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Firstly, we analyze the deficiencies ofexisting ones, extend domains of fuzzy logical operators from the cap product withtwo sets to them with arbitrary (finite, enumerable, or non-enumerable) number ofsets to form two general mappings so as to conform them to the fuzzy conjunctionand infimum, and to fuzzy disjunction and supremum, respectively, and fuzzify theset inclusion of fuzzy sets. Similar to the classical adjunction, the concept of fuzzyadjunction is introduced.

首先分析了原有"模糊形态学"框架的缺陷，进而将模糊合取和模糊析取算子的定义域从两个集合的笛卡儿乘积推广到任意多个集合的笛卡儿乘积，形成两个一般的映射，使得模糊合取与下确界、模糊析取与上确界保持一致，以此将模糊集的包含关系一般化。

First, for the Rayleigh-Ritz method, a general formula is derived in this paper for elastic vibration problem in which the admissible function can be taken in arbitrary form.

第一，利用Rayleigh—Ritz法推导了取任意形式容许函数时弹性体振动问题的一般解。

In Chapter 2, we will use one-parameter subgroups of the Lie group to develop a version ofNewton\'s method on an arbitrary Lie group for maps from a Lie group to its correspondingLie algebra which is independent of affine connections on the Lie group and differs fromNewton\'s method for sections on Riemannian manifolds.

在第2章中，我们主要研究关于从李群到其李代数的映射f的Newton法的收敛性。利用李群的单参数子群来给出关于映射f的Newton迭代点列，此Newton迭代与李群上的联络无关，也与第1章中研究的Newton迭代不同。

First, we introduce and discuss the various methods of multivariate polynomial interpolation in the literature. Based on this study, we state multivariate Lagrange interpolation over again from algebraic geometry viewpoint:Given different interpolation nodes A1,A2 .....,An in the affine n-dimensional space Kn, and accordingly function values fi(i = 1,..., m), the question is how to find a polynomial p K[x1, x2,...,xn] satisfying the interpolation conditions:where X=(x1,X2,....,xn). Similarly with univariate problem, we have provedTheorem If the monomial ordering is given, a minimal ordering polynomial satisfying conditions (1) is uniquely exsisted.Such a polynomial can be computed by the Lagrange-Hermite interpolation algorithm introduced in chapter 2. Another statement for Lagrange interpolation problem is:Given monomials 1 ,2 ,.....,m from low degree to high one with respect to the ordering, some arbitrary values fi(i= 1,..., m), find a polynomial p, such thatIf there uniquely exists such an interpolation polynomial p{X, the interpolation problem is called properly posed.

文中首先对现有的多元多项式插值方法作了一个介绍和评述，在此基础上我们从代数几何观点重新讨论了多元Lagrange插值问题：给定n维仿射空间K~n中两两互异的点A_1，A_2，…，A_m，在结点A_i处给定函数值f_i(i=1，…，m)，构造多项式p∈K[X_1，X_2，…，X_n]，满足Lagrange插值条件：p=f_i，i=1，…，m (1)其中X=(X_1，X_2，…，X_n)，与一元情形相似地，本文证明了定理满足插值条件(1)的多项式存在，并且按"序"最低的多项式是唯一的，上述多项式可利用第二章介绍的Lagrange-Hermite插值算法求出，Lagrange插值另一种描述是：按序从低到高给定单项式ω_1，ω_2，…，ω_m，对任意给定的f_1，f_2，…，f_m，构造多项式p，满足插值条件：p=sum from i=1 to m=Ai=f_i，i=1，…，m (2)如果插值多项式p存在且唯一，则称插值问题适定。

The ability to communicate arbitrary ideas through thin air via sound waves is a complex and fascinating process.

人类是利用空气及声波的传导，来沟通彼此的意念及想法，而这种能力本身，则是一套复杂且迷人的过程。

Ambrose and St. Augustine in their beautiful works on the Psalms rather spiritualize, or moralize, than allegorize, and their imaginative interpretations are chiefly of events, actions, numbers, etc. But almost all allegorical interpretation is so arbitrary and depends so much on the caprice of the exegete that it is difficult to conciliate it with reverence, however one may he dazzled by the beauty of much of it.

刘汉铨和圣奥古斯丁在其美丽的作品的诗篇，而spiritualize ，或说教，而不是allegorize ，他们的想像力的解释是主要的事件，行动，数字等，但几乎所有的寓言是这样的解释，而且取决於任意这麼多的任性的exegete ，这是很难调解与崇敬，但是人们可以通过他眼花缭乱的美丽多了。

An analytical coupled water-energy balance equation at arbitrary time acale was derived by dimension analysis and mathematical derivation.

本文通过量纲分析和数学推导得到能用于任意时间尺度上的水热耦合平衡方程，该方程含有唯一参数。

An analytical coupled water-energy balance equation at arbitrary time acale was derived by dimension analysis and mathematical derivation. The equation has only one parameter.

本文通过量纲分析和数学推导得到能用于任意时间尺度上的水热耦合平衡方程，该方程含有唯一参数。

This semi-analytical method is accurate and widely used. It overcomes the limitation of the analytical solution such as conformal mapping. The position of cracks can be arbitrary.

这种解析数值相结合求解应力强度因子的方法，充分利用了解析方法精度高和数值方法适用性广的特点，同时又克服了保角变换等解析解的局限，各裂纹位置可以是任意的。

Here, rather than the remote entanglement we use the local entanglement, which canbe prepared in an easier way than the remote entanglement, to enhance the remoteentanglement; Based on generalized measurement, a general method for theconcentration of a known non-maximally entangled state for arbitrary dimensionalsystem is proposed only by introducing a qubit ancilla, which will greatly simplifythe total concentration process.

这里我们只需要利用局域纠缠而非远程纠缠便可提高远程纠缠；对于任意维系统己知非最大纠缠态的浓缩，可以仅仅通过引入一个二能级系统作为辅助来实现。

Hanna: That's over now, isn't it?

都结束了，对吗

You must be ill. You look so pale.

你一定是病了，你的脸色苍白。