上有界的

Brown and Halmos [3] showed that the only compact Toeplitz operator on the Hardy space is the zero operator and a Toeplitz operator is bounded on the Hardy space if and only if its symbol is bounded.

Brown和Halmos[3]证明了经典Hardy空间上只有零算子才是紧的Toeplitz算子，并且证明了Toeplitz算子有界等价于它的符号是有界的。

Theorem 1 (Boundedness and max-min theorem) If is a continuous function on a closed interval, then has an absolute maximum and an absolute minimum on the interval.

定理 1 (有界性与最大值最小值定理)在闭区间上连续的函数在该区间上有界且一定能取得它的最大值和最小值。

In chapter three, we prove that there exist solutions to the Ky Fan variation inequality, as the set-valued mappings are defined on spheres in infinite dimensional Banach spaces or odd dimensional Euclidean spaces, following from these theorems, we obtain some fixed point theorems for set-valued mappings defined on a sphere. When G is an approximate compact convex subset of E, or G is a almost quasi-convex set-valued mapping, we prove that there exist solutions to and type generalized Ky Fan variation inequality, following these theorems, we prove several best approximation theorems and coincidence theorems involving two set-valued mappings and two different spaces. In chapter four, we first present a new Simplicial algorithm for computing the Leray - Schauder fixed points, the algorithm can solve the set-valued nonlinear complementarily problem. We give a condition to guarantee the computation proceeding in a bounded region. We present integer-labeling algorithms for computing fixed points of some set-valued mappings, the best approximation points and solutions to a kind of set-valued variation inequalities.

第四章给出了计算定义在非凸集上的非自映射的Leray-Schauder不动点的算法，而现有的不动点算法都是计算凸集的上半连续集值自映射的不动点；给出了保证计算有界的一个充分条件，我们的条件大大弱于Mdrrill条件，我们的算法也可用来计算Eaves不动点；给出了集值非线性互补问题存在解的一个充分条件，此时可利用Leray-Schauder不动点算法来求解；向量标号算法以往是计算集值映射不动点的唯一有效算法，我们给出用整数标号算法计算一类集值映射的Kakutani 不动点的算法；定义在紧凸集上的连续映射不一定有不动点，但一定有最近点，最近点是不动点概念的推广，我们给出了计算最近点的算法；集值映射变分不等式尚无有效的求解算法，我们给出求解一类集值映射变分不等式的算法。

Property 1 (Boundedness and max-min theorem) Let be a continuous function of several variables on a closed region D, then has an absolute maximum and an absolute minimum on the region.

性质1（有界性与最大值最小值定理）在有界闭区域D上的多元留恋许函数，必定在D上有界且一定能取得它的最大值和最小值。

On the basis of this, we estaiblish a theory of the space of bounded linear operators on F* spaces.

在此基础上，建立F~*空间上有界线性算子空间的理论。

As the inversion and compression attack includes the question relating to solving thepreimage from image of function, a subdivision algorithm for solving preimage of function withbounded derivative based on Lagrange mean value theorem is proposed firstly.

由于逆推压缩攻击方法涉及混沌映射的求逆，针对复杂混沌映射求逆困难的问题，运用拉格朗日中值定理，我们提出了求解导函数有界的函数原象的分割算法，给出了使分割算法计算复杂性上界达到最小的充分条件。

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。

In Chapter 5,we obtain a boundedness results of commutators of convolution opera-tors on MK_~according to the corresponding boundedness of the sublinear operators on the homogeneous Morrey-Herz spaces.

第五章通过次线性算子在齐次Morrey-Herz空间上的有界性导出了一个卷积型交换子在MK_~上有界的结果。

After the addition of carbon, the microhardness in the grain boundary is about 150 MPa higher than that in the grain. The fracture type of pure Mo is intergranular fracture, the grain boundary strength of Mo-C alloy is strengthened due to the addition of carbon and the fracture type is changed to transgranular fracture, transcrystalline fracture and a large number of cleavage planes and steps are observed on the bending fracture surface. But if the carbon content is too high, the carbide precipitation will be severe at grain boundaries and is deleterioated to the ductility of the alloy.

添加碳后，合金的晶界显微硬度比合金的晶内显微硬度高150MPa左右；纯钼的断裂方式以沿晶断裂为主，添加碳后，由于合金的晶界强度得到提高，合金的断裂方式变成以穿晶断裂为主，断口上有大量的解理面，具有明显的河流状花样和解理台阶，且塑性越好的断口，撕裂岭也越多；当添加的碳含量偏高时，在晶界处产生了大颗粒的碳化物，这种大颗粒的碳化物会降低合金的塑性。

Definition Let be a smooth surface in three-space, and let be bounded on .

定义 设是三维空间中的有向光滑曲面，在上有界。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。