The Analytic and Formal Normal Form for the Nilpotent Singularity

We study orbital normal forms for analytic planar vector fields with nilpotent singularity. We show that the Takens normal form is analytic. In the case of generalized cusp we present the complete formal orbital normal form; it contains functional moduli. We interprete the coefficients of these moduli in terms of the hidden holonomy group.     

On a Conjecture of Groves for Modules Over Infinite Nilpotent Groups

We show that a conjecture of Groves for modules over nilpotent groups of class 2 holds for the codimension 2 case with certain assumptions.     

On the Structure of Specht Modules

Most of our notation is taken from James [7], where furtherdetails of the representation theory of the symmetric groupsmay be found; note, however, that we write functions on theleft. Let n be a non-negative integer, and a partition of n. Saythat two -tableaux are row equivalent if one can be obtainedfrom the other by permuting the entries within each row, anddefine column equivalence similarly. Let _row and _col denotethese relations.     

Idempotent and Nilpotent Submodules of Multiplication Modules

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.     

利用有限域上一类幂零阵的标准形构造Cartesian认证码

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.     

基于2-Jordan型幂零阵的Cartesian认证码的构造

利用群在集合上的共轭作用,给出了一种由2 -Jordan型幂零矩阵构造Cartesian认证码的新方案,计算了相关参数,并在假定编码规则按等概率分布选取时分析了模仿攻击和替换攻击成功的概率.     

On Transformations into Normal Form

We discuss a certain class of transformations of an ordinary differential equation into normal form near a stationary point. These transformations preserve certain special structures and one can derive from them a universally applicable algorithm to determine a normal form up to any desired degree. The algorithm can be programmed with the help of any computer algebra system. In the same manner, transformations into normal form on an invariant manifold are discussed.     

On the Direct Computation of the Byrnes-Isidori Normal Form

If an input-affine single-input single-output control system has a well-defined relative degree, there exists a (local) change of coordinates transforming the system into the Byrnes-Isidori normal form. By this transformation, the system is decomposed into two subsystems. The first subsystem can be linearized by feedback, and the second subsystem does not directly depend on the input. To achieve this structure, the last components of the transformation have to fulfill a partial differential equation. Practically, the appropriate choice of these coordinates is usually left to the (experienced) user. In this note, the author suggests a direct calculation based on the computation of flows of certain vector fields. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Dimension of Nilpotent Algebras

The Eggert conjecture claims that a finite commutative algebra R over a field of prime characteristic p has the property dim Rdim R⁽¹), where R(1) is the subspace of R spanned by the pth powers of elements of R. We obtain results related to this conjecture and results on nilpotent algebras of rather high nilpotency class.     

广义半交换环的幂零结构

研究了广义半交换环的幂零结构,定义了一类新的环类,即幂零$\alpha$-半交换环.说明了$\alpha$-半交换环与半交换环, $\alpha$-半交换环和$\alpha$-刚性环等环密切相关,通过构造反例说明了幂零$\alpha$-半交换环未必是$\alpha$-半交换环.研究了幂零$\alpha$-半交换环的各种性质,推广和统一了与环的半交换性质有关的若干结论.     

Normal Forms of Modules over Admissible Algebras with Formal Two-ray Modules

The aim of the paper is to classify the indecomposable modules and describe the Auslander-Reiten sequences for the admissible algebras with formal two-ray modules.     

On Co-Hopfian Nilpotent Groups

Co-Hopfian finitely generated torsion-free nilpotent groupsare characterized in this paper in terms of their Lie algebraautomorphisms. Many examples of such groups are also constructed.2000 Mathematics Subject Classification 17B30, 20F18 (primary).     

Multiplicativity of the Smith Normal Form

New conditions are derived which are sufficient to make the Smith Normal Form a Multiplicative matrix function.     

关于幂零群的π-孤立子群

从群的上、下中心列出发，给出了涉及幂零群的π—孤立子群的一些结果。     

On finding a worst-case optimal Fourth Normal Form database decomposition

Although several Fourth Normal Form (4NF) decomposition algorithms have been published, the problem's computational complexity remains an open question. Part of the difficulty is that no one has established an upper bound on the size of a 4NF decomposition scheme for a given number of attributes. We prove such an upper bound and we present an algorithm which is worst-case optimal: it never produces a 4NF decomposition scheme that is larger than the upper bound.     

关于合同变换矩阵的一种结构形式

对于任意非零的对称矩阵A,B∈P~(n×n),介绍A,B在P中合同的一个充分必要条件,并基于该条件构造满足P~TAP=B的所有可逆矩阵P∈P~(n×n)是目标.针对■和■这两种常见情形,获得了P的明确结构.     

On Ordering the Groups with Nilpotent Commutant

We prove that every group with nilpotent commutant, having an abelian normal subgroup such that the factor by this subgroup is nilpotent, is preorderable if and only if the group is -torsion-free. An example is exhibited of a nonorderable -torsion-free group with two-step nilpotent radical. This example demonstrates that for the variety of groups with nilpotent commutant the absence of -torsion in a group is not a sufficient condition for orderability.     

一类带有非退化不变对称双线性函数的幂零李代数

本文证明了一类带有非退化不变对称双线性函数、幂零指数为Ｎ的李代数 满足：定理１．（ｉ）ｃ（ｇ）＝ｇ~Ｎ; （ｉｉ）ｄｉｍｃ（ｇ）＝ｌ．ｌ是ｇ的生成元个数．定理３给出了这类李代数结构的充分且必要条件．