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.

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.

Bogomolov Multipliers for Some p-groups of Nilpotency Class 2

The Bogomolov multiplier B0 （G） of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier is an obstruction to Noether＇s problem. We show that if G is a central product of G1 and G2, regarding Ki ≤ Z（Gi）,i = 1,2, and θ ： G1 →G2 is a group homomorphism such that its restriction θ｜K1 ： K1 → K2 is an isomorphism, then the triviality of Bo（G1/K1）, Bo（G1） and B0（G2） implies the triviality of Bo（G）. We give a positive answer to Noether＇s problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 （with the usual requirement for the roots of unity）.

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.

Homotopy epimorphisms preserve nilpotency

Recall that f: X→Y is a homotopy epimorphism (monomorphism) if given u, v: Y→W(u, v:W→X), u(?)of(?)v(?)f implies u(?)v(f(?)u(?)f(?)v implies u(?)v). In ref. [1], Lin and Shen

On Local Nilpotency of the Normal Subgroups of a Group

A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a group G is said to have property v if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP（G） is the Hirsch-Plotkin radical of G, and φf （G） is the intersection of all the maximal subgroups of finite index in G （here φf（G） = G if no such maximal subgroups exist）. It is shown that a group G has property μ if and only if HP（G/φf（G）） = HP（G）/φf（G） and that the class of groups with property v is a proper subclass of that of groups with property it. Also, the structure of the normal subgroups of a group： nilpotency with the minimal condition, is investigated.

ON REDUCIBILITY OF THE SELF-HOMOTOPY EQUIVALENCES OF WEDGE SPACES

Reducibility of the self-homotopy equivalences of wedge spaces is studied and some conditions implying the reducibility are obtained.

Successive Approximation of p-Class Towers

Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.

Some additive results for the generalized Drazin inverse in a Banach algebra

Let a, b be two generalized Drazin invertible elements in a Banach algebra. An explicit expression for the generalized Drazin inverse of the sum a + b in terms of a,b,a^d,b^d is given. The generalized Drazin inverse for the sum of two elements in a Banach algebra is studied by means of the system of idempotents. It is first proved that a + b∈A^(qnil) under the condition that a,b∈A^(qnil),aba = 0 and ab^2= 0 and then the explicit expressions for the generalized Drazin inverse of the sum a + b under some newconditions are given. Also, some known results are extended.

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．

The Solvable Lie Algebras with Filiform R_n Nilradicals

In this paper we explicitly determine automorphism group of filiform Lie algebra Rn to find the indecomposable solvable Lie algebras with filiform Lie algebra Rn nilradicals.We also prove that the indecomposable solvable Lie algebras with filiform Rn nilradicals is complete.

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.

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.

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.

On Skew Power-serieswise nil-Armendariz Rings

For a ring endomorphism α, in this paper we introduce the notion of s-power- serieswise nil-Armendariz rings, which are a generalization of α-power-serieswise Armendariz rings. A number of properties of this generalization are established, and the extensions of α- power-serieswise nil-Armendariz rings are investigated. Which generalizes the corresponding results of nil-Armendariz rings and power-serieswise nil-Armendariz rings.

Thermal Stresses and Theorem on Decomposition

The thermal expansion strain is considered as a special case of eigenstrain.The authors proved the theorem on decomposition of eigenstrain existing in a body into two constituents:Impotent eigenstrain(not causing stress in any point of a body)and nilpotent eigenstrain(not causing strain in any point of a body).According to this theorem,the thermal stress can be easily found through the nilpotent eigenstrain.If the eigenstrain is an impotent one,the thermal stress vanishes.In this case,the eigenstrain must be compatible.The authors suggest a new approach to measure of eigenstrain incompatibility and hence to estimate of thermal stresses.

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.