, Fourier multipliers;- multivariable operator theory, model theory, commutant lifting theorems, coisometric realizations;- Hankel operators and forms;- operator algebras;- the Bellman function approach in singular integrals and harmonic analysis, singular integral operators and integral representations;- approximation in holomorphic spaces.

卷包括：主要课程-定期微分算子和延迟方程谱分析，稳定控制器，傅立叶乘数；-多变量算子理论，模型论，换位解除定理，coisometric变现；- Hankel算子和形式；-算子代数；-在奇异积分和谐波分析，奇异积分算子和积分表示贝尔曼函数方法；-在全纯空间近似。

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。

Chapter 1 gives the background,current research process of relatedproblems and summarizes this thesis\'s work.In chapter 2,we study the Brownian motion with holding and jumping on the boundary.We use the resolvent method to obtain the infinitesimal generator because the domain of the infinitesimal generator is essentially the same as the range of the resolvent.Knowledge of this range and of the differential operator determines uniquely the infinitesimal generator.Since the semigroup generated by the DHJ is not strongly continuous,to use the nice property of strongly continuous semigroup in analytic theory,in chapter 3 we show that the dual is strongly continuous and derive ergodicity through spectral radius formulas and finally obtain the ergodic theorem by duality. In chapter 4,we discuss a class of a more general process---one dimensional Feller diffusion proposed by W.Feller in 1954.The Feller diffusion allows the possibility of jumps from boundary to boundary,not only from boundary to the interior.We give the stationary distribution of this process.

具体地，本文的结构如下：第一章给出了问题产生的背景，研究现状及本文的主要工作；第二章研究了在边界上逗留后随机跳的布朗运动，我(来源：3dABC论文网www.abclunwen.com)们用预解算子的方法得到其无穷小生成元，因为无穷小生成元的定义域本质上就是预解算子的值域，知道这个值域和微分算子形式就能唯一地决定无穷小生成元；由于DHJ过程产生的半群不是强连续的，为利用强连续半群的一些漂亮性质，在第三章中我们证明其对偶半群是强连续的，然后由谱半径公式得到遍历性并且最后由对偶得到遍历定理；第四章讨论了Feller在1954年引入的更广的一类过程----一维Feller扩散过程，Feller扩散过程允许有从边界到边界的跳发生，即不仅仅局限于从边界到内部的跳，在这一章中，我们给出了一维Feller扩散过程的平稳分布；在第五章，我们讨论了一些相关的问题，给出了DHJ过程对应的PDE问题及特征值与收敛速度的关系。

In section one, we introduce some background of the topic, in section two we review some basic and recent results about the structure and hierarchies of the computably enumerable degrees which are closely related to our topic- the algebraic structure of the plus cupping Turing degrees, in section three, we outline the basic principles of the priority tree argument, one of the main frameworks and tools of theorem proving in computability theory, and in section four, we prove a new result concerning the algebraic structure of the plus cupping Turing degrees that there exist two computably enumerable degrees a, b such that a, b ? PC, and the join a V b of a and b is high.

度结合为0′。本篇论文分为4个部分：第一部分介绍了这个领域的一些背景知识；第二部分主要回顾了前人在研究可计算枚举度的结构和层谱时所取得的一些基本和最新结果，这些结果与我们的主题—加杯图灵度的代数结构密切相关；在第三部分中，我们概要的描述了优先树方法的基本原理，此方法是可计算性理论中定理证明的一个重要框架和工具；第四部分证明了一个加杯图灵度代数结构的新结果：存在两个可计算枚举度a，b，满足a，b∈PC，而且a和b的并a∨b是一个高度。

The main results are as follows: the relations between local fractional integrated semigroups and the corresponding Cauchy problem, global fractional integrated semigroups and regularized semigroups are given; introduction of the notion of regularized resolvent families, and the generation theorem and analyticity criterions for regularized resolvent families are obtained; the spectral inclusions between fractional resolvent family and its generator, and the approximation for fractional resolvent families in the cases of generators approximation and fractional orders approximation; elliptic operators with variable coefficients generating fractional resolvent family on L^2 by using numerical range techniques; and the L^p theory for elliptic operators with real coefficients highest order are obtained by Sobolev''s inequalities and the a priori estimates for elliptic operators; and a kind of coercive differential operators generates fractional regularized resolvent family by applying the Fourier multiplier method, functional calculus and some basic properties of Mittag-Leffler functions.

主要结论是：给出了局部分数次积分半群和相应的Cauchy问题的关系以及分数次积分半群和正则半群的关系；引入了正则预解族的概念，并给出了其生成定理和解析生成法则；给出了分数次预解族与其生成元的谱包含关系，并研究了在生成元逼近和分数阶逼近两种情况下相应的预解族的逼近问题；利用数值域方法证明了具变系数的椭圆算子在L^2上生成分数次预解族；利用Sobolev不等式和椭圆算子的先验估计证明了具变系数的椭圆算子在其最高项系数为实数时在L^p上生成分数次预解族；运用Fourier乘子理论、泛函演算和Mittag-Leffler函数证明了一类强制微分算子可以生成分数次正则预解族，并给出了该预解族的范数估计。

With the knowledge of linear algebra, general formula for arbitrary order Taylor finite difference has been derived. In this paper, the analysis of spectral property and computational error of arbitrary order TFD is presented in detail. Then it is applied into the numerical solution of Maxwell's equations. Stability condition and numerical dispersion relation are derived.

本文借助于采样定理，详细分析了不同阶Taylor中心差分格式的谱特性以及计算误差，并将任意阶Taylor中心差分格式用于数值求解麦克斯韦方程中，严格导出了稳定性条件和数值色散关系的表达式，引入了新的误差定义来衡量算法的好坏。

Starting from the sampling theorem,the fast Fourier transform computation of angular spectrum diffraction formula and its inverse operation are studied.

从取样定理出发，对角谱衍射公式及逆运算的快速傅里叶变换进行了研究。

The spectral mapping theorem about integrated semigroup is also obtained.

进而得到了 n次积分半群的谱映射定理。

Based on up_ and down_going wave decomposition and elastic wave power reciprocity theorem, this paper derives, in the lateral velocity varying case, linear prediction operators expression and the generalized spectral factorization equation for reflection dataset.

本文基于上下行波分解及弹性波互易定理，导出横向变速介质条件下线性预测算子的表达式和反射数据的广义谱分解方程。

Then the propagation of an arbitrary nonideal optical beam is analyzed. In terms of the moment theorem, the performance of modes of BOR is analyzed from both points of view of geometry optics and diffractive optics. Comparing BOR with the resonators using plane mirror and spherical mirror or using plane mirror and aspherical mirror by finite element method, the design procedure for BOR is gave.

然后以衍射的角谱理论为基础，用矩量定理讨论了任意光束的传播规律，从几何光学和衍射光学的角度完整地分析了二元光学谐振腔的模式特征，用有限元方法数值对二元光学谐振腔、非球面腔和球面腔进行衍射分析，比较了三种腔型的差异，给出了二元光学谐振腔的设计方法。

With Death guitarist Schuldiner adopting vocal duties, the band made a major impact on the scene.

随着死亡的吉他手Schuldiner接受主唱的职务，乐队在现实中树立了重要的影响。

But he could still end up breakfasting on Swiss-government issue muesli because all six are accused of nicking around 45 million pounds they should have paid to FIFA.

不过他最后仍有可能沦为瑞士政府&议事餐桌&上的一道早餐，因为这所有六个人都被指控把本应支付给国际足联的大约4500万英镑骗了个精光。