On Engel's Theorem for Leibniz Algebras

Engel's Theorem has been generalised to Leibniz algebras by Ayupov and Omirov, and in a stronger form by Patsourakos. I give a simpler proof of their results.     

The Subgroup Normalizer Problem for Integral Group Rings of Some Nilpotent and Metacyclic Groups

For a group G and a subgroup H of G, this article discusses the normalizer of H in the units of a group ring RG. We prove that H is only normalized by the “obvious” units, namely products of elements of G normalizing H and units of RG centralizing H, provided H is cyclic. Moreover, we show that the normalizers of all subgroups of certain nilpotent and metacyclic groups in the corresponding group rings are as small as possible. These classes contain all dihedral groups, all finite nilpotent groups, and all finite groups with all Sylow subgroups being cyclic.     

Character tables of commutative hecke algebras associated with exceptional weyl groups

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups when they are commutative. In the case when Weyl groups are of classical type, they are already known in [D.1] and [D.2]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with exceptional Weyl groups.     

A Characterization of Root Classes of Groups

We prove that a class of groups is root in a sense of K. W. Gruenberg if, and only if, it is closed under subgroups and Cartesian wreath products. Using this result we obtain a condition which is sufficient for the generalized free product of two nilpotent groups to be residual solvable.     

Nilpotent Elements and Idempotents in Commutative Group Rings #

Flatness properties of acts over monoids and their connection with monoid amalgamation have been investigated for almost four decades and a substantial literature on the subject has now appeared. Analogous research, concerning the action of partially ordered monoids on partially ordered sets and its relation to pomonoid amalgamation, was begun in 1980s in two articles by S. M. Fakhruddin. The subject then remained dormant until the recent past when several articles on flatness in the setting of ordered monoids acting on posets (briefly, S-posets) appeared. It has now been established that the introduction of order results in severe restrictions as far as absolute flatness is concerned. Also, after formulating in the ordered context the Representation Extension and Right Congruence Extension Properties, first used by T. E. Hall to study semigroup amalgams, the authors recently observed in their article [4 Bulman-Fleming , S. , Nasir , S. ( 2010 ). Examples concerning absolute flatness and amalgamation in pomonoids . Semigroup Forum 80 : 272 – 292 .[Crossref], [Web of Science ®] , [Google Scholar]] that inverse monoids, though being amalgamation bases in the class of all monoids, may fail to possess this property when put in the ordered scenario. The purpose of the present article is to formulate an ordered version of the Strong Representation Extension Property of Hall and to explore its connections with absolute flatness and amalgamation of pomonoids. This enables us to prove that pogroups are (strong) amalgamation bases in the class of all pomonoids.     

Multivalued sheaf constructions

In this note, we modify the“classical”theory of sheaves, in such a way that it includes applications where more than one restriction map between sections is needed.     

一类具有13个参数的7次系统的极限环分支

研究了一类具有幂零奇点的7次多项式微分系统的极限环分支与中心问题.借助于数学软件MATHEMATICA,推导出系统在原点的前14个拟Lyapunov常数,从而得到了系统的原点为中心的充要条件,证明了系统在3阶幂零奇点处可以分支出14个极限环,给出了7次李雅谱诺夫系统在3阶幂零奇点处的环性数的下界.     

具有幂零临界点的Linard系统的中心-焦点判定

本文研究了一类具有幂零临界点的Linard系统的中心-焦点判定.利用Cherkas方法,得到系统的广义Lyapunov常数,分析了系统奇点稳定性与中心条件,推广了文[6]对于初等临界点中心焦点判定的结果.     

Zero-divisor semigroups and refinements of a star graph

Let G be a refinement of a star graph with center c. Let be the subgraph of G induced on the vertex set V(G)?{c or end vertices adjacent to c}. In this paper, we completely determine the structure of commutative zero-divisor semigroups S whose zero-divisor graph G=Γ(S) and S satisfy one of the following properties: (1) has at least two connected components, (2) is a cycle graph C_n of length n≥5, (3) is a path graph P_n with n≥6, (4) S is nilpotent and Γ(S) is a finite or an infinite star graph. For any finite or infinite cardinal number n≥2, we prove that for any nilpotent semigroup S with zero element 0, S⁴=0 if Γ(S) is a star graph K_1,n. We prove that there is exactly one nilpotent semigroup S such that S³≠0 and Γ(S)≅K_1,n. For several classes of finite graphs G which are refinements of a star graph, we also obtain formulas to count the number of non-isomorphic corresponding semigroups.     

The Paley–Wiener theorem for certain nilpotent Lie groupsJean

We generalize the classical Paley–Wiener theorem to special types of connected, simply connected, nilpotent Lie groups: First we consider nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. Then we consider nilpotent Lie groups such that the co-adjoint orbits of all the elements of a dense subset of the dual of the Lie algebra 𝔤^* are flat (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)