幂零的

This paper researches linear maps preserving orthogonality,obtains the linear maps preserving othrogonality,it is either a rank -declining or rank -keeping map or a map with nilpotent element in its codomain.

利用分块成向量的方法证明了MnMn(F为域F上所有n×n矩阵构成的乘法半群上的n×n拟正交矩阵组至多含有n个矩阵，利用方程组的解的理论证明了Mn中与给定矩阵A构成两两拟正交矩阵组的矩阵个数不超过n-Rank+1，从而得到Mn上保持拟正交性的线性映射φ要么是降秩的或者保秩的映射，要么φ的值域中含有幂零元。

Through the operation of formal power series, we set up the expansion of Ar, a popularizing of the common polynomial theorem, where r is a non-zero real number.

通过形式幂级数A的幂运算，建立了Ar的展开式，这里r为非零实数，这是常见的多项式定理的推广。

It comes from "googol", the number 1 followed by 100 zeros.

"google"一词源于单词"googol"，即10的100次幂，写出的形式为数字1后跟100个零，表示数量极大。

On the basis, three equivalent statements are obtained. Let S be a semigroup with left central idempotents, then (1) S is a quasi-right semigroup;(2) S is a quasi-completely regular, and RegS is an ideal;(3) S is a nil-extension of strong semilattice of right semigroup.

在此基础上得到了3个等价命题：若S为具有左中心幂等元半群，则(1) S为拟右半群；(2) S为拟完全正则的，RegS为S的理想；(3) S为右群强半格的诣零理想扩张。

In this paper, the author gives sufficient and necessary condition of solvable group using the weak c-normality of maximal and Sylow subgroup and gets some results about supersolvability and p-nilpotentlity of groups

作者研究了弱c-正规的性质并用Sylow子群和极大子群的弱c-正规性来确定一些群的结构，给出了一个群为可解群的充分必要条件，同时由极大子群、Sylow子群的弱c-正规性得到群G p-幂零、超可解的有关定理。

On the basis of the definition of matrix traces , this paper discusses their characteristics at first and then according to the norm of the F of square matrix and Cauchy-Schwarz inequality gives how to prove the zero matrix, unsimilar matrix, number cloth matrix, column matrix idempotent matrix and non-equality matrix.

根据矩阵迹的定义，首先给出了矩阵迹的性质，然后依据方阵的F—范数定义Cauchy—Schwarz不等式，给出了零矩阵，不相似矩阵，数幂矩阵，列矩阵，幂等矩阵及矩阵不等式的证法。

By using properties of quasi-regular semigroups and left central idempotents, some statements are proved. Let S be a quasi-right semigroup, then (1) S is a quasi-completely regular semigroup;(2) RegS is a completely regular semigroup;(3) R(superscript *) is the smallest semilattice congruence on S;(4) Each R-class T(subscript α) on RegS is a right group;(5) T(subscript α)G(subscript α)×E(subscript α), where G(subscript α) is a group, E(subscript α) is a right zero semigroup.

利用拟正则半群和左中心幂等元的性质，证明了S为拟右半群时，(1) S为拟完全正则半群；(2) RegS为完全正则半群；(3) R为S上的最小半格同余；(4) RegS上的每个R-类T为右群；(5) TG×E，其中G为群，E为右零半群。

Special morphisms such as monomorphisms, epimorphisms, sections, retractions, extremal monomorphisms, extremal epimorphisms, constant morphisms, coconstant morphisms, zero morphisms and special objects such as initial objects, terminal objects and zero objects are studied and their characterization is obtained.

较为系统地讨论了范畴Quant的性质，考察了范畴Quant中单态射、满态射、截节、收缩、极端单态射、极端满态射、常值态射、余常值态射、零态射等特殊态射和始对象、终对象等特殊对象，给出了它们的具体刻划，得到了范畴Quant是良幂的，余良幂点化范畴。

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