定理的

In this paper, by computing the Laplace of the square of the length of the second fundamental form and introducing a self-adjoint operator and using Stokes Theorem and Hopf Theorem, we obtained some pinching theorems and rigidity theorems for hypersurfaces and submanifolds in hyperbolic space.

本论文通过计算双曲空间中子流形的第二基本形式模长平方的拉普拉斯和引进一个新的自共轭二阶算子，利用Stokes定理和Hopf定理得到了子流形的一些拼挤定理和刚性定理。

In thisthesis, we first extend the vanishing theorem due to Lawson, Simons and Xinto the case of compact submanifolds of a hyperbolic space. Thus, by using thenew vanishing theorem for homology groups, we prove the topological spheretheorem for complete submanifolds in a hyperbolic space. Hence we generalizethe Shiohama-Xu topological sphere theorem.

本文进一步将Lawson-Simons-Xin同调群消没定理拓广到双曲空间中紧致子流形的情形，并运用这一新的同调群消没定理证明了双曲空间中完备子流形的拓扑球面定理，从而推广了Shiohama-Xu的拓扑球面定理。

The paper consists of six chapters. Chapter One is the theoretical background, we review the course of mechanical theorem proving and briefly introduce the main mechanical theorem prover of the last decade, Among them, we stress on the Boyer-Moore theorem prover and its throretical foundation a computational logic. In Chapter Two, we discuss two theorem proving tactics. One is the rippling-out tactic, which is a tactic for manipulating the induction conclusion to enable the induction hypothesis to be used in its proof, because it extremely restricts the search space, so it has high efficiency.

全文共分六章，第一章是理论背景知识，回顾了定理机器证明的历程，并对近十年来的主要定理证明器作了扼要的介绍，其中，着重介绍了计算逻辑及Boyer-Moore定理证明系统；第二章讨论了两种定理证明策略，一种是波动策略，它是一种在证明中通过处理归纳结论来激活归纳假设的策略，由于极大的限制了搜索空间，因此搜索效率高。

In our thesis, contents are organized as following In Chapter 1 we present our topic's internal and overseas research situations, theoretical and practical significance, and introduce the research object and contents, and the main contributions of this dissertation. Chapter 2 reviews the development of the stability results for nonlinear systems and some relevant recent results, which include Lyapunov and LaSalle-Yoshizawa theorems for nonlinear systems, and stochastic edition for stochastic nonlinear systems. Sontag's formula for systems affine in control is presented in the frame of CLF. The concepts of disturbance attenuation and the inverse optimality are also explained in this Chapter. In chapter 3 we present the solvable theorem of inverse optimal gain assignment problem, design the inverse optimal controller and the inverse optimal tracking controller for strict-feedback nonlinear continuous systems with unknown time-varing bounded disturbances and constant unknown parameters using an adaptive backstepping algorithm, which are nonlinear, continuous and are easier to realize. These designs are fully systematic and the algorithm can be directly coded in symbolic software. The results of simulation show the effectiveness of the control algorithms.

论文的结构如下：在第1章中，给出了本文研究课题的研究现状、理论意义和实际应用，并介绍了本文的研究对象、研究内容以及主要贡献；在第2章中，针对确定性非线性系统和随机非线性系统，分别介绍了Lyapunov定理、LaSalle-Yoshizawa定理及其随机版本；对仿射系统，在控制Lyapunov函数框架下，给出了Sontag公式；同时给出了非线性系统扰动抑制和逆最优控制问题的基本概念；在第3章中，针对具有未知时变有界扰动和未知定常参数的一类不确定非线性系统，给出并证明了逆最优增益配置可解定理，使用自适应Backstepping算法和均值定理，系统地设计了自适应逆最优控制器和逆最优跟踪器，这种设计方法可同时获得逆最优控制策略和自适应律，简单明了，仿真结果表明该控制算法的有效性，并给出了性能估计。

The theorem of mean has the Lagrange theorem of mean and the Cauchy theorem of mean, they are prove the inequality the powerful tool.

中值定理有Lagrange中值定理和Cauchy中值定理，它们都是证明不等式的有力工具。

The sine, law of cosines's several sine law and the law of cosines are three angular corner relations and three jiao functions plants using abstract the is most common, the most important two theorems.

正弦定理和余弦定理是三角形边角关系及三角函数种最常见、最重要的两个定理。

On the other hand , illuminated by close relation between harmonic maps and sumanifold theory, we have also researched in-depth about some relative problems in submanifolds theory which include: the estimate of heat kernel and eigenvalue and it geometric application of submanifolds with or without boundary; many kinds of pinching problem of submanifolds which include the rigidity theorem, topological sphere theorem, differential sphere theorem and topological finite theorem.

另外，鉴于调和映照与子流形之间的密切关系，我们还要深入研究子流形几何中与之相关的一些问题，主要包括：带边与不带边子流形的热核与特征值估计及其几何应用；子流形的刚性定理、拓扑球定理、微分球面定理、拓扑有限性定理等各类Pinching问题。

By applying existence theorems of maximal elements for a family of GB-majorized mappings in a product space of G-convex spaces, some coincidence theorem, Fan-Browder type fixed point theorem and some existence theorems of solutions for a system of minimax inequalities are proved under noncompact setting of G-convex spaces.

通过应用G-凸空间的乘积空间内一族GB-优化映象的极大元的存在定理，在G-凸空间的非紧设置下证明了某些重合点定理，Fan-Browder型不动点定理和极小极大不等式组的解的存在性定理。

In the study of the first problem, by using the fixed-point index theorems, we get the existence of boundary value problem in neither superlinear nor sublinear condition.

在第二种形式边值问题的研究中，主要利用Krasnoselskii不动点定理，得到了一些正解存在性定理及正解不存在性定理。

The sine, law of cosines's several sine law and the law of cosines are three angular corner relations and three jiao functions plants using abstract the is most common, the most important two theorems.

帮我把这段文字翻译为英文，谢谢，非常感谢摘要正弦定理和余弦定理是三角形边角关系及三角函数种最常见、最重要的两个定理。

We have no common name for a mime of Sophron or Xenarchus and a Socratic Conversation; and we should still be without one even if the imitation in the two instances were in trimeters or elegiacs or some other kind of verse--though it is the way with people to tack on 'poet' to the name of a metre, and talk of elegiac-poets and epic-poets, thinking that they call them poets not by reason of the imitative nature of their work, but indiscriminately by reason of the metre they write in.

索夫农 、森那库斯和苏格拉底式的对话采用的模仿没有一个公共的名称；三音步诗、挽歌体或其他类型的诗的模仿也没有——人们把&诗人&这一名词和格律名称结合到一起，称之为挽歌体诗人或者史诗诗人，他们被称为诗人，似乎只是因为遵守格律写作，而非他们作品的模仿本质。

The relationship between communicative competence and grammar teaching should be that of the ends and the means.

交际能力和语法的关系应该是目标与途径的关系。