矩阵方程AX=A+X有正定解和幂零解的充要条件

给出了矩阵方程AX=A+X有解、实对称解、正定解和幂零解的充要条件.

阶为偶数的超可解群与幂零解

证明了生成关系为ａｎ＝ｂ２＝ｃ２＝ｅ，（ａｂ）２＝（ｂｃ）２＝ｅ，ａｃ＝ｃａ的三元生成群为超可解群．

Schur Forms and Normal-Nilpotent Decompositions

Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.

直接有限环的新刻画

从一道数学竞赛题的解法出发,得出直接有限环的一些等价刻画,证明了如下结论:1)R为直接有限环当且仅当对任意a,b∈R,当ab=a+b时,有ab=ba;2)R为直接有限环当且仅当对任意a∈R,1-a左可逆时,方程ax=a+x有解;3)R为直接有限环当且仅当对任意a∈R,方程ax=a+x有解时,方程x^(2)-(ax^(2)+x^(2)a)+ax^(2)a=1有解.

The Nilpotency of Lie Triple Algebras

In this paper, we mainly concerned about the nilpotence of Lie triple algebras.We give the definition of nilpotence of the Lie triple algebra and obtained that if Lie triplealgebra is nilpotent, then its standard enveloping Lie algebra is nilpotent.

On Pronormal Minimal Subgroups of Finite Groups

By using pronormal minimal subgroups and weak left Engel elements ofprime order of the normalizers of Sylow subgroups of a finite group G, we obtain somesufficient conditions for G to be p-nilpotent, nilpotent and supersolvable respectively,which generalize some known results.

ON SOLVABLE COMPLETE LIE SUPERALGEBRAS

The authors discuss the properties of solvable complete Lie superalgebra, proving that solvable Lie superalgebras of maximal rank are complete.

COMPLETE LIE ALGEBRAS WITH l-STEP NILPOTENT RADICALS

The authors first give a necessary and sufficient condition for some solvable Lie algebras with l-step nilpotent radicals to be complete, and then construct a new class of infinite dimensional complete Lie algebras by using the modules of simple Lie algebras. The quotient algebras of this new constructed Lie algebras are non-solvable complete Lie algebras with l-step nilpotent radicals.

Minimal inner-Σ-Ω-groups and their applications

We introduce and study the minimal inner-Σ-Ω-groups and the minimal outer-Σ-■-groups.Then we give some applications and obtain some interesting results,including characterizations of nilpotent,supersolvable,solvable,and p-closed groups in terms of the join of two conjugate cyclic subgroups having the same property.

On conjugacy class sizes of primary and biprimary elements of a finite group

Let m, n 〉 1 be two coprime integers. In this paper, we prove that a finite solvable group is nilpotent if the set of the conjugacy class sizes of its primary and biprimary elements is {1, rn, n, mn}.