On the Nilpotence of Malcev Algebras

We investigate the nilpotence of a Malcev algebra M and of its standard enveloping Lie algebra L(M) = M(?)D(M, M). The main result shows that an ideal A of M is nilpotent in M if and only if the corresponding ideal I(A) = A(?)D(A, M) is nilpotent in L(M).

On the Strong Nilpotence Problem

In this paper, we describe the structure of quadratic homogeneous polynomial maps F = X ＋ H with JH3 = 0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F = X ＋ H is linearly triangularizable.

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

Let a, b be two generalized Drazin invertibleelements in a Banach algebra. An explicit expression for thegeneralized Drazin inverse of the sum a ＋ b in terms of a, b,as, bd is given. The generalized Drazin inverse for the sum oftwo elements in a Banach algebra is studied by means of thesystem of idempotents. It is first proved that a ＋ b ∈ Aqnll underthe condition that a, b ∈ Aqnil, aba = 0 and ab^2 = 0 and then theexplicit expressions for the generalized Drazin inverse of thesum a ＋ b under some new conditions are given. Also, someknown results are extended.

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 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.

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.

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.

The Solvable Lie Algebras with Filiform R_n Nilradicals

在这份报纸，我们明确地决定丝状的谎言代数学 Rn 的自守组发现有丝状的谎言代数学 Rn nilradicals.We 的不能分解的可解决的谎言代数学也证明有丝状的 Rn nilradicals 的不能分解的可解决的谎言代数学是完全的。

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.

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.

Multiplicative Jordan Decomposition in Integral Group Ring of Group K_8×C_5

In this article,we present the multiplicative Jordan decomposition in integral group ring of group K8 × C5,where K8 is the quaternion group of order 8.Thus,we give a positive answer to the question raised by Hales A W,Passi I B S and Wilson L E in the paper 'The multiplicative Jordan decomposition in group rings II.

Optimal control of quadratic functionals for affine nonlinear systems

In this paper we analyze the optimal control problem for a class of afflne nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.

On Some Varieties of Soluble Lie Algebras

In this paper, we study a class of soluble Lie algebras with variety relations that the commutator of m and n is zero. The aim of the paper is to consider the relationship between the Lie algebra L with the variety relations and the Lie algebra L which satisfies the permutation variety relations for the permutation ψ of {3, ··· , k}.

Exact Solution of a Linear Difference Equation in a Finite Number of Steps

An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

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.

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.

Enveloping Lie Algebras of Low Dimensional Leibniz Algebras

We calculate the enveloping Lie algebras of Leibniz algebras of dimensions two and three. We show how these Lie algebras could be used to distinguish non-isomorphic (nilpotent) Leibniz algebras of low dimension in some cases. These results could be used to associate geometric objects (loop spaces) to low dimensional Leibniz algebras.