多面体截面和小覆盖的闭子流形

设π:Mn→Pn是Pn上的小覆盖,S是Pn的任意一个n-1维截面.给出了π-1(S)是n-1维闭子流形(或者两个相互同胚n-1维闭于流形的不交并),以及π-1(S)是n-1维伪流形的充要条件.     

关于极小子流形全纯量曲率的一个空隙定理

本文证明了欧氏空间RN中的n-维（n>2）完备极小子流形,若其全纯量曲率小于（π/46~（1/2）n,则必是n-维平面.此结论改进了文[2,6]中的结果.     

2011年清华金秋营数学试题及解答

1.求sinπnsin2πn…sin(n-1)πn的值.解设ε=cosπn+isinπn(i为虚数单位),则1,ε,ε2,…,ε2(n-1)为x2n-1=0的根,且sinkπn=εk-ε-k2i=ε2k-12iεk,所以sinπnsin2πn…sin(n-1)πn=(ε2-1)(ε4-1)…[ε2(n-1)-1]2n-1in-1ε12n(n-1)()n-1(2)(4)…[2(n-1)]     

关于极小子流形全纯量曲率的一个空隙定理

本文证明了欧氏空间RN中的n-维(n＞2)完备极小子流形,若其全纯量曲率小于(π/4@√6)n,则必是n-维平面.此结论改进了文[2,6]中的结果.     

常曲率空间中具正Ricci曲率的子流形

设S~(n+p)(1)是一单位球面,M~n是浸入S~(n+p)(1)的具有非零平行平均曲率向量的n维紧致子流形.证明了当n≥4,p≥2时,如果M~n的Ricci曲率不小于(n-2)(1+H~2),则M~n是全脐的或者M~n的Ricci曲率等于(n-2)(1+H~2),进而M~n的几何分类被完全给出.     

关于积的三个三角恒等式的推广

一易证下列三个恒等式成立: (1)sinθsin(θ+π/ 3)sin(θ+2π/ 3) =sin3θ/4; (2)cosθcos(θ+π/3)cos(θ+2π/3) =-1/4cos3θ; (3)tgθtg(θ+π/3)tg(θ+2π/3) =-tg3θ。本文把上述三个恒等式予以推广,其一般形式为: (Ⅰ) multiply form j=1 to n sin(θ+(j-1)/nπ)=sinnθ/2~(n-1); (Ⅱ) multiply form j=1 to n cos(θ+(j-1)/nπ) =(-1)~(n-2) sinnθ/2~(n/1) (n为偶数), (-1)~(n-1)~2 cosnθ/2~(n-1)(n为奇数);     

单位球面中具平行单位平均曲率向量的子流形

设M是n-维闭黎曼流形,等距浸入(n+p)-维单位球空间S^n+p,具有平行的单位平均曲率向量。若S≤min{2n/3,2(n-1)^1/2},其中S是M的第二基本形式长度的平方,则M是S^n+p的一个(n+1)-维全测地子流形Sⁿ⁺¹中的超曲面。     

一个与Wallis不等式有关的单调减少数列

证明了{n (64 n~3+16 n~2+72n+15)/64 n~3-16 n~2+72n-15~(1/2) integral from 0 to π/2 sin~nxdx}为严格单调减少数列,且极限为π/2~(1/2),因而得π(64 n~3-16 n~2+72n-15)/2n 64 n~3+16 n~2(+72n+15)~(1/2)integral from 0 to π/2 sin~nxdxπ(64 n~3+208 n~2+296n+167)/2 n(+1)(64 n~3+176 n~2+232n+105)~(1/2),将Wallis不等式改进为512 n~3-64 n~2+144n-15/πn (512 n~3+64 n~2+144n+15)~(1/2)2(n-1)!!/2(n)!!512 n~3+832 n~2+592n+167/(πn+0.5)(512 n~3+704 n~2+464n+105)~(1/2).     

一个有极限的单调减少数列

证明了{(n(4n+1)/4n-1)~(1/2)∫π/20 sin~nxdx}为严格单调减少数列,且极限为(π/2)~(1/2),因而得(π(4n-1)/2n(4n+1))~(1/2)∫π/20 sin~nxdx (π(4 n+5)/2(n+1)(4n+3))~(1/2).     

On the Approximation of Continuous Functions by Pal-Type Interpolation Polynomials

Let us denote by -1=x_n相似文献   

双曲空间中完备子流形的特征值估计

研究了$(n+p)$维双曲空间$\mathbb{H}^{n+p}$中完备非紧子流形的第一特征值的上界.特别地,证明了$\mathbb{H}^{n+p}$中具有平行平均曲率向量$H$和无迹第二基本形式有限$L^q(q\geq n)$范数的完备子流形的第一特征值不超过$\frac{(n-1)^2(1-|H|^2)}{4}$,和$\mathbb{H}^{n+1}(n\leq5)$中具有常平均曲率向量$H$和无迹第二基本形式有限$L^q(2(1-\sqrt{\frac{2}{n}})相似文献   

平衡子欧拉符号图及符号线图

符号图$S=(S^u,\sigma)$是以$S^u$作为底图并且满足$\sigma: E(S^u)\rightarrow\{+,-\}$. 设$E^-(S)$表示$S$的负边集. 如果$S^u$是欧拉的(或者分别是子欧拉的, 欧拉的且$|E^-(S)|$是偶数, 则$S$是欧拉符号图(或者分别是子欧拉符号图, 平衡欧拉符号图). 如果存在平衡欧拉符号图$S''$使得$S''$由$S$生成, 则$S$是平衡子欧拉符号图. 符号图$S$的线图$L(S)$也是一个符号图, 使得$L(S)$的点是$S$中的边, 其中$e_ie_j$是$L(S)$中的边当且仅当$e_i$和$e_j$在$S$中相邻,并且$e_ie_j$是$L(S)$中的负边当且仅当$e_i$和$e_j$在$S$中都是负边. 本文给出了两个符号图族$S$和$S''$,它们应用于刻画平衡子欧拉符号图和平衡子欧拉符号线图. 特别地, 本文证明了符号图$S$是平衡子欧拉的当且仅当$\not\in S$, $S$的符号线图是平衡子欧拉的当且仅当$S\not\in S''$.     

Complex specializations of the reduced Gassner representation of the pure braid group

We will give a necessary and sufficient condition for the specialization of the reduced Gassner representation to be irreducible. It will be shown that for , is irreducible if and only if .

     

分数次型 Marcinkiewicz 积分在 Hardy 空间中的有界性

The authors in the paper proved that if Ω is homogeneous of degree zero and satisfies some certain logarithmic type Lipschitz condition,then the fractional type Marcinkiewicz Integral μ Ω,α is an operator of type （H˙ K n（1-1/q 1 ）,p q 1 ,˙ K n（1-1/q 1 ）,p q 2 ） and of type （H 1 （R n ）,L n/（n-α） ）.     

Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

This paper gives $n$-dimensional analogues of the Apollonian circle packings in Parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of $n$-dimensional Descartes configurations, which consist of $n+2$ mutually touching spheres. We work in the space $M_D^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, augmented curvature-center coordinates, as those $(n+2) \times (n+2)$ real matrices $W$ with $W^T Q_{D,n} W = Q_{W,n}$ where $Q_{D,n} = x_1^2 + \cdots + x_{n+2}^2 - ({1}/{n})(x_1 +\cdots + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + \cdots + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. On the parameter space $M_D^n$ of augmented curvature-center matrices, the group ${\it Aut}(Q_{D,n})$ acts on the left and ${\it Aut}(Q_{W,n})$ acts on the right. Both these groups are isomorphic to the $(n+2)$-dimensional Lorentz group $O(n+1,1)$, and give two different "geometric" actions. The right action of ${\it Aut}(Q_{W,n})$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^n$ while the left action of ${\it Aut}(Q_{D,n})$ is defined only on the parameter space $M_D^n$. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in ${\it Aut}(Q_{D,n})$, with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions there exist rational Apollonian cluster ensembles (all curvatures are rational) and strongly rational Apollonian sphere ensembles (all augmented curvature-center coordinates are rational).     

Higher Order Turán Inequalities

The celebrated Turán inequalities , where denotes the Legendre polynomial of degree , are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities , which hold for the Maclaurin coefficients of the real entire function in the Laguerre-Pólya class, .

     

Small Covers over a Product Space

A small cover is a closed manifold $M^{n}$ with a locally standard $(\Bbb{Z}_{2})^{n}$-action such that its orbit space is a simple convex polytope $P^{n}$. Let $\Delta^{n}$ denote an $n$-simplex and $P(m)$ an $m$-gon. This paper gives formulas for calculating the number of D-J equivalent classes and equivariant homeomorphism classes of orientable small covers over the product space $\Delta^{n_1}\times \Delta^{n_2} \times P(m)$, where $n_1$ is odd.     

Möbius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in Sn+1

Let x : M~n→ S~(n+1) be an immersed hypersurface in the(n + 1)-dimensional sphere S~(n+1). If, for any points p, q ∈ Mn, there exists a Mbius transformation φ :S~(n+1)→ S~(n+1) such that φox(Mn~) = x(M~n) and φ ox(p) = x(q), then the hypersurface is called a Mbius homogeneous hypersurface. In this paper, the Mbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mbius transformation.     

Subsurface 1-distance of the Handleb o dy

For a handlebody H with ?H = S, let F  S be an essential connected subsurface of S. Let C(S) be the curve complex of S, AC(F) be the arc and curve complex of F, D(H)  C(S) be the disk complex of H and πF(D(H))  AC(F) be the image of D(H) in AC(F). We introduce the definition of subsurface 1-distance between the 1-simplices of AC(F) and show that under some hypothesis, πF(D(H))comes within subsurface 1-distance at most 4 of every 1-simplex of AC(F).