单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

自由群的直积的检验元素的注记

本文研究了自由群的直积的检验元素,通过对直积的自同态的分解,得到了直积中的元素为检验元素的充分必要条件,改进了O'neill和Turner的结果.此外,构造了两类具体的检验元素.     

乘方的指数与其幂的位数的关系

为了探究乘方的指数与其幂的位数的关系,定义了几个有关的新概念,并且证明了两个关于乘方以及进制进位的定理,由此建立起关于乘方以及进制进位的理论体系,其中包括进位理论中判定乘方的指数与其幂的位数是否存在周期规律的判别法,以及进位规律的求解法和四条相关的性质.     

逆周期的交换半群生成的簇的子簇

In this paper, all subvarieties of the varieties Ak （k ∈N） generated by aperiodic commutative semigroups are characterized. Based on this characterization, the structure of lattice of subvarieties of Ak is investigated.     

Lindelf的工作与Chaplygin的工作的互补性

运用Vakonomic模型导出Lindel f方程 ,表明Lindel f的工作与Vakonomic模型相吻合 ;运用Chetaev模型导出Chaplygin方程 ,表明Chaplygin的工作与Chetaev模型相吻合·　在此基础上 ,通过改进Chaplygin方程和Lindel f方程的表示形式 ,实现了从Lindel f方程向Chaplygin方程的合理过渡和从Chaplygin方程向Lindel f方程的合理的过渡·　最后 ,给出一个典型实例·　结果表明 ,正如Vako nomic模型与Chetaev模型是互补的一样 ,Lindel f的工作与Chaplygin的工作也是互补的·     

一元二次方程根的判别式的应用

<正>一元二次方程根的判别式是初中数学的重要内容,本文以近年中考中所考查的题型为例,归纳整理如下,供同仁们参考.一、求待定字母的取值范围(1)已知方程根的情况,求待定字母的取值范围例1若关于x的方程(k-1)x2+2(k)^(1/2)x+1=0有两个不相等的实数根.求k的獉獉取值范围.析解由题意"方程有两个不相等的实数獉獉根"可知:该方程是一元二次方程,且Δ>0,即     

树的最大特征值的上界的一个注记

设T是一个树,V是T的顶点集．记dv是υ∈V的度,△是T的最大顶点度．设υ∈V且dw=1．记k=ew+1,这里ew是w的excentricity．设δj′= max{dυ:dist(υ,w)=j},j=1,2,…,k-2,我们证明和这里μ1(T)和λ1(T)分别是T的Laplacian矩阵和邻接矩阵的最大特征值．特别地,记δo′=2．     

不同的公理不同的几何

一位语文老师在博客上写反思.他在教《自相矛盾》这则寓言时,有学生站起来反驳:有这么傻的人么?太不可信了吧.面对这样大胆质疑的学生,这位老师不知如何处理为好! 现在的学生比以前更敢想敢做了.     

抛物线上的点列构成的多边形的面积问题

引理设Pn,Pn＋d,Pn＋2d,…,Pn＋（k-1）d是抛物线y=ax^2＋vx＋c（a,b,c为常数,a≠0）上的点列（注：本文用只表示函数图象上横坐标为t的点）,则k边形Pn＋dPn＋2d…Pn＋（k-1）d的面积S与n无关（即与点列的起始位置无关）,     

别样的思考 同样的精彩

人们喜欢数学竞赛题是因为它能引发广阔的思考空间,能训练发散思维,引导人们去寻求各种不同的解法;但是我认为它给我的精彩不仅在于有多少种解法,而在于它能引发我从不同的角度来拓展问题,从而享受其中的乐趣.以下以第六届"祖冲之杯"初中数学邀请赛     

Equivalent Separating Curves on Closed Surface of Genus 2

Let {A,B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to (A, B) if it is disjoint from A∪B and it cuts F into two once-punctured tori X, Y with A(?)X, B(?)Y. Letγbe a simple closed curve on F which is disjoint from A∪B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hnγ(C):n∈Z} is the set containing all the simple closed curves on F which is separating with respect to (A,B), where hγis the Dehn twist alongγon F. This also shows how two simple closed curves on F which are separating with respect to (A,B) are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.     

Haken 3-Manifolds in Small Covers*

The authors prove that a 3-dimensional small cover M is a Haken manifold if and only if M is aspherical or equivalently the underlying simple polytope is a flag polytope. In addition, they find that M being Haken is also equivalent to the existence of a Riemannian metric with non-positive sectional curvature on M.     

区域创新系统协同演化机制与优化设计研究

为揭示区域创新系统协同演化机制的本质与规律,在对区域创新系统自组织特征予以描述的基础上,构建区域创新系统协同演化的哈肯模型,选用基础共性技术创新数据与制度创新数据进行实证研究,运用Minitab软件设计系统演化方案并提出优化对策。研究结果表明:基础共性技术创新代表量是区域创新系统动态演化的序参量;基础共性技术创新与制度创新在区域创新系统演化过程中具有协同效应,但协同度欠佳;制度创新递增的正反馈机制在区域创新系统演化过程中的信噪比最大,基础共性技术创新对制度创新推动作用的反映最灵敏;当信噪比最大的参数λ₂<0,形成制度创新递增的正反馈机制,同时调控参数使得a、b、λ₁均小于0且尽量保持a的绝对值最大,区域创新系统将会实现最优演化水平。     

Incompressible surfaces in punctured-torus bundles

We derive in this paper the classification up to isotopy of the incompressible surfaces in hyperbolic 3-manifolds which fiber over the circle with fiber a once-punctured torus. From this classification it follows that most of the 3-manifolds obtained by compactifying these bundles via a circle at infinity are closed hyperbolic 3-manifolds which contain 1.0 incompressible surfaces, i.e., are not Haken manifolds.     

Bounded Classes in the Cohomology of Manifolds

We discuss the image of a natural homomorphism from the bounded cohomology to the ordinary cohomology of a manifold. We give a necessary and sufficient condition for a Haken 3-manifold M to have the property that any class in the nth cohomology of M is bounded if n > 1.The first author was partially supported by JSPS.     

The Heegaard Genus of Amalgamated 3-Manifolds

Let M and M′ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that Mand M′ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is ‘sufficiently complicated’, the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of Mand M′ and the genus of their common boundary. Here, a homeomorphism is ‘sufficiently complicated’ if it is the composition of a homeomorphism from the boundary ofM to some surface S, followed by a sufficiently high power of a pseudo-Anosov onS, followed by a homeomorphism to the boundary of M′. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.     

球面2型超曲面

本文在无先念条件下,证明了Ｃｈｅｎ［１］关于２型超曲面的一个定理．     

A generalization of the classical sphere theorem

In this paper, we prove a sphere theorem for Riemannian manifolds with partially positive curvature which generalizes the classical sphere theorem.

     

On the Modular Figure and Ford-Spheres

In §1 we study Figure 1 for variable h. In §2 and §3 this is combined with Ford-circles (Figure 2) and the choice h = 0 (Figure 3). In §4 a limit is evaluated. In §5 we treat Figure 5 (“Ford-circles inside the unit-circle”) and a technical application. In §6 contains various modular figures (Figures 7, 8, 9). §7 treats synchronous rotation of Figure 1 and Eoing from R² to R³; the rotating upper half-plane of C corresponds then to the unit-sphere. In §8 a sphere-packing on C is introduced. Finally in §9 a sphere-packing inside the unit-sphere arises.