引理

When the nonlinearity is an odd function, multiple positive solutions for the sec-ond order self-adjoint difference equation with the Neumann boundary conditionΔu(0)= 0,Δu = 0 are obtained by applying Clark's theorem.

另一方面，当非线性项非负且在0~+和+∞是渐近线性的情况下，应用山路引理研究了方程在混合边值条件u(0)= 0，Δu = 0下正解的存在性。

A discrete weighted Hardy inequality is studied, and by proving a refinement lemma, the characterization of the weight sequence ｛ω_｝_(n≥1) in order that the weighted l~-Hardy inequality holds for all nonnegative and nonincreasing sequence c_ is given.

本文研究离散的加权Hardy不等式，通过建立权序列的加细引理，给出了对任何非负非增数列 lp-加权Hardy不等式成立时权序列{ωn}n≥1的特征刻划。

The sufficient conditions to guarantee robust exponential stability for the closed-loop systems are obtained based on Linear Matrix Inequality, moreover, the approach of design of robust H# output feedback controller is introduced.3 The problem of reliable H# control for a class of Lur"e systems with polytopic uncertainties is studied when all control components are operational as well as when some control components experience failures, the design approach of reliable H# controller is also obtained.4 Considering a class of uncertain Lur"e singular system with time-delays, the problem of robust stability is investigated based on Barbalats Lemma and nonsingular linear transformation of model reduction.

针对一类不确定Lur'e奇异时滞系统，基于Barbalat引理以及非奇异降阶变换，讨论了Lur'e奇异时滞系统的鲁棒稳定问题，提出了鲁棒H_∞状态浙江大学博士学位论文反馈控制器的设计方法。5、针对一类具有饱和执行器的不确定Lur'e奇异系统，提出了时滞依赖的鲁棒稳定与鲁棒二次镇定的充分条件，所的结果不需要参数的整定。6、针对任意协方差有界的有色噪声，讨论了一类不确定Lur'e奇异系统的鲁棒凡滤波问题，得出了鲁棒几滤波器设计的充分条件。7、针对一类不确定Lur'e时滞系统，通过线性变换，把它转换成奇异系统，根据奇异系统鲁棒控制理论所得出的鲁棒稳定与鲁棒镇定控制器设计的充分条件具有很小的保守性。

Later, we estimate our variational functional to get a nontrivial solution of the new equation and so the second solution for is obtained.

然后我们利用函数平移将原来的非齐次边界问题转化为奇次边界问题，验证了其对应的变分泛函满足不带条件的山路引理的两个条件，并给出了泛函临界点存在的一个充分条件，最后对具体的变分泛函进行估计，得到了新方程非平凡解的存在性结果，从而得到了原方程第二个正解的存在性结果。

Firstly, the eigenvalue problem of a class of second order elliptic equation with critical potential and indefinite weights is considered. Then, using critical point theory, Trudinger-Moser inequality and the properties of the first eigenvalue, we prove the existence of a nontrivial solution for a class of nonlinear elliptic with critical potentialand indefinite weights in R~2. Secondly, we prove the existence of nontrivial solutions for a class of subcritical and critical elliptic systems with indefinite part in R~2 byusing a generalized linking theorem, Trudinger-Moser inequality and concentration-compactness principle.5. The existence of at least three weak solutions for discrete boundary value problem is established by using a three critical point theorem introduced by Ricceri.

首先，讨论了R~2中一类带不定权且含临界位势的二阶椭圆型方程的特征值问题，并借此特征值问题的第一特征值性质，利用山路引理及Trudinger-Moser不等式，证明了R~2中一类带不定权且含临界位势的非线性椭圆型方程非平凡解的存在性；其次，利用广义环绕定理，Trudinger-Moser不等式及集中列紧原理，得到了R~2上一类具有强不定部分的半线性椭圆型方程组在非线性项分别为次临界增长和临界增长情形下非平凡解的存在性。

In order to simplify the process of solving dynamic programming equation, the lemmas and corollaries concerning properties of the optimal value function in up-peak of elevator are presented and proved; the structure of optimal dispatching policy in up-peak is constructed, and conclusion that the optimal dispatching policy in up-peak is a threshold policy is obtained.

为了简化求解动态规划方程，提出并证明了电梯上高峰优化值函数特性的相关引理和推论，构造了电梯上高峰最优调度策略的结构，得出了上高峰优化调度策略是基于阈值的策略的结论。

Then we point out the defects in the original definition of PI reasoning, and give the revised one.

最后对PI归结的完备性定理证明所需要的引理给出了两种简化证明。

We give left and right prediction of jump detection on basis of orthonormal polynomial. In theory analysis, we first give some theorems, then to proof the approximation, which is in a jump point detection by nonlinear prediction, can arrive one to number of samples. This result is considerable well. In the meantime, we study the approximate theory of measurement of jump point.

在理论分析中，我们首先给出了几个引理，然后证明了非线性预测法在检测单个突变点的逼近程度达到样本数的分之一，这样的结果是比较好的。

In the course of proving,we extend two lemmas and use the relating results of partial sum increment of Independent r.v.

在证明过程中，首先推广了两个引理，并用到了部分和增量的有关结果。

The stuff in between, including the internal Lemma, is there only to prove there exists an injective map from the group of rational points on an elliptic curve to the group of points over each finite field of prime order not dividing 2D where D is the discriminant.

证明中间的引理，主要是证明椭圆曲线上有限阶有理点到有限域上椭圆曲线的点之间的嵌入映射的存在性，其中点的阶不整除两倍的判别式2D。

For a big chunk of credit-card losses; the number of filings (and thus charge-off rates) would be rising again, whether

年美国个人破产法的一个改动使得破产登记急速下降，而后引起了信用卡大规模的亏损。

Eph. 4:23 And that you be renewed in the spirit of your mind

弗四23 而在你们心思的灵里得以更新