Constructing infra-nilmanifolds admitting an Anosov diffeomorphism

In this paper we establish an algebraic characterization of those infra-nilmanifolds modeled on a free c-step nilpotent Lie group and with an abelian holonomy group admitting an Anosov diffeomorphism. We also develop a new method for constructing examples of infra-nilmanifolds having an Anosov diffeomorphism.     

A note on blocks with dihedral defect groups

We prove that the 2-regular subspace of the center of a block with a dihedral defect group is multiplicatively closed. Received: 25 April 2008     

Subgroups generated by small classes in finite groups

Let be the subgroup of generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if is nilpotent, then has nilpotence class at most . Using a slight variation on Mann's methods, we obtain results that do not require us to assume that is nilpotent. We show that if is supersolvable, then is nilpotent with class at most , and in general, the Fitting subgroup of has class at most .

     

Allow problems concerning spectral properties of sign pattern matrices: A survey

An n×n sign pattern matrix has entries in {+,-,0}. This paper surveys the following problems concerning spectral properties of sign pattern matrices: sign patterns that allow all possible spectra (spectrally arbitrary sign patterns); sign patterns that allow all inertias (inertially arbitrary sign patterns); sign patterns that allow nilpotency (potentially nilpotent sign patterns); and sign patterns that allow stability (potentially stable sign patterns). Relationships between these four classes of sign patterns are given, and several open problems are identified.     

On potentially nilpotent double star sign patterns

A matrix whose entries come from the set {+, −, 0} is called a sign pattern matrix, or sign pattern. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by (m, 2), is introduced. We determine all potentially nilpotent sign patterns in (3, 2) and (5, 2), and prove that one sign pattern in (3, 2) is potentially stable. Supported by youth scientific funds of the Education Department of Jiangxi Province (No. GJJ09460), Natural Science Foundation of Jiangxi Normal University (No.2058), National Natural Science Foundation of China (No.10601001).     

关于有限群两个幂零子群积的问题

众所周知,有限群的两个幂零子群的积不一定是幂零的．本文研究了Ｅｎｇｅｌ条件对两个幂零子群的影响,得到两个幂零子群的积为幂零群的几个充分条件。     

VARIATIONS ON A THEME BY EULER

1.IntroductionAHallliltolliansystemofdifferentialequationsonRZnisgivedbyP~~H,(P,q),q=HP(P,q),(1)wherep=(pl,'.,P.),q=(ql,',q.)eR"arethegeneralizedcoordinatesandmolllentarespectivelyandH(P,q)istheellergyofthesystem.Thesystem(1)canberewrittenasthecompactf…     

Roe–Strichartz theorem on two-step nilpotent Lie groupsa

Strichartz characterized eigenfunctions of the Laplacian on Euclidean spaces by boundedness conditions which generalized a result of Roe for the one-dimensional case. He also proved an analogous statement for the sub-Laplacian on the Heisenberg groups. In this paper, we extend this result to connected, simply connected two-step nilpotent Lie groups.