单调性

For a given Sturm-Liouville equation with positive leading coefficient function, by use of the monotonicity results of its natural loops, limit of boundary conditions and eigenvalues in the space of self-adjoint boundary conditions, a proof of the equalities among eigenvalues for coupled boundary conditions and those for separated boundary conditions is made.

内蒙古工业大学理学院副教授 2。对于给定的首相系数函数为正的 Sturm-Liouville方程，利用在自伴边界条件空间上自然圈、边界条件的极限和特征值的单调性的几个结果，给出了在偶合边界条件与分离边界条件下的特征值间的一些等式的证明。

In chapter two, under non-Lipschitz condition, the existence and uniqueness of the solution of the second kind of BSDE is researched, based on it, the stability of the solution is proved; In chapter three, under non-Lipschitz condition, the comparison theorem of the solution of the second kind of BSDE is proved and using the monotone iterative technique , the existence of minimal and maximal solution is constructively proved; in chapter four, on the base of above results, we get some results of the second kind of BSDE which partly decouple with SDE, which include that the solution of the BSDE is continuous in the initial value of SDE and the application to optimal control and dynamic programming. At the end of this section, the character of the corresponding utility function has been discussed, e.g monotonicity, concavity and risk aversion; in chapter 5, for the first land of BSDE ,using the monotone iterative technique , the existence of minimal and maximal solution is proved and other characters and applications to utility function are studied.

首先，第二章在非Lipschitz条件下，研究了第二类方程的解的存在唯一性问题，在此基础上，又证明了解的稳定性；第三章在非Lipschitz条件下，证明了第二类BSDE解的比较定理，并在此基础上，利用单调迭代的方法，构造性证明了最大、最小解的存在性；第四章在以上的一些理论基础之上，得到了相应的与第二类倒向随机微分方程耦合的正倒向随机微分方程系统的一些结果，主要包括倒向随机微分方程的解关于正向随机微分方程的初值是具有连续性的，得到了最优控制和动态规划的一些结果，在这一章的最后还讨论了相应的效用函数的性质，如，效用函数的单调性、凹性以及风险规避性等；第五章，针对第一类倒向随机微分方程，运用单调迭代方法，证明了最大和最小解的存在性，并研究了解的其它性质及在效用函数上的应用。

Key words:Almost monotone approximation,Boundary value problems,Non-linear elliptic systems

在第五章，我们提出了具有拟单调性的非线性椭圆型方程组边值问题的几乎单调逼近。

The monotone property of multivariate function proved that the maximum stress strength point exists on the inner surface of

应用多元函数单调性理论，证明了在三轴应力条件下管体中最大应力强度值出现在套管内壁。

Measured in equi intensity cantilever,the result indicates that not only the coupling ratio of the coupler is sensitive to strain,but also the function is monotone.

在悬臂梁上进行的应变实验表明，该型耦合器的耦合比不但对应变敏感，而且有单调性。

This paper describes the application of a quadratic function, namely : the use of quadratic function for Inequality, discuss other monotone function with the most value, the real roots distribution, series of comprehensive, quadratic function embodied in the high school math widely used.

本文主要介绍二次函数的应用，即：利用二次函数解不等式，讨论其他函数的单调性与最值、实根分布，数列中的应用等综合问题，体现了二次函数在高中数学中的广泛应用。

However,To prove Inequality with elementary method,we often create complex computational process. The second ,we will take full advantage of the knowledge of calculus Inquiry Testimony of inequality,and concluded the higher mathematics to prove Inequality several main method and its application conditions.Constructors in the context of the use of the monotone function,Calculus value theorem,function and the most extreme value,integral, it can be a very effective solution to the inequality problem proof. At last,we summed up several convenient and simple way to prove Inequality.It will be play a great role in our problem Solving.

但是用初等方法证明往往会造成复杂的运算过程，本文接着充分利用微积分的知识探究不等式的证明方法，并指出微分学和积分学在不等式的证明的具体应用，那就是在构造函数的背景下运用函数的单调性、微积分中值定理、函数的极值和最值、定积分，那么就可以十分有效地解决不等式中的证明问题，从而归纳出几种方便而又简捷的方法，这样对我们解题将会起到很大的作用。

Engineering design problems have two characteristics: the design variables are often monotonic in the objective function and constraints. Moreover, the objective function and constraints are often implicit functions which cannot be expressed explicitly in terms of design variables.

工程最佳化设计问题通常有两个特性：设计变数在目标函数与限制条件常具有单调性，且其目标函数与限制条件常为内隐式函数，也就是不能以设计变数来表示其外显形式。

This paper gives a mathematical model for the growth phenomena,and discusses its additive property,monotonicity and stability.

本文给出了生长过程的数学模型，讨论了生长过程的可加性、单调性和简单生长过程的稳定性。

Thus the study of this monotonicity develops rapid in recent 20 years.

因而最近二十年，对系统周期轨的单调性的研究发展得很快。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失