下界

This paper discusses the arithmetic anamnesis of bubblesort and quicksort and gives some ameliorate method.

探讨了冒泡排序和快速排序两种排序方法的算法思想、改进方法以及存在的一些问题，给出了理论上比较次数最少的插入归并排序，得出排序算法时间复杂度的下界和上界。

In the fourth chapter, firstly it introduces the Boolean functions algebraic expressions of the 2-value clock-controlled stop-and-go generator and Gunther generator. It reveals the balanced property of the two kinds of Boolean functions, and studies the Walsh cycle spectrum and the autocorrelation function. It also obtains the coincidence rate of their output sequences with affine sum of some bits of input sequences, and analyzes their ability of resisting the best affine approximation cryptanalysis and differential cryptanalysis. Secondly, we properly present a new definition of the Best Affine Approximation, namely BAA on the Boolean vector functions, followed by the spectral characteristic of such defined BAA attacks through using the decomposition formula of the union distribution for random variables. A lower bound of such BAA attacks is proposed. Finally, we also study the spectral characteristic of the second kind of nonlinearity of Boolean vector functions, followed by a higher bound of such nonlinearity. Furthermore, the limited relationship between the second kind of nonlinearity of Boolean vector functions and the linear structure of the linear combination of every component is analyzed.

在第四章中，首先给出了2值密钥流"停走生成器"和"衮特生成器"中实际存在的布尔函数的代数表示，揭示了这两类布尔函数的平衡性，随后研究了它们的Walsh循环谱和自相关函数等，得到了它们的输出序列与输入序列中的某些bit的仿射项的符合率，分析了它们抵抗最佳仿射逼近攻击和差分攻击的能力；其次，我们合理地给出了布尔向量函数最佳仿射逼近的新定义，利用布尔随机变量联合分布的分解式考察了相应的谱特征，并给出了布尔向量函数与其最佳仿射逼近的符合率的一个下界；最后，我们还考察了布尔向量函数第二类非线性度的谱特征，给出了布尔向量函数第二类非线性度的一个上界，并揭示了布尔向量函数第二类非线性度与其各个分量的线性和的线性结构之间存在的制约关系。

The proposed method has the merit that the numerical characteristics of stochastic temperature field response can be obtained by analyzing the random temperature field just in one time.3. Perturbed numerical algorithm of nonprobabilistic convex set theoretical models on the temperature fieldThe uncertain parameters of physical parameters and initial boundary conditions of heat conduction are described by the convex model. The perturbation formulas of the upper and lower bounds of temperature field response with unknown-but-bounded parameters are given via the combination of matrix perturbation theory and the convex set theory model.4. Numerical analysis for transient temperature field with interval parametersConsidering the uncertainties of the transient heat transfer, the physical parameters and initial boundary conditions are regarded as interval variables.

该方法具有只进行一次随机温度场分析便可以获得其响应的数字特征的优点。3、温度场的非概率凸集合理论模型的摄动数值解法将结构导热的物理参数、温度场的初始和边界条件等不确定性参数以凸模型加以描述，基于矩阵摄动理论和处理不确定问题的凸集合理论模型的结合，导出有界不确定参数瞬态温度场响应所在集合的上、下界摄动计算公式。4、具有区间参数的瞬态温度场数值分析考虑结构瞬态热传导问题的不确定性，将结构各物理参数和温度的初、边值条件均视为区间变量。

When the information of the bent function exceeds the lower bound, we can find it.

在对bent函数的研究过程中，我们希望找到一个下界，当我们知道bent函数的信息量超过这个下界时，我们就能得到这个函数。

We investigate the lower bounds of concurrence for tripartite, and even multipartite quantum mixed states in Sec.

在本文的第四部分，通过寻找Concurrence与推广的部分转置之间的关系，我们给出了三体及多体量子态Concurrence的一个下界，这个下界是易于计算的。

In chapter three, we study the lower orientable strong radius and strong diameterof the Cartesian product of graphs and prove that: srad(G_1×G_2)= 2r(G_1×G_2),sdiam(G_1×G_2)≤min{sdiam(G_1)+sdiam(G_2), 2(G_1×G_2), 4r(G_1×G_2)}. Furthermore,we establish three sufficient conditions for sdiam(G_1×G_2)= 2d(G_1×G_2)holds and determine the values of the lower orientable strong diameters of somespecial graphs. Moreover, we give the exact value of SDIAM, a lowerbound for SDIAM, an upper and lower bound for SRAD andSRAD, respectively.

在第三章，研究了笛卡尔乘积图G_1×G_2的最小强半径，证明了如下结果：srad(G_1×G_2)=2r(G_1×G_2)，sdiam(G_1×G_2)≤min{sdiam(G_1)+sdiam(G_2)，2d(G_1×G_2)，4r(G_1×G_2)；给出sdiam(G_1×G_2)=2d(G_1×G_2)成立的三个充分条件，并由所给出的充分条件确定了一些特殊笛卡尔乘积图的最小强直径的值；确定了SDIAM的确切值，SDIAM的下界，SRAD和SRAD的上、下界。

In this paper,lower bounds on the spectral radius of adjacency matrices of trees and perfect trees are discussed by algebraic method and edge switching of graph,and all trees which reach lower bounds on the spectral radius are obtained.

利用代数方法、图的边变换，以及树的邻接矩阵谱与L ap lacian谱的关系，研究树和完美树的邻接矩阵谱半径和L ap lacian谱半径的下界，给出达到下界的所有极树，得到的新结果改进了文献[2]的结论。

Compare to existing results for general graph, new lower bound on the size of the minimum feedback vertex set of-star graph is given based on its special structure.

本文主要讨论了关于组合星图反馈结点集的下界问题，在基于组合星图的性质和已有相关结果的基础上，给出了组合星图反馈结点集的新的下界。

In many cases,even it is very difficult to find out good bounds of the crossing numbers of graphy .

Johnson已经证明确定图的交叉数是一个NP完全问题(见文献[4])，因为其难度，我们能够确定交叉数的图类非常少，在许多情况下，即使找出图的交叉数的一个好的上界或下界也是非常困难。

So to find a good lower bound is an very important issue in the theory of uniform designs.The present paper aims at obtaining some new lower bounds for the symmetric L_2-discrepancy on some kind of symmetric factorials and asymmetric factorials.

在本文中我们针对某些类型的对称和非对称因子设计得到了对称化L_2—偏差的新的下界，利用这些下界我们不仅可以衡量给定设计的均匀性程度，还可以帮助我们构造均匀设计。

Do you know, i need you to come back

你知道吗，我需要你回来

Yang yinshu、Wang xiangsheng、Li decang,The first discovery of haemaphysalis conicinna.

1〕 杨银书，王祥生，李德昌。安徽省首次发现嗜群血蜱。