下界集

It is proved that if'sparse NP complete sets under polynomial-time Turing reductions exist'then 'SAT is polynomial-time non-adaptively search reducible to decision', and that if 'P is not equal to NP'then either'SAT is not polynomial-time non-adaptively search reducible to decision'or'SAT is not polynomial-time truth-table reducible to bounded approximable sets', and that if'P is not equal to NP'then'sparse complete sets for NP under polynomial-time disjunctive reductions do not exist'.

因为用现有的证明技术不可能绝对地解决这个假设，本文研究了这个假设与其他关于SAT结构性质的假设之间的关系，证明了如果'NP有多项式时间图灵归约下的稀疏完全集'则'SAT是多项式时间并行地搜索归约为判定'，以及如果假设'P不等于NP'，则要么'SAT不是多项式时间并行地搜索归约为判定'，要么'SAT不能用多项式时间真值表归约归约为有界可近似集'。

Finally, the infimum is acquired of Hausdorff dimensions of its polar sets by means of constructing a Cantor-type set to connect Hausdorff dimension and capacity.

同时通过一个特殊的Cantor型集的构造将维数与容度巧妙地结合起来，得到了广义Brownian Sheet极集的Hausdorff维数的下确界。

Chapter 4,based on strong efficiency rather than efficiency or weak effi-ciency,discusses the similar problem to chapter 3.The concepts of strongsupremum and strong infimum for the set in finite-dimensionalEuclidean space are given,s-conjugate functions and s-subdifferentials theory is developed,by means of which,the duality problem ofmulitobjectivre programming for strong efficient solutions or called absoluteoptimal solutions is analysed,and a new duality theory-s-conjugateduality theory is proposed.

第四章，基于强有效性而不是有效性和弱有效性，给出了类似于第三章的结果，提出了有限维欧氏空间中集合的强上、下确界等概念，研究了点集映象的强共轭函数和强次微分问题，建立了点集映象的s-共轭函数及其s-次微分理论，并借助于这一理论，讨论了多目标规划关于绝对最优解或称强有效解的对偶性问题，建立了基于强有效性的s-共轭对偶理论。

The problem that under what conditions the - resolvent operator of a maximal TJ - monotone set-valued mapping is a Lipschitz continuous single-valued mapping on whole space, which also answers the open problem mentioned above, is studied on finite dimensional Euclidean spaces.

借助于图收敛理论证明了有限维欧氏空间中的两个极大单调集值映射的和映射在较弱条件下仍是极大单调集值映射，并在此基础上讨论了一般集值变分包含问题解集的凸性、闭性和有界性。

If P is a convex lattice polygon, and its pair-wise disjoint family of translates are also lattice polygons, whose union contains every lattice point, it is called a TLP .

若P为凸格点多边形，其内部边界不交平移覆盖平面格点集，则称之为可平移格点多边形TLP。记顶点数为v的TLP的内格点数的下确界为i ，得出i(5)=i(6)=1，i(7)=i(8)=4。

It was considered that karstification is the base of reservior formation, and lithology and faulting control the formation and development of internal reservior.

其发育规律为：岩溶作用是下古生界储集层发育的基础；岩性和断裂作用决定内幕储层的形成与发育。

Based on the surjection of maximal monotone operators, we prove that some semilinear heat equations with weak Lipschitz nonlinear terms are exactly controllable by L〓 0, T: H〓(Ω control; Based on the critical point theorem of coercive convex functionals, we prove that some parabolic equations with Lipschitz nonlinear terms are globally approximately controllable and finite-dimensional exactly controllable by controls acting on mobile supports, and finally, we extend the results to parabolic equations in R〓.

首先利用极大单调算子的满射原理证明了半线性热方程在较弱Lipschitz条件下可通过L〓0.T；H〓(Ω控制实现精确能控。然后通过构造强制凸泛函并利用其临界点理论证明了带Lipschitz非线性项的抛物型方程可通过作用在在移动支集上的控制来实现整体近似能控与有限维精确能控，最后将整体近似能控与有限维精确能控结论推广到无界区域R〓上。

The method attempts to transform the original problem into a series of unconstraint optimization problems.

在这些研究中，均假定在对数障碍函数是严格凸的及解集是非空有界的情况下，证明了方法的全局收敛性。

Assuming that there is a plural Z and , with the following iterative calculation : Z = Z2 +, because the value of different , after several iteration number Z after the amplitude may tend to infinity. May also maintained sector, Mandebo Lott who is set to make Z maintained sector pool put on the complex plane make the distribution of images, as demonstrated above in the extremely complex structure.

假设有复数Z和μ，用下式迭代计算：Z=Z2＋μ，由于μ的取值不同，经过若干次数迭代以后Z的幅值可能趋向无穷，也可能保持有界，曼德勃罗特集就是那些使Z保持有界的μ的集合，把μ在复平面上的分布作成图像，就像上面演示的那样具有极其复杂的结构。

Modeling and stability analysis problem of linear switched system are emphasized in this part and main results include:(1) Through investigation in the projection of 〓 on x, where x denotes continuous state vector, a series of sufficient conditions for stability, asymptotic stability and unstability of linear switched system's equilibrium in the sense of Lyapunov are achieved. And also, a sufficient and necessary condition for state bounded stability of linear switched system is received.(2) For a class of periodical linear switched system whose switching sets are a group of ordinal lines starting from the original point, it's proved that state bounded stability can be defined by the running characteristic in a single period (3) Based on linear switched system, the concept of linear switched system network is put forward. Linear switched system network is a complex network interconnected by many linear switched systems.

这部分着重研究线性切换系统的建模与稳定性分析问题，主要成果包括：（1）通过考察〓在x上的投影，得到线性切换系统的平衡点在Lyapunov意义下稳定、渐近稳定及不稳定的一系列充分条件，并得到了线性切换系统状态有界稳定的一个充要条件；（2）证明了对于一类切换集为从原点出发的一组射线的周期线性切换系统，其状态有界稳定可根据一个周期内的系统运行特性来判定；（3）在线性切换系统的基础上提出了线性切换系统网络的概念，线性切换系统网络是由多个线性切换系统交互作用而形成的复杂网络。

In the negative and interrogative forms, of course, this is identical to the non-emphatic forms.

。但是，在否定句或疑问句里，这种带有"do"的方法表达的效果却没有什么强调的意思。

Go down on one's knees;kneel down

屈膝跪下。。。下跪祈祷