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31 Chapter 3 Number Systems and Boolean Algebra 3.2 Boolean Algebra Table 3-2 Distributivity Idempotency Absorption laws 分配律同一律吸收律 a=ab+ac a+= a+a=a aa=a a+ab=a a=a'=a'b''=a'+b' DeMorgan's laws德摩根定理计算机专业英语 3-32 Chapter 3 Number Systems and Boolean Algebra 3.2 Boolean Algebra Since a finite set of n elements has exactly 2n subsets, and it can be shown that the finite Boolean algebras are precisely the finite set algebras, each finite Boolean algebra consists of exactly 2n elements for some integer n.

由于n个元素的有限集有且只有个子集由于个元素的有限集有且只有2n个子集，而且很显然有限布个元素的有限集有且只有个子集，尔代数一定是有限集合代数，所以对某个整数n而言而言，尔代数一定是有限集合代数，所以对某个整数而言，每个有限布尔代数也有且只有2n个元素。例如，上文定义的集合T的限布尔代数也有且只有个元素。

In order to solve parabolic equation of heat conduction with interval parameters, the regions of space are discretized by finite elements and the regions of time are discretized by finite difference. The interval finite element method based on the element is established via the combination of interval analysis and the traditional finite element method.

对具有区间参数的热传导抛物型方程的求解，在空间域上利用有限单元离散，在时间域上利用差分离散，将区间分析和常规的有限元法相结合，建立了求解不确定温度场的基于单元的区间有限元方法。

The main problems researched in the dissertation and achievements are as follows: The dissertation explains the science background and the theoretical and actual meaning of the task. The progress and the problems of the flow imaging technique are briefly explained. The advantages of electrical capacitance tomography are discussed. The paper discusses the formation and operating principle of ECT, and builds up the definite mathematical model by detail theoretical analyse, and establishes the Finite Element Model of the system by changing the finite boundary value problem of ECT into equivalence variation problem according to variation principle. On the basis of the FEM of the system, the paper builds the finite element model of the capacitance sensitivity field distribution and programs its finite element simulation software and proves the validity of the FEM of the ECT from different aspects. With the FEM and the qualitative analyse of the effect of each structural parameter of the capacitance transducer on its capability, this paper optimizes the structural parameters of the capacitance transducer, and obviously improves its capability. The compare of the simulated result with the static experimental result shows that the error of them is less than 0.9%.

其主要研究内容包括：阐述课题研究的学术背景及其理论与实际意义，概括总结出流动层析成像技术的发展现状和存在的问题，指出电容层析成像技术的优点；讨论电容层析成像系统的组成及工作原理，进行详尽的理论分析，以变分原理为基础，将ECT的有限元边值问题转换为等价的变分问题，从而建立系统的有限元模型；在系统有限元模型的基础上，建立敏感场的数学模型，编制有限元的仿真软件，并从不同角度证明电容层析成像系统有限元模型的正确性；利用有限元模型，定性地分析电容传感器各结构参数对其性能的影响，并对传感器结构参数进行优化设计，使其性能得到明显的改善。8电极ECT系统的仿真和静态实验结果的比较表明，它们之间的相对误差小于0。

By means of finite element analysis of ANSYS, this paper would analyze the behavior of the connections with beam-end horizontal haunch such as strength and ductility .The main contests are listed as follows: Firstly, build up the finite element model. Through comparing the finite element result to the test result, it proves feasible in finite element model to get the behavior of beam-column connections.

本文通过ANSYS有限元分析，对梁端翼缘扩大型节点的承载性能，延性等方面进行了研究，主要内容包括： 1建立有限元模型，并将有限元分析结果与试验结果进行对比，验证有限元分析的可行性。

This paper systematically provides an overview of recent developments in small punch test technology including empirical and computational methods for determining elasto-plastic parameters from the measured data of small punch test. The computational methods involve finite element and parameter method, inverse finite element procedure, finite element and inverse methodology, inverse identification based on artificial neural networks, finite element optimization, shape deformation and hybrid inverse procedure et al.

本文系统综述小冲杆试验的测量技术及从测量数据来确定材料弹塑性参数的各种经验方法和计算方法，例如有限元分析和参数法、反向有限元法、有限元和反方法、反向识别和人工神经网络、有限元优化和试验变形形状以及杂交反方法等。

Then we found bijections between quadratic form finite automata and input memory linear automata which lead to a way to produce all weakly invertible or weak inverse quadratic form finite automata. In the last section of chapter three, we extend these results to a more general class of nonlinear finite automata (called qusi-quadratic form finite automata).

然后我们建立了建立全体二次型有限自动机与全体输入存储线性有限自动机之间的的一个一一对应L，由于L保持弱逆性（3.20定理），由此我们得到了产生全体可逆二次型有限自动机的方法：要产生一个具有某种可逆性的二次型有限自动机只需产生一个具有同种可逆性的输入存储线性有限自动机，然后作其二次型提升即可。

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 paper studies on weak invertibility of a linear finite automata based on its matrix model,gets a full and essential condition on which a linear finite automata weakly and invertibly delays finite steps,and also gives an algorithm whether or not a linear finite automata has weak invertibility .

主要讨论了基于矩阵模型表示的线性有限自动机的弱可逆性，利用矩阵模型得出了线性有限自动机延迟有限步弱可逆的充分必要条件，并且给出了线性有限自动机是否具有弱可逆的算法。

With an example, the paper uses finite element program ANSYS to establish three calculating model of the steel-concrete composite open-web floor: the combination finite element method of spatial beam elements and shell elements, the finite element method of shell elements and the combination finite element method of shell elements and entity element.

结合一算例，利用ANSYS有限元软件，建立了组合空腹楼盖的三种计算模型：空间梁单元与壳单元的混合元模型、空间壳单元模型、壳单元和块体单元模型三种理论计算方法，对这三种计算模型作分析对比，得到了静荷载作用下该类结构的内力分布规律及其他一些有价值的结论。

First, this article introduced 2 grade of rotary kiln principle of work and the equipment, to rolled the circle mechanics behavior to carry on the analysis, proposed - rolled the circle - under roller's rotary kiln overall modelling based on the tube body - kiln liner - big tooth ring, to rolled the circle - under roller's contact stress to carry on the finite element analysis the method, and obtained rolled the circle contact stress the size and the distribution rule, finally indicated that rolled the circle contact stress mainly to have two main peak values and the sudden change in one week, appeared is rolling the circle and the under roller contact area; Then, rolls the circle the stress to regard as is the principal stress unknown plane stress test, the use quite mature modern test technology, proposed rolls the circle stress test the principle and the plan, and to rolled the circle the stress to carry on the scene test, the stress which the scene test analysis result and the finite element analysis calculated has compared, had proven the whole method modelling's finite element analysis method's rationality and rolled the circle - under roller contact stress the result accuracy; Finally, uses ANSYS APDL the re-development technology, in view of 2 grades of rotary kilns, rolled the circle - under roller contact stress the overall finite element analytic method to carry on the re-development.

首先，本文介绍了2档回转窑工作原理和装置，对滚圈的力学行为进行了分析，提出了基于筒体-窑衬-大齿圈-滚圈-托轮的回转窑整体建模，对滚圈-托轮的接触应力进行有限元分析的方法，并得到了滚圈接触应力的大小和分布规律，结果表明滚圈接触应力在一周内主要存在两个主要的峰值和突变，都出现在滚圈与托轮接触区域；然后，把滚圈的受力看成是主应力未知的平面应力的测试，利用比较成熟的现代测试技术，提出了滚圈应力测试的原理和方案，并对滚圈的应力进行了现场测试，现场测试分析结果与有限元分析计算出的应力相比较，证明了整体法建模的有限元分析方法的合理性和滚圈-托轮接触应力的结果的正确性；最后，利用ANSYS的APDL的二次开发技术，针对2档回转窑，就滚圈-托轮接触应力的整体有限元分析法进行了二次开发。

It is also known as one of the most poisonous naturally occurring substances.

它也被称为一个最自然发生的有毒物质。

The greatest stress is found at the location on the cross section where V is the largest.

最大应力出现在横截面上V为最大的地方。