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   

$K_{5}\times S_{n}$ 的交叉数

把完全图$K_{5}$的五个顶点与另外$n$个顶点都联边得到一类特殊的图$H_{n}$.文中证明了$H_{n}$的交叉数为$Z(5,n)+2n+\lfloor \frac{n}{2}\rfloor+1$,并在此基础上证明了$K_{5}$与星$K_{1,n}$的笛卡尔积的交叉数为$Z(5,n)+5n+\lfloor\frac{n}{2} \rfloor+1$.     

W6*Sn的交叉数

早在20世纪50年代,Zarankiewicz 猜想完全2-部图K_{m,n}(m\leq n)的交叉数为\lfloor\frac{m}{2}\rfloor\times \lfloor\frac{m-1}{2}\rfloor\times\lfloor\frac{n}{2}\rfloor\times\lfloor\frac{n-1}{2}\rfloor (对任意实数x,\lfloor x\rfloor表示不超过x的最大整数). 目前这一猜想的正确性只证明了当m\leq6时成立. 假定著名的Zarankiewicz的猜想对m=7的情形成立,确定了6-轮W_{6}与星S_{n}的笛卡尔积图的交叉是 cr(W_{6}\times S_{n})=9\lfloor\frac{n}{2}\rfloor\times\lfloor\frac{n-1}{2}\rfloor+2n+5\lfloor\frac{n}{2}\rfloor.     

Powers of the Catalan Generating Function and Lagrange''s 1770 Trinomial Equation Series

The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. They are named for Eugene Catalan who studied them as early as 1838. They were also found by Leonhard Euler (1758), Nicholas von Fuss (1795), and Andreas von Segner (1758). The Catalan numbers have the binomial generating function $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation $x^3 - 3x + 1 =0$, the numbers $\frac{1}{2k+1}\binom{3k}{k}$ analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.     

一类具有奇异灵敏度的logistic趋化系统解的渐近行为

本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.     

K2,4×Sn的交叉数

Garey和Johnson证明了确定图的交叉数是一个NP-完全问题.确定了笛卡尔积图$K_{2,4}\times S_{n}$的交叉数是$Z(6,n)+4n.$ 当$m\geq 5,$猜想${\rm cr}(K_{2,m}\timesS_{n})={\rm cr}(K_{2,m,n})+n\lfloor\frac{m}{2}\rfloor\lfloor\frac{m-1}{2}\rfloor$.     

在Banach空间中推广的 Roper-Suffridge算子(III)

假定 $X$ 是具有范数$\|\cdot\|$的复 Banach 空间, $n$ 是一个满足 $\dim X\geq n\geq2$的正整数. 本文考虑由下式定义的推广的Roper-Suffridge算子 $\Phi_{n,\beta_2, \gamma_2, \ldots , \beta_{n+1}, \gamma_{n+1}}(f)$: \begin{equation} \begin{array}{lll} \Phi _{n, \beta_2, \gamma_2, \ldots, \beta_{n+1},\gamma_{n+1}}(f)(x) &;\hspace{-3mm}=&;\hspace{-3mm}\dl\he{j=1}{n}\bigg(\frac{f(x^*_1(x))}{x^*_1(x)})\bigg)^{\beta_j}(f''(x^*_1(x))^{\gamma_j}x^*_j(x) x_j\\ &;&;+\bigg(\dl\frac{f(x^*_1(x))}{x^*_1(x)}\bigg)^{\beta_{n+1}}(f''(x^*_1(x)))^{\gamma_{n+1}}\bigg(x-\dl\he{j=1}{n}x^*_j(x) x_j\bigg),\nonumber \end{array} \end{equation} 其中 $x\in\Omega_{p_1, p_2, \ldots, p_{n+1}}$, $\beta_1=1, \gamma_1=0$ 和 \begin{equation} \begin{array}{lll} \Omega_{p_1, p_2, \ldots, p_{n+1}}=\bigg\{x\in X: \dl\he{j=1}{n}| x^*_j(x)|^{p_j}+\bigg\|x-\dl\he{j=1}{n}x^*_j(x)x_j\bigg\|^{p_{n+1}}<1\bigg\},\nonumber \end{array} \end{equation} 这里 $p_j>1 \,( j=1, 2,\ldots, n+1$), 线性无关族 $\{x_1, x_2, \ldots, x_n \}\subset X $ 与 $\{x^*_1, x^*_2, \ldots, x^*_n \}\subset X^* $ 满足 $x^*_j(x_j)=\|x_j\|=1 (j=1, 2, \ldots, n)$ 和 $x^*_j(x_k)=0 \, (j\neq k)$, 我们选取幂函数的单值分支满足 $(\frac{f(\xi)}{\xi})^{\beta_j}|_{\xi=0}= 1$ 和 $(f''(\xi))^{\gamma_j}|_{\xi=0}=1, \, j=2, \ldots , n+1$. 本文将证明: 对某些合适的常数$\beta_j, \gamma_j$, 算子$\Phi_{n,\beta_2, \gamma_2, \ldots, \beta_{n+1}, \gamma_{n+1}}(f)$ 在$\Omega_{p_1, p_2, \ldots , p_{n+1}}$上保持$\alpha$阶的殆$\beta$型螺形映照和 $\alpha$阶的$\beta$型螺形映照.     

The Non-Existence of Limit Cycle of Some Quadratic Differential System

In this paper, we consider the system $\frac {dx}{dt}=-y+lx^2+5axy,\frac {dy}{dt}=x(1+ax+3ly)$ (1) which has a fine focus of -the third order O(0,0) . We prove the following Theorem. No limit cycle of system (1) can exist in the neighborhood of O(0,0). In fact, using a few transformations, we can change system (1) into the Van Der Pol's system $\frac{dx}{dt}=-y,\frac {dy}{dt}=-g(x)-f(x)y$(2) where $g(x)=x^-3(x-1)[3x(x+4)+\delta(6x-1)]$ $f(x)=-x^-2(x-1)(x+4),\delta=(5a^2-3l^2)/l^2$ Hence the problem of the existence or non-existence of limit oyole for system (1) around O changes into that for system (2) around (1, 0) . Again through Филиппов transformation, we arrive at the following system $\frac {dy}{dz}=\frac {1}{F_1(z)-y},\frac {dy}{dz}=\frac {1}{F_2(z)-y}$ where $\int_1^x {f(\xi)d \xi=F(x)=F_1(z),(x>1)}$ $\int_1^x {f(\xi)d \xi=F(x)=F_2(z),(0相似文献   

逼近已知函数微商的广义Lanczos算法

给出逼近已知函数微商的广义Lanczos 算法, 构造了一列逼近算子$D_{h}^{n}$以提高稳定近似解的收敛速率. 当$n=2$时, 逼近精度达到$O(\delta^{6 \over 7})$, 而对一般的自然数$n$逼近精度为$O(\delta^{\frac{2n+2}{2n+3}})$, 这里$\delta$是近似函数的误差界.     

Clifford 环面 Sm(\sqrt{\frac{m}{n}})×Sn-m(\sqrt{\frac{n-m}{n}})的刚性定理

\small\zihao{-5}\begin{quote}{\heiti 摘要:} 设$M$为$n+1$维单位球面$S^{n+1}(1)$中的一个极小闭超曲面，如果 $ n \le S \le n+\frac{2}{3}$, 则有 $S=n$ 且 $M$ 与某一Clifford 环面 $S^m(\sqrt{m/n}) \times S^{n-m}(\sqrt{(n-m)/n})$等距.     

Generalized Ejiri''s Rigidity Theorem for Submanifolds in Pinched Manifolds

Let $M^{n}(n\geq4)$ be an oriented compact submanifold with parallel mean curvature in an $(n+p)$-dimensional complete simply connected Riemannian manifold $N^{n+p}$. Then there exists a constant $\delta(n,p)\in(0,1)$ such that if the sectional curvature of $N$ satisfies $\ov{K}_{N}\in[\delta(n,p), 1]$, and if $M$ has a lower bound for Ricci curvature and an upper bound for scalar curvature, then $N$ is isometric to $S^{n+p}$. Moreover, $M$ is either a totally umbilic sphere $S^n\big(\frac{1}{\sqrt{1+H^2}}\big)$, a Clifford hypersurface $S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)\times S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)$ in the totally umbilic sphere $S^{n+1}\big(\frac{1}{\sqrt{1+H^2}}\big)$ with $n=2m$, or $\mathbb{C}P^{2}\big(\frac{4}{3}(1+H^2)\big)$ in $S^7\big(\frac{1}{\sqrt{1+H^2}}\big)$. This is a generalization of Ejiri''s rigidity theorem.     

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.     

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).     

与$\zeta(2k 1)$函数有关的两类快速收敛级数

Values of new series sum（（（2n-1）!ζ（2n））/（2n ＋ 2k）!）α2n from n=1 to ∞,sum（（（2n-1）!ζ（2n））/（2n＋2k ＋1）!）β2n from n=1 to ∞ are given concerning ζ（2k ＋ 1）,where k is a positive integer,α can be taken as 1,1/2,1/3,2/3,1/4,3/4,1/6,5/6 and β can be taken as 1,1/2.Some previous results are included as special cases in the present paper and new series converges more rapidly than those exsiting results for α = 1/3,or α = 1/4,or α = 1/6.     

由圈长分布确定的二部图~$K_{n,n r}-A\ (|A|\leq 3)$

The cycle length distribution of a graph G of order n is a sequence （c1 （G）,…, cn （G））, where ci （G） is the number of cycles of length i in G. In general, the graphs with cycle length distribution （c1（G） ,…,cn（G）） are not unique. A graph G is determined by its cycle length distribution if the graph with cycle length distribution （c1 （G）,…, cn （G）） is unique. Let Kn,n＋r be a complete bipartite graph and A lohtaib in E（Kn,n＋r）. In this paper, we obtain： Let s 〉 1 be an integer. （1） If r = 2s, n 〉 s（s - 1） ＋ 2｜A｜, then Kn,n＋r - A （A lohtain in E（Kn,n＋r）,｜A｜ ≤ 3） is determined by its cycle length distribution; （2） If r = 2s ＋ 1,n 〉 s^2 ＋ 2｜A｜, Kn,n＋r - A （A lohtain in E（Kn,n＋r）, ｜A｜ ≤3） is determined by its cycle length distribution.     

偶图$K_{n,n+7}-A(|A|\leq3)$的圈长分布唯一性

阶为$n$的图$G$的圈长分布是序列($c_1,c_2,\ldots,c_n$), 其中$c_i$是图$G$中长为$i$的圈数.本文得到如下结果: 设$A\subseteq E(K_{n,n+7})$,在以下情况, 图 $G$ 由其圈长分布唯一确定.(1) $G=K_{n,n+7}$(n\geq10)$;(2) $G=K_{n,n+7}-A$ $(|A|=1,n\geq12)$;(3)$G=K_{n,n+7}-A$(|A|=2,n\geq14)$;(4)$G=K_{n,n+7}-A$ $(|A|=3     

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.     

On the Tangent Bundle of a Hypersurface in a Riemannian Manifold

Let(Mn, g) and(Nn+1, G) be Riemannian manifolds. Let TMn and TNn+1 be the associated tangent bundles. Let f :(Mn,g) →(Nn+1,G) be an isometrical immersion with g = f*G, F =(f, df) :(TMn, ■) →(TNn+1, Gs) be the isometrical immersion with ■= F*Gs where (df)x: TxM → Tf(x)N for any x ∈M is the differential map, and Gs be the Sasaki metric on TN induced from G. This paper deals with the geometry of TMn as a submanifold of TNn+1 by the moving frame method. The authors firstly study the extrinsic geometry of TMn in TNn+1. Then the integrability of the induced almost complex structure of TM is discussed.