Discussion on the Complex Structure of Nilpotent Lie Groups Gk

Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group Gk, defined by the nilpotent Lie algebra g/ak, where g is the Lie algebra of G, and ak is an ideal of g. Then, J gives rise to an almost complex structure Jk on Gk. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure Jk on the nilpotent Lie group Gk, and Jk is also nilpotent.

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.

THE CLASSIFICATION OF NILPOTENT LEIBNIZ 3-ALGEBRAS

The notions of the nilpotent and the strong-nilpotent Leibniz 3-algebras are defined. And the three dimensional two-step nilpotent, strong-nilpotent Leibniz 3-algebras are classified.

一种构造高维Nilpotent正规型的方法

正规型方法是一种有效的简化一类非线性方程的方法．今提出了一种简便的代数方法去构造高维非线性系统的Ｎｉｌｐｏｔｅｎｔ范式．通过引入一系列简单的变换和代数运算，而无需求解任何偏微分方程，即可得到高维非线性系统的Ｎｉｌｐｏｔｅｎｔ范式．以四维非线性系统为例介绍这个方法，该方法完全适用于分析高于四维的非线性系统的Ｎｉｌｐｏｔｅｎｔ范式．

Using Normal Form of Nilpotent Matrices over Finite Fields to Construct Cartesian Authentication Codes

Let Fq be a finite field. In this paper, a construction of Cartesian au-thentication codes from the normal form of a class of nilpotent matrices over the field Fq is presented. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the probabilities PI and PS, of a successful im-personation attack and of a successful substitution attack respectively, of these codes are also computed.

Finitely Generated Torsion-free Nilpotent Groups Admitting an Automorphism of Prime Order

Let G be a finitely generated torsion-free nilpotent group and a an automorphism of prime order p of G. If the map ψ ： G → G defined by g^φ = [g, α] is surjective, then the nilpotent class of G is at most h（p）, where h（p） is a function depending only on p. In particular, if α^3 = 1, then the nilpotent class of G is at most 2.

Linear Operators That Strongly Preserve Nilpotent Matrices over Antinegative Semirings

Let S be an antinegative commutative semiring without zero divisors and Mn（S） be the semiring of all n × n matrices over S. For a linear operator L on Mn（S）, we say that L strongly preserves nilpotent matrices in Mn（S） if for any A ∈ Mn（S）, A is nilpotent if and only if L（A） is nilpotent. In this paper, the linear operators that strongly preserve nilpotent matrices over S are characterized.

Generalized Derivations with Skew Nilpotent Values on Lie Ideals

This article gives characterizations of generalized derivations with skew nilpotent values on noncommutative Lie ideals of a prime ring. The results simultaneously generalize the ones of Herstein, Lee and Carini et al.

Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center

In this paper we deal with a cubic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have 5 limit cycles by using bifurcation theory.

A characteristic condition of finite nilpotent group

This paper gives a characteristic condition of finite nilpotent group under the assumption that all minimal subgroups of G are well-suited in G.

The Hypoellipticity for a Class of Nonhomogeneous Convolution Operator on the Nilpotent Lie Group H^n×R^k

In this paper,we use the method of representation of Lie group to study a class of nonhomoge- neous convolution operator on the nilpotent Lie group H^M×R^k,and give a criteerion of their hypoellipticity.

A Note on Nilpotent Operators

We find that a bounded linear operator T on a complex Hilbert space H satisfies the norm relation |||T|na|| =2q, for any vector a in H such that q≤(||Ta||-4-1||Ta||2)≤1.A partial converse to Theorem 1 by Haagerup and Harpe in [1] is suggested. We establish an upper bound for the numerical radius of nilpotent operators.

Herz Spaces on Nilpotent Lie Groups and Its Applications

In this paper, we define Herz type spaces on nilpotent Lie groups and give the boundedess of heat kernel and Riesz transform on Herz spaces.

Two Theorems about Nilpotent Subgroup

In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theorems about nilpotent subgroup.

GROUPS WHOSE PROPER QUOTIENTS ARE FINITE—BY—NILPOTENT

A group G is said to an FNc-group if the （c+1）th term γc+1G of its lower central series is finite（or equivalently if it is finite-by-nilpotent of class≤c）.A group G is called a JNFNc-group if all itsproper quotients are FNc-groupe but G itself is not.The structure of JNFA-groups（JNFNc-groupswith c=1） has been described in a joint work of D.J.S Robinson and the author（see J.Algebra,（2）118（1988）,346～368）.Now we consider JNFNc-groups with non-trivial Fitting subgroup forarbitrary c and give a complete description of groups of this type.

Finite Groups in Which Every Maximal Subgroup Is Nilpotent or a TI-Subgroup or Has Order p'

We obtain a complete characterization of the structure of a finite group G in which every maximal subgroup is nilpotent or a TI-subgroup or has order p'for any fixed prime divisor p of|G|.Moreover,we show that there exists at most one prime divisor q of|G|such that G is neither q-nilpotent nor q-closed.

On the Product of Two Nilpotent Subgroups of a Finite Group

It is known that the product of two nilpotent subgroups of a finite group is not necessarily nilpotent.In this paper, we study the influence of the Engel condition on the product of two nilpotent subgroups. Our results generalize some well-known results.

Subalgebras of Nilpotent Matrices

In this note we describe explicitly the subalgebras of nilpotent matrices and obtain some interesting results.

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.

The Automorphism Groups of Complete Lie Algebras with Commutative Nilpotent Radical

The automorphism groups of finite-dimensional complete Lie algebras withcommutative nilpotent radical over complex field C are given.