On the Class-number of the Maximal Real Subfield of a Cyclotomic Field

For any square-free positive integer m, let H(m) be the class-number of the field , where ζ_m is a primitive m-th root of unity. We show that if m = {3(8 g + 5)}² ? 2 is a square-free integer, where g is a positive integer, then H(4 m) > 1. Similar result holds for a square-free integer m = {3(8 g +7)}² ?2, where g is a positive integer. We also show that n|H(4 m) for certain positive integers m and n.     

On the Genus of a Finite Classical Group

Let G be a finite group acting faithfully and transitively ona set of size m, and let E = {x₁, ..., x_k} be a generatingset for G with x₁x₂...x_k = 1. If x G has cycles of length r₁,..., r_l in its action on , define . Then the genus g = g(G, , E) is defined by 1991 Mathematics Subject Classification 20B25, 20G40,30F99.     

On the Mislin Genus of Symplectic Groups

In this paper we give lower bounds for the Mislin genus of thesymplectic groups Sp(m). This result appears to be the exactanalogue of Zabrodsky's theorem concerning the special unitarygroups SU(n). It is achieved by the determination of the stablegenus of the quasi-projective quaternionic spaces QH(m), followingthe approach of McGibbon. It leads to a symplectic version ofZabrodsky's conjecture, saying that these lower bounds are infact the exact cardinality of the genus sets. The genus of Sp(2)is well known to contain exactly two elements. We show thatthe genus of Sp(3) has exactly 32 elements and see that theconjecture is true in these two cases. Independently, we also show that any homotopy type in the genusof Sp(m) fibers over the sphere S⁴m–1 with fiber in thegenus of Sp(m–1), and that any homotopy type in the genusof SU(n) fibers over the sphere S²n–1 with fiber in thegenus of SU(n–1). Moreover, these fibrations are principalwith respect to some appropriate loop structures on the fibers.These constructions permit us to produce particular spaces realizingthe lower bounds obtained. 2000 Mathematics Subject Classification55P60 (primary), 55P15, 55R35 (secondary)     

On a Universal Mapping Class Group of Genus Zero

The aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed a universal mapping class group of genus zero, denoted The latter is a nontrivial extension of the Thompson group V (acting on the Cantor set) by an inductive limit of pure mapping class groups of all genus zero surfaces. We prove that is a finitely presented group, and give an explicit presentation of it.     

关于嵌入图的最大亏格

不依赖图的其它参数, 而主要依据图嵌入在定向曲面上的有关嵌入性质, 该文研究图的最大亏格.     

关于3-正则图的平均亏格

一个图G的2－因子F是一个使得每个点v在F中的度dF(v)=2的G的生成子图。易知F中的每个圈是点不交的。如果F中每个圈的长度为4，我们说G有四边形2－因子F。我们首先在3－正则图上定义了3种扩张运算，然后讨论这些运算对平均亏格的影响。运用扩张运算，我们研究了含有四边形2－因子的3－正则图的平均亏格，得到了3－正则图的平均亏格与最大亏格之间的关系。     

On Lifting Fibrations of Genus One

We investigate the problem of lifting fibrations of genus one on algebraic surfaces of Kodaira dimension zero. We prove that fibrations on the following surfaces lift: Enriques surfaces, K3 surfaces covering Enriques surfaces, certain hyperelliptic, and quasi-hyperelliptic surfaces.     

On the Limit Cycles Bifurcating from a Quadratic Reversible Center of Genus One

The paper is concerned with the bifurcation of limit cycles in perturbations of a quadratic reversible system with a center of genus one. By studying the properties of the auxiliary curve and centroid curve defined by the Abelian integrals, we have proved that under small quadratic perturbations, at most two limit cycles arise from the period annulus surrounding the quadratic reversible center, and the bound is sharp. This partially verifies Conjecture 1 given in Gautier et al. (Discrete Contin Dyn Syst 25:511–535, 2009).     

On the Energy of a Unit Vector Field

The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M T ₁ M, where the unit tangent bundle T ₁ M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M=S ⁵,S ⁷,..., and an estimate on the index is obtained.     

On the Genus of the Nilpotent Graphs of Finite Groups

The nilpotent graph of a group G is a simple graph whose vertex set is G?nil(G), where nil(G) = {y ∈ G | ? x, y ? is nilpotent ? x ∈ G}, and two distinct vertices x and y are adjacent if ? x, y ? is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.     

关于图的最大亏格的下界

主要考虑2-边和3-边连通图的最大亏格下界情形,推广了J.Chen等(1996)的主要结果,证明了     

关于直径为４的图的最大亏格

该文证明了如下结果：设犌为直径为４的简单图,若犌不含３阶完全子图犓３,则犌的Ｂｅｔｔｉ亏数ξ（犌）≤４,因此有犌的最大亏格γ犕（犌）≥ １２β（犌）－２．     

两类子域码的重量分布（英文）

子域码是一类特殊的线性码.线性码由于其有效的编码及译码算法,在电子消费产品、数据存储系统和通信系统中有广泛的应用.然而,确定线性码的重量分布通常是困难的工作.本文给出了两类二元子域码C₍₍H(f)_<sub>1))~((2))和C₍₍H(f)_<sub>2))~((2))及其对偶码的重量分布,其中f₁(x)=x~4,f₂(x)=x~6+x~4+x~2.     

域同态的扩张的个数

设h:F→Ω是域的同态,E是F的有限次扩域,本文讨论了使h到E上的扩张的个数取得最大值的条件.     

数域上的线性无关性

本文给出一组复数在数域上线性无关的一个判别法则及它对于超越数论的一个应用．     

On the Number of Genus Embeddings of Complete Bipartite Graphs

In this paper, on the basis of joint tree model introduced by Liu, by dividing the associated surfaces into segments layer by layer, we show that there are at least ${C_{1}\cdot C_{2}^{\frac{m}{2}}\cdot C_{3}^{\frac{n}{2}}(m-1)^{m-\frac{1}{2}}(n-1)^{n-\frac{1}{2}}}$ distinct genus embeddings for complete bipartite graph K _m,n , where C ₁, C ₂, and C ₃ are constants depending on the residual class of m modular 4 and that of n modular 4.     

On a Relative Fourier–Mukai Transform on Genus One Fibrations

We study relative Fourier–Mukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and non-projective. Grothendieck duality is used to prove a skew–commutativity relation between this equivalence of categories and certain duality functors. We use our results to explicitly construct examples of semi-stable sheaves on degenerating families of elliptic curves.     

关于内亏格1的弱可约SD-分解(英)

设(M；H_1,H_2；F_0)为带边3-流形M的一个SD-分解．称该分解为可约的(或弱可约的)若存在本质圆片D_1■H)_1,D_2■H_2使得■D_1,■D_2■F_0并且■D_1=■D_2(或■D_1∩■D_2=■)．称(M；H_1,H_2；F_0)为内亏格1若F_0为穿孔环面．本文主要结果：一个弱可约的内亏格1的SD-分解或是可约的或是双经的．     

On Sums and Products in a Field

We study sums and products in a field. Let F be a field with ch(F) ≠ 2, where ch(F) is the characteristic of F. For any integer k ? 4, we show that any x ∈ F can be written as a₁ + … + a_k with a₁, …, _ak ∈ F and a₁… a_k = 1, and that for any α ∈ F {0} we can write every x ∈ F as a₁ … a_k with a₁, …, a_k ∈ F and a₁ + … + a_k = α. We also prove that for any x ∈ F and k ∈ {2, 3, …} there are a₁, …, a_2k ∈ F such that a₁ + … + a_2k = x = a₁ … a_2k.