幂

In 1999, Nicholson further introduced the concept of strong clean rings as follows: A ring R is said to be strongly clean if every element of R is the sum of an idempotent element and an invertible element that commute with each other.

W.K.Nicholson进一步引入了强clean 环的定义：一个环R 被称为强clean环，如果R中的任意一个元素均可以表示成一个幂等元和一个单位和的形式，并且幂等元和单位可换。

Therefore, in order to offer reference to Readers, based on idempotent matrix, involutory matrix, nilpotent matrix, diagonal matrix, the main character of special matrix are proved in this paper after the Defined and algorithm of eigenvalue of matrix .for example , some problems of the eigenvalues of matrix are solved in a special method based on the eigenvalues of matrix .

为此， 本文除了介绍矩阵特征值的定义和算法外，还围绕幂等矩阵、幂零矩阵、对角矩阵、等特殊矩阵给出了其主要性质并加以证明，同时还介绍了一些特殊矩阵的特征值的算法，例如：本文利用矩阵的特征值，对与矩阵的特征值相关的一些典型问题给出了较好的处理方法。

Finally, after defining an evaluation module of the corresponding Loop superalgebra (Section 4), two major results of the paper -Theorem 4.land Theorem 4.2 are proved: Theorem 4.1 reduces the irreducibility of the tensor product of finitely many evaluation modules to the irreducibility of the tensor product of finitely many irreducible modules of a nilpotent Lie superalgebra; Theorem 4.2 gives a criterion for the tensor product of such modules to be irreducible.

第4节在定义了相应的Loop超代数的赋值模之后，证明了本文的两个主要结论：定理4.1和定理4.2。定理4.1将有限多个赋值模其张量积的不可约性归结为一幂零李超代数的限多个不可约模其张量积的不可约性；定理4.2利用不可约指标给出了一幂零李超代数的限多个不可约模其张量积仍不可约的判别准则。

Based on the outcome of Xu Yang and Qin Keyun about lattice implication algebra and lattice-valued prepositional logic LP with truth-value in a lattice implication algebra, the author studied the properties of lattice implication algebra and the α-automated reasoning method based on α-resolution principle of LP. The specific contents are as follows: The Study of Lattice Implication Algebra On the basis of previous results of lattice implication algebra, this part consists of the following three points: 1. Some properties of lattice implication algebra L were discussed, and some important results were given if L was a complete lattice implication algebra. 2. The properties of left idempotent elements of lattice implication algebras were discussed, and the conclusion that lattice implication algebra L was equals of the directed sum of the range and dual kernel of a left map constructed by a left idempotent element was proved. 3. The properties of the filters of lattice implication algebra were discussed, the theorem was shown that they satisfy the hypothetical syllogism and substitute theorem of the propositional logic. 4. The concept of weak niters of lattice implication algebras and their properties and structures are discussed. It is proved that all weak filters of a lattice implication algebra form a topology and the the implication isomorphism betweem two lattice implication algebras is a topological mapping between their topological spaces. The Study of α-automated reasoning method based on the lattice-valued propositional logic LP In this part, the author given an a-automated reasoning method based on the lattice-valued propositional logic LP.

本文基于徐扬和秦克云的关于格蕴涵代数和以格蕴涵代数为真值域的格值命题逻辑系统LP的研究工作，对格蕴涵代数以及格值命题逻辑系统LP中基于α-归结原理的自动推理方法进行了系统深入的研究，主要有以下两方面的研究成果：一、关于格蕴涵代数的研究 1、对格蕴涵代数的格论性质进行了研究，得到了当L为完备格蕴涵代数时，关于∨，∧，→运算的一些结果； 2、对格蕴涵代数的左幂等元进行了研究，证明了格蕴涵代数L可以分解为任何一个左幂等元所对应的左映射的像集合与其对偶核的直和； 3、对格蕴涵代数的滤子的性质进行了研究，证明了滤子的结构相似于逻辑学中的Hypothetical syllogism规则和替换定理； 4、给出了格蕴涵代数中弱滤子的概念，对弱滤子的性质个结构进行了研究，证明了格蕴涵代数的全体弱滤子构成一个拓扑结构，格蕴涵代数之间的蕴涵同构是相应的拓扑空间之间的拓扑映射。

The paper provides the theorem of the limit of combination of power and exponent function ,which gives us a new method to work out the limit of this kind of function.

给出了幂指函数极限定理，对求幂指函数极限的方法进行了讨论，并给出一些应用实例。

Based on the analysis of some aspects, such as the approximative algorithm of FOC, the Tustin transform theory and its generating function formula's character, the convergence guarantee of binomial power function by Maclaurin expanding, and the consideration of the limitation of conventional methods, an improved method is proposed to compute the numerical evalution of FOC using PSE and Tustin transform and is further applied to solving the linear FOS.

以分数阶算子近似方法的分析研究为基础，基于Tustin变换理论及其用于分数阶算子的离散生成函数公式特点，利用二项式幂函数的Maclaurin展开能够保证收敛的特性，考虑常用算法的局限性，提出了一种改进的基于幂级数展开和Tustin变换的分数阶运算方法，并应用于线性分数阶系统的求解，给出了递推算法的详细推导。

Based on the observation and analysis of social communication network, it is noticed that in social communication network, the transfer of message is optimized.

基于P2P网络的幂律分布和小世界特性，通过对复杂网络幂律特性产生机制的分析并借鉴人际传播中谣言传播机制，提出了一种结合择优连接机制和谣言传播中兴趣衰减机制的信息资源搜索方法。

Op1 + op2 The addition operator will add two numbers. op1 - op2 The subtraction operator will subtract two numbers. op1 * op2 The multiplication operator will multiply two numbers. op1 / op2 The division operator will divide two numbers. op1 % op2 The modulus operator will return the remainder of the division of two integer operands. op1 xx op2 The exponentiation operator will raise op1 to the power of op2.++op1 The pre-increpment operator will increase the value of op1 first, then assign it. op1++ The post-increment operator will increase the value of op1 after it is assigned.--op1 The pre-decrement operator will decrease the value of op1 before it is assigned. op1-- The post-decrement operator will decrease the value of op1 after it is assigned.

op1 + op2 对两个数值做加法操作 op1 - op2 对两个数值做减法操作 op1 * op2 对两个数值做乘法操作 op1 / op2 对两个数值做除法操作 op1 % op2 求两个整型数值的余数 op1 xx op2 求幂操作：求 op1 的 op2 次幂++op1 前加操作： op1 的值先增加，然后将值赋给自身 op1++后加操作： op1 的值先赋给自身，再增加值--op1 前减操作： op1 的值先减少，然后将值赋给自身 op1—后减操作： op1 的值先赋给自身，再减少值

Nilpotency of a table algebra depends on that of their table subsets and quotient subsets, which is not fully corresponding to extension problems of nilpotent groups.

同时会发现该条件在表代数幂零性的判别方面并不完全对应于幂零群的中心扩张。

In the present paper,we investigate solvable radicals and Hopkins nilpotent radicals for Lie triple systems and prove that both radicals are invariant under actions of deriva- tions.

本文讨论李三系的可解根基和Hopking幂零的某些性质及导子作用下的不变性，讨论了李三系次理想的某些性质，证明了李三系为幂零的当且仅当每个子系都是次理想。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。